Stacks Project

Stacks Project

Version 7835fb9, compiled on Jul 24, 2012.

The following people have contributed to this work: Dan Abramovich, Jarod Alper, Dima Arinkin, Bhargav Bhatt, Mark Behrens, Pieter Belmans, Ingo Blechschmidt, David Brown, Kestutis Cesnavicius, Nava Chitrik, Fraser Chiu, Johan Commelin, Brian Conrad, Peadar Coyle, Rankeya Datta, Aise Johan de Jong, Matt DeLand, Daniel Disegni, Joel Dodge, Alexander Palen Ellis, Andrew Fanoe, Maxim Fedorchuck, Cameron Franc, Lennart Galinat, Martin Gallauer, Luis Garcia, Alberto Gioia, Xue Hang, Philipp Hartwig, Florian Heiderich, Jeremiah Heller, Kristen Hendricks, Fraser Hiu, Yuhao Huang, Christian Kappen, Timo Keller, Keenan Kidwell, Andrew Kiluk, Lars Kindler, Emmanuel Kowalski, Daniel Krashen, Min Lee, Tobi Lehman, Max Lieblich, Hsing Liu, Zachary Maddock, Sonja Mapes, Akhil Mathew, Yusuf Mustopa, Josh Nichols-Barrer, Thomas Nyberg, Catherine O’Neil, Martin Olsson, Brian Osserman, Thanos Papaioannou, Peter Percival, Alex Perry, Bjorn Poonen, Thibaut Pugin, You Qi, Fred Rohrer, Matthieu Romagny, Joe Ross, Julius Ross, David Rydh, Beren Sanders, Rene Schoof, Jaakko Seppala, Chung-chieh Shan, Jason Starr, Abolfazl Tarizadeh, John Tate, Titus Teodorescu, Michael Thaddeus, Ravi Vakil, Theo van den Bogaart, Kevin Ventullo, Hendrik Verhoek, Jonathan Wang, Ian Whitehead, Amnon Yekutieli, Fan Zhou, David Zureick-Brown.

2

Copyright (C) 2005 -- 2012 Johan de Jong Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

Contents Chapter 1. Introduction 1.1. Overview 1.2. Attribution 1.3. Other chapters

43 43 43 44

Chapter 2. Conventions 2.1. Comments 2.2. Set theory 2.3. Categories 2.4. Algebra 2.5. Notation 2.6. Other chapters

45 45 45 45 45 45 45

Chapter 3. Set Theory 3.1. Introduction 3.2. Everything is a set 3.3. Classes 3.4. Ordinals 3.5. The hierarchy of sets 3.6. Cardinality 3.7. Cofinality 3.8. Reflection principle 3.9. Constructing categories of schemes 3.10. Sets with group action 3.11. Coverings of a site 3.12. Abelian categories and injectives 3.13. Other chapters

47 47 47 47 48 48 48 49 49 50 54 55 57 57

Chapter 4. Categories 4.1. Introduction 4.2. Definitions 4.3. Opposite Categories and the Yoneda Lemma 4.4. Products of pairs 4.5. Coproducts of pairs 4.6. Fibre products 4.7. Examples of fibre products 4.8. Fibre products and representability 4.9. Pushouts 4.10. Equalizers 4.11. Coequalizers

59 59 59 63 64 64 65 66 66 67 68 68

3

4

CONTENTS

4.12. 4.13. 4.14. 4.15. 4.16. 4.17. 4.18. 4.19. 4.20. 4.21. 4.22. 4.23. 4.24. 4.25. 4.26. 4.27. 4.28. 4.29. 4.30. 4.31. 4.32. 4.33. 4.34. 4.35. 4.36. 4.37. 4.38. 4.39.

Initial and final objects Limits and colimits Limits and colimits in the category of sets Connected limits Finite limits and colimits Filtered colimits Cofiltered limits Limits and colimits over partially ordered sets Essentially constant systems Exact functors Adjoint functors Monomorphisms and Epimorphisms Localization in categories Formal properties 2-categories (2, 1)-categories 2-fibre products Categories over categories Fibred categories Inertia Categories fibred in groupoids Presheaves of categories Presheaves of groupoids Categories fibred in sets Categories fibred in setoids Representable categories fibred in groupoids Representable 1-morphisms Other chapters

Chapter 5. Topology 5.1. Introduction 5.2. Basic notions 5.3. Bases 5.4. Connected components 5.5. Irreducible components 5.6. Noetherian topological spaces 5.7. Krull dimension 5.8. Codimension and catenary spaces 5.9. Quasi-compact spaces and maps 5.10. Constructible sets 5.11. Constructible sets and Noetherian spaces 5.12. Characterizing proper maps 5.13. Jacobson spaces 5.14. Specialization 5.15. Submersive maps 5.16. Dimension functions 5.17. Nowhere dense sets 5.18. Miscellany 5.19. Other chapters

69 69 71 72 73 76 78 78 81 83 83 85 85 95 97 99 100 106 107 112 114 120 122 123 125 127 128 131 133 133 133 133 134 135 137 138 139 139 142 143 144 147 149 151 152 153 154 155

CONTENTS

5

Chapter 6. Sheaves on Spaces 6.1. Introduction 6.2. Basic notions 6.3. Presheaves 6.4. Abelian presheaves 6.5. Presheaves of algebraic structures 6.6. Presheaves of modules 6.7. Sheaves 6.8. Abelian sheaves 6.9. Sheaves of algebraic structures 6.10. Sheaves of modules 6.11. Stalks 6.12. Stalks of abelian presheaves 6.13. Stalks of presheaves of algebraic structures 6.14. Stalks of presheaves of modules 6.15. Algebraic structures 6.16. Exactness and points 6.17. Sheafification 6.18. Sheafification of abelian presheaves 6.19. Sheafification of presheaves of algebraic structures 6.20. Sheafification of presheaves of modules 6.21. Continuous maps and sheaves 6.22. Continuous maps and abelian sheaves 6.23. Continuous maps and sheaves of algebraic structures 6.24. Continuous maps and sheaves of modules 6.25. Ringed spaces 6.26. Morphisms of ringed spaces and modules 6.27. Skyscraper sheaves and stalks 6.28. Limits and colimits of presheaves 6.29. Limits and colimits of sheaves 6.30. Bases and sheaves 6.31. Open immersions and (pre)sheaves 6.32. Closed immersions and (pre)sheaves 6.33. Glueing sheaves 6.34. Other chapters

157 157 157 157 158 159 160 161 163 163 165 165 166 167 167 168 169 170 172 173 174 175 178 180 181 184 185 187 188 188 189 197 201 203 205

Chapter 7. Commutative Algebra 7.1. Introduction 7.2. Conventions 7.3. Basic notions 7.4. Snake lemma 7.5. Finite modules and finitely presented modules 7.6. Ring maps of finite type and of finite presentation 7.7. Finite ring maps 7.8. Colimits 7.9. Localization 7.10. Internal Hom 7.11. Tensor products 7.12. Tensor algebra

207 207 207 207 209 210 211 212 213 217 222 223 228

6

CONTENTS

7.13. 7.14. 7.15. 7.16. 7.17. 7.18. 7.19. 7.20. 7.21. 7.22. 7.23. 7.24. 7.25. 7.26. 7.27. 7.28. 7.29. 7.30. 7.31. 7.32. 7.33. 7.34. 7.35. 7.36. 7.37. 7.38. 7.39. 7.40. 7.41. 7.42. 7.43. 7.44. 7.45. 7.46. 7.47. 7.48. 7.49. 7.50. 7.51. 7.52. 7.53. 7.54. 7.55. 7.56. 7.57. 7.58. 7.59. 7.60.

Base change Miscellany Cayley-Hamilton The spectrum of a ring Local rings Nakayama’s lemma Open and closed subsets of spectra Connected components of spectra Glueing functions More glueing results Total rings of fractions Irreducible components of spectra Examples of spectra of rings A meta-observation about prime ideals Images of ring maps of finite presentation More on images Noetherian rings Curiosity Hilbert Nullstellensatz Jacobson rings Finite and integral ring extensions Normal rings Going down for integral over normal Flat modules and flat ring maps Going up and going down Transcendence Algebraic elements of field extensions Separable extensions Geometrically reduced algebras Separable extensions, continued Perfect fields Geometrically irreducible algebras Geometrically connected algebras Geometrically integral algebras Valuation rings More Noetherian rings Length Artinian rings Homomorphisms essentially of finite type K-groups Graded rings Proj of a graded ring Blow up algebras Noetherian graded rings Noetherian local rings Dimension Applications of dimension theory Support and dimension of modules

229 230 232 233 238 239 240 241 242 245 248 248 250 253 255 258 260 262 263 264 272 276 279 280 286 289 290 291 293 295 296 297 301 303 303 305 308 311 312 313 316 317 321 322 324 326 329 330

CONTENTS

7.61. 7.62. 7.63. 7.64. 7.65. 7.66. 7.67. 7.68. 7.69. 7.70. 7.71. 7.72. 7.73. 7.74. 7.75. 7.76. 7.77. 7.78. 7.79. 7.80. 7.81. 7.82. 7.83. 7.84. 7.85. 7.86. 7.87. 7.88. 7.89. 7.90. 7.91. 7.92. 7.93. 7.94. 7.95. 7.96. 7.97. 7.98. 7.99. 7.100. 7.101. 7.102. 7.103. 7.104. 7.105. 7.106. 7.107. 7.108.

Associated primes Symbolic powers Relative assassin Weakly associated primes Embedded primes Regular sequences and depth Quasi-regular sequences Ext groups and depth An application of Ext groups Tor groups and flatness Functorialities for Tor Projective modules Finite projective modules Open loci defined by module maps Faithfully flat descent for projectivity of modules Characterizing flatness Universally injective module maps Descent for finite projective modules Transfinite d´evissage of modules Projective modules over a local ring Mittag-Leffler systems Inverse systems Mittag-Leffler modules Interchanging direct products with tensor Coherent rings Examples and non-examples of Mittag-Leffler modules Countably generated Mittag-Leffler modules Characterizing projective modules Ascending properties of modules Descending properties of modules Completion Criteria for flatness Base change and flatness Flatness criteria over Artinian rings What makes a complex exact? Cohen-Macaulay modules Cohen-Macaulay rings Catenary rings Regular local rings Epimorphisms of rings Pure ideals Rings of finite global dimension Regular rings and global dimension Homomorphisms and dimension The dimension formula Dimension of finite type algebras over fields Noether normalization Dimension of finite type algebras over fields, reprise

7

333 336 336 339 343 344 346 349 352 353 358 358 360 363 364 364 366 372 373 375 376 378 378 383 387 389 391 392 394 394 396 401 408 408 411 414 416 417 418 420 423 425 428 430 432 433 435 437

8

CONTENTS

7.109. Dimension of graded algebras over a field 7.110. Generic flatness 7.111. Around Krull-Akizuki 7.112. Factorization 7.113. Orders of vanishing 7.114. Quasi-finite maps 7.115. Zariski’s Main Theorem 7.116. Applications of Zariski’s Main Theorem 7.117. Dimension of fibres 7.118. Algebras and modules of finite presentation 7.119. Colimits and maps of finite presentation 7.120. More flatness criteria 7.121. Openness of the flat locus 7.122. Openness of Cohen-Macaulay loci 7.123. Differentials 7.124. The naive cotangent complex 7.125. Local complete intersections 7.126. Syntomic morphisms 7.127. Smooth ring maps 7.128. Formally smooth maps 7.129. Smoothness and differentials 7.130. Smooth algebras over fields 7.131. Smooth ring maps in the Noetherian case 7.132. Overview of results on smooth ring maps ´ 7.133. Etale ring maps 7.134. Local homomorphisms 7.135. Integral closure and smooth base change 7.136. Formally unramified maps 7.137. Conormal modules and universal thickenings 7.138. Formally ´etale maps 7.139. Unramified ring maps 7.140. Henselian local rings 7.141. Serre’s criterion for normality 7.142. Formal smoothness of fields 7.143. Constructing flat ring maps 7.144. The Cohen structure theorem 7.145. Nagata and Japanese rings 7.146. Ascending properties 7.147. Descending properties 7.148. Geometrically normal algebras 7.149. Geometrically regular algebras 7.150. Geometrically Cohen-Macaulay algebras 7.151. Colimits and maps of finite presentation, II 7.152. Other chapters Chapter 8. Brauer groups 8.1. Introduction 8.2. Noncommutative algebras 8.3. Wedderburn’s theorem

439 440 444 449 450 453 456 461 462 465 467 474 479 481 484 488 494 501 508 515 521 522 526 529 530 542 542 544 545 548 549 555 568 571 574 575 579 588 591 594 595 597 597 601 603 603 603 603

CONTENTS

8.4. 8.5. 8.6. 8.7. 8.8. 8.9.

Lemmas on algebras The Brauer group of a field Skolem-Noether The centralizer theorem Splitting fields Other chapters

Chapter 9. Sites and Sheaves 9.1. Introduction 9.2. Presheaves 9.3. Injective and surjective maps of presheaves 9.4. Limits and colimits of presheaves 9.5. Functoriality of categories of presheaves 9.6. Sites 9.7. Sheaves 9.8. Families of morphisms with fixed target 9.9. The example of G-sets 9.10. Sheafification 9.11. Injective and surjective maps of sheaves 9.12. Representable sheaves 9.13. Continuous functors 9.14. Morphisms of sites 9.15. Topoi 9.16. G-sets and morphisms 9.17. More functoriality of presheaves 9.18. Cocontinuous functors 9.19. Cocontinuous functors and morphisms of topoi 9.20. Cocontinuous functors which have a right adjoint 9.21. Localization 9.22. Glueing sheaves 9.23. More localization 9.24. Localization and morphisms 9.25. Morphisms of topoi 9.26. Localization of topoi 9.27. Localization and morphisms of topoi 9.28. Points 9.29. Constructing points 9.30. Points and and morphisms of topoi 9.31. Localization and points 9.32. 2-morphisms of topoi 9.33. Morphisms between points 9.34. Sites with enough points 9.35. Criterion for existence of points 9.36. Exactness properties of pushforward 9.37. Almost cocontinuous functors 9.38. Sheaves of algebraic structures 9.39. Pullback maps 9.40. Topologies 9.41. The topology defined by a site

9

604 606 607 608 608 610 613 613 613 614 614 615 617 619 620 623 625 630 630 632 633 634 636 637 638 640 643 644 647 648 650 653 659 660 662 666 669 670 672 673 674 675 677 681 683 686 687 690

10

CONTENTS

9.42. 9.43. 9.44. 9.45. 9.46.

Sheafification in a topology Topologies and sheaves Topologies and continuous functors Points and topologies Other chapters

692 695 696 696 696

Chapter 10. Homological Algebra 10.1. Introduction 10.2. Basic notions 10.3. Abelian categories 10.4. Extensions 10.5. Additive functors 10.6. Localization 10.7. Serre subcategories 10.8. K-groups 10.9. Cohomological delta-functors 10.10. Complexes 10.11. Truncation of complexes 10.12. Homotopy and the shift functor 10.13. Filtrations 10.14. Spectral sequences 10.15. Spectral sequences: exact couples 10.16. Spectral sequences: differential objects 10.17. Spectral sequences: filtered differential objects 10.18. Spectral sequences: filtered complexes 10.19. Spectral sequences: double complexes 10.20. Injectives 10.21. Projectives 10.22. Injectives and adjoint functors 10.23. Inverse systems 10.24. Exactness of products 10.25. Differential graded algebras 10.26. Other chapters

699 699 699 699 704 706 708 711 713 715 717 721 723 726 731 732 733 734 737 739 742 743 744 745 748 749 749

Chapter 11. Derived Categories 11.1. Introduction 11.2. Triangulated categories 11.3. The definition of a triangulated category 11.4. Elementary results on triangulated categories 11.5. Localization of triangulated categories 11.6. Quotients of triangulated categories 11.7. The homotopy category 11.8. Cones and termwise split sequences 11.9. Distinguished triangles in the homotopy category 11.10. Derived categories 11.11. The canonical delta-functor 11.12. Triangulated subcategories of the derived category 11.13. Filtered derived categories 11.14. Derived functors in general

751 751 751 751 754 761 766 772 772 778 781 783 785 786 789

CONTENTS

11.15. 11.16. 11.17. 11.18. 11.19. 11.20. 11.21. 11.22. 11.23. 11.24. 11.25. 11.26. 11.27. 11.28. 11.29. 11.30.

Derived functors on derived categories Higher derived functors Injective resolutions Projective resolutions Right derived functors and injective resolutions Cartan-Eilenberg resolutions Composition of right derived functors Resolution functors Functorial injective embeddings and resolution functors Right derived functors via resolution functors Filtered derived category and injective resolutions Ext groups Unbounded complexes K-injective complexes Bounded cohomological dimension Other chapters

Chapter 12. More on Algebra 12.1. Introduction 12.2. A comment on the Artin-Rees property 12.3. Fitting ideals 12.4. Computing Tor 12.5. Derived tensor product 12.6. Derived change of rings 12.7. Tor independence 12.8. Spectral sequences for Tor 12.9. Products and Tor 12.10. Formal glueing of module categories 12.11. Lifting 12.12. Auto-associated rings 12.13. Flattening stratification 12.14. Flattening over an Artinian ring 12.15. Flattening over a closed subset of the base 12.16. Flattening over a closed subsets of source and base 12.17. Flattening over a Noetherian complete local ring 12.18. Descent flatness along integral maps 12.19. Torsion and flatness 12.20. Flatness and finiteness conditions 12.21. Blowing up and flatness 12.22. Completion and flatnes 12.23. The Koszul complex 12.24. Koszul regular sequences 12.25. Regular ideals 12.26. Local complete intersection maps 12.27. Cartier’s equality and geometric regularity 12.28. Geometric regularity 12.29. Topological rings and modules 12.30. Formally smooth maps of topological rings 12.31. Some results on power series rings

11

796 799 802 807 809 811 812 813 815 817 817 825 828 831 832 834 837 837 837 838 840 841 844 844 845 846 848 856 861 863 864 864 865 867 868 870 871 875 876 877 880 886 887 889 890 893 894 899

12

CONTENTS

12.32. 12.33. 12.34. 12.35. 12.36. 12.37. 12.38. 12.39. 12.40. 12.41. 12.42. 12.43. 12.44. 12.45. 12.46. 12.47. 12.48. 12.49. 12.50.

Geometric regularity and formal smoothness Regular ring maps Ascending properties along regular ring maps Permanence of properties under completion Permanence of properties under henselization Field extensions, revisited The singular locus Regularity and derivations Formal smoothness and regularity G-rings Excellent rings Pseudo-coherent modules Tor dimension Perfect complexes Characterizing perfect complexes Relatively finitely presented modules Relatively pseudo-coherent modules Pseudo-coherent and perfect ring maps Other chapters

901 906 907 907 908 911 914 915 917 919 924 924 931 934 938 942 945 951 952

Chapter 13. Smoothing Ring Maps 13.1. Introduction 13.2. Colimits 13.3. Singular ideals 13.4. Presentations of algebras 13.5. The lifting problem 13.6. The lifting lemma 13.7. The desingularization lemma 13.8. Warmup: reduction to a base field 13.9. Local tricks 13.10. Separable residue fields 13.11. Inseparable residue fields 13.12. The main theorem 13.13. The approximation property for G-rings 13.14. Other chapters

953 953 954 955 957 962 964 967 970 971 973 975 981 981 983

Chapter 14. Simplicial Methods 14.1. Introduction 14.2. The category of finite ordered sets 14.3. Simplicial objects 14.4. Simplicial objects as presheaves 14.5. Cosimplicial objects 14.6. Products of simplicial objects 14.7. Fibre products of simplicial objects 14.8. Pushouts of simplicial objects 14.9. Products of cosimplicial objects 14.10. Fibre products of cosimplicial objects 14.11. Simplicial sets 14.12. Products with simplicial sets

985 985 985 987 988 989 990 991 991 992 992 992 993

CONTENTS

14.13. 14.14. 14.15. 14.16. 14.17. 14.18. 14.19. 14.20. 14.21. 14.22. 14.23. 14.24. 14.25. 14.26. 14.27. 14.28. 14.29.

Hom from simplicial sets into cosimplicial objects Internal Hom Hom from simplicial sets into simplicial objects Splitting simplicial objects Skelet and coskelet functors Augmentations Left adjoints to the skeleton functors Simplicial objects in abelian categories Simplicial objects and chain complexes Dold-Kan Dold-Kan for cosimplicial objects Homotopies Homotopies in abelian categories Homotopies and cosimplicial objects More homotopies in abelian categories A homotopy equivalence Other chapters

13

995 996 996 1001 1005 1011 1012 1016 1020 1023 1026 1027 1029 1030 1031 1035 1038

Chapter 15. Sheaves of Modules 15.1. Introduction 15.2. Pathology 15.3. The abelian category of sheaves of modules 15.4. Sections of sheaves of modules 15.5. Supports of modules and sections 15.6. Closed immersions and abelian sheaves 15.7. A canonical exact sequence 15.8. Modules locally generated by sections 15.9. Modules of finite type 15.10. Quasi-coherent modules 15.11. Modules of finite presentation 15.12. Coherent modules 15.13. Closed immersions of ringed spaces 15.14. Locally free sheaves 15.15. Tensor product 15.16. Flat modules 15.17. Flat morphisms of ringed spaces 15.18. Symmetric and exterior powers 15.19. Internal Hom 15.20. Koszul complexes 15.21. Invertible sheaves 15.22. Localizing sheaves of rings 15.23. Other chapters

1041 1041 1041 1041 1043 1044 1045 1046 1047 1047 1049 1052 1054 1056 1057 1058 1060 1061 1062 1063 1065 1065 1067 1068

Chapter 16.1. 16.2. 16.3. 16.4. 16.5.

1071 1071 1071 1072 1073 1074

16. Modules on Sites Introduction Abelian presheaves Abelian sheaves Free abelian presheaves Free abelian sheaves

14

CONTENTS

16.6. 16.7. 16.8. 16.9. 16.10. 16.11. 16.12. 16.13. 16.14. 16.15. 16.16. 16.17. 16.18. 16.19. 16.20. 16.21. 16.22. 16.23. 16.24. 16.25. 16.26. 16.27. 16.28. 16.29. 16.30. 16.31. 16.32. 16.33. 16.34. 16.35. 16.36.

Ringed sites Ringed topoi 2-morphisms of ringed topoi Presheaves of modules Sheaves of modules Sheafification of presheaves of modules Morphisms of topoi and sheaves of modules Morphisms of ringed topoi and modules The abelian category of sheaves of modules Exactness of pushforward Exactness of lower shriek Global types of modules Intrinsic properties of modules Localization of ringed sites Localization of morphisms of ringed sites Localization of ringed topoi Localization of morphisms of ringed topoi Local types of modules Tensor product Internal Hom Flat modules Flat morphisms Invertible modules Modules of differentials Stalks of modules Skyscraper sheaves Localization and points Pullbacks of flat modules Locally ringed topoi Lower shriek for modules Other chapters

Chapter 17. Injectives 17.1. Introduction 17.2. Abelian groups 17.3. Modules 17.4. Projective resolutions 17.5. Modules over noncommutative rings 17.6. Baer’s argument for modules 17.7. G-modules 17.8. Abelian sheaves on a space 17.9. Sheaves of modules on a ringed space 17.10. Abelian presheaves on a category 17.11. Abelian Sheaves on a site 17.12. Modules on a ringed site 17.13. Embedding abelian categories 17.14. Grothendieck’s AB conditions 17.15. Injectives in Grothendieck categories 17.16. K-injectives in Grothendieck categories

1074 1075 1076 1077 1078 1078 1079 1080 1081 1083 1084 1086 1087 1088 1090 1091 1093 1094 1098 1099 1101 1104 1104 1105 1108 1110 1111 1111 1112 1117 1118 1119 1119 1119 1120 1121 1121 1121 1125 1126 1126 1127 1128 1130 1131 1133 1134 1136

CONTENTS

17.17. 17.18.

Additional remarks on Grothendieck abelian categories Other chapters

15

1139 1141

Chapter 18. Cohomology of Sheaves 18.1. Introduction 18.2. Topics 18.3. Cohomology of sheaves 18.4. Derived functors 18.5. First cohomology and torsors 18.6. Locality of cohomology 18.7. Projection formula 18.8. Mayer-Vietoris ˇ ˇ 18.9. The Cech complex and Cech cohomology 18.10. Cech cohomology as a functor on presheaves 18.11. Cech cohomology and cohomology 18.12. The Leray spectral sequence 18.13. Functoriality of cohomology 18.14. The base change map 18.15. Cohomology and colimits 18.16. Vanishing on Noetherian topological spaces ˇ 18.17. The alternating Cech complex ˇ 18.18. Locally finite coverings and the Cech complex ˇ 18.19. Cech cohomology of complexes 18.20. Flat resolutions 18.21. Derived pullback 18.22. Cohomology of unbounded complexes 18.23. Producing K-injective resolutions 18.24. Other chapters

1143 1143 1143 1143 1144 1145 1146 1148 1149 1150 1151 1154 1158 1160 1162 1163 1165 1167 1170 1171 1179 1181 1182 1183 1184

Chapter 19. Cohomology on Sites 19.1. Introduction 19.2. Topics 19.3. Cohomology of sheaves 19.4. Derived functors 19.5. First cohomology and torsors 19.6. First cohomology and extensions 19.7. First cohomology and invertible sheaves 19.8. Locality of cohomology 19.9. The Cech complex and Cech cohomology 19.10. Cech cohomology as a functor on presheaves 19.11. Cech cohomology and cohomology 19.12. Cohomology of modules 19.13. Limp sheaves 19.14. The Leray spectral sequence 19.15. The base change map 19.16. Cohomology and colimits 19.17. Flat resolutions 19.18. Derived pullback 19.19. Cohomology of unbounded complexes

1187 1187 1187 1187 1188 1189 1190 1191 1192 1194 1195 1199 1201 1203 1205 1206 1207 1209 1212 1213

16

CONTENTS

19.20. 19.21. 19.22. 19.23.

Producing K-injective resolutions Spectral sequences for Ext Derived lower shriek Other chapters

1214 1217 1217 1219

Chapter 20. Hypercoverings 20.1. Introduction 20.2. Hypercoverings 20.3. Acyclicity 20.4. Covering hypercoverings 20.5. Adding simplices 20.6. Homotopies 20.7. Cech cohomology associated to hypercoverings 20.8. Cohomology and hypercoverings 20.9. Hypercoverings of spaces 20.10. Other chapters

1221 1221 1221 1224 1226 1229 1230 1232 1234 1236 1239

Chapter 21. Schemes 21.1. Introduction 21.2. Locally ringed spaces 21.3. Open immersions of locally ringed spaces 21.4. Closed immersions of locally ringed spaces 21.5. Affine schemes 21.6. The category of affine schemes 21.7. Quasi-Coherent sheaves on affines 21.8. Closed subspaces of affine schemes 21.9. Schemes 21.10. Immersions of schemes 21.11. Zariski topology of schemes 21.12. Reduced schemes 21.13. Points of schemes 21.14. Glueing schemes 21.15. A representability criterion 21.16. Existence of fibre products of schemes 21.17. Fibre products of schemes 21.18. Base change in algebraic geometry 21.19. Quasi-compact morphisms 21.20. Valuative criterion for universal closedness 21.21. Separation axioms 21.22. Valuative criterion of separatedness 21.23. Monomorphisms 21.24. Functoriality for quasi-coherent modules 21.25. Other chapters

1241 1241 1241 1242 1243 1245 1247 1250 1254 1254 1255 1257 1258 1259 1261 1264 1266 1268 1270 1272 1273 1276 1281 1281 1282 1284

Chapter 22.1. 22.2. 22.3. 22.4.

1287 1287 1287 1289 1290

22. Constructions of Schemes Introduction Relative glueing Relative spectrum via glueing Relative spectrum as a functor

CONTENTS

22.5. 22.6. 22.7. 22.8. 22.9. 22.10. 22.11. 22.12. 22.13. 22.14. 22.15. 22.16. 22.17. 22.18. 22.19. 22.20. 22.21. 22.22.

17

Affine n-space Vector bundles Cones Proj of a graded ring Quasi-coherent sheaves on Proj Invertible sheaves on Proj Functoriality of Proj Morphisms into Proj Projective space Invertible sheaves and morphisms into Proj Relative Proj via glueing Relative Proj as a functor Quasi-coherent sheaves on relative Proj Functoriality of relative Proj Invertible sheaves and morphisms into relative Proj Twisting by invertible sheaves and relative Proj Projective bundles Other chapters

1293 1294 1294 1295 1301 1302 1305 1307 1311 1314 1316 1317 1323 1324 1325 1326 1327 1329

Chapter 23. Properties of Schemes 23.1. Introduction 23.2. Constructible sets 23.3. Integral, irreducible, and reduced schemes 23.4. Types of schemes defined by properties of rings 23.5. Noetherian schemes 23.6. Jacobson schemes 23.7. Normal schemes 23.8. Cohen-Macaulay schemes 23.9. Regular schemes 23.10. Dimension 23.11. Catenary schemes 23.12. Serre’s conditions 23.13. Japanese and Nagata schemes 23.14. The singular locus 23.15. Quasi-affine schemes 23.16. Characterizing modules of finite type and finite presentation 23.17. Flat modules 23.18. Locally free modules 23.19. Locally projective modules 23.20. Extending quasi-coherent sheaves 23.21. Gabber’s result 23.22. Sections with support in a closed 23.23. Sections of quasi-coherent sheaves 23.24. Ample invertible sheaves 23.25. Affine and quasi-affine schemes 23.26. Quasi-coherent sheaves and ample invertible sheaves 23.27. Finding suitable affine opens 23.28. Other chapters

1331 1331 1331 1332 1333 1334 1336 1337 1339 1339 1340 1341 1342 1343 1345 1345 1346 1347 1347 1348 1348 1353 1355 1357 1360 1364 1365 1366 1368

18

CONTENTS

Chapter 24. Morphisms of Schemes 24.1. Introduction 24.2. Closed immersions 24.3. Immersions 24.4. Closed immersions and quasi-coherent sheaves 24.5. Supports of modules 24.6. Scheme theoretic image 24.7. Scheme theoretic closure and density 24.8. Dominant morphisms 24.9. Birational morphisms 24.10. Rational maps 24.11. Surjective morphisms 24.12. Radicial and universally injective morphisms 24.13. Affine morphisms 24.14. Quasi-affine morphisms 24.15. Types of morphisms defined by properties of ring maps 24.16. Morphisms of finite type 24.17. Points of finite type and Jacobson schemes 24.18. Universally catenary schemes 24.19. Nagata schemes, reprise 24.20. The singular locus, reprise 24.21. Quasi-finite morphisms 24.22. Morphisms of finite presentation 24.23. Constructible sets 24.24. Open morphisms 24.25. Submersive morphisms 24.26. Flat morphisms 24.27. Flat closed immersions 24.28. Generic flatness 24.29. Morphisms and dimensions of fibres 24.30. Morphisms of given relative dimension 24.31. The dimension formula 24.32. Syntomic morphisms 24.33. Conormal sheaf of an immersion 24.34. Sheaf of differentials of a morphism 24.35. Smooth morphisms 24.36. Unramified morphisms ´ 24.37. Etale morphisms 24.38. Relatively ample sheaves 24.39. Very ample sheaves 24.40. Ample and very ample sheaves relative to finite type morphisms 24.41. Quasi-projective morphisms 24.42. Proper morphisms 24.43. Projective morphisms 24.44. Integral and finite morphisms 24.45. Universal homeomorphisms 24.46. Finite locally free morphisms 24.47. Generically finite morphisms

1371 1371 1371 1372 1374 1375 1377 1379 1380 1382 1382 1384 1385 1386 1389 1391 1393 1395 1397 1399 1399 1400 1405 1407 1408 1409 1409 1412 1413 1415 1416 1418 1420 1424 1426 1432 1438 1442 1447 1448 1451 1455 1455 1458 1461 1464 1464 1467

CONTENTS

24.48. 24.49. 24.50. 24.51.

Normalization Zariski’s Main Theorem (algebraic version) Universally bounded fibres Other chapters

19

1470 1476 1478 1481

Chapter 25. Cohomology of Schemes 25.1. Introduction 25.2. Cech cohomology of quasi-coherent sheaves 25.3. Vanishing of cohomology 25.4. Derived category of quasi-coherent modules 25.5. Quasi-coherence of higher direct images 25.6. Cohomology and base change, I 25.7. Colimits and higher direct images 25.8. Cohomology and base change, II 25.9. Ample invertible sheaves and cohomology 25.10. Cohomology of projective space 25.11. Coherent sheaves on locally Noetherian schemes 25.12. Coherent sheaves on Noetherian schemes 25.13. Depth 25.14. Devissage of coherent sheaves 25.15. Finite morphisms and affines 25.16. Coherent sheaves and projective morphisms 25.17. Chow’s Lemma 25.18. Higher direct images of coherent sheaves 25.19. The theorem on formal functions 25.20. Applications of the theorem on formal functions 25.21. Cohomology and base change, III 25.22. Other chapters

1483 1483 1483 1485 1486 1487 1490 1491 1491 1494 1496 1502 1504 1506 1506 1511 1513 1516 1518 1519 1524 1525 1526

Chapter 26. Divisors 26.1. Introduction 26.2. Associated points 26.3. Morphisms and associated points 26.4. Embedded points 26.5. Weakly associated points 26.6. Morphisms and weakly associated points 26.7. Relative assassin 26.8. Relative weak assassin 26.9. Effective Cartier divisors 26.10. Relative effective Cartier divisors 26.11. The normal cone of an immersion 26.12. Regular ideal sheaves 26.13. Regular immersions 26.14. Relative regular immersions 26.15. Meromorphic functions and sections 26.16. Relative Proj 26.17. Blowing up 26.18. Strict transform 26.19. Admissible blowups

1529 1529 1529 1531 1531 1532 1533 1534 1535 1536 1539 1543 1545 1548 1551 1557 1562 1566 1570 1574

20

CONTENTS

26.20.

Other chapters

1575

Chapter 27. Limits of Schemes 27.1. Introduction 27.2. Directed limits of schemes with affine transition maps 27.3. Absolute Noetherian Approximation 27.4. Limits and morphisms of finite presentation 27.5. Finite type closed in finite presentation 27.6. Descending relative objects 27.7. Characterizing affine schemes 27.8. Variants of Chow’s Lemma 27.9. Applications of Chow’s lemma 27.10. Universally closed morphisms 27.11. Limits and dimensions of fibres 27.12. Other chapters

1577 1577 1577 1579 1586 1587 1591 1596 1597 1599 1603 1606 1607

Chapter 28. Varieties 28.1. Introduction 28.2. Notation 28.3. Varieties 28.4. Geometrically reduced schemes 28.5. Geometrically connected schemes 28.6. Geometrically irreducible schemes 28.7. Geometrically integral schemes 28.8. Geometrically normal schemes 28.9. Change of fields and locally Noetherian schemes 28.10. Geometrically regular schemes 28.11. Change of fields and the Cohen-Macaulay property 28.12. Change of fields and the Jacobson property 28.13. Algebraic schemes 28.14. Closures of products 28.15. Schemes smooth over fields 28.16. Types of varieties 28.17. Groups of invertible functions 28.18. Uniqueness of base field 28.19. Other chapters

1609 1609 1609 1609 1610 1613 1619 1623 1624 1625 1626 1629 1629 1629 1630 1631 1633 1634 1636 1638

Chapter 29. Chow Homology and Chern Classes 29.1. Introduction 29.2. Determinants of finite length modules 29.3. Periodic complexes 29.4. Symbols 29.5. Lengths and determinants 29.6. Application to tame symbol 29.7. Setup 29.8. Cycles 29.9. Cycle associated to a closed subscheme 29.10. Cycle associated to a coherent sheaf 29.11. Preparation for proper pushforward

1641 1641 1641 1648 1656 1660 1666 1667 1668 1669 1670 1671

CONTENTS

29.12. 29.13. 29.14. 29.15. 29.16. 29.17. 29.18. 29.19. 29.20. 29.21. 29.22. 29.23. 29.24. 29.25. 29.26. 29.27. 29.28. 29.29. 29.30. 29.31. 29.32. 29.33. 29.34. 29.35. 29.36. 29.37. 29.38. 29.39. 29.40. 29.41. 29.42.

Proper pushforward Preparation for flat pullback Flat pullback Push and pull Preparation for principal divisors Principal divisors Two fun results on principal divisors Rational equivalence Properties of rational equivalence Different characterizations of rational equivalence Rational equivalence and K-groups Preparation for the divisor associated to an invertible sheaf The divisor associated to an invertible sheaf Intersecting with Cartier divisors Cartier divisors and K-groups Blowing up lemmas Intersecting with effective Cartier divisors Commutativity Gysin homomorphisms Relative effective Cartier divisors Affine bundles Projective space bundle formula The Chern classes of a vector bundle Intersecting with chern classes Polynomial relations among chern classes Additivity of chern classes The splitting principle Chern classes and tensor product Todd classes Grothendieck-Riemann-Roch Other chapters

21

1671 1673 1674 1676 1677 1678 1680 1681 1683 1685 1688 1691 1692 1693 1697 1699 1705 1711 1712 1715 1715 1716 1719 1720 1723 1724 1726 1727 1728 1728 1728

Chapter 30. Topologies on Schemes 30.1. Introduction 30.2. The general procedure 30.3. The Zariski topology 30.4. The ´etale topology 30.5. The smooth topology 30.6. The syntomic topology 30.7. The fppf topology 30.8. The fpqc topology 30.9. Change of topologies 30.10. Change of big sites 30.11. Other chapters

1731 1731 1731 1732 1737 1743 1745 1748 1751 1754 1755 1756

Chapter 31.1. 31.2. 31.3.

1757 1757 1757 1759

31. Descent Introduction Descent data for quasi-coherent sheaves Descent for modules

22

CONTENTS

31.4. 31.5. 31.6. 31.7. 31.8. 31.9. 31.10. 31.11. 31.12. 31.13. 31.14. 31.15. 31.16. 31.17. 31.18. 31.19. 31.20. 31.21. 31.22. 31.23. 31.24. 31.25. 31.26. 31.27. 31.28. 31.29. 31.30. 31.31. 31.32. 31.33. 31.34. 31.35. 31.36. 31.37.

Fpqc descent of quasi-coherent sheaves 1764 Descent of finiteness properties of modules 1766 Quasi-coherent sheaves and topologies 1767 Parasitic modules 1775 Derived category of quasi-coherent modules 1777 Fpqc coverings are universal effective epimorphisms 1778 Descent of finiteness properties of morphisms 1779 Local properties of schemes 1783 Properties of schemes local in the fppf topology 1784 Properties of schemes local in the syntomic topology 1785 Properties of schemes local in the smooth topology 1785 Variants on descending properties 1786 Germs of schemes 1787 Local properties of germs 1787 Properties of morphisms local on the target 1788 Properties of morphisms local in the fpqc topology on the target 1790 Properties of morphisms local in the fppf topology on the target 1797 Application of fpqc descent of properties of morphisms 1797 Properties of morphisms local on the source 1798 Properties of morphisms local in the fpqc topology on the source 1799 Properties of morphisms local in the fppf topology on the source 1799 Properties of morphisms local in the syntomic toplogy on the source1800 Properties of morphisms local in the smooth topology on the source1800 Properties of morphisms local in the ´etale topology on the source 1800 Properties of morphisms ´etale local on source-and-target 1801 Properties of morphisms of germs local on source-and-target 1807 Descent data for schemes over schemes 1810 Fully faithfulness of the pullback functors 1813 Descending types of morphisms 1818 Descending affine morphisms 1820 Descending quasi-affine morphisms 1820 Descent data in terms of sheaves 1821 Descent in terms of simplicial schemes 1822 Other chapters 1825

Chapter 32. Adequate Modules 32.1. Introduction 32.2. Conventions 32.3. Adequate functors 32.4. Higher exts of adequate functors 32.5. Adequate modules 32.6. Parasitic adequate modules 32.7. Derived categories of adequate modules, I 32.8. Pure extensions 32.9. Higher exts of quasi-coherent sheaves on the big site 32.10. Derived categories of adequate modules, II 32.11. Other chapters

1827 1827 1827 1828 1835 1841 1846 1848 1850 1853 1854 1855

Chapter 33.

1857

More on Morphisms

CONTENTS

33.1. 33.2. 33.3. 33.4. 33.5. 33.6. 33.7. 33.8. 33.9. 33.10. 33.11. 33.12. 33.13. 33.14. 33.15. 33.16. 33.17. 33.18. 33.19. 33.20. 33.21. 33.22. 33.23. 33.24. 33.25. 33.26. 33.27. 33.28. 33.29. 33.30. 33.31. 33.32. 33.33. 33.34. 33.35. 33.36. 33.37. 33.38. 33.39. 33.40. 33.41. Chapter 34.1. 34.2. 34.3. 34.4. 34.5. 34.6.

Introduction Thickenings First order infinitesimal neighbourhood Formally unramified morphisms Universal first order thickenings Formally ´etale morphisms Infinitesimal deformations of maps Infinitesimal deformations of schemes Formally smooth morphisms Smoothness over a Noetherian base Pushouts in the category of schemes Openness of the flat locus Crit`ere de platitude par fibres Normal morphisms Regular morphisms Cohen-Macaulay morphisms Slicing Cohen-Macaulay morphisms Generic fibres Relative assassins Reduced fibres Irreducible components of fibres Connected components of fibres Connected components meeting a section Dimension of fibres Limit arguments ´ Etale neighbourhoods Slicing smooth morphisms Finite free locally dominates ´etale ´ Etale localization of quasi-finite morphisms Application to the structure of quasi-finite morphisms Application to morphisms with connected fibres Application to the structure of finite type morphisms Application to the fppf topology Closed points in fibres Stein factorization Descending separated locally quasi-finite morphisms Pseudo-coherent morphisms Perfect morphisms Local complete intersection morphisms Exact sequences of differentials and conormal sheaves Other chapters 34. More on Flatness Introduction A remark on finite type versus finite presentation Lemmas on ´etale localization The local structure of a finite type module One step d´evissage Complete d´evissage

23

1857 1857 1858 1859 1862 1867 1870 1873 1877 1881 1882 1886 1887 1890 1891 1893 1895 1898 1903 1905 1907 1912 1915 1918 1919 1922 1924 1928 1929 1932 1938 1941 1943 1944 1950 1954 1955 1959 1961 1967 1967 1969 1969 1969 1969 1972 1975 1980

24

CONTENTS

34.7. Translation into algebra 34.8. Localization and universally injective maps 34.9. Completion and Mittag-Leffler modules 34.10. Projective modules 34.11. Flat finite type modules, Part I 34.12. Flat finitely presented modules 34.13. Flat finite type modules, Part II 34.14. Examples of relatively pure modules 34.15. Impurities 34.16. Relatively pure modules 34.17. Examples of relatively pure sheaves 34.18. A criterion for purity 34.19. How purity is used 34.20. Flattening functors 34.21. Flattening stratifications 34.22. Flattening stratification over an Artinian ring 34.23. Flattening a map 34.24. Flattening in the local case 34.25. Flat finite type modules, Part III 34.26. Universal flattening 34.27. Blowing up and flatness 34.28. Other chapters

1984 1986 1988 1989 1991 1997 2003 2007 2009 2012 2014 2015 2019 2022 2027 2028 2029 2030 2033 2034 2038 2044

Chapter 35. Groupoid Schemes 35.1. Introduction 35.2. Notation 35.3. Equivalence relations 35.4. Group schemes 35.5. Examples of group schemes 35.6. Properties of group schemes 35.7. Properties of group schemes over a field 35.8. Actions of group schemes 35.9. Principal homogeneous spaces 35.10. Equivariant quasi-coherent sheaves 35.11. Groupoids 35.12. Quasi-coherent sheaves on groupoids 35.13. Quasi-coherent modules on simplicial schemes 35.14. Groupoids and simplicial schemes 35.15. Colimits of quasi-coherent modules 35.16. Groupoids and group schemes 35.17. The stabilizer group scheme 35.18. Restricting groupoids 35.19. Invariant subschemes 35.20. Quotient sheaves 35.21. Separation conditions 35.22. Finite flat groupoids, affine case 35.23. Finite flat groupoids 35.24. Descent data give equivalence relations 35.25. An example case

2047 2047 2047 2047 2049 2050 2052 2053 2056 2057 2058 2059 2061 2062 2064 2067 2070 2070 2072 2073 2074 2077 2078 2083 2084 2085

CONTENTS

35.26.

Other chapters

25

2086

Chapter 36. More on Groupoid Schemes 36.1. Introduction 36.2. Notation 36.3. Useful diagrams 36.4. Sheaf of differentials 36.5. Properties of groupoids 36.6. Comparing fibres 36.7. Cohen-Macaulay presentations 36.8. Restricting groupoids 36.9. Properties of groupoids on fields 36.10. Morphisms of groupoids on fields 36.11. Slicing groupoids ´ 36.12. Etale localization of groupoids 36.13. Other chapters

2087 2087 2087 2087 2088 2088 2091 2092 2093 2095 2101 2104 2108 2110

´ Chapter 37. Etale Morphisms of Schemes 37.1. Introduction 37.2. Conventions 37.3. Unramified morphisms 37.4. Three other characterizations of unramified morphisms 37.5. The functorial characterization of unramified morphisms 37.6. Topological properties of unramified morphisms 37.7. Universally injective, unramified morphisms 37.8. Examples of unramified morphisms 37.9. Flat morphisms 37.10. Topological properties of flat morphisms ´ 37.11. Etale morphisms 37.12. The structure theorem ´ 37.13. Etale and smooth morphisms 37.14. Topological properties of ´etale morphisms 37.15. Topological invariance of the ´etale topology 37.16. The functorial characterization ´ 37.17. Etale local structure of unramified morphisms ´ 37.18. Etale local structure of ´etale morphisms 37.19. Permanence properties 37.20. Other chapters

2113 2113 2113 2113 2115 2117 2118 2119 2120 2121 2122 2123 2125 2126 2127 2127 2129 2129 2130 2131 2133

Chapter 38.1. 38.2. 38.3. 38.4. 38.5. 38.6. 38.7. 38.8. 38.9.

´ 38. Etale Cohomology Introduction Which sections to skip on a first reading? Prologue The ´etale topology Feats of the ´etale topology A computation Nontorsion coefficients Sheaf theory Presheaves

2135 2135 2135 2135 2136 2137 2137 2139 2139 2139

26

CONTENTS

38.10. Sites 38.11. Sheaves 38.12. The example of G-sets 38.13. Sheafification 38.14. Cohomology 38.15. The fpqc topology 38.16. Faithfully flat descent 38.17. Quasi-coherent sheaves 38.18. Cech cohomology 38.19. The Cech-to-cohomology spectral sequence 38.20. Big and small sites of schemes 38.21. The ´etale topos 38.22. Cohomology of quasi-coherent sheaves 38.23. Examples of sheaves 38.24. Picard groups 38.25. The ´etale site ´ 38.26. Etale morphisms ´ 38.27. Etale coverings 38.28. Kummer theory 38.29. Neighborhoods, stalks and points 38.30. Points in other topologies 38.31. Supports of abelian sheaves 38.32. Henselian rings 38.33. Stalks of the structure sheaf 38.34. Functoriality of small ´etale topos 38.35. Direct images 38.36. Inverse image 38.37. Functoriality of big topoi 38.38. Functoriality and sheaves of modules 38.39. Comparing big and small topoi 38.40. Recovering morphisms 38.41. Push and pull 38.42. Property (A) 38.43. Property (B) 38.44. Property (C) 38.45. Topological invariance of the small ´etale site 38.46. Closed immersions and pushforward 38.47. Integral universally injective morphisms 38.48. Big sites and pushforward 38.49. Exactness of big lower shriek ´ 38.50. Etale cohomology 38.51. Colimits 38.52. Stalks of higher direct images 38.53. The Leray spectral sequence 38.54. Vanishing of finite higher direct images 38.55. Schemes ´etale over a point 38.56. Galois action on stalks 38.57. Cohomology of a point

2140 2141 2141 2142 2143 2144 2146 2148 2149 2152 2152 2155 2155 2157 2158 2159 2159 2160 2161 2165 2171 2172 2174 2175 2177 2177 2178 2180 2181 2182 2183 2188 2189 2190 2192 2194 2195 2197 2198 2199 2201 2201 2202 2202 2203 2204 2205 2207

CONTENTS

38.58. 38.59. 38.60. 38.61. 38.62. 38.63. 38.64. 38.65. 38.66. 38.67. 38.68. 38.69. 38.70. 38.71. 38.72. 38.73. 38.74. 38.75. 38.76. 38.77. 38.78. 38.79. 38.80. 38.81. 38.82. 38.83. 38.84. 38.85. 38.86. 38.87. 38.88. 38.89. 38.90. 38.91. 38.92. 38.93. 38.94. 38.95. 38.96. 38.97. 38.98. Chapter 39.1. 39.2. 39.3. 39.4. 39.5. 39.6.

Cohomology of curves Brauer groups Higher vanishing for the multiplicative group Picards groups of curves Constructible sheaves Extension by zero Higher vanishing for torsion sheaves The trace formula Frobenii Traces Why derived categories? Derived categories Filtered derived category Filtered derived functors Application of filtered complexes Perfectness Filtrations and perfect complexes Characterizing perfect objects Lefschetz numbers Preliminaries and sorites Proof of the trace formula Applications On l-adic sheaves L-functions Cohomological interpretation List of things which we should add above Examples of L-functions Constant sheaves The Legendre family Exponential sums Trace formula in terms of fundamental groups Fundamental groups Profinite groups, cohomology and homology Cohomology of curves, revisited Abstract trace formula Automorphic forms and sheaves Counting points Precise form of Chebotarov How many primes decompose completely? How many points are there really? Other chapters 39. Crystalline Cohomology Introduction Divided powers Divided power rings Extending divided powers Divided power polynomial algebras Divided power envelope

27

2209 2209 2212 2214 2216 2217 2219 2222 2222 2226 2226 2227 2228 2229 2230 2230 2231 2232 2233 2237 2240 2243 2243 2244 2245 2248 2248 2248 2250 2251 2252 2252 2254 2255 2257 2258 2261 2261 2262 2263 2264 2267 2267 2267 2271 2273 2275 2277

28

CONTENTS

39.7. Some explicit divided power thickenings 39.8. Compatibility 39.9. Affine crystalline site 39.10. Module of differentials 39.11. Divided power schemes 39.12. The big crystalline site 39.13. The crystalline site 39.14. Sheaves on the crystalline site 39.15. Crystals in modules 39.16. Sheaf of differentials 39.17. Two universal thickenings 39.18. The de Rham complex 39.19. Connections 39.20. Cosimplicial algebra 39.21. Notes on Rlim 39.22. Crystals in quasi-coherent modules 39.23. General remarks on cohomology 39.24. Cosimplicial preparations 39.25. Divided power Poincar´e lemma 39.26. Cohomology in the affine case 39.27. Two counter examples 39.28. Applications 39.29. Some further results 39.30. Pulling back along αp -covers 39.31. Frobenius action on crystalline cohomology 39.32. Other chapters

2281 2282 2283 2286 2292 2293 2296 2298 2300 2301 2303 2304 2304 2306 2307 2310 2315 2316 2318 2319 2322 2324 2325 2331 2336 2338

Chapter 40. Algebraic Spaces 40.1. Introduction 40.2. General remarks 40.3. Representable morphisms of presheaves 40.4. Lists of useful properties of morphisms of schemes 40.5. Properties of representable morphisms of presheaves 40.6. Algebraic spaces 40.7. Fibre products of algebraic spaces 40.8. Glueing algebraic spaces 40.9. Presentations of algebraic spaces 40.10. Algebraic spaces and equivalence relations 40.11. Algebraic spaces, retrofitted 40.12. Immersions and Zariski coverings of algebraic spaces 40.13. Separation conditions on algebraic spaces 40.14. Examples of algebraic spaces 40.15. Change of big site 40.16. Change of base scheme 40.17. Other chapters

2341 2341 2341 2342 2343 2345 2347 2348 2349 2351 2351 2356 2358 2359 2360 2364 2365 2368

Chapter 41. Properties of Algebraic Spaces 41.1. Introduction 41.2. Conventions

2369 2369 2369

CONTENTS

41.3. 41.4. 41.5. 41.6. 41.7. 41.8. 41.9. 41.10. 41.11. 41.12. 41.13. 41.14. 41.15. 41.16. 41.17. 41.18. 41.19. 41.20. 41.21. 41.22. 41.23. 41.24. 41.25. 41.26. 41.27. 41.28. 41.29. 41.30. 41.31. 41.32. 41.33.

Separation axioms Points of algebraic spaces Quasi-compact spaces Special coverings Properties of Spaces defined by properties of schemes Dimension at a point Reduced spaces The schematic locus Obtaining a scheme Points on quasi-separated spaces Noetherian spaces ´ Etale morphisms of algebraic spaces Spaces and fpqc coverings The ´etale site of an algebraic space Points of the small ´etale site Supports of abelian sheaves The structure sheaf of an algebraic space Stalks of the structure sheaf Dimension of local rings Local irreducibility Regular algebraic spaces Sheaves of modules on algebraic spaces ´ Etale localization Recovering morphisms Quasi-coherent sheaves on algebraic spaces Properties of modules Locally projective modules Quasi-coherent sheaves and presentations Morphisms towards schemes Quotients by free actions Other chapters

Chapter 42. Morphisms of Algebraic Spaces 42.1. Introduction 42.2. Conventions 42.3. Properties of representable morphisms 42.4. Immersions 42.5. Separation axioms 42.6. Surjective morphisms 42.7. Open morphisms 42.8. Submersive morphisms 42.9. Quasi-compact morphisms 42.10. Universally closed morphisms 42.11. Valuative criteria 42.12. Valuative criterion for universal closedness 42.13. Valuative criterion of separatedness 42.14. Monomorphisms 42.15. Pushforward of quasi-coherent sheaves 42.16. Closed immersions

29

2369 2370 2374 2375 2377 2378 2379 2380 2382 2383 2384 2385 2388 2390 2397 2402 2403 2404 2405 2405 2407 2407 2408 2410 2414 2417 2418 2419 2421 2421 2422 2425 2425 2425 2425 2426 2427 2432 2433 2435 2436 2438 2442 2446 2447 2448 2450 2451

30

CONTENTS

42.17. Closed immersions and quasi-coherent sheaves 42.18. Supports of modules 42.19. Universally injective morphisms 42.20. Affine morphisms 42.21. Quasi-affine morphisms 42.22. Types of morphisms ´etale local on source-and-target 42.23. Morphisms of finite type 42.24. Points and geometric points 42.25. Points of finite type 42.26. Quasi-finite morphisms 42.27. Morphisms of finite presentation 42.28. Flat morphisms 42.29. Flat modules 42.30. Generic flatness 42.31. Relative dimension 42.32. Morphisms and dimensions of fibres 42.33. Syntomic morphisms 42.34. Smooth morphisms 42.35. Unramified morphisms ´ 42.36. Etale morphisms 42.37. Proper morphisms 42.38. Integral and finite morphisms 42.39. Finite locally free morphisms 42.40. Normalization of algebraic spaces 42.41. Separated, locally quasi-finite morphisms 42.42. Applications 42.43. Universal homeomorphisms 42.44. Other chapters

2453 2455 2457 2460 2461 2461 2464 2466 2468 2471 2473 2476 2479 2481 2482 2483 2486 2487 2489 2492 2494 2495 2497 2498 2499 2501 2502 2503

Chapter 43. Decent Algebraic Spaces 43.1. Introduction 43.2. Conventions 43.3. Universally bounded fibres 43.4. Finiteness conditions and points 43.5. Conditions on algebraic spaces 43.6. Reasonable and decent algebraic spaces 43.7. Points and specializations 43.8. Schematic locus 43.9. Points on spaces 43.10. Reduced singleton spaces 43.11. Decent spaces 43.12. Valuative criterion 43.13. Relative conditions 43.14. Monomorphisms 43.15. Other chapters

2505 2505 2505 2505 2507 2512 2515 2517 2518 2520 2522 2525 2525 2527 2531 2531

Chapter 44. Cohomology of Algebraic Spaces 44.1. Introduction 44.2. Conventions

2533 2533 2533

CONTENTS

44.3. 44.4. 44.5. 44.6. 44.7. 44.8. 44.9. 44.10. 44.11. 44.12. 44.13. 44.14. 44.15. 44.16.

Derived category of quasi-coherent modules Higher direct images Colimits and cohomology ˇ The alternating Cech complex Higher vanishing for quasi-coherent sheaves Vanishing for higher direct images Cohomology and base change, I Coherent modules on locally Noetherian algebraic spaces Coherent sheaves on Noetherian spaces Devissage of coherent sheaves Limits of coherent modules Vanishing cohomology Finite morphisms and affines Other chapters

31

2533 2533 2535 2536 2540 2542 2543 2544 2546 2547 2551 2552 2556 2557

Chapter 45. Limits of Algebraic Spaces 45.1. Introduction 45.2. Conventions 45.3. Morphisms of finite presentation 45.4. Limits of algebraic spaces 45.5. Descending relative objects 45.6. More on limits 45.7. Absolute Noetherian approximation 45.8. Applications 45.9. Characterizing affine spaces 45.10. Other chapters

2559 2559 2559 2559 2564 2566 2568 2569 2571 2572 2574

Chapter 46.1. 46.2. 46.3. 46.4. 46.5. 46.6. 46.7. 46.8. 46.9.

2575 2575 2575 2576 2577 2578 2578 2579 2579 2579

46. Topologies on Algebraic Spaces Introduction The general procedure Fpqc topology Fppf topology Syntomic topology Smooth topology ´ Etale topology Zariski topology Other chapters

Chapter 47. Descent and Algebraic Spaces 47.1. Introduction 47.2. Conventions 47.3. Descent data for quasi-coherent sheaves 47.4. Fpqc descent of quasi-coherent sheaves 47.5. Descent of finiteness properties of modules 47.6. Fpqc coverings 47.7. Descent of finiteness properties of morphisms 47.8. Descending properties of spaces 47.9. Descending properties of morphisms 47.10. Descending properties of morphisms in the fpqc topology

2581 2581 2581 2581 2583 2583 2585 2586 2586 2587 2589

32

CONTENTS

47.11. 47.12. 47.13. 47.14. 47.15. 47.16. 47.17. 47.18. 47.19.

Descending properties of morphisms in the fppf topology 2597 Properties of morphisms local on the source 2598 Properties of morphisms local in the fpqc topology on the source 2599 Properties of morphisms local in the fppf topology on the source 2599 Properties of morphisms local in the syntomic toplogy on the source2600 Properties of morphisms local in the smooth topology on the source2600 Properties of morphisms local in the ´etale topology on the source 2600 Properties of morphisms smooth local on source-and-target 2600 Other chapters 2603

Chapter 48. More on Morphisms of Spaces 48.1. Introduction 48.2. Conventions 48.3. Radicial morphisms 48.4. Conormal sheaf of an immersion 48.5. Sheaf of differentials of a morphism 48.6. Topological invariance of the ´etale site 48.7. Thickenings 48.8. First order infinitesimal neighbourhood 48.9. Formally smooth, ´etale, unramified transformations 48.10. Formally unramified morphisms 48.11. Universal first order thickenings 48.12. Formally ´etale morphisms 48.13. Infinitesimal deformations of maps 48.14. Infinitesimal deformations of algebraic spaces 48.15. Formally smooth morphisms 48.16. Pushouts in the category of algebraic spaces 48.17. Openness of the flat locus 48.18. Crit`ere de platitude par fibres 48.19. Slicing Cohen-Macaulay morphisms 48.20. The structure of quasi-finite morphisms 48.21. Regular immersions 48.22. Pseudo-coherent morphisms 48.23. Perfect morphisms 48.24. Local complete intersection morphisms 48.25. Exact sequences of differentials and conormal sheaves 48.26. Other chapters

2605 2605 2605 2605 2607 2609 2613 2615 2619 2620 2624 2626 2631 2633 2635 2635 2641 2646 2647 2651 2652 2652 2654 2655 2656 2659 2659

Chapter 49.1. 49.2. 49.3. 49.4.

49. Quot and Hilbert Spaces Introduction Conventions When is a morphism an isomorphism? Other chapters

2661 2661 2661 2661 2666

Chapter 50.1. 50.2. 50.3. 50.4.

50. Algebraic Spaces over Fields Introduction Conventions Geometric components Schematic locus

2667 2667 2667 2667 2668

CONTENTS

50.5. 50.6.

Spaces smooth over fields Other chapters

33

2669 2669

Chapter 51. Stacks 51.1. Introduction 51.2. Presheaves of morphisms associated to fibred categories 51.3. Descent data in fibred categories 51.4. Stacks 51.5. Stacks in groupoids 51.6. Stacks in setoids 51.7. The inertia stack 51.8. Stackification of fibred categories 51.9. Stackification of categories fibred in groupoids 51.10. Inherited topologies 51.11. Gerbes 51.12. Functoriality for stacks 51.13. Stacks and localization 51.14. Other chapters

2671 2671 2671 2673 2675 2678 2679 2682 2682 2686 2687 2689 2692 2700 2701

Chapter 52. Formal Deformation Theory 52.1. Introduction 52.2. Notation and Conventions 52.3. The category CΛ 52.4. The category CbΛ 52.5. Categories cofibered in groupoids 52.6. Prorepresentable functors and predeformation categories 52.7. Formal objects and completion categories 52.8. Smooth morphisms 52.9. Schlessinger’s conditions 52.10. Tangent spaces of functors 52.11. Tangent spaces of predeformation categories 52.12. Versal formal objects 52.13. Minimal versal formal objects 52.14. Miniversal formal objects and tangent spaces 52.15. Rim-Schlessinger conditions and deformation categories 52.16. Lifts of objects 52.17. Schlessinger’s theorem on prorepresentable functors 52.18. Infinitesimal automorphisms 52.19. Groupoids in functors on an arbitrary category 52.20. Groupoids in functors on CΛ 52.21. Smooth groupoids in functors on CΛ 52.22. Deformation categories as quotients of groupoids in functors 52.23. Presentations of categories cofibered in groupoids 52.24. Presentations of deformation categories 52.25. Remarks regarding minimality 52.26. Change of residue field 52.27. Other chapters

2703 2703 2705 2705 2711 2714 2716 2717 2721 2726 2732 2734 2737 2741 2744 2747 2751 2754 2754 2758 2760 2761 2762 2763 2764 2765 2768 2770

Chapter 53.

2773

Groupoids in Algebraic Spaces

34

CONTENTS

53.1. 53.2. 53.3. 53.4. 53.5. 53.6. 53.7. 53.8. 53.9. 53.10. 53.11. 53.12. 53.13. 53.14. 53.15. 53.16. 53.17. 53.18. 53.19. 53.20. 53.21. 53.22. 53.23. 53.24. 53.25. 53.26. 53.27. 53.28. 53.29.

Introduction Conventions Notation Equivalence relations Group algebraic spaces Properties of group algebraic spaces Examples of group algebraic spaces Actions of group algebraic spaces Principal homogeneous spaces Equivariant quasi-coherent sheaves Groupoids in algebraic spaces Quasi-coherent sheaves on groupoids Crystals in quasi-coherent sheaves Groupoids and group spaces The stabilizer group algebraic space Restricting groupoids Invariant subspaces Quotient sheaves Quotient stacks Functoriality of quotient stacks The 2-cartesian square of a quotient stack The 2-coequalizer property of a quotient stack Explicit description of quotient stacks Restriction and quotient stacks Inertia and quotient stacks Gerbes and quotient stacks Quotient stacks and change of big site Separation conditions Other chapters

2773 2773 2773 2774 2775 2775 2776 2777 2778 2779 2780 2781 2783 2785 2786 2787 2788 2789 2791 2793 2795 2796 2797 2799 2801 2802 2803 2804 2805

Chapter 54. More on Groupoids in Spaces 54.1. Introduction 54.2. Notation 54.3. Useful diagrams 54.4. Properties of groupoids 54.5. Comparing fibres 54.6. Restricting groupoids 54.7. Properties of groups over fields and groupoids on fields 54.8. The finite part of a morphism 54.9. Finite collections of arrows 54.10. The finite part of a groupoid ´ 54.11. Etale localization of groupoid schemes 54.12. Other chapters

2807 2807 2807 2807 2808 2809 2809 2810 2813 2820 2821 2822 2826

Chapter 55.1. 55.2. 55.3. 55.4.

2829 2829 2829 2829 2832

55. Bootstrap Introduction Conventions Morphisms representable by algebraic spaces Properties of maps of presheaves representable by algebraic spaces

CONTENTS

55.5. 55.6. 55.7. 55.8. 55.9. 55.10. 55.11. 55.12. 55.13.

Bootstrapping the diagonal Bootstrap Finding opens Slicing equivalence relations Quotient by a subgroupoid Final bootstrap Applications Algebraic spaces in the ´etale topology Other chapters

35

2833 2835 2836 2838 2839 2841 2843 2846 2848

Chapter 56. Examples of Stacks 2851 56.1. Introduction 2851 56.2. Notation 2851 56.3. Examples of stacks 2851 56.4. Quasi-coherent sheaves 2851 56.5. The stack of finitely generated quasi-coherent sheaves 2852 56.6. Algebraic spaces 2854 56.7. The stack of finite type algebraic spaces 2855 56.8. Examples of stacks in groupoids 2857 56.9. The stack associated to a sheaf 2857 56.10. The stack in groupoids of finitely generated quasi-coherent sheaves2857 56.11. The stack in groupoids of finite type algebraic spaces 2857 56.12. Quotient stacks 2857 56.13. Classifying torsors 2858 56.14. Quotients by group actions 2862 56.15. The Picard stack 2865 56.16. Examples of inertia stacks 2866 56.17. Finite Hilbert stacks 2866 56.18. Other chapters 2868 Chapter 57. Quotients of Groupoids 57.1. Introduction 57.2. Conventions and notation 57.3. Invariant morphisms 57.4. Categorical quotients 57.5. Quotients as orbit spaces 57.6. Coarse quotients 57.7. Topological properties 57.8. Invariant functions 57.9. Good quotients 57.10. Geometric quotients 57.11. Other chapters

2871 2871 2871 2871 2872 2874 2882 2883 2883 2884 2884 2884

Chapter 58.1. 58.2. 58.3. 58.4. 58.5.

2887 2887 2887 2887 2888 2888

58. Algebraic Stacks Introduction Conventions Notation Representable categories fibred in groupoids The 2-Yoneda lemma

36

CONTENTS

58.6. 58.7. 58.8. 58.9. 58.10. 58.11. 58.12. 58.13. 58.14. 58.15. 58.16. 58.17. 58.18. 58.19. 58.20.

Representable morphisms of categories fibred in groupoids Split categories fibred in groupoids Categories fibred in groupoids representable by algebraic spaces Morphisms representable by algebraic spaces Properties of morphisms representable by algebraic spaces Stacks in groupoids Algebraic stacks Algebraic stacks and algebraic spaces 2-Fibre products of algebraic stacks Algebraic stacks, overhauled From an algebraic stack to a presentation The algebraic stack associated to a smooth groupoid Change of big site Change of base scheme Other chapters

2889 2890 2891 2891 2894 2897 2898 2900 2901 2902 2905 2908 2909 2910 2911

Chapter 59. Sheaves on Algebraic Stacks 59.1. Introduction 59.2. Conventions 59.3. Presheaves 59.4. Sheaves 59.5. Computing pushforward 59.6. The structure sheaf 59.7. Sheaves of modules 59.8. Representable categories 59.9. Restriction 59.10. Restriction to algebraic spaces 59.11. Quasi-coherent modules 59.12. Stackification and sheaves 59.13. Quasi-coherent sheaves and presentations 59.14. Quasi-coherent sheaves on algebraic stacks 59.15. Cohomology 59.16. Injective sheaves ˇ 59.17. The Cech complex ˇ 59.18. The relative Cech complex 59.19. Cohomology on algebraic stacks 59.20. Higher direct images and algebraic stacks 59.21. Comparison 59.22. Change of topology 59.23. Other chapters

2913 2913 2913 2914 2916 2918 2920 2921 2922 2922 2924 2927 2930 2931 2933 2934 2935 2937 2939 2945 2946 2948 2948 2951

Chapter 60.1. 60.2. 60.3. 60.4. 60.5. 60.6. 60.7.

2953 2953 2953 2953 2954 2956 2959 2961

60. Criteria for Representability Introduction Conventions What we already know Morphisms of stacks in groupoids Limit preserving on objects Formally smooth on objects Surjective on objects

CONTENTS

60.8. Algebraic morphisms 60.9. Spaces of sections 60.10. Relative morphisms 60.11. Restriction of scalars 60.12. Finite Hilbert stacks 60.13. The finite Hilbert stack of a point 60.14. Finite Hilbert stacks of spaces 60.15. LCI locus in the Hilbert stack 60.16. Bootstrapping algebraic stacks 60.17. Applications 60.18. When is a quotient stack algebraic? 60.19. Algebraic stacks in the ´etale topology 60.20. Other chapters

37

2962 2963 2965 2968 2970 2974 2977 2978 2981 2982 2983 2985 2986

Chapter 61. Artin’s axioms 61.1. Introduction 61.2. Conventions 61.3. Predeformation categories 61.4. Pushouts and stacks 61.5. The Rim-Schlessinger condition 61.6. Deformation categories 61.7. Change of field 61.8. Tangent spaces 61.9. Formal objects 61.10. Approximation 61.11. Versality 61.12. Axioms 61.13. Limit preserving 61.14. Openness of versality 61.15. Axioms for functors 61.16. Algebraic spaces 61.17. Algebraic stacks 61.18. Infinitesimal deformations 61.19. Obstruction theories 61.20. Naive obstruction theories 61.21. A dual notion 61.22. Examples of deformation problems 61.23. Other chapters

2989 2989 2989 2989 2991 2992 2993 2994 2995 2997 3001 3003 3005 3006 3007 3009 3011 3012 3013 3017 3019 3023 3026 3026

Chapter 62.1. 62.2. 62.3. 62.4. 62.5. 62.6. 62.7. 62.8. 62.9.

3029 3029 3029 3030 3035 3039 3040 3040 3042 3043

62. Properties of Algebraic Stacks Introduction Conventions and abuse of language Properties of morphisms representable by algebraic spaces Points of algebraic stacks Surjective morphisms Quasi-compact algebraic stacks Properties of algebraic stacks defined by properties of schemes Monomorphisms of algebraic stacks Immersions of algebraic stacks

38

CONTENTS

62.10. 62.11. 62.12.

Reduced algebraic stacks Residual gerbes Other chapters

3050 3051 3055

Chapter 63. Morphisms of Algebraic Stacks 63.1. Introduction 63.2. Conventions and abuse of language 63.3. Properties of diagonals 63.4. Separation axioms 63.5. Inertia stacks 63.6. Higher diagonals 63.7. Quasi-compact morphisms 63.8. Noetherian algebraic stacks 63.9. Open morphisms 63.10. Submersive morphisms 63.11. Universally closed morphisms 63.12. Types of morphisms smooth local on source-and-target 63.13. Morphisms of finite type 63.14. Points of finite type 63.15. Special presentations of algebraic stacks 63.16. Quasi-finite morphisms 63.17. Flat morphisms 63.18. Morphisms of finite presentation 63.19. Gerbes 63.20. Stratification by gerbes 63.21. Existence of residual gerbes 63.22. Smooth morphisms 63.23. Other chapters

3057 3057 3057 3057 3060 3065 3068 3069 3071 3071 3072 3072 3073 3076 3077 3080 3086 3090 3091 3093 3098 3100 3101 3101

Chapter 64. Cohomology of Algebraic Stacks 64.1. Introduction 64.2. Conventions and abuse of language 64.3. Notation 64.4. Pullback of quasi-coherent modules 64.5. The key lemma 64.6. Locally quasi-coherent modules 64.7. Flat comparison maps 64.8. Parasitic modules 64.9. Quasi-coherent modules, I 64.10. Pushforward of quasi-coherent modules 64.11. The lisse-´etale and the flat-fppf sites 64.12. Quasi-coherent modules, II 64.13. Derived categories of quasi-coherent modules 64.14. Derived pushforward of quasi-coherent modules 64.15. Derived pullback of quasi-coherent modules 64.16. Other chapters

3103 3103 3103 3103 3104 3104 3106 3108 3112 3114 3115 3119 3125 3129 3132 3133 3134

Chapter 65. Introducing Algebraic Stacks 65.1. Why read this?

3135 3135

CONTENTS

65.2. 65.3. 65.4. 65.5. 65.6. 65.7. 65.8.

Preliminary The moduli stack of elliptic curves Fibre products The definition A smooth cover Properties of algebraic stacks Other chapters

Chapter 66. Examples 66.1. Introduction 66.2. Noncomplete completion 66.3. Noncomplete quotient 66.4. Completion is not exact 66.5. The category of complete modules is not abelian 66.6. Regular sequences and base change 66.7. A Noetherian ring of infinite dimension 66.8. Local rings with nonreduced completion 66.9. A non catenary Noetherian local ring 66.10. Non-quasi-affine variety with quasi-affine normalization 66.11. A locally closed subscheme which is not open in closed 66.12. Pushforward of quasi-coherent modules 66.13. A nonfinite module with finite free rank 1 stalks 66.14. A finite flat module which is not projective 66.15. A projective module which is not locally free 66.16. Zero dimensional local ring with nonzero flat ideal 66.17. An epimorphism of zero-dimensional rings which is not surjective 66.18. Finite type, not finitely presented, flat at prime 66.19. Finite type, flat and not of finite presentation 66.20. Topology of a finite type ring map 66.21. Pure not universally pure 66.22. A formally smooth non-flat ring map 66.23. A formally ´etale non-flat ring map 66.24. A formally ´etale ring map with nontrivial cotangent complex 66.25. Ideals generated by sets of idempotents and localization 66.26. Non flasque quasi-coherent sheaf associated to injective module 66.27. A non-separated flat group scheme 66.28. A non-flat group scheme with flat identity component 66.29. A non-separated group algebraic space over a field 66.30. Specializations between points in fibre ´etale morphism 66.31. A torsor which is not an fppf torsor 66.32. Stack with quasi-compact flat covering which is not algebraic 66.33. Limit preserving on objects, not limit preserving 66.34. A non-algebraic classifying stack 66.35. Sheaf with quasi-compact flat covering which is not algebraic 66.36. Sheaves and specializations 66.37. Sheaves and constructible functions 66.38. The lisse-´etale site is not functorial 66.39. Derived pushforward of quasi-coherent modules 66.40. A big abelian category

39

3135 3136 3137 3138 3139 3140 3141 3143 3143 3143 3144 3145 3146 3146 3148 3148 3149 3150 3152 3152 3153 3153 3153 3156 3156 3156 3158 3158 3159 3160 3161 3161 3162 3163 3163 3164 3164 3165 3165 3166 3167 3167 3168 3169 3170 3172 3172 3173

40

CONTENTS

66.41.

Other chapters

3174

Chapter 67. Exercises 67.1. Algebra 67.2. Colimits 67.3. Additive and abelian categories 67.4. Flat ring maps 67.5. The Spectrum of a ring 67.6. Localization 67.7. Nakayama’s Lemma 67.8. Length 67.9. Singularities 67.10. Hilbert Nullstellensatz 67.11. Dimension 67.12. Catenary rings 67.13. Fraction fields 67.14. Transcendence degree 67.15. Finite locally free modules 67.16. Glueing 67.17. Going up and going down 67.18. Fitting ideals 67.19. Hilbert functions 67.20. Proj of a ring 67.21. Cohen-Macaulay rings of dimension 1 67.22. Infinitely many primes 67.23. Filtered derived category 67.24. Regular functions 67.25. Sheaves 67.26. Schemes 67.27. Morphisms 67.28. Tangent Spaces 67.29. Quasi-coherent Sheaves 67.30. Proj and projective schemes 67.31. Morphisms from surfaces to curves 67.32. Invertible sheaves ˇ 67.33. Cech Cohomology 67.34. Divisors 67.35. Differentials 67.36. Schemes, Final Exam, Fall 2007 67.37. Schemes, Final Exam, Spring 2009 67.38. Schemes, Final Exam, Fall 2010 67.39. Schemes, Final Exam, Spring 2011 67.40. Schemes, Final Exam, Fall 2011 67.41. Other chapters

3177 3177 3178 3179 3180 3180 3182 3183 3183 3184 3184 3185 3185 3185 3185 3186 3187 3187 3188 3188 3189 3191 3193 3194 3196 3196 3198 3199 3200 3202 3203 3204 3205 3206 3207 3209 3211 3212 3214 3214 3216 3217

Chapter 68.1. 68.2. 68.3.

3219 3219 3219 3219

68. A Guide to the Literature Short introductory articles Classic references Books and online notes

CONTENTS

68.4. 68.5. 68.6. 68.7. 68.8.

Related references on foundations of stacks Papers in the literature Stacks in other fields Higher stacks Other chapters

41

3220 3221 3231 3231 3231

Chapter 69. Desirables 69.1. Introduction 69.2. Conventions 69.3. Sites and Topoi 69.4. Stacks 69.5. Simplicial methods 69.6. Cohomology of schemes 69.7. Deformation theory a la Schlessinger 69.8. Definition of algebraic stacks 69.9. Examples of schemes, algebraic spaces, algebraic stacks 69.10. Properties of algebraic stacks 69.11. Lisse ´etale site of an algebraic stack 69.12. Things you always wanted to know but were afraid to ask 69.13. Quasi-coherent sheaves on stacks 69.14. Flat and smooth 69.15. Artin’s representability theorem 69.16. DM stacks are finitely covered by schemes 69.17. Martin Olson’s paper on properness 69.18. Proper pushforward of coherent sheaves 69.19. Keel and Mori 69.20. Add more here 69.21. Other chapters

3233 3233 3233 3233 3233 3233 3234 3234 3234 3234 3235 3235 3235 3235 3235 3235 3235 3235 3236 3236 3236 3236

Chapter 70. Coding Style 70.1. List of style comments 70.2. Other chapters

3239 3239 3241

Chapter 71.1. 71.2. 71.3. 71.4. 71.5. 71.6.

71. Obsolete Introduction Lemmas related to ZMT Formally smooth ring maps Devissage of coherent sheaves Very reasonable algebraic spaces Other chapters

3243 3243 3243 3245 3245 3246 3247

Chapter 72.1. 72.2. 72.3. 72.4. 72.5. 72.6. 72.7. 72.8.

72. GNU Free Documentation License APPLICABILITY AND DEFINITIONS VERBATIM COPYING COPYING IN QUANTITY MODIFICATIONS COMBINING DOCUMENTS COLLECTIONS OF DOCUMENTS AGGREGATION WITH INDEPENDENT WORKS TRANSLATION

3249 3249 3251 3251 3251 3253 3253 3254 3254

42

CONTENTS

72.9. TERMINATION 72.10. FUTURE REVISIONS OF THIS LICENSE 72.11. ADDENDUM: How to use this License for your documents 72.12. Other chapters Chapter 73.1. 73.2. 73.3.

73. Auto generated index Alphabetized definitions Definitions listed per chapter Other chapters

Bibliography

3254 3254 3255 3255 3257 3257 3281 3302 3305

CHAPTER 1

Introduction 1.1. Overview Besides the book by Laumon and Moret-Bailly, see [LMB00], and the work (in progress) by Fulton et al, we think there is a place for an open source textbook on algebraic stacks and the algebraic geometry that is needed to define them. The Stacks Project attempts to do this by building the foundations starting with commutative algebra and proceeding via the theory of schemes and algebraic spaces to a comprehensive foundation for the theory of algebraic stacks. We expect this material to be read online as a key feature are the hyperlinks giving quick access to internal references spread over many different pages. If you use an embedded pdf or dvi viewer in your browser, the cross file links should work. This project is a collaborative effort and we encourage you to help out. Please email any typos or errors you find while reading or any suggestions, additional material, or examples you have to [email protected]. You can download a tarball containing all source files, extract, run make, and use a dvi or pdf viewer locally. Please feel free to edit the LaTeX files and email your improvements. 1.2. Attribution The scope of this work is such that it is a daunting task to attribute correctly and succinctly all of those mathematicians whose work has led to the development of the theory we try to explain here. We hope eventually to generate enough community interest to find contributors willing to write sections with historical remarks for each and every chapter. Those who contributed to this work are listed on the title page of the book version of this work and online. Here we would like to name a selection of major contributions: (1) (2) (3) (4) (5) (6) (7) (8)

Jarod Alper wrote Guide to Literature. ´ Bhargav Bhatt wrote the initial version of Etale Morphisms of Schemes. Bhargav Bhatt wrote the initial version of More on Algebra, Section 12.10. Algebra, Section 7.26 and Injectives, Section 17.6 are from The CRing Project, courtesy of Akhil Mathew. Alex Perry wrote the material on projective modules, Mittag-Leffler modules, including the proof of Algebra, Theorem 7.90.5. Alex Perry wrote Formal Deformation Theory. Thibaut Pugin, Zachary Maddock and Min Lee took course notes which ´ formed the basis for Etale Cohomology. David Rydh has contributed many helpful comments, pointed out several mistakes, helped out in an essential way with the material on residual 43

44

1. INTRODUCTION

gerbes, and was the originator for the material in More on Groupoids in Spaces, Sections 54.8 and 54.11. 1.3. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology

(39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 2

Conventions 2.1. Comments The philosophy behind the conventions used in writing these documents is to choose those conventions that work. 2.2. Set theory We use Zermelo-Fraenkel set theory with the axiom of choice. See [Kun83]. We do not use universes (different from SGA4). We do not stress set-theoretic issues, but we make sure everything is correct (of course) and so we do not ignore them either. 2.3. Categories A category C consists of a set of objects and, for each pair of objects, a set of morphisms between them. In other words, it is what is called a “small” category in other texts. We will use “big” categories (categories whose objects form a proper class) as well, but only those that are listed in Categories, Remark 4.2.2. 2.4. Algebra In these notes a ring is a commutative ring with a 1. Hence the category of rings has an initial object Z and a final object {0} (this is the unique ring where 1 = 0). Modules are assumed unitary. See [Eis95]. 2.5. Notation The natural integers are elements of N = {1, 2, 3, . . .}. The integers are elements of Z = {. . . , −2, −1, 0, 1, 2, . . .}. The field of rational numbers is denoted Q. The field of real numbers is denoted R. The field of complex numbers is denoted C. 2.6. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra

(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 45

Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings

46

2. CONVENTIONS

(21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44)

Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces

(48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 3

Set Theory 3.1. Introduction We need some set theory every now and then. We use Zermelo-Fraenkel set theory with the axiom of choice (ZFC) as described in [Kun83] and [Jec02]. 3.2. Everything is a set Most mathematicians think of set theory as providing the basic foundations for mathematics. So how does this really work? For example, how do we translate the sentence “X is a scheme” into set theory? Well, we just unravel the definitions: A scheme is a locally ringed space such that every point has an open neighbourhood which is an affine scheme. A locally ringed space is a ringed space such that every stalk of the structure sheaf is a local ring. A ringed space is a pair (X, OX ) consisting of a topological space X and a sheaf of rings OX on it. A topological space is a pair (X, τ ) consisting of a set X and a set of subsets τ ⊂ P(X) satisfying the axioms of a topology. And so on and so forth. So how, given a set S would we recognize whether it is a scheme? The first thing we look for is whether the set S is an ordered pair. This is defined (see [Jec02], page 7) as saying that S has the form (a, b) := {{a}, {a, b}} for some sets a, b. If this is the case, then we would take a look to see whether a is an ordered pair (c, d). If so we would check whether d ⊂ P(c), and if so whether d forms the collection of sets for a topology on the set c. And so on and so forth. So even though it would take a considerable amount of work to write a complete formula φscheme (x) with one free variable x in set theory that expresses the notion “x is a scheme”, it is possible to do so. The same thing should be true for any mathematical object. 3.3. Classes Informally we use the notion of a class. Given a formula φ(x, p1 , . . . , pn ), we call C = {x : φ(x, p1 , . . . , pn )} a class. A class is easier to manipulate than the formula that defines it, but it is not strictly speaking a mathematical object. For example, if R is a ring, then we may consider the class of all R-modules (since after all we may translate the sentence “M is an R-module” into a formula in set theory, which then defines a class). A proper class is a class which is not a set. In this way we may consider the category of R-modules, which is a “big” category— in other words, it has a proper class of objects. Similarly, we may consider the “big” category of schemes, the “big” category of rings, etc. 47

48

3. SET THEORY

3.4. Ordinals A set T is transitive if x ∈ T implies x ⊂ T . A set α is an ordinal if it is transitive and well-ordered by ∈. In this case, we define α + 1 = α ∪ {α}, which is another ordinal called the successor of α. An ordinal α is called a successor ordinal if there exists an ordinal β such that α = β + 1. The smallest ordinal is ∅ which is also denoted 0. If α is not 0, and not a successor ordinal, then α is called a limit ordinal and we have [ α= γ. γ∈α

The first limit ordinal is ω and it is also the first infinite ordinal. The collection of all ordinals is a proper class. It is well-ordered by ∈ in the following sense: any nonempty set (or even class) of ordinals has a least element. Given a set A of S ordinals, we define the supremum of A to be supα∈A α = α∈A α. It is the least ordinal bigger or equal to all α ∈ A. Given any well ordered set (S, ≥), there is a unique ordinal α such that (S, ≥) ∼ = (α, ∈); this is called the order type of the well ordered set. 3.5. The hierarchy of sets We define, by transfinite induction, V0 = ∅, Vα+1 = P (Vα ) (power set), and for a limit ordinal α, [ Vα = Vβ . β α. + You can use this to define ℵ1 = ℵ+ 0 , ℵ2 = ℵ1 , etc, and in fact you can define ℵα for any ordinal α by transfinite induction. The addition of cardinals κ, λ is denoted κ ⊕ λ; it is the cardinality of κ q λ. The multiplication of cardinals κ, λ is denoted κ ⊗ λ; it is the cardinality of κ × λ. It is uninteresting since if κ and λ are infinite cardinals, then κ ⊗ λ = max(κ, λ). The exponentiation of cardinals κ, λ is denoted κλ ; it is the cardinality of the set of (set) maps S from λ to κ. Given any set K of cardinals, the supremum of K is supκ∈K κ = κ∈K κ, which is also a cardinal.

3.8. REFLECTION PRINCIPLE

49

3.7. Cofinality A cofinal subset S of a partially ordered set T is a subset S ⊂ T such that ∀t ∈ T ∃s ∈ S(t ≤ s). Note that a subset of a well-ordered set is a well-ordered set (with induced ordering). Given an ordinal α, the cofinality cf(α) of α is the least ordinal β which occurs as the order type of some cofinal subset of α. The cofinality of an ordinal is always a cardinal (this is clear from the definition). Hence alternatively we can define the cofinality of α as the least cardinality of a cofinal subset of α. Lemma 3.7.1. Suppose that Tβ = colimα |S|. Proof. For each element s ∈ S pick a αs < β and an element ts ∈ Tαs which maps to ϕ(s) in T . By assumption α = sups∈S αs is strictly smaller than β. Hence the map ϕα : S → Tα which assigns to s the image of ts in Tα is a solution.  The following is essentially Grothendieck’s argument for the existence of ordinals with arbitrarily large cofinality which he used to prove the existence of enough injectives in certain abelian categories, see [Gro57]. Proposition 3.7.2. Let κ be a cardinal. Then there exists an ordinal whose cofinality is bigger than κ. Proof. If κ is finite, then ω = cf(ω) works. Let us thus assume that κ is infinite. Consider the smallest ordinal α whose cardinality is strictly greater than κ. We claim that cf(α) > κ. Note that α is a limit ordinal, since if α = β + 1, then |α| = |β| (because α and β are infinite) and this contradicts the minimality of α. (Of course α is also a cardinal, but we do not need this.) To get a contradiction suppose S ⊂ α is a cofinal subset with |S| ≤ κ. For β ∈ S, i.e., β < α, we have |β| ≤Sκ by minimality of α. As α is a limit ordinal and S cofinal in α we obtain α = β∈S β. Hence |α| ≤ |S| ⊗ κ ≤ κ ⊗ κ ≤ κ which is a contradiction with our choice of α.  3.8. Reflection principle Some of this material is in the chapter of [Kun83] called “Easy consistency proofs”. Let φ(x1 , . . . , xn ) be a formula of set theory. Let us use the convention that this notation implies that all the free variables in φ occur among x1 , . . . , xn . Let M be a set. The formula φM (x1 , . . . , xn ) is the formula obtained from φ(x1 , . . . , xn ) by replacing all the ∀x and ∃x by ∀x ∈ M and ∃x ∈ M , respectively. So the formula φ(x1 , x2 ) = ∃x(x ∈ x1 ∧ x ∈ x2 ) is turned into φM (x1 , x2 ) = ∃x ∈ M (x ∈ x1 ∧ x ∈ x2 ). The formula φM is called the relativization of φ to M . Theorem 3.8.1. See [Jec02, Theorem 12.14] or [Kun83, Theorem 7.4]. Suppose given φ1 (x1 , . . . , xn ), . . . , φm (x1 , . . . , xn ) a finite collection of formulas of set theory. Let M0 be a set. There exists a set M such that M0 ⊂ M and ∀x1 , . . . , xn ∈ M , we have ∀i = 1, . . . , m, φM i (x1 , . . . , xn ) ⇔ ∀i = 1, . . . , m, φi (x1 , . . . , xn ). In fact we may take M = Vα for some limit ordinal α.

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We view this theorem as saying the following: Given any x1 , . . . , xn ∈ M the formulas hold with the bound variables ranging through all sets if and only if they hold for the bound variables ranging through elements of Vα . This theorem is a meta-theorem because it deals with the formulas of set theory directly. It actually says that given the finite list of formulas φ1 , . . . , φm with at most free variables x1 , . . . , xn the sentence ∀M0 ∃M, M0 ⊂ M ∀x1 , . . . , xn ∈ M M φ1 (x1 , . . . , xn ) ∧ . . . ∧ φm (x1 , . . . , xn ) ↔ φM 1 (x1 , . . . , xn ) ∧ . . . ∧ φm (x1 , . . . , xn ) is provable in ZFC. In other words, whenever we actually write down a finite list of formulas φi , we get a theorem. It is somewhat hard to use this theorem in “ordinary mathematics” since the meaning of the formulas φM i (x1 , . . . , xn ) is not so clear! Instead, we will use the idea of the proof of the reflection principle to prove the existence results we need directly. 3.9. Constructing categories of schemes We will discuss how to apply this to produce, given an initial set of schemes, a “small” category of schemes closed under a list of natural operations. Before we do so, we introduce the size of a scheme. Given a scheme S we define size(S) = max(ℵ0 , κ1 , κ2 ), where we define the cardinal numbers κ1 and κ2 as follows: (1) We let κ1 be the cardinality of the set of affine opens of S. (2) We let κ2 be the supremum of all the cardinalities of all Γ(U, OS ) for all U ⊂ S affine open. Lemma 3.9.1. For every cardinal κ, there exists a set A such that every element of A is a scheme and such that for every scheme S with Size(S) ≤ κ, there is an element X ∈ A such that X ∼ = S (isomorphism of schemes). Proof. Omitted. Hint: think about how any scheme is isomorphic to a scheme obtained by glueing affines.  We denote Bound the function which to each cardinal κ associates (3.9.1.1)

Bound(κ) = max{κℵ0 , κ+ }.

We could make this function grow much more rapidly, e.g., we could set Bound(κ) = κκ , and the result below would still hold. For any ordinal α, we denote Schα the full subcategory of category of schemes whose objects are elements of Vα . Here is the result we are going to prove. Lemma 3.9.2. With notations size, Bound and Schα as above. Let S0 be a set of schemes. There exists a limit ordinal α with the following properties: (1) We have S0 ⊂ Vα ; in other words, S0 ⊂ Ob(Schα ). (2) For any S ∈ Ob(Schα ) and any scheme T with size(T ) ≤ Bound(size(S)), there exists a scheme S 0 ∈ Ob(Schα ) such that T ∼ = S0. (3) For any countable diagram1 category I and any functor F : I → Schα , the limit limI F exists in Schα if and only if it exists in Sch and moreover, in this case, the natural morphism between them is an isomorphism. 1Both the set of objects and the morphism sets are countable. In fact you can prove the lemma with ℵ0 replaced by any cardinal whatsoever in (3) and (4).

3.9. CONSTRUCTING CATEGORIES OF SCHEMES

51

(4) For any countable diagram category I and any functor F : I → Schα , the colimit colimI F exists in Schα if and only if it exists in Sch and moreover, in this case, the natural morphism between them is an isomorphism. Proof. We define, by transfinite induction, a function f which associates to every ordinal an ordinal as follows. Let f (0) = 0. Given f (α), we define f (α + 1) to be the least ordinal β such that the following hold: (1) We have α + 1 ≤ β and f (α) ≤ β. (2) For any S ∈ Ob(Schf (α) ) and any scheme T with size(T ) ≤ Bound(size(S)), there exists a scheme S 0 ∈ Ob(Schβ ) such that T ∼ = S0. (3) For any countable diagram category I and any functor F : I → Schf (α) , if the limit limI F or the colimit colimI F exists in Sch, then it is isomorphic to a scheme in Schβ . To see β exists, we argue as follows. Since Ob(Schf (α) ) is a set, we see that κ = supS∈Ob(Schf (α) ) Bound(size(S)) exists and is a cardinal. Let A be a set of schemes obtained starting with κ as in Lemma 3.9.1. There is a set CountCat of countable categories such that any countable category is isomorphic to an element of CountCat. Hence in (3) above we may assume that I is an element in CountCat. This means that the pairs (I, F ) in (3) range over a set. Thus, there exists a set B whose elements are schemes such that for every (I, F ) as in (3), if the limit or colimit exists, then it is isomorphic to an element in B. Hence, if we pick any β such that A ∪ B ⊂ Vβ and β > max{α + 1, f (α)}, then (1)–(3) hold. Since every nonempty collection of ordinals has a least element, we see that f (α + 1) is well defined. Finally, if α is a limit ordinal, then we set f (α) = supα0 β0 with cofinality cf(β1 ) > ω = ℵ0 . This is possible since the cofinality of ordinals gets arbitrarily large, see Proposition 3.7.2. We claim that α = f (β1 ) is a solution to the problem posed in the lemma. The first property of the lemma holds by our choice of β1 > β0 above. Since β1 is a limit ordinal (as its cofinality is infinite), we get f (β1 ) = supβ κ. This is possible since the cofinality of ordinals gets arbitrarily large, see Proposition 3.7.2. We claim that the pair κ, α = f (β2 ) is a solution to the problem posed in the lemma. The first and third property of the lemma holds by our choices of κ, β2 > β1 > β0 above. Since β2 is a limit ordinal (as its cofinality is infinite) we get f (β2 ) = supβ 1. Since I is connected there exist indices i1 , i2 and j0 and morphisms a : xi1 → yj0 and b : xi2 → yj0 . Consider the category I 0 = {x} q {x1 , . . . , x ˆi1 , . . . , x ˆi2 , . . . xn } q {y1 , . . . , ym } with MorI 0 (x, yj ) = MorI (xi1 , yj ) q MorI (xi2 , yj ) and all other morphism sets the same as in I. For any functor M : I → C we can construct a functor M 0 : I 0 → C by setting M 0 (x) = M (xi1 ) ×M (a),M (yj ),M (b) M (xi2 )

4.16. FINITE LIMITS AND COLIMITS

75

and for a morphism f 0 : x → yj corresponding to, say, f : xi1 → yj we set M 0 (f ) = M (f ) ◦ pr1 . Then the functor M has a limit if and only if the functor M 0 has a limit (proof omitted). Hence by induction we reduce to the case n = 1. If n = 1, then the limit of any M : I → C is the successive equalizer of pairs of maps x1 → yj hence exists by assumption.  Lemma (1) (2) (3)

4.16.3. Let C be a category. The following are equivalent: Nonempty finite limits exist in C. Products of pairs and equalizers exist in C. Products of pairs and fibre products exist in C.

Proof. Since products of pairs, fibre products, and equalizers are limits with nonempty index categories we see that (1) implies both (2) and (3). Assume (2). Then finite nonempty products and equalizers exist. Hence by Lemma 4.13.10 we see that finite nonempty limits exist, i.e., (1) holds. Assume (3). If a, b : A → B are morphisms of C, then the equalizer of a, b is (A ×a,B,b A) ×(pr1 ,pr2 ),A×A,∆ A. Thus (3) implies (2), and the lemma is proved. Lemma (1) (2) (3)



4.16.4. Let C be a category. The following are equivalent: Finite limits exist in C. Finite products and equalizers exist. The category has a final object and fibred products exist.

Proof. Since products of pairs, fibre products, equalizers, and final objects limits over finite index categories we see that (1) implies both (2) and (3). Lemma 4.13.10 above we see that (2) implies (1). Assume (3). Note that product A × A is the fibre product over the final object. If a, b : A → B morphisms of C, then the equalizer of a, b is

are By the are

(A ×a,B,b A) ×(pr1 ,pr2 ),A×A,∆ A. Thus (3) implies (2) and the lemma is proved.



Lemma 4.16.5. Let C be a category. The following are equivalent: (1) Nonempty connected finite colimits exist in C. (2) Coequalizers and pushouts exist in C. Proof. Omitted. Hint: This is dual to Lemma 4.16.2. Lemma (1) (2) (3)

4.16.6. Let C be a category. The following are equivalent: Nonempty finite colimits exist in C. Coproducts of pairs and coequalizers exist in C. Coproducts of pairs and pushouts exist in C.

Proof. Omitted. Hint: This is the dual of Lemma 4.16.3. Lemma (1) (2) (3)





4.16.7. Let C be a category. The following are equivalent: finite colimits exist in C, finite coproducts and coequalizers exist in C, and C has an initial object and pushouts exist.

Proof. Omitted. Hint: This is dual to Lemma 4.16.4.



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4.17. Filtered colimits Colimits are easier to compute or describe when they are over a filtered diagram. Here is the definition. Definition 4.17.1. We say that a diagram M : I → C is directed, or filtered if the following conditions hold: (1) the category I has at least one object, (2) for every pair of objects x, y of I there exists an object z and morphisms x → z, y → z, and (3) for every pair of objects x, y of I and every pair of morphisms a, b : x → y of I there exists a morphism c : y → z of I such that M (c ◦ a) = M (c ◦ b) as morphisms in C. We say that an index category I is directed, or filtered if id : I → I is filtered (in other words you erase the M in part (3) above.) We observe that any diagram with filtered index category is filtered, and this is how filtered colimits usually come about. In fact, if M : I → C is a filtered diagram, then we can factor M as I → I 0 → C where I 0 is a filtered index category1 such that colimI M exists if and only if colimI 0 M 0 exists in which case the colimits are canonically isomorphic. Suppose that M : I → Sets is a filtered diagram. In this case we may describe the equivalence relation in the formula a colimI M = ( Mi )/ ∼ i∈I

simply as follows mi ∼ mi0 ⇔ ∃i00 , φ : i → i00 , φ0 : i0 → i00 , M (φ)(mi ) = M (φ0 )(mi0 ). In other words, two elements are equal in the colimit if and only if the “eventually become equal”. Lemma 4.17.2. Let I and J be index categories. Assume that I is filtered and J is finite. Let M : J × I → Sets, (i, j) 7→ Mi,j be a diagram of diagrams of sets. In this case colimi limj Mi,j = limj colimi Mi,j . In particular, colimits over I commute with finite products, fibre products, and equalizers of sets. Proof. Omitted.



Instead of giving the easy proof of the lemma we give a counter example to the case where J is infinite. Namely, let I consist of N = {1, 2, 3, . . .} with a unique morphism i → i0 whenever i ≤ i0 . Let J consist of the discrete category N = {1, 2, 3, . . .} (only morphisms are identities). Let Mi,j = {1, 2, . . . , i} with obvious inclusion maps Mi,j → Mi0 ,j when i ≤ i0 . In this case colimi Mi,j = N and hence Y limj colimi Mi,j = N = NN j

1Namely, let I 0 have the same objects as I but where Mor 0 (x, y) is the quotient of Mor (x, y) I I by the equivalence relation which identifies a, b : x → y if M (a) = M (b).

4.17. FILTERED COLIMITS

On the other hand limj Mi,j =

77

Q

Mi,j and hence [ colimi limj Mi,j = {1, 2, . . . , i}N j

i

which is smaller than the other limit. Lemma 4.17.3. Let I be an index category, i.e., a category. Assume (1) for every pair of morphisms a : w → x and b : w → y in I there exists an object z and morphisms c : x → z and d : y → z such that c ◦ a = d ◦ b, and (2) for every pair of morphisms a, b : x → y there exists a morphism c : y → z such that c ◦ a = c ◦ b. Then I is a (possibly empty) union of disjoint filtered index categories Ij . Proof. If I is the empty category, then the lemma is true. Otherwise, we define a relation on objects of I by saying that x ∼ y if there exists a z and morphisms x → z and y → z. This is an equivalence relation by the first assumption of the lemma. Hence Ob(I) is a disjoint union of equivalence classes. Let Ij be the full subcategories corresponding to these equivalence classes. The rest is clear from the definitions.  Lemma 4.17.4. Let I be an index category satisfying the hypotheses of Lemma 4.17.3 above. Then colimits over I commute with fibre products and equalizers in sets (and more generally with connected finite nonempty limits). ` Proof. By Lemma 4.17.3 we may write I = Ij with each Ij filtered. By Lemma 4.17.2 we see that colimits of Ij commute with equalizers and fibred products. Thus it suffices to show that equalizers and fibre products commute with coproducts in the category of sets (including empty coproducts). In other words, given a set J and sets Aj , Bj , Cj and set maps Aj → Bj , Cj → Bj for j ∈ J we have to show that a a a ( Aj ) ×(`j∈J Bj ) ( Cj ) = Aj × B j C j j∈J

j∈J

and given aj , a0j : Aj → Bj that a a Equalizer( aj , j∈J

j∈J

a0j ) =

j∈J

a j∈J

Equalizer(aj , a0j )

This is true even if J = ∅. Details omitted.



Definition 4.17.5. Let I, J be filtered index categories. Let H : I → J be a functor. We say I is cofinal in J if (1) for all y ∈ Ob(J ) there exists a x ∈ Ob(I) and a morphism y → H(x), and (2) for all x1 , x2 ∈ Ob(I) and any ϕ : H(x1 ) → H(x2 ) there exists x12 ∈ Ob(I) and morphisms x1 → x12 , x2 → x12 such that H(x12 ) : d H(x1 ) commutes.

ϕ

/ H(x2 )

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Lemma 4.17.6. Let I, J be filtered index categories. Let H : I → J be a functor. Assume I is cofinal in J . Then for every diagram M : J → C we have a canonical isomorphism colimI M ◦ H = colimJ M if either side exists. Proof. Omitted.

 4.18. Cofiltered limits

Limits are easier to compute or describe when they are over a cofiltered diagram. Here is the definition. Definition 4.18.1. We say that a diagram M : I → C is codirected or cofiltered if the following conditions hold: (1) the category I has at least one object, (2) for every pair of objects x, y of I there exists an object z and morphisms z → x, z → y, and (3) for every pair of objects x, y of I and every pair of morphisms a, b : x → y of I there exists a morphism c : w → x of I such that M (a ◦ c) = M (b ◦ c) as morphisms in C. We say that an index category I is codirected, or cofiltered if id : I → I is cofiltered (in other words you erase the M in part (3) above.) We observe that any diagram with cofiltered index category is cofiltered, and this is how this situation usually occurs. Here is an example of why cofiltered limits of sets are “easier” than general ones: If M : I → Sets is a cofiltered diagram, and all the Mi are finite nonempty, then limi Mi is nonempty. The same does not hold for a general limit of finite nonempty sets. 4.19. Limits and colimits over partially ordered sets A special case of diagrams is given by systems over partially ordered sets. Definition 4.19.1. Let (I, ≥) be a partially ordered set. Let C be a category. (1) A system over I in C, sometimes called a inductive system over I in C is given by objects Mi of C and for every i ≤ i0 a morphism fii0 : Mi → Mi0 such that fii = id and such that fii00 = fi0 i00 ◦ fii0 whenever i ≤ i0 ≤ i00 . (2) An inverse system over I in C, sometimes called a projective system over I in C is given by objects Mi of C and for every i ≥ i0 a morphism fii0 : Mi → Mi0 such that fii = id and such that fii00 = fi0 i00 ◦ fii0 whenever i ≥ i0 ≥ i00 . (Note reversal of inequalities.) We will say (Mi , fii0 ) is a (inverse) system over I to denote this. The maps fii0 are sometimes called the transition maps. In other words a system over I is just a diagram M : I → C where I is the category with objects I and a unique arrow i → i0 if and only i ≤ i0 . And an inverse system is a diagram M : I opp → C. From this point of view we could take (co)limits of any (inverse) system over I. However, it is customary to take only colimits of systems

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79

over I and only limits of inverse systems over I. More precisely: Given a system (Mi , fii0 ) over I the colimit of the system (Mi , fii0 ) is defined as colimi∈I Mi = colimI M, i.e., as the colimit of the corresponding diagram. Given a inverse system (Mi , fii0 ) over I the limit of the inverse system (Mi , fii0 ) is defined as limi∈I Mi = limI opp M, i.e., as the limit of the corresponding diagram. Definition 4.19.2. With notation as above. We say the system (resp. inverse system) (Mi , fii0 ) is a directed system (resp. directed inverse system) if the partially ordered set I is directed: I is nonempty and for all i1 , i2 ∈ I there exists i ∈ I such that i1 ≤ i and i2 ≤ i. In this case the colimit is sometimes (unfortunately) called the “direct limit”. We will not use this last terminology. It turns out that diagrams over a filtered category are no more general than directed systems in the following sense. Lemma 4.19.3. Let I be a filtered index category. There exists a directed partially ordered set (I, ≥) and a system (xi , ϕii0 ) over I in I with the following properties: (1) For every category C and every diagram M : I → C with values in C, denote (M (xi ), M (ϕii0 )) the corresponding system over I. If colimi∈I M (xi ) exists then so does colimI M and the transformation θ : colimi∈I M (xi ) −→ colimI M of Lemma 4.13.7 is an isomorphism. (2) For every category C and every diagram M : I opp → C in C, denote (M (xi ), M (ϕii0 )) the corresponding inverse system over I. If limi∈I M (xi ) exists then so does limI M and the transformation θ : limI opp M −→ limi∈I M (xi ) of Lemma 4.13.8 is an isomorphism. Proof. Consider quadruples (S, A, x, {fs }s∈S ) with the following properties (1) S is a finite set of objects of I, (2) A is a finite set of arrows of I such that each a ∈ A is an arrow a : s(a) → t(a) with s(a), t(a) ∈ S, (3) x is an object of I, and (4) fs : s → x is a morphism of I such that for all a ∈ A we have ft(a) ◦ a = fs(a) . Given such a quadruple i = (S, A, x, {fs }s∈S ) we denote Si = S, Ai = A, xi = x, and fs,i = fs for s ∈ Si . We also set S˜i = Si ∪ {xi } and A˜i = Ai ∪ {fs,i , s ∈ Si }. Let I be the set of all such quadruples. We define a relation on I by the rule i ≤ i0 ⇔ S˜i ⊂ Si0 and A˜i ⊂ Ai0 It is obviously a partial ordering on I. Note that if i ≤ i0 , then there is a given morphism ϕii0 : xi → xi0 namely fxi ,i0 because xi ∈ Si0 . Hence we have a system over I in I by taking (xi , ϕii0 ). We claim that this system satisfies all the conditions of the lemma.

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First we show that I is a directed partially ordered set. Note that I is nonempty since ({x}, ∅, x, {idx }) is a quadruple where x is any object of I, and I is not empty according to Definition 4.17.1. Suppose that i, i0 ∈ I. Consider the set of objects S = Si ∪ Si0 ∪ {xi , xi0 } of I. This is a finite set. According to Definition 4.17.1 and a simple induction argument there exists an object x0 of I such that for each s ∈ S there is a morphism fs0 : s → x0 . Consider the set of arrows A = Ai ∪ Ai0 ∪ {fs,i , s ∈ Si } ∪ {fs,i0 , s ∈ Si0 }. This is a finite set of arrows whose source and target are elements of S. According to Definition 4.17.1 and a simple induction argument there exists a morphism f : x0 → x such that for all a ∈ A we have 0 0 ◦ a = f ◦ fs(a) f ◦ ft(a)

as morphisms into x. Hence we see that (S, A, x, {f ◦ fs0 }s∈S ) is a quadruple which is ≥ i and ≥ i0 in the partial ordering defined above. This proves I is directed. Next, we prove the statement about colimits. Let C be a category. Let M : I → C be a functor. Denote (M (xi ), M (ϕii0 )) the corresponding system over I. Below we will write Mi = M (xi ) for clarity. Assume K = colimi∈I M (xi ) exists. We will verify that K is also the colimit of the diagram M . Recall that for every object x of I the quadruple ix = ({x}, ∅, x, {idx }) is an element of I. By definition of a colimit there is a morphism M (x) = Mix −→ K Let ϕ : x → x0 be a morphism of I. The quadruples ix , ix0 and iϕ = ({x, x0 }, {idx , idx0 , ϕ}, x0 , {ϕ, idx0 }) are elements of I. Moreover, ix ≤ iϕ and ix0 ≤ iϕ . Thus the diagram M (x) = Mix

/ M (x0 ) = Miϕ o

M (x0 ) = Mix0

(  v K is commutative in C. Since the left pointing horizontal arrow is the identity morphism on M (x0 ) by our definition of ϕix0 iϕ we see that the morphisms M (x) → K so defined satisfy condition (1) of Definition 4.13.2. Finally we have to verify condition (2) of Definition 4.13.2. Suppose that W is an object of C and suppose that we are given morphisms wx : M (x) → W such that for all morphisms a of I we have ws(a) = wt(a) ◦ a. In this case, set wi = wxi for a quadruple i = (Si , Ai , xi , {fs,i }s∈Si ). Note that the condition on the maps wx in particular guarantees that wi0 = wi ◦ M (ϕii0 ) if i ≤ i0 in I. Because K is the colimit of the system (M (xi ), M (ϕii0 ) we obtain a unique morphism K → W compatible with the maps wi and the given morphisms Mi → K. This proves the statement about colimits of the lemma. We omit the proof of the statement about limits. (Hint: You can change it into a statement about colimits by considering the opposite category of C.) 

4.20. ESSENTIALLY CONSTANT SYSTEMS

81

4.20. Essentially constant systems Let M : I → C be a diagram in a category C. Assume the index category I is filtered. In this case there are three successively stronger notions which pick out an object X of C. The first is just X = colimi∈I Mi . Then X comes equipped with projection morphisms Mi → X. A stronger condition would be to require that X is the colimit and that there exists an i ∈ I and a morphism X → Mi such that the composition X → Mi → X is idX . A stronger condition is the following. Definition 4.20.1. Let M : I → C be a diagram in a category C. (1) Assume the index category I is filtered. We say M is essentially constant with value X if X = colimi Mi and there exists an i ∈ I and a morphism X → Mi such that (a) X → Mi → X is idX , and (b) for all j there exist k and morphisms i → k and j → k such that the morphism Mj → Mk equals the composition Mj → X → Mi → Mk . (2) Assume the index category I is cofiltered. We say M is essentially constant with value X if X = limi Mi and there exists an i ∈ I and a morphism Mi → X such that (a) X → Mi → X is idX , and (b) for all j there exist k and morphisms k → i and k → j such that the morphism Mk → Mj equals the composition Mk → Mi → X → Mj . Which of the two versions is meant will be clear from context. If there is any confusion we will distinguish between these by saying that the first version means M is essentially constant as an ind-object, and in the second case we will say it is essentially constant as an pro-object. This terminology is further explained in Remarks 4.20.3 and 4.20.4. In fact we will often use the terminology “essentially constant system” which formally speaking is only defined for systems over directed partially ordered sets. Definition 4.20.2. Let C be a category. A directed system (Mi , fii0 ) is an essentially constant system if M viewed as a functor I → C defines an essentially constant diagram. A directed inverse system (Mi , fii0 ) is an essentially constant inverse system if M viewed as a functor I opp → C defines an essentially constant inverse diagram. If (Mi , fii0 ) is an essentially constant system and the morphisms fii0 are monomorphisms, then for all i ≤ i0 sufficiently large the morphisms fii0 are isomorphisms. In general this need not be the case however. An example is the system Z2 → Z2 → Z2 → . . . with maps given by (a, b) 7→ (a + b, 0). L This system is essentially constant with value Z. A non-example is to let M = n≥0 Z and to let S : M → M be the shift operator (a0 , a1 , . . .) 7→ (a1 , a2 , . . .). In this case the system M → M → M → . . . with transition maps S has colimit 0, and a map 0 → M but the system is not essentially constant.

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Remark 4.20.3. Let C be a category. There exists a big category Ind-C of indobjects of C. Namely, if F : I → C and G : J → C are filtered diagrams in C, then we can define MorInd-C (F, G) = limi colimj MorC (F (i), G(j)). There is a canonical functor C → Ind-C which maps X to the constant system on X. This is a fully faithful embedding. In this language one sees that a diagram F is essentially constant if and only F is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here. Remark 4.20.4. Let C be a category. There exists a big category Pro-C of proobjects of C. Namely, if F : I → C and G : J → C are cofiltered diagrams in C, then we can define MorPro-C (F, G) = limj colimi MorC (F (i), G(j)). There is a canonical functor C → Pro-C which maps X to the constant system on X. This is a fully faithful embedding. In this language one sees that a diagram F is essentially constant if and only F is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here. Lemma 4.20.5. Let C be a category. Let M : I → C be a diagram with filtered (resp. cofiltered) index category I. Let F : C → D be a functor. If M is essentially constant as an ind-object (resp. pro-object), then so is F ◦ M : I → D. Proof. If X is a value for M , then it follows immediately from the definition that F (X) is a value for F ◦ M .  Lemma 4.20.6. Let C be a category. Let M : I → C be a diagram with filtered index category I. The following are equivalent (1) M is an essentially constant ind-object, and (2) X = colimi Mi exists and for any W in C the map colimi MorC (W, Mi ) −→ MorC (W, X) is bijective. Proof. Assume (2) holds. Then idX ∈ MorC (X, X) comes from a morphism X → Mi for some i, i.e., X → Mi → X is the identity. Then both maps MorC (W, X) −→ colimi MorC (W, Mi ) −→ MorC (W, X) are bijective for all W where the first one is induced by the morphism X → Mi we found above, and the composition is the identity. This means that the composition colimi MorC (W, Mi ) −→ MorC (W, X) −→ colimi MorC (W, Mi ) is the identity too. Setting W = Mj and starting with idMj in the colimit, we see that Mj → X → Mi → Mk is equal to Mj → Mk for some k large enough. This proves (1) holds. The proof of (1) ⇒ (2) is omitted.  Lemma 4.20.7. Let C be a category. Let M : I → C be a diagram with cofiltered index category I. The following are equivalent (1) M is an essentially constant pro-object, and

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83

(2) X = limi Mi exists and for any W in C the map colimi MorC (Mi , W ) −→ MorC (X, W ) is bijective. Proof. Assume (2) holds. Then idX ∈ MorC (X, X) comes from a morphism Mi → X for some i, i.e., X → Mi → X is the identity. Then both maps MorC (X, W ) −→ colimi MorC (Mi , W ) −→ MorC (X, W ) are bijective for all W where the first one is induced by the morphism Mi → X we found above, and the composition is the identity. This means that the composition colimi MorC (Mi , W ) −→ MorC (X, W ) −→ colimi MorC (Mi , W ) is the identity too. Setting W = Mj and starting with idMj in the colimit, we see that Mk → Mi → X → Mj is equal to Mk → Mj for some k large enough. This proves (1) holds. The proof of (1) ⇒ (2) is omitted.  4.21. Exact functors Definition 4.21.1. Let F : A → B be a functor. (1) Suppose all finite limits exist in A. We say F is left exact if it commutes with all finite limits. (2) Suppose all finite colimits exist in A. We say F is right exact if it commutes with all finite colimits. (3) We say F is exact if it is both left and right exact. Lemma 4.21.2. Let F : A → B be a functor. Suppose all finite limits exist in A, see Lemma 4.16.4. The following are equivalent: (1) F is left exact, (2) F commutes with finite products and equalizers, and (3) F transforms a final object of A into a final object of B, and commutes with fibre products. Proof. Lemma 4.13.10 shows that (2) implies (1). Suppose (3) holds. The fibre product over the final object is the product. If a, b : A → B are morphisms of A, then the equalizer of a, b is (A ×a,B,b A) ×(pr1 ,pr2 ),A×A,∆ A. Thus (3) implies (2). Finally (1) implies (3) because the empty limit is a final object, and fibre products are limits.  4.22. Adjoint functors Definition 4.22.1. Let C, D be categories. Let u : C → D and v : D → C be functors. We say that u is a left adjoint of v, or that v is a right adjoint to u if there are bijections MorD (u(X), Y ) −→ MorC (X, v(Y )) functorial in X ∈ Ob(C), and Y ∈ Ob(D).

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In other words, this means that there is a given isomorphism of functors C opp ×D → Sets from MorD (u(−), −) to MorC (−, v(−)). For any object X of C we obtain a morphism X → v(u(X)) corresponding to idu(X) . Similarly, for any object Y of D we obtain a morphism u(v(Y )) → Y corresponding to idv(Y ) . These maps are called the adjunction maps. The adjunction maps are functorial in X and Y . Moreover, if α : u(X) → Y and β : X → v(Y ) are morphisms, then the following are equivalent (1) α and β correspond to each other via the bijection of the definition, v(β)

(2) β is the composition X → v(u(X)) −−−→ v(Y ), and u(α)

(3) α is the composition u(X) −−−→ u(v(Y )) → Y . In this way one can refomulate the notion of adjoint functors in terms of adjunction maps. Lemma 4.22.2. Let u be a left adjoint to v as in Definition 4.22.1. Then (1) u is fully faithful ⇔ id ∼ = v ◦ u. (2) v is fully faithful ⇔ u ◦ v ∼ = id. Proof. Assume u is fully faithful. We have to show the adjunction map X → v(u(X)) is an isomorphism. Let X 0 → v(u(X)) be any morphism. By adjointness this corresponds to a morphism u(X 0 ) → u(X). By fully faithfulness of u this corresponds to a morphism X 0 → X. Thus we see that X → v(u(X)) defines a bijection Mor(X 0 , X) → Mor(X 0 , v(u(X))). Hence it is an isomorphism. Conversely, if id ∼ = v ◦ u then u has to be fully faithful, as v defines an inverse on morphism sets. Part (2) is dual to part (1).



Lemma 4.22.3. Let u be a left adjoint to v as in Definition 4.22.1. (1) Suppose that M : I → C is a diagram, and suppose that colimI M exists in C. Then u(colimI M ) = colimI u ◦ M . In other words, u commutes with (representable) colimits. (2) Suppose that M : I → D is a diagram, and suppose that limI M exists in D. Then v(limI M ) = limI v ◦ M . In other words v commutes with representable limits. Proof. A morphism from a colimit into an object is the same as a compatible system of morphisms from the constituents of the limit into the object, see Remark 4.13.4. So MorD (u(colimi∈I Mi ), Y )

= MorC (colimi∈I Mi , v(Y )) = limi∈I opp MorC (Mi , v(Y )) = limi∈I opp MorD (u(Mi ), Y )

proves that u(colimi∈I Mi ) is the colimit we are looking for. A similar argument works for the other statement.  Lemma 4.22.4. Let u be a left adjoint of v as in Definition 4.22.1. (1) If C has finite colimits, then u is right exact. (2) If D has finite limits, then v is left exact. Proof. Obvious from the definitions and Lemma 4.22.3.



4.24. LOCALIZATION IN CATEGORIES

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4.23. Monomorphisms and Epimorphisms Definition 4.23.1. Let C be a category, and let f : X → Y be a morphism of C. (1) We say that f is a monomorphism if for every object W and every pair of morphisms a, b : W → X such that f ◦ a = f ◦ b we have a = b. (2) We say that f is an epimorphism if for every object W and every pair of morphisms a, b : Y → W such that a ◦ f = b ◦ f we have a = b. Example 4.23.2. In the category of sets the monomorphisms correspond to injective maps and the epimorphisms correspond to surjective maps. 4.24. Localization in categories The basic idea of this section is given a category C and a set of arrows to construct a functor F : C → S −1 C such that all elements of S become invertible in S −1 C and such that F is universal among all functors with this property. References for this section are [GZ67, Chapter I, Section 2] and [Ver96, Chapter II, Section 2]. Definition 4.24.1. Let C be a category. A set of arrows S of C is called a left multiplicative system if it has the following properties: LMS1 The identity of every object of C is in S and the composition of two composable elements of S is in S. LMS2 Every solid diagram /Y X g

s

t

 Z

f

 /W

with t ∈ S can be completed to a commutative dotted square with s ∈ S. LMS3 For every pair of morphisms f, g : X → Y and t ∈ S with target X such that f ◦ t = g ◦ t there exists a s ∈ S with source Y such that s ◦ f = s ◦ g. A set of arrows S of C is called a right multiplicative system if it has the following properties: RMS1 The identity of every object of C is in S and the composition of two composable elements of S is in S. RMS2 Every solid diagram /Y X g

s

t

 Z

f

 /W

with s ∈ S can be completed to a commutative dotted square with t ∈ S. RMS3 For every pair of morphisms f, g : X → Y and s ∈ S with source Y such that s ◦ f = s ◦ g there exists a t ∈ S with target X such that f ◦ t = g ◦ t. A set of arrows S of C is called a multiplicative system if it is both a left multiplicative system and a right multiplicative system. In other words, this means that MS1, MS2, MS3 hold, where MS1 = LMS1 = RMS1, MS2 = LMS2 + RMS2, and MS3 = LMS3 + RMS3. These conditions are useful to construct the categories S −1 C as follows.

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Left calculus of fractions. Let C be a category and let S be a left multiplicative system. We define a new category S −1 C as follows (we verify this works in the proof of Lemma 4.24.2): (1) We set Ob(S −1 C) = Ob(C). (2) Morphisms X → Y of S −1 C are given by pairs (f : X → Y 0 , s : Y → Y 0 ) with s ∈ S up to equivalence. (Think of this as s−1 f : X → Y .) (3) Two pairs (f1 : X → Y1 , s1 : Y → Y1 ) and (f2 : X → Y2 , s2 : Y → Y2 ) are said to be equivalent if there exists a third pair (f3 : X → Y3 , s3 : Y → Y3 ) and morphisms u : Y1 → Y3 and v : Y2 → Y3 of C fitting into the commutative diagram

f1 f3

X

> Y1 _ u

 / Y3 o O v

f2

Y2



s1 s3

Y

s2

(4) The composition of the equivalence classes of the pairs (f : X → Y 0 , s : Y → Y 0 ) and (g : Y → Z 0 , t : Z → Z 0 ) is defined as the equivalence class of a pair (h ◦ f : X → Z 00 , u ◦ t : Z → Z 00 ) where h and u ∈ S are chosen to fit into a commutative diagram Y

/ Z0

g

u

s

 Y0

 / Z 00

h

which exists by assumption. Lemma 4.24.2. Let C be a category and let S be a left multiplicative system. (1) The relation on pairs defined above is an equivalence relation. (2) The composition rule given above is well defined on equivalence classes. (3) Composition is associative and hence S −1 C is a category. Proof. Proof of (1). Let us say two pairs p1 = (f1 : X → Y1 , s1 : Y → Y1 ) and p2 = (f2 : X → Y2 , s2 : Y → Y2 ) are elementary equivalent if there exists a morphism a : Y1 → Y2 of C such that a ◦ f1 = f2 and a ◦ s1 = s2 . Diagram: X

f1

/ Y1 o

s1

Y

a

X

f2

 / Y2 o

s2

Y

Let us denote this property by saying p1 Ep2 . Note that pEp and aEb, bEc ⇒ aEc. Part (1) claims that the relation p ∼ p0 ⇔ ∃q : pEq∧p0 Eq is an equivalence relation. A simple formal argument, using the properties of E above shows that it suffices to prove p2 Ep1 , p2 Ep3 ⇒ p1 ∼ p2 . Thus suppose that we are given a commutative

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87

diagram f1 f3

X

> YO 1 _ a31

/ Y3 o

s3

a32

f2

  Y2

s1

Y

s2

with si ∈ S. First we apply LMS2 to get a commutative diagram Y

s3

s34

s1

 Y1

/ Y3

a14

 / Y4

with s34 ∈ S. Then we have s34 ◦ s2 = a14 ◦ a31 ◦ s2 . Hence by LMS3 there exists a morphism s44 : Y4 → Y40 , s44 ∈ S such that s44 ◦ s34 = s44 ◦ a14 ◦ a31 . Hence after replacing Y4 by Y40 , a14 by s44 ◦ a14 , and s24 by s44 ◦ s24 we may assume that s34 = a14 ◦ a31 . Next, we apply LMS2 to get a commutative diagram Y3

s34

s45

a32

 Y2

/ Y4

a25

 / Y5

with s45 ∈ S. Thus we obtain a pair p5 = (s45 ◦ s34 ◦ f3 : X → Y5 , s45 ◦ s34 ◦ s3 : Y → Y5 ) and the morphisms s45 ◦ a14 : Y1 → Y5 and a25 : Y2 → Y5 show that indeed p1 Ep5 and p2 Ep5 as desired. Proof of (2). Let p = (f : X → Y 0 , s : Y → Y 0 ) and q = (g : Y → Z 0 , t : Z → Z 0 ) be pairs as in the definition of composition above. To compose we have to choose a diagram g / Z0 Y u2

s

 Y0

h2

 / Z2

We first show that the equivalence class of the pair r2 = (h2 ◦f : X → Z2 , u2 ◦t : Z → Z2 ) is independent of the choice of (Z2 , h2 , u2 ). Namely, suppose that (Z3 , h3 , u3 ) is another choice with corresponding composition r3 = (h3 ◦ f : X → Z3 , u3 ◦ t : Z → Z3 ). Then by LMS2 we can choose a diagram Z0

u3

u2

 Z2

/ Z3 u34

h24

 / Z4

with u34 ∈ S. Hence we obtain a pair r4 = (h24 ◦ h2 ◦ f : X → Z4 , u34 ◦ u3 ◦ t : Z → Z4 ) and the morphisms h24 : Z2 → Z4 and u34 : Z3 → Z4 show that we have r2 Er4 and r3 Er4 as desired. Thus it now makes sense to define p ◦ q as the equivalence class of all possible pairs r obtained as above.

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To finish the proof of (2) we have to show that given pairs p1 , p2 , q such that p1 Ep2 then p1 ◦ q = p2 ◦ q and q ◦ p1 = q ◦ p2 whenever the compositions make sense. To do this, write p1 = (f1 : X → Y1 , s1 : Y → Y1 ) and p2 = (f2 : X → Y2 , s2 : Y → Y2 ) and let a : Y1 → Y2 be a morphism of C such that f2 = a ◦ f1 and s2 = a ◦ s1 . First assume that q = (g : Y → Z 0 , t : Z → Z 0 ). In this case choose a commutative diagram as the one on the left Y

g

s2

 Y2

/ Z0 u

h

 / Z 00

Y ⇒

/ Z0

g

s1

 Y1

u

h◦a

 / Z 00

which implies the diagram on the right is commutative as well. Using these diagrams we see that both compositions are the equivalence class of (h◦a◦f1 : X → Z 00 , u◦t : Z → Z 00 ). Thus p1 ◦ q = p2 ◦ q. The proof of the other case, in which we have to show q ◦ p1 = q ◦ p2 , is omitted. Proof of (3). We have to prove associativity of composition. Consider a solid diagram Z

W

Y

 / Z0

X

 / Y0

 / Z 00

 / W0

 / Y 00

 / Z 000

which gives rise to three composable pairs. Using LMS2 we can choose the dotted arrows making the squares commutative and such that the vertical arrows are in S. Then it is clear that the composition of the three pairs is the equivalence class of the pair (W → Z 000 , Z → Z 000 ) gotten by composing the horizontal arrows on the bottom row and the vertical arrows on the right column.  We can “write any finite collection of morphisms with the same target as fractions with common denominator”. Lemma 4.24.3. Let C be a category and let S be a left multiplicative system of morphisms of C. Given any finite collection gi : Xi → Y of morphisms of S −1 C we can find an element s : Y → Y 0 of S and fi : Xi → Y 0 such that gi is the equivalence class of the pair (fi : Xi → Y 0 , s : Y → Y 0 ). Proof. For each i choose a representative (Xi → Yi , si : Y → Yi ). The lemma follows if we can find a morphism s : Y → Y 0 in S such that for each i there is a morphism ai : Yi → Y 0 with ai ◦ si = s. If we have two indices i = 1, 2, then we

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89

can do this by completing the square Y

/ Y2

s2

s1

 Y1

t2

a1

 / Y0

with t2 ∈ S as is possible by Definition 4.24.1. Then s = t2 ◦ s1 ∈ S works. If we have n > 2 morphisms, then we use the above trick to reduce to the case of n − 1 morphisms, and we win by induction.  There is an easy characterization of equality of morphisms if they have the same denominator. Lemma 4.24.4. Let C be a category and let S be a left multiplicative system of morphisms of C. Let A, B : X → Y be morphisms of S −1 C which are the equivalence classes of (f : X → Y 0 , s : Y → Y 0 ) and (g : X → Y 0 , s : Y → Y 0 ). Then A = B if and only if there exists a morphism a : Y 0 → Y 00 with a ◦ s ∈ S and such that a ◦ f = a ◦ g. Proof. The equality of A and B means that there exists a commutative diagram 0

>Y ` f h

X

s

u

 /Zo O

v

g

Y0

~

t

Y s

with t ∈ S. In particular u ◦ s = v ◦ s. Hence by LMS3 there exists a s0 : Z → Y 00 in S such that s0 ◦ u = s0 ◦ v. Setting a equal to this common value does the job.  Remark 4.24.5. Let C be a category. Let S be a left multiplicative system. Given an object Y of C we denote Y /S the category whose objects are s : Y → Y 0 with s ∈ S and whose morphisms are commutative diagrams Y s

Y0

t

~

/ Y 00

a

where a : Y 0 → Y 00 is arbitrary. We claim that the category Y /S is filtered (see Definition 4.17.1). Namely, LMS1 implies that idY : Y → Y is in Y /S hence Y /S is nonempty. LMS2 implies that given s1 : Y → Y1 and s2 : Y → Y2 we can find a diagram Y s2 / Y2 s1

 Y1

t

a

 / Y3

with t ∈ S. Hence s1 : Y → Y1 and s2 : Y → Y2 both map to t ◦ s2 : Y → Y3 in Y /S. Finally, given two morphisms a, b from s1 : Y → Y1 to s2 : Y → Y2 in S/Y

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4. CATEGORIES

we see that a ◦ s1 = b ◦ s1 hence by LMS3 there exists a t : Y2 → Y3 such that t ◦ a = t ◦ b. Now the combined results of Lemmas 4.24.3 and 4.24.4 tell us that (4.24.5.1)

MorS −1 C (X, Y ) = colim(s:Y →Y 0 )∈Y /S MorC (X, Y 0 )

This formula expressing morphisms in S −1 C as a filtered colimit of morphisms in C is occasionally useful. Lemma 4.24.6. Let C be a category and let S be a left multiplicative system of morphisms of C. (1) The rules X 7→ X and (f : X → Y ) 7→ (f : X → Y, idY : Y → Y ) define a functor Q : C → S −1 C. (2) For any s ∈ S the morphism Q(s) is an isomorphism in S −1 C. (3) If G : C → D is any functor such that G(s) is invertible for every s ∈ S, then there exists a unique functor H : S −1 C → D such that H ◦ Q = G. Proof. Parts (1) and (2) are clear. To see (3) just set H(X) = G(X) and set H((f : X → Y 0 , s : Y → Y 0 )) = H(s)−1 ◦ H(f ). Details omitted.  Lemma 4.24.7. Let C be a category and let S be a left multiplicative system of morphisms of C. The localization functor Q : C → S −1 C commutes with finite colimits. Proof. This is clear from (4.24.5.1), Remark 4.13.4, and Lemma 4.17.2.



Lemma 4.24.8. Let C be a category. Let S be a left multiplicative system. If f : X → Y , f 0 : X 0 → Y 0 are two morphisms of C and if Q(X)

/ Q(X 0 )

a

Q(f 0 )

Q(f )

 Q(Y )

 / Q(Y 0 )

b

is a commutative diagram in S −1 C, then there exists a morphism f 00 : X 00 → Y 00 in C and a commutative diagram X

g

/ X 00 o

h



f 00

f

 Y

s

/ Y 00 o

X0 f0

t

 Y0

in C with s, t ∈ S and a = s−1 g, b = t−1 h. Proof. We choose maps and objects in the following way: First write a = s−1 g for some s : X 0 → X 00 in S and h : X → X 00 . By LMS2 we can find t : Y 0 → Y 00 in S and f 00 : X 00 → Y 00 such that X

g

/ X 00 o

f



 Y

s

f 00

Y 00 o

X0 f0

t

 Y0

commutes. Now in this diagram we are going to repeatedly change our choice of f 00

t

X 00 −−→ Y 00 ← −Y0

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91

by postcomposing both t and f 00 by a morphism d : Y 00 → Y 000 with the property that d◦t ∈ S. According to Remark 4.24.5 we may after such a replacement assume that there exists a morphism h : Y → Y 00 such that b = t−1 h. At this point we have everything as in the lemma except that we don’t know that the left square of the diagram commutes. However, we do know that Q(f 00 g) = Q(hf ) in S −1 D because the right square commutes, the outer square commutes in S −1 D by assumption, and because Q(s), Q(t) are isomorphisms. Hence using Lemma 4.24.4 we can find a morphism d : X 000 → X 00 in S (!) such that df 00 g = dhf . Hence we make one more replacement of the kind described above and we win.  Right calculus of fractions. Let C be a category and let S be a right multiplicative system. We define a new category S −1 C as follows (we verify this works in the proof of Lemma 4.24.9): (1) We set Ob(S −1 C) = Ob(C). (2) Morphisms X → Y of S −1 C are given by pairs (f : X 0 → Y, s : X 0 → X) with s ∈ S up to equivalence. (Think of this as f s−1 : X → Y .) (3) Two pairs (f1 : X1 → Y, s1 : X1 → X) and (f2 : X2 → Y, s2 : X2 → X) are said to be equivalent if there exists a third pair (f3 : X3 → Y, s3 : X3 → X) and morphisms u : X3 → X1 and v : X3 → X2 of C fitting into the commutative diagram s1

~

X`o

s3

s2

XO 1 u

X3

f1

/Y >

f3

v

 X2

f2

(4) The composition of the equivalence classes of the pairs (f : X 0 → Y, s : X 0 → X) and (g : Y 0 → Z, t : Y 0 → Y ) is defined as the equivalence class of a pair (g ◦ h : X 00 → Z, s ◦ u : X 00 → X) where h and u ∈ S are chosen to fit into a commutative diagram X 00

h

/ Y0

f

 /Y

u

 X0

t

which exists by assumption. Lemma (1) (2) (3)

4.24.9. Let C be a category and let S be a right multiplicative system. The relation on pairs defined above is an equivalence relation. The composition rule given above is well defined on equivalence classes. Composition is associative and hence S −1 C is a category.

Proof. This lemma is dual to Lemma 4.24.2. It follows formally from that lemma by replacing C by its opposite category in which S is a left multiplicative system.  We can “write any finite collection of morphisms with the same source as fractions with common denominator”.

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4. CATEGORIES

Lemma 4.24.10. Let C be a category and let S be a right multiplicative system of morphisms of C. Given any finite collection gi : X → Yi of morphisms of S −1 C we can find an element s : X 0 → X of S and fi : X 0 → Yi such that gi is the equivalence class of the pair (fi : X 0 → Yi , s : X 0 → X). Proof. This lemma is the dual of Lemma 4.24.3 and follows formally from that lemma by replacing all categories in sight by their opposites.  There is an easy characterization of equality of morphisms if they have the same denominator. Lemma 4.24.11. Let C be a category and let S be a right multiplicative system of morphisms of C. Let A, B : X → Y be morphisms of S −1 C which are the equivalence classes of (f : X 0 → Y, s : X 0 → X) and (g : X 0 → Y, s : X 0 → X). Then A = B if and only if there exists a morphism a : X 00 → X 0 with s ◦ a ∈ S and such that f ◦ a = g ◦ a. Proof. This is dual to Lemma 4.24.4.



Remark 4.24.12. Let C be a category. Let S be a right multiplicative system. Given an object X of C we denote S/X the category whose objects are s : X 0 → X with s ∈ S and whose morphisms are commutative diagrams X0

/ X 00

a s

X

}

t

where a : X 0 → X 00 is arbitrary. The category S/X is cofiltered (see Definition 4.18.1). (This is dual to the corresponding statement in Remark 4.24.5.) Now the combined results of Lemmas 4.24.10 and 4.24.11 tell us that (4.24.12.1)

MorS −1 C (X, Y ) = colim(s:X 0 →X)∈(S/X)opp MorC (X 0 , Y )

This formula expressing morphisms in S −1 C as a filtered colimit of morphisms in C is occasionally useful. Lemma 4.24.13. Let C be a category and let S be a right multiplicative system of morphisms of C. (1) The rules X 7→ X and (f : X → Y ) 7→ (f : X → Y, idX : X → X) define a functor Q : C → S −1 C. (2) For any s ∈ S the morphism Q(s) is an isomorphism in S −1 C. (3) If G : C → D is any functor such that G(s) is invertible for every s ∈ S, then there exists a unique functor H : S −1 C → D such that H ◦ Q = G. Proof. This lemma is the dual of Lemma 4.24.6 and follows formally from that lemma by replacing all categories in sight by their opposites.  Lemma 4.24.14. Let C be a category and let S be a right multiplicative system of morphisms of C. The localization functor Q : C → S −1 C commutes with finite limits. Proof. This is clear from (4.24.12.1), Remark 4.13.4, and Lemma 4.17.2.



4.24. LOCALIZATION IN CATEGORIES

93

Lemma 4.24.15. Let C be a category. Let S be a right multiplicative system. If f : X → Y , f 0 : X 0 → Y 0 are two morphisms of C and if Q(X)

/ Q(X 0 )

a

Q(f 0 )

Q(f )

 Q(Y )

 / Q(Y 0 )

b

is a commutative diagram in S −1 C, then there exists a morphism f 00 : X 00 → Y 00 in C and a commutative diagram Xo

s

X 00

t



f 00

f

 Y o

g

f0

h

Y 00

/ X0  / Y0

in C with s, t ∈ S and a = gs−1 , b = ht−1 . Proof. This lemma is dual to Lemma 4.24.8 but we can also prove it directly as follows. We choose maps and objects in the following way: First write b = ht−1 for some t : Y 00 → Y in S and h : Y 00 → Y 0 . By RMS2 we can find s : X 00 → X in S and f 00 : X 00 → Y 00 such that Xo

s

X 00

t



f

 Y o

X0

f 00

f0

h

Y 00

 / Y0

commutes. Now in this diagram we are going to repeatedly change our choice of s

f 00

X← − X 00 −−→ Y 00 by precomposing both s and f 00 by a morphism d : X 000 → X 00 with the property that s ◦ d ∈ S. According to Remark 4.24.12 we may after such a replacement assume that there exists a morphism g : X 00 → X 0 such that a = gs−1 . At this point we have everything as in the lemma except that we don’t know that the right square of the diagram commutes. However, we do know that Q(f 0 g) = Q(hf 00 ) in S −1 D because the left square commutes, the outer square commutes in S −1 D by assumption, and because Q(s), Q(t) are isomorphisms. Hence using Lemma 4.24.11 we can find a morphism d : X 000 → X 00 in S (!) such that f 0 gd = hf 00 d. Hence we make one more replacement of the kind described above and we win.  Multiplicative systems and two sided calculus of fractions. If S is a multiplicative system then left and right calculus of fractions given canonically isomorphic categories. Lemma 4.24.16. Let C be a category and let S be a multiplicative system. The category of left fractions and the category of right fractions S −1 C are canonically isomorphic. Proof. Denote Clef t , Cright the two categories of fractions. By the universal properties of Lemmas 4.24.6 and 4.24.13 we obtain functors Clef t → Cright and Cright → Clef t . By the uniqueness of these functors they are each others inverse. 

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Definition 4.24.17. Let C be a category and let S be a multiplicative system. We say S is saturated if, in addition to MS1, MS2, MS3 we also have MS4 Given three composable morphisms f, g, h, if f g, gh ∈ S, then g ∈ S. Note that a saturated multiplicative system contains all isomorphisms. Moreover, if f, g, h are composable morphisms in a category and f g, gh are isomorphisms, then g is an isomorphism (because then g has both a left and a right inverse, hence is invertible). Lemma 4.24.18. Let C be a category and let S be a multiplicative system. Denote Q : S → S −1 C the localization functor. The set Sˆ = {f ∈ Arrows(C) | Q(f ) is an isomorphism} is equal to S 0 = {f ∈ Arrows(C) | there exist g, h such that gf, f h ∈ S} and is the smallest saturated multiplicative system containing S. In particular, if S is saturated, then Sˆ = S. Proof. It is clear that S ⊂ S 0 ⊂ Sˆ because elements of S 0 map to morphisms in S −1 C which have both left and right inverses. Note that S 0 satisfies MS4, and that ˆ Sˆ satisfies MS1. Next, we prove that S 0 = S. ˆ Let s−1 g = ht−1 be the inverse morphism in S −1 C. (We may use Let f ∈ S. both left fractions and right fractions to describe morphisms in S −1 C, see Lemma 4.24.16.) The relation idX = s−1 gf in S −1 C means there exists a commutative diagram 0

=X a gf

X

f

s

u

0

idX

 / X 00 o O !

v

X

s0

}

X

idX

for some morphisms f 0 , u, v and s0 ∈ S. Hence ugf = s0 ∈ S. Similarly, using that idY = f ht−1 one proves that f hw ∈ S for some w. We conclude that f ∈ S 0 . Thus ˆ Provided we prove that S 0 = Sˆ is a multiplicative system it is now clear S 0 = S. that this implies that S 0 = Sˆ is the smallest saturated system containing S. Our remarks above take care of MS1 and MS4, so to finish the proof of the lemma ˆ Let us check that we have to show that LMS2, RMS2, LMS3, RMS3 hold for S. ˆ LMS2 holds for S. Suppose we have a solid diagram X

g

s

t

 Z

/Y

f

 /W

4.25. FORMAL PROPERTIES

95

ˆ Pick a morphism a : Z → Z 0 such that at ∈ S. Then we can use LMS2 with t ∈ S. for S to find a commutative diagram X

g

/Y

t

 Z

s

a

 Z0

f0

 /W

and setting f = f 0 ◦ a we win. The proof of RMS2 is dual to this. Finally, suppose given a pair of morphisms f, g : X → Y and t ∈ Sˆ with target X such that f t = gt. Then we pick a morphism b such that tb ∈ S. Then f tb = gtb which implies by LMS3 for S that there exists an s ∈ S with source Y such that sf = sg as desired. The proof of RMS3 is dual to this.  4.25. Formal properties In this section we discuss some formal properties of the 2-category of categories. This will lead us to the definition of a (strict) 2-category later. Let us denote Ob(Cat) the class of all categories. For every pair of categories A, B ∈ Ob(Cat) we have the “small” category of functors Fun(A, B). Composition of transformation of functors such as F 00

A

 F0

t0



t

#

/ B composes to A


BO above

f

g

/T ? gf =h

A

Later we would like to make assertions such as “any category fibred in groupoids over C is equivalent to a split one”, or “any category fibred in groupoids whose fibre categories are setlike is equivalent to a category fibred in sets”. The notion of equivalence depends on the 2-category we are working with. Definition 4.32.6. Let C be a category. The 2-category of categories fibred in groupoids over C is the sub 2-category of the 2-category of fibred categories over C (see Definition 4.30.8) defined as follows: (1) Its objects will be categories p : S → C fibred in groupoids. (2) Its 1-morphisms (S, p) → (S 0 , p0 ) will be functors G : S → S 0 such that p0 ◦ G = p (since every morphism is strongly cartesian G automatically preserves them). (3) Its 2-morphisms t : G → H for G, H : (S, p) → (S 0 , p0 ) will be morphisms of functors such that p0 (tx ) = idp(x) for all x ∈ Ob(S). Note that every 2-morphism is automatically an isomorphism! Hence this is actually a (2, 1)-category and not just a 2-category. Here is the obligatory lemma on 2-fibre products. Lemma 4.32.7. Let C be a category. The 2-category of categories fibred in groupoids over C has 2-fibre products, and they are described as in Lemma 4.29.3. Proof. By Lemma 4.30.9 the fibre product as described in Lemma 4.29.3 is a fibred category. Hence it suffices to prove that the fibre categories are groupoids,

4.32. CATEGORIES FIBRED IN GROUPOIDS

117

see Lemma 4.32.2. By Lemma 4.29.4 it is enough to show that the 2-fibre product of groupoids is a groupoid, which is clear (from the construction in Lemma 4.28.4 for example).  Lemma 4.32.8. Let p : S → C and p0 : S 0 → C be categories fibred in groupoids, and suppose that G : S → S 0 is a functor over C. (1) Then G is faithful (resp. fully faithful, resp. an equivalence) if and only if for each U ∈ Ob(C) the induced functor GU : SU → SU0 is faithful (resp. fully faithful, resp. an equivalence). (2) If G is an equivalence, then G is an equivalence in the 2-category of categories fibred in groupoids over C. Proof. Let x, y be objects of S lying over the same object U . Consider the commutative diagram MorS (x, y)

G p

' v MorC (U, U )

/ MorS 0 (G(x), G(y)) p0

From this diagram it is clear that if G is faithful (resp. fully faithful) then so is each GU . Suppose G is an equivalence. For every object x0 of S 0 there exists an object x of S such that G(x) is isomorphic to x0 . Suppose that x0 lies over U 0 and x lies over U . Then there is an isomorphism f : U 0 → U in C, namely, p0 applied to the isomorphism x0 → G(X). By the axioms of a category fibred in groupoids there exists an arrow f ∗ x → x of S lying over f . Hence there exists an isomorphism α : x0 → G(f ∗ x) such that p0 (α) = idU 0 (this time by the axioms for S 0 ). All in all we conclude that for every object x0 of S 0 we can choose a pair (ox0 , αx0 ) consisting of an object ox0 of S and an isomorphism αx0 : x0 → G(ox0 ) with p(αx0 ) = idp0 (x0 ) . From this point on we proceed as usual (see proof of Lemma 4.2.19) to produce an inverse functor F : S 0 → S, by taking x0 7→ ox0 and ϕ0 : x0 → y 0 to the unique arrow 0 ϕϕ0 : ox0 → oy0 with αx−1 0 ◦ G(ϕϕ0 ) ◦ αy 0 = ϕ . With these choices F is a functor over C. We omit the verification that G ◦ F and F ◦ G are 2-isomorphic (in the 2-category of categories fibred in groupoids over C). Suppose that GU is faithful (resp. fully faithful) for all U ∈ Ob(C). To show that G is faithful (resp. fully faithful) we have to show for any objects x, y ∈ Ob(S) that G induces an injection (resp. bijection) between MorS (x, y) and MorS 0 (G(x), G(y)). Set U = p(x) and V = p(y). It suffices to prove that G induces an injection (resp. bijection) between morphism x → y lying over f to morphisms G(x) → G(y) lying over f for any morphism f : U → V . Now fix f : U → V . Denote f ∗ y → y a pullback. Then also G(f ∗ y) → G(y) is a pullback. The set of morphisms from x to y lying over f is bijective to the set of morphisms between x and f ∗ y lying over idU . (By the second axiom of a category fibred in groupoids.) Similarly the set of morphisms from G(x) to G(y) lying over f is bijective to the set of morphisms between G(x) and G(f ∗ y) lying over idU . Hence the fact that GU is faithful (resp. fully faithful) gives the desired result. Finally suppose for all GU is an equivalence for all U , so it is fully faithful and essentially surjective. We have seen this implies G is fully faithful, and thus to

118

4. CATEGORIES

prove it is an equivalence we have to prove that it is essentially surjective. This is clear, for if z 0 ∈ Ob(S 0 ) then z 0 ∈ Ob(SU0 ) where U = p0 (z 0 ). Since GU is essentially surjective we know that z 0 is isomorphic, in SU0 , to an object of the form GU (z) for some z ∈ Ob(SU ). But morphisms in SU0 are morphisms in S 0 and hence z 0 is isomorphic to G(z) in S 0 .  Lemma 4.32.9. Let C be a category. Let p : S → C and p0 : S 0 → C be categories fibred in groupoids. Let G : S → S 0 be a functor over C. Then G is fully faithful if and only if the diagonal ∆G : S −→ S ×G,S 0 ,G S is an equivalence. Proof. By Lemma 4.32.8 it suffices to look at fibre categories over an object U of C. An object of the right hand side is a triple (x, x0 , α) where α : G(x) → G(x0 ) is a morphism in SU0 . The functor ∆G maps the object x of SU to the triple (x, x, idG(x) ). Note that (x, x0 , α) is in the essential image of ∆G if and only if α = G(β) for some morphism β : x → x0 in SU (details omitted). Hence in order for ∆G to be an equivalence, every α has to be the image of a morphism β : x → x0 , and also every two distinct morphisms β, β 0 : x → x0 have to given distinct morphisms G(β), G(β 0 ). This proves one direction of the lemma. We omit the proof of the other direction.  Lemma 4.32.10. Let C be a category. Let Si , i = 1, 2, 3, 4 be categories fibred in groupoids over C. Suppose that ϕ : S1 → S2 and ψ : S3 → S4 are equivalences over C. Then MorCat/C (S2 , S3 ) −→ MorCat/C (S1 , S4 ),

α 7−→ ψ ◦ α ◦ ϕ

is an equivalence of categories. Proof. This is a generality and holds in any 2-category.



Lemma 4.32.11. Let C be a category. If p : S → C is fibred in groupoids, then so is the inertia fibred category IS → C. Proof. Clear from the construction in Lemma 4.31.1 or by using (from the same lemma) that IS → S ×∆,S×C S,∆ S is an equivalence and appealing to Lemma 4.32.7.  Lemma 4.32.12. Let C be a category. Let U ∈ Ob(C). If p : S → C is a category fibred in groupoids and p factors through p0 : S → C/U then p0 : S → C/U is fibred in groupoids. Proof. We have already seen in Lemma 4.30.10 that p0 is a fibred category. Hence it suffices to prove the fibre categories are groupoids, see Lemma 4.32.2. For V ∈ Ob(C) we have a SV = S(f :V →U ) f :V →U

where the left hand side is the fibre category of p and the right hand side is the disjoint union of the fibre categories of p0 . Hence the result.  Lemma 4.32.13. Let p : S → C be a category fibred in groupoids. Let x → y and z → y be morphisms of S. If p(x) ×p(y) p(z) exists, then x ×y z exists and p(x ×y z) = p(x) ×p(y) p(z).

4.32. CATEGORIES FIBRED IN GROUPOIDS

Proof. Follows from Lemma 4.30.11.

119



Lemma 4.32.14. Let C be a category. Let F : X → Y be a 1-morphism of categories fibred in groupoids over C. There exists a factorization X → X 0 → Y by 1-morphisms of categories fibred in groupoids over C such that X → X 0 is an equivalence over C and such that X 0 is a category fibred in groupoids over Y. Proof. Denote p : X → C and q : Y → C the structure functors. We construct X 0 explicitly as follows. An object of X 0 is a quadruple (U, x, y, f ) where x ∈ Ob(XU ), y ∈ Ob(YU ) and f : F (x) → y is an isomorphism in YU . A morphism (a, b) : (U, x, y, f ) → (U 0 , x0 , y 0 , f 0 ) is given by a : x → x0 and b : y → y 0 with p(a) = q(b) and such that f 0 ◦ F (a) = b ◦ f . In other words X 0 = X ×F,Y,id Y with the construction of the 2-fibre product from Lemma 4.29.3. By Lemma 4.32.7 we see that X 0 is a category fibred in groupoids over C and that X 0 → Y is a morphism of categories over C. As functor X → X 0 we take x 7→ (p(x), x, F (x), idF (x) ) on objects and (a : x → x0 ) 7→ (a, F (a)) on morphisms. It is clear that the composition X → X 0 → Y equals F . We omit the verification that X → X 0 is an equivalence of fibred categories over C. Finally, we have to show that X 0 → Y is a category fibred in groupoids. Let b : y 0 → y be a morphism in Y and let (U, x, y, f ) be an object of X 0 lying over y. Because X is fibred in groupoids over C we can find a morphism a : x0 → x lying over U 0 = q(y 0 ) → q(y) = U . Since Y is fibred in groupoids over C and since both F (x0 ) → F (x) and y 0 → y lie over the same morphism U 0 → U we can find f 0 : F (x0 ) → y 0 lying over idU 0 such that f ◦ F (a) = b ◦ f 0 . Hence we obtain (a, b) : (U 0 , x0 , y 0 , f 0 ) → (U, x, y, f ). This verifies the first condition (1) of Definition 4.32.1. To see (2) let (a, b) : (U 0 , x0 , y 0 , f 0 ) → (U, x, y, f ) and (a0 , b0 ) : (U 00 , x00 , y 00 , f 00 ) → (U, x, y, f ) be morphisms of X 0 and let b00 : y 0 → y 00 be a morphism of Y such that b0 ◦ b00 = b. We have to show that there exists a unique morphism a00 : x0 → x00 such that f 00 ◦ F (a00 ) = b00 ◦ f 0 and such that (a0 , b0 ) ◦ (a00 , b00 ) = (a, b). Because X is fibred in groupoids we know there exists a unique morphism a00 : x0 → x00 such that a0 ◦a00 = a and p(a00 ) = q(b00 ). Because Y is fibred in groupoids we see that F (a00 ) is the unique morphism F (x0 ) → F (x00 ) such that F (a0 ) ◦ F (a00 ) = F (a) and q(F (a00 )) = q(b00 ). The relation f 00 ◦ F (a00 ) = b00 ◦ f 0 follows from this and the given relations f ◦ F (a) = b ◦ f 0 and f ◦ F (a0 ) = b0 ◦ f 00 .  Lemma 4.32.15. Let C be a category. Let F : X → Y be a 1-morphism of categories fibred in groupoids over C. Assume we have a 2-commutative diagram X0 o

a f

X

/ X 00

b F

 ~ Y

g

where a and b are equivalences of categories over C and f and g are categories fibred in groupoids. Then there exists an equivalence h : X 00 → X 0 of categories over Y such that h ◦ b is 2-isomorphic to a as 1-morphisms of categories over C. If the diagram above actually commutes, then we can arrange it so that h ◦ b is 2-isomorphic to a as 1-morphisms of categories over Y. Proof. We will show that both X 0 and X 00 over Y are equivalent to the category fibred in groupoids X ×F,Y,id Y over Y, see proof of Lemma 4.32.14. Choose a

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quasi-inverse b−1 : X 00 → X in the 2-category of categories over C. Since the right triangle of the diagram is 2-commutative we see that X o F

b−1

 Yo

X 00  Y

g

is 2-commutative. Hence we obtain a 1-morphism c : X 00 → X ×F,Y,id Y by the universal property of the 2-fibre product. Moreover c is a morphism of categories over Y (!) and an equivalence (by the assumption that b is an equivalence, see Lemma 4.28.7). Hence c is an equivalence in the 2-category of categories fibred in groupoids over Y by Lemma 4.32.8. We still have to construct a 2-isomorphism between c ◦ b and the functor d : X → X ×F,Y,id Y, x 7→ (p(x), x, F (x), idF (x) ) constructed in the proof of Lemma 4.32.14. Let α : F → g ◦ b and β : b−1 ◦ b → id be 2-isomorphisms between 1-morphisms of categories over C. Note that c ◦ b is given by the rule x 7→ (p(x), b−1 (b(x)), g(b(x)), αx ◦ F (βx )) on objects. Then we see that (βx , αx ) : (p(x), x, F (x), idF (x) ) −→ (p(x), b−1 (b(x)), g(b(x)), αx ◦ F (βx )) is a functorial isomorphism which gives our 2-morphism d → b ◦ c. Finally, if the diagram commutes then αx is the identity for all x and we see that this 2-morphism is a 2-morphism in the 2-category of categories over Y.  4.33. Presheaves of categories In this section we compare the notion of fibred categories with the closely related notion of a “presheaf of categories”. The basic construction is explained in the following example. Example 4.33.1. Let C be a category. Suppose that F : C opp → Cat is a functor to the 2-category of categories, see Definition 4.26.5. For f : V → U in C we will suggestively write F (f ) = f ∗ for the functor from F (U ) to F (V ). From this we can construct a fibred category SF over C as follows. Define Ob(SF ) = {(U, x) | U ∈ Ob(C), x ∈ Ob(F (U ))}. For (U, x), (V, y) ∈ Ob(SF ) we define MorSF ((V, y), (U, x)) = {(f, φ) | f ∈ MorC (V, U ), φ ∈ MorF (V ) (y, f ∗ x)} a = MorF (V ) (y, f ∗ x) f ∈MorC (V,U )

In order to define composition we use that g ∗ ◦f ∗ = (f ◦g)∗ for a pair of composable morphisms of C (by definition of a functor into a 2-category). Namely, we define the composition of ψ : z → g ∗ y and φ : y → f ∗ x to be g ∗ (φ) ◦ ψ. The functor pF : SF → C is given by the rule (U, x) 7→ U . Let us check that this is indeed a fibred category. Given f : V → U in C and (U, x) a lift of U , then we claim

4.33. PRESHEAVES OF CATEGORIES

121

(f, idf ∗ x ) : (V, f ∗ x) → (U, x) is a strongly cartesian lift of f . We have to show a h in the diagram on the left determines (h, ν) on the right: VO h

W

/U ?

f

g

(f,idf ∗ x )

(V, f ∗ x) O

/ (U, x) :

(h,ν) (g,ψ)

(W, z)

Just take ν = ψ which works because f ◦ h = g and hence g ∗ x = h∗ f ∗ x. Moreover, this is the only lift making the diagram (on the right) commute. Definition 4.33.2. Let C be a category. Suppose that F : C opp → Cat is a functor to the 2-category of categories. We will write pF : SF → C for the fibred category constructed in Example 4.33.1. A split fibred category is a fibred category isomorphic (!) over C to one of these categories SF . Lemma 4.33.3. Let C be a category. Let S be a fibred category over C. Then S is split if and only if for some choice of pullbacks (see Definition 4.30.5) the pullback functors (f ◦ g)∗ and g ∗ ◦ f ∗ are equal. Proof. This is immediate from the definitions.



Lemma 4.33.4. Let p : S → C be a fibred category. There exists a functor F : C → Cat such that S is equivalent to SF in the 2-category of fibred categories over C. In other words, every fibred category is equivalent to a split one. Proof. Let us make a choice of pullbacks (see Definition 4.30.5). By Lemma 4.30.6 we get pullback functors f ∗ for every morphism f of C. We construct a new category S 0 as follows. The objects of S 0 are pairs (x, f ) consisting of a morphism f : V → U of C and an object x of S over U , i.e., x ∈ Ob(SU ). The functor p0 : S 0 → C will map the pair (x, f ) to the source of the morphism f , in other words p0 (x, f : V → U ) = V . A morphism ϕ : (x1 , f1 : V1 → U1 ) → (x2 , f2 : V2 → U2 ) is given by a pair (ϕ, g) consisting of a morphism g : V1 → V2 and a morphism ϕ : f1∗ x1 → f2∗ x2 with p(ϕ) = g. It is no problem to define the composition law: (ϕ, g) ◦ (ψ, h) = (ϕ ◦ ψ, g ◦ h) for any pair of composable morphisms. There is a natural functor S → S 0 which simply maps x over U to the pair (x, idx ). At this point we need to check that p0 makes S 0 into a fibred category over C, and we need to check that S → S 0 is an equivalence of categories over C which maps strongly cartesian morphisms to strongly cartesian morphisms. We omit the verifications. Finally, we can define pullback functors on S 0 by setting g ∗ (x, f ) = (x, f ◦ g) on objects if g : V 0 → V and f : V → U . On morphisms (ϕ, idV ) : (x1 , f1 ) → (x2 , f2 ) between morphisms in SV0 we set g ∗ (ϕ, idV ) = (g ∗ ϕ, idV 0 ) where we use the unique identifications g ∗ fi∗ xi = (fi ◦g)∗ xi from Lemma 4.30.6 to think of g ∗ ϕ as a morphism from (f1 ◦g)∗ x1 to (f2 ◦g)∗ x2 . Clearly, these pullback functors g ∗ have the property that g1∗ ◦ g2∗ = (g2 ◦ g1 )∗ , in other words S 0 is split as desired. 

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4.34. Presheaves of groupoids In this section we compare the notion of categories fibred in groupoids with the closely related notion of a “presheaf of groupoids”. The basic construction is explained in the following example. Example 4.34.1. This example is the analogue of Example 4.33.1, for “presheaves of groupoids” instead of “presheaves of categories”. The output will be a category fibred in groupoids instead of a fibred category. Suppose that F : C opp → Groupoids is a functor to the category of groupoids, see Definition 4.26.5. For f : V → U in C we will suggestively write F (f ) = f ∗ for the functor from F (U ) to F (V ). We construct a category SF fibred in groupoids over C as follows. Define Ob(SF ) = {(U, x) | U ∈ Ob(C), x ∈ Ob(F (U ))}. For (U, x), (V, y) ∈ Ob(SF ) we define MorSF ((V, y), (U, x)) = {(f, φ) | f ∈ MorC (V, U ), φ ∈ MorF (V ) (y, f ∗ x)} a = MorF (V ) (y, f ∗ x) f ∈MorC (V,U )

In order to define composition we use that g ∗ ◦f ∗ = (f ◦g)∗ for a pair of composable morphisms of C (by definition of a functor into a 2-category). Namely, we define the composition of ψ : z → g ∗ y and φ : y → f ∗ x to be g ∗ (φ) ◦ ψ. The functor pF : SF → C is given by the rule (U, x) 7→ U . The condition that F (U ) is a groupoid for every U guarantees that SF is fibred in groupoids over C, as we have already seen in Example 4.33.1 that SF is a fibred category, see Lemma 4.32.2. But we can also prove conditions (1), (2) of Definition 4.32.1 directly as follows: (1) Lifts of morphisms exist since given f : V → U in C and (U, x) an object of SF over U , then (f, idf ∗ x ) : (V, f ∗ x) → (U, x) is a lift of f . (2) Suppose given solid diagrams as follows VO h

W

/U ?

f

g

(V, y) O

(f,φ)

/ (U, x) ;

(h,ν) (g,ψ)

(W, z)

Then for the dotted arrows we have ν = (h∗ φ)−1 ◦ ψ so given h there exists a ν which is unique by uniqueness of inverses. Definition 4.34.2. Let C be a category. Suppose that F : C opp → Groupoids is a functor to the 2-category of groupoids. We will write pF : SF → C for the category fibred in groupoids constructed in Example 4.34.1. A split category fibred in groupoids is a category fibred in groupoids isomorphic (!) over C to one of these categories SF . Lemma 4.34.3. Let p : S → C be a category fibred in groupoids. There exists a functor F : C → Groupoids such that S is equivalent to SF over C. In other words, every category fibred in groupoids is equivalent to a split one. Proof. Make a choice of pullbacks (see Definition 4.30.5). By Lemmas 4.30.6 and 4.32.2 we get pullback functors f ∗ for every morphism f of C. We construct a new category S 0 as follows. The objects of S 0 are pairs (x, f ) consisting of a morphism f : V → U of C and an object x of S over U , i.e.,

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x ∈ Ob(SU ). The functor p0 : S 0 → C will map the pair (x, f ) to the source of the morphism f , in other words p0 (x, f : V → U ) = V . A morphism ϕ : (x1 , f1 : V1 → U1 ) → (x2 , f2 : V2 → U2 ) is given by a pair (ϕ, g) consisting of a morphism g : V1 → V2 and a morphism ϕ : f1∗ x1 → f2∗ x2 with p(ϕ) = g. It is no problem to define the composition law: (ϕ, g) ◦ (ψ, h) = (ϕ ◦ ψ, g ◦ h) for any pair of composable morphisms. There is a natural functor S → S 0 which simply maps x over U to the pair (x, idx ). At this point we need to check that p0 makes S 0 into a category fibred in groupoids over C, and we need to check that S → S 0 is an equivalence of categories over C. We omit the verifications. Finally, we can define pullback functors on S 0 by setting g ∗ (x, f ) = (x, f ◦ g) on objects if g : V 0 → V and f : V → U . On morphisms (ϕ, idV ) : (x1 , f1 ) → (x2 , f2 ) between morphisms in SV0 we set g ∗ (ϕ, idV ) = (g ∗ ϕ, idV 0 ) where we use the unique identifications g ∗ fi∗ xi = (fi ◦g)∗ xi from Lemma 4.32.2 to think of g ∗ ϕ as a morphism from (f1 ◦g)∗ x1 to (f2 ◦g)∗ x2 . Clearly, these pullback functors g ∗ have the property that g1∗ ◦ g2∗ = (g2 ◦ g1 )∗ , in other words S 0 is split as desired.  We will see an alternative proof of this lemma in Section 4.38. 4.35. Categories fibred in sets Definition 4.35.1. A category is called discrete if the only morphisms are the identity morphisms. A discrete category has only one interesting piece of information: its set of objects. Thus we sometime confuse discrete categories with sets. Definition 4.35.2. Let C be a category. A category fibred in sets, or a category fibred in discrete categories is a category fibred in groupoids all of whose fibre categories are discrete. We want to clarify the relationship between categories fibred in sets and presheaves (see Definition 4.3.3). To do this it makes sense to first make the following definition. Definition 4.35.3. Let C be a category. The 2-category of categories fibred in sets over C is the sub 2-category of the category of categories fibred in groupoids over C (see Definition 4.32.6) defined as follows: (1) Its objects will be categories p : S → C fibred in sets. (2) Its 1-morphisms (S, p) → (S 0 , p0 ) will be functors G : S → S 0 such that p0 ◦ G = p (since every morphism is strongly cartesian G automatically preserves them). (3) Its 2-morphisms t : G → H for G, H : (S, p) → (S 0 , p0 ) will be morphisms of functors such that p0 (tx ) = idp(x) for all x ∈ Ob(S). Note that every 2-morphism is automatically an isomorphism. Hence this 2-category is actually a (2, 1)-category. Here is the obligatory lemma on the existence of 2-fibre products. Lemma 4.35.4. Let C be a category. The 2-category of categories fibred in sets over C has 2-fibre products. More precisely, the 2-fibre product described in Lemma 4.29.3 returns a category fibred in sets if one starts out with such. Proof. Omitted.



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Example 4.35.5. This example is the analogue of Examples 4.33.1 and 4.34.1 for presheaves instead of “presheaves of categories”. The output will be a category fibred in sets instead of a fibred category. Suppose that F : C opp → Sets is a presheaf. For f : V → U in C we will suggestively write F (f ) = f ∗ : F (U ) → F (V ). We construct a category SF fibred in sets over C as follows. Define Ob(SF ) = {(U, x) | U ∈ Ob(C), x ∈ Ob(F (U ))}. For (U, x), (V, y) ∈ Ob(SF ) we define MorSF ((V, y), (U, x)) = {f ∈ MorC (V, U ) | f ∗ x = y} Composition is inherited from composition in C which works as g ∗ ◦ f ∗ = (f ◦ g)∗ for a pair of composable morphisms of C. The functor pF : SF → C is given by the rule (U, x) 7→ U . As every fibre category SF,U is discrete with underlying set F (U ) and we have already see in Example 4.34.1 that SF is a category fibred in groupoids, we conclude that SF is fibred in sets. Lemma 4.35.6. Let C be a category. The only 2-morphisms between categories fibred in sets are identities. In other words, the 2-category of categories fibred in sets is a category. Moreover, there is an equivalence of categories     the category of presheaves the category of categories ↔ of sets over C fibred in sets over C The functor from left to right is the construction F → SF discussed in Example 4.35.5. The functor from right to left assigns to p : S → C the presheaf of objects U 7→ Ob(SU ). Proof. The first assertion is clear, as the only morphisms in the fibre categories are identities. Suppose that p : S → C is fibred in sets. Let f : V → U be a morphism in C and let x ∈ Ob(SU ). Then there is exactly one choice for the object f ∗ x. Thus we see that (f ◦ g)∗ x = g ∗ (f ∗ x) for f, g as in Lemma 4.32.2. It follows that we may think of the assignments U 7→ Ob(SU ) and f 7→ f ∗ as a presheaf on C.  Here is an important example of a category fibred in sets. Example 4.35.7. Let C be a category. Let X ∈ Ob(C). Consider the representable presheaf hX = MorC (−, X) (see Example 4.3.4). On the other hand, consider the category p : C/X → C from Example 4.2.13. The fibre category (C/X)U has as objects morphisms h : U → X, and only identities as morphisms. Hence we see that under the correspondence of Lemma 4.35.6 we have hX ←→ C/X. In other words, the category C/X is canonically equivalent to the category ShX associated to hX in Example 4.35.5. For this reason it is tempting to define a “representable” object in the 2-category of categories fibred in groupoids to be a category fibred in sets whose associated presheaf is representable. However, this is would not be a good definition for use since we prefer to have a notion which is invariant under equivalences. To make this precise we study exactly which categories fibred in groupoids are equivalent to categories fibred in sets.

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4.36. Categories fibred in setoids Definition 4.36.1. Let us call a category a setoid4 if it is a groupoid where every object has exactly one automorphism: the identity. If C is a set with an equivalence relation ∼, then we can make a setoid C as follows: Ob(C) = C and MorC (x, y) = ∅ unless x ∼ y in which case we set MorC (x, y) = {1}. Transitivity of ∼ means that we can compose morphisms. Conversely any setoid category defines an equivalence relation on its objects (isomorphism) such that you recover the category (up to unique isomorphism – not equivalence) from the procedure just described. Discrete categories are setoids. For any setoid C there is a canonical procedure to make a discrete category equivalent to it, namely one replaces Ob(C) by the set of isomorphism classes (and adds identity morphisms). In terms of sets endowed with an equivalence relation this corresponds to taking the quotient by the equivalence relation. Definition 4.36.2. Let C be a category. A category fibred in setoids is a category fibred in groupoids all of whose fibre categories are setoids. Below we will clarify the relationship between categories fibred in setoids and categories fibred in sets. Definition 4.36.3. Let C be a category. The 2-category of categories fibred in setoids over C is the sub 2-category of the category of categories fibred in groupoids over C (see Definition 4.32.6) defined as follows: (1) Its objects will be categories p : S → C fibred in setoids. (2) Its 1-morphisms (S, p) → (S 0 , p0 ) will be functors G : S → S 0 such that p0 ◦ G = p (since every morphism is strongly cartesian G automatically preserves them). (3) Its 2-morphisms t : G → H for G, H : (S, p) → (S 0 , p0 ) will be morphisms of functors such that p0 (tx ) = idp(x) for all x ∈ Ob(S). Note that every 2-morphism is automatically an isomorphism. Hence this 2-category is actually a (2, 1)-category. Here is the obligatory lemma on the existence of 2-fibre products. Lemma 4.36.4. Let C be a category. The 2-category of categories fibred in setoids over C has 2-fibre products. More precisely, the 2-fibre product described in Lemma 4.29.3 returns a category fibred in setoids if one starts out with such. Proof. Omitted.



Lemma 4.36.5. Let C be a category. Let S be a category over C. (1) If S → S 0 is an equivalence over C with S 0 fibred in sets over C, then (a) S is fibred in setoids over C, and (b) for each U ∈ Ob(C) the map Ob(SU ) → Ob(SU0 ) identifies the target as the set of isomorphism classes of the source. (2) If p : S → C is a category fibred in setoids, then there exists a category fibred in sets p0 : S 0 → C and an equivalence can : S → S 0 over C. 4A set on steroids!?

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Proof. Let us prove (2). An object of the category S 0 will be a pair (U, ξ), where U ∈ Ob(C) and ξ is an isomorphism class of objects of SU . A morphism (U, ξ) → (V, ψ) is given by a morphism x → y, where x ∈ ξ and y ∈ ψ. Here we identify two morphisms x → y and x0 → y 0 if they induce the same morphism U → V , and if for some choices of isomorphisms x → x0 in SU and y → y 0 in SV the compositions x → x0 → y 0 and x → y → y 0 agree. By construction there are surjective maps on objects and morphisms from S → S 0 . We define composition of morphisms in S 0 to be the unique law that turns S → S 0 into a functor. Some details omitted.  Thus categories fibred in setoids are exactly the categories fibred in groupoids which are equivalent to categories fibred in sets. Moreover, an equivalence of categories fibred in sets is an isomorphism by Lemma 4.35.6. Lemma 4.36.6. Let C be a category. The construction of Lemma 4.36.5 part (2) gives a functor     the 2-category of categories the category of categories F : −→ fibred in setoids over C fibred in sets over C (see Definition 4.26.5). This functor is an equivalence in the following sense: (1) for any two 1-morphisms f, g : S1 → S2 with F (f ) = F (g) there exists a unique 2-isomorphism f → g, (2) for any morphism h : F (S1 ) → F (S2 ) there exists a 1-morphism f : S1 → S2 with F (f ) = h, and (3) any category fibred in sets S is equal to F (S). In particular, defining Fi ∈ PSh(C) by the rule Fi (U ) = Ob(Si,U )/ ∼ =, we have . MorCat/C (S1 , S2 ) 2-isomorphism = MorPSh(C) (F1 , F2 ) More precisely, given any map φ : F1 → F2 there exists a 1-morphism f : S1 → S2 which induces φ on isomorphism classes of objects and which is unique up to unique 2-isomorphism. Proof. By Lemma 4.35.6 the target of F is a category hence the assertion makes sense. The construction of Lemma 4.36.5 part (2) assigns to S the category fibred in sets whose value over U is the set of isomorphism classes in SU . Hence it is clear that it defines a functor as indicated. Let f, g : S1 → S2 with F (f ) = F (g) be as in (1). For each object U of C and each object x of S1,U we see that f (x) ∼ = g(x) by assumption. As S2 is fibred in setoids there exists a unique isomorphism tx : f (x) → g(x) in S2,U . Clearly the rule x 7→ tx gives the desired 2-isomorphism f → g. We omit the proofs of (2) and (3). To see the final assertion use Lemma 4.35.6 to see that the right hand side is equal to MorCat(C) (F (S1 ), F (S2 )) and apply (1) and (2) above.  Here is another characterization of categories fibred in setoids among all categories fibred in groupoids. Lemma 4.36.7. Let C be a category. Let p : S → C be a category fibred in groupoids. The following are equivalent: (1) p : S → C be a category fibred in setoids, and (2) the canonical 1-morphism IS → S, see (4.31.2.1), is an equivalence (of categories over C).

4.37. REPRESENTABLE CATEGORIES FIBRED IN GROUPOIDS

127

Proof. Assume (2). The category IS has objects (x, α) where x ∈ S, say with p(x) = U , and α : x → x is a morphism in SU . Hence if IS → S is an equivalence over C then every pair of objects (x, α), (x, α0 ) are isomorphic in the fibre category of IS over U . Looking at the definition of morphisms in IS we conclude that α, α0 are conjugate in the group of automorphisms of x. Hence taking α0 = idx we conclude that every automorphism of x is equal to the identity. Since S → C is fibred in groupoids this implies that S → C is fibred in setoids. We omit the proof of (1) ⇒ (2).  Lemma 4.36.8. Let C be a category. The construction of Lemma 4.36.6 which associates to a category fibred in setoids a presheaf is compatible with products, in the sense that the presheaf associated to a 2-fibre product X ×Y Z is the fibre product of the presheaves associated to X , Y, Z. Proof. Let U ∈ Ob(C). The lemma just says that ∼ Ob((X ×Y Z)U )/ ∼ = equals Ob(XU )/ ∼ = ×Ob(YU )/∼ = Ob(ZU )/ = the proof of which we omit. (But note that this would not be true in general if the category YU is not a setoid.)  4.37. Representable categories fibred in groupoids Here is our definition of a representable category fibred in groupoids. As promised this is invariant under equivalences. Definition 4.37.1. Let C be a category. A category fibred in groupoids p : S → C is called representable if there exists an object X of C and an equivalence j : S → C/X (in the 2-category of groupoids over C). The usual abuse of notation is to say that X represents S and not mention the equivalence j. We spell out what this entails. Lemma 4.37.2. Let C be a category. Let p : S → C be a category fibred in groupoids. (1) S is representable if and only if the following conditions are satisfied: (a) S is fibred in setoids, and (b) the presheaf U 7→ Ob(SU )/ ∼ = is representable. (2) If S is representable the pair (X, j), where j is the equivalence j : S → C/X is uniquely determined up to isomorphism. Proof. The first assertion follows immediately from Lemma 4.36.5. For the second, suppose that j 0 : S → C/X 0 is a second such pair. Choose a 1-morphism t0 : C/X 0 → S such that j 0 ◦ t0 ∼ = idC/X 0 and t0 ◦ j 0 ∼ = idS . Then j ◦ t0 : C/X 0 → C/X is an equivalence. Hence it is an isomorphism, see Lemma 4.35.6. Hence by the Yoneda Lemma 4.3.5 (via Example 4.35.7 for example) it is given by an isomorphism X 0 → X.  Lemma 4.37.3. Let C be a category. Let X , Y be categories fibred in groupoids over C. Assume that X , Y are representable by objects X, Y of C. Then . MorCat/C (X , Y) 2-isomorphism = MorC (X, Y ) More precisely, given φ : X → Y there exists a 1-morphism f : X → Y which induces φ on isomorphism classes of objects and which is unique up to unique 2isomorphism.

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Proof. By Example 4.35.7 we have C/X = ShX and C/Y = ShY . By Lemma 4.36.6 we have . MorCat/C (X , Y) 2-isomorphism = MorPSh(C) (hX , hY ) By the Yoneda Lemma 4.3.5 we have MorPSh(C) (hX , hY ) = MorC (X, Y ).



4.38. Representable 1-morphisms Let C be a category. In this section we explain what it means for a 1-morphism between categories fibred in groupoids over C to be representable. Note that the 2-category of categories fibred in groupoids over C is a “full” sub 2-category of the 2-category of categories over C (see Definition 4.32.6). Hence if S, S 0 are fibred in groupoids over C then MorCat/C (S, S 0 ) denotes the category of 1-morphisms in this 2-category (see Definition 4.29.1). These are all groupoids, see remarks following Definition 4.32.6. Here is the 2category analogue of the Yoneda lemma. Lemma 4.38.1 (2-Yoneda lemma). Let S → C be fibred in groupoids. Let U ∈ Ob(C). The functor MorCat/C (C/U, S) −→ SU given by G 7→ G(idU ) is an equivalence. Proof. Make a choice of pullbacks for S (see Definition 4.30.5). We define a functor SU −→ MorCat/C (C/U, S) as follows. Given x ∈ Ob(SU ) the associated functor is (1) on objects: (f : V → U ) 7→ f ∗ x, and (2) on morphisms: the arrow (g : V 0 /U → V /U ) maps to the composition (αg,f )x

(f ◦ g)∗ x −−−−−→ g ∗ f ∗ x → f ∗ x where αg,f is as in Lemma 4.32.2. We omit the verification that this is an inverse to the functor of the lemma.



Remark 4.38.2. We can use the 2-Yoneda lemma to give an alternative proof of Lemma 4.34.3. Let p : S → C be a category fibred in groupoids. We define a contravariant functor F from C to the category of groupoids as follows: for U ∈ Ob(C) let F (U ) = MorCat/C (C/U, S). If f : U → V the induced functor C/U → C/V induces the morphism F (f ) : F (V ) → F (U ). Clearly F is a functor. Let S 0 be the associated category fibred in groupoids from Example 4.34.1. There is an obvious functor G : S 0 → S over C given by taking the pair (U, x), where U ∈ Ob(C) and x ∈ F (U ), to x(idU ) ∈ S. Now Lemma 4.38.1 implies that for each U , GU : SU0 = F (U ) = MorCat/C (C/U, S) → SU is an equivalence, and thus G equivalence between S and S 0 by Lemma 4.32.8.

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Let C be a category. Let X , Y be categories fibred in groupoids over C. Let U ∈ Ob(C). Let F : X → Y and G : C/U → Y be 1-morphisms of categories fibred in groupoids over C. We want to describe the 2-fibre product /X

(C/U ) ×Y X  C/U

G

 /Y

F

Let y = G(idU ) ∈ YU . Make a choice of pullbacks for Y (see Definition 4.30.5). Then G is isomorphic to the functor (f : V → U ) 7→ f ∗ y, see Lemma 4.38.1 and its proof. We may think of an object of (C/U )×Y X as a quadruple (V, f : V → U, x, φ), see Lemma 4.29.3. Using the description of G above we may think of φ as an isomorphism φ : f ∗ y → F (x) in YV . Lemma 4.38.3. In the situation above the fibre category of (C/U ) ×Y X over an object f : V → U of C/U is the category described as follows: (1) objects are pairs (x, φ), where x ∈ Ob(XV ), and φ : f ∗ y → F (x) is a morphism in YV , (2) the set of morphisms between (x, φ) and (x0 , φ0 ) is the set of morphisms ψ : x → x0 in XV such that F (ψ) = φ0 ◦ φ−1 . Proof. See discussion above.



Lemma 4.38.4. Let C be a category. Let X , Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. Let G : C/U → Y be a 1-morphism. Then (C/U ) ×Y X −→ C/U is a category fibred in groupoids. Proof. We have already seen in Lemma 4.32.7 that the composition (C/U ) ×Y X −→ C/U −→ C is a category fibred in groupoids. Then the lemma follows from Lemma 4.32.12.  Definition 4.38.5. Let C be a category. Let X , Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. We say F is representable, or that X is relatively representable over Y, if for every U ∈ Ob(C) and any G : C/U → X the category fibred in groupoids (C/U ) ×Y X −→ C/U is representable over C/U . Lemma 4.38.6. Let C be a category. Let X , Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. If F is representable then every one of the functors FU : XU −→ YU between fibre categories is faithful. Proof. Clear from the description of fibre categories in Lemma 4.38.3 and the characterization of representable fibred categories in Lemma 4.37.2. 

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Lemma 4.38.7. Let C be a category. Let X , Y be categories fibred in groupoids over C. Let F : X → Y be a 1-morphism. Make a choice of pullbacks for Y. Assume (1) each functor FU : XU −→ YU between fibre categories is faithful, and (2) for each U and each y ∈ YU the presheaf (f : V → U ) 7−→ {(x, φ) | x ∈ XV , φ : f ∗ y → F (x)}/ ∼ = is a representable presheaf on C/U . Then F is representable. Proof. Clear from the description of fibre categories in Lemma 4.38.3 and the characterization of representable fibred categories in Lemma 4.37.2.  Before we state the next lemma we point out that the 2-category of categories fibred in groupoids is a (2, 1)-category, and hence we know what it means to say that it has a final object (see Definition 4.28.1). And it has a final object namely id : C → C. Thus we define 2-products of categories fibred in groupoids over C as the 2-fibred products X × Y := X ×C Y. With this definition in place the following lemma makes sense. Lemma 4.38.8. Let C be a category. Let S → C be a category fibred in groupoids. Assume C has products of pairs of objects and fibre products. The following are equivalent: (1) The diagonal S → S × S is representable. (2) For every U in C, any G : C/U → S is representable. Proof. Suppose the diagonal is representable, and let U, G be given. Consider any V ∈ Ob(C) and any G0 : C/V → S. Note that C/U × C/V = C/U × V is representable. Hence the fibre product (C/U × V ) ×(S×S) S

/S

 C/U × V

 / S ×S

(G,G0 )

is representable by assumption. This means there exists W → U × V in C, such that /S C/W   / S ×S C/U × C/V is cartesian. This implies that C/W ∼ = C/U ×S C/V (see Lemma 4.28.11) as desired. Assume (2) holds. Consider any V ∈ Ob(C) and any (G, G0 ) : C/V → S × S. We have to show that C/V ×S×S S is representable. What we know is that C/V ×G,S,G0 C/V is representable, say by a : W → V in C/V . The equivalence C/W → C/V ×G,S,G0 C/V followed by the second projection to C/V gives a second morphism a0 : W → V . Consider W 0 = W ×(a,a0 ),V ×V V . There exists an equivalence C/W 0 ∼ = C/V ×S×S S

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namely C/W 0

∼ = C/W ×(C/V ×C/V ) C/V
∼ = C/V ×(G,S,G0 ) C/V ×(C/V ×C/V ) C/V ∼ = C/V ×(S×S) S

(for the last isomorphism see Lemma 4.28.12) which proves the lemma.



Biographical notes: Parts of this have been taken from Vistoli’s notes [Vis]. 4.39. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

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CHAPTER 5

Topology 5.1. Introduction Basic topology will be explained in this document. A reference is [Eng77]. 5.2. Basic notions The following notions are considered basic and will not be defined, and or proved. This does not mean they are all necessarily easy or well known. (1) X is a topological space, (2) x ∈ X is a point, (3) x ∈ X is a closed point, (4) f : X1 → X2 is continuous, (5) a neighbourhood of x ∈ X is any subset E ⊂ X which contains an open subset that S contains x, (6) U : U = i∈I Ui is an open covering of U (note: we allow any Ui to be empty and we even allow, in case U is empty, the empty set for I), (7) S the open covering V is S a refinement of the open covering U (if V : V = j∈J Vj and U : U = i∈I Ui this means each Vj is completely contained in one of the Ui ), (8) {Ei }i∈I is a fundamental system of neighbourhoods of x in X, (9) a topological space X is called Hausdorff or separated if and only if for every distinct pair of points x, y ∈ X there exist disjoint opens U, V ⊂ X such that x ∈ U , y ∈ V , (10) the product of two topological spaces, (11) the fibre product X ×Y Z of a pair of continuous maps f : X → Y and g :Z →Y, (12) etc. 5.3. Bases Definition 5.3.1. Let X be a topological space. A collection of subsets B of X is called a base for the topology on X or a basis for the topology on X if the following conditions hold: (1) Every element B ∈ B is open in X. (2) For every open U ⊂ X and every x ∈ U , there exists an element B ∈ B such that x ∈ B ⊂ U . Lemma 5.3.2. S Let X be a topological space. Let B be a basis for the topology on X. Let U : U = i Ui be an open covering of U ⊂ X. There exists an open covering S U = Vj which is a refinement of U such that each Vj is an element of the basis B. 133

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Proof. Omitted.

 5.4. Connected components

Definition 5.4.1. Let X be a topological space. ` (1) We say X is connected if whenever X = T1 T2 with Ti ⊂ X open and closed, then either T1 = ∅ or T2 = ∅. (2) We say T ⊂ X is a connected component of X if T is a maximal connected subset of X. The empty space is connected. Lemma 5.4.2. Let f : X → Y be a continuous map of topological spaces. If E ⊂ X is a connected subset, then f (E) ⊂ Y is connected as well. Proof. Omitted.



Lemma 5.4.3. Let X be a topological space. If T ⊂ X is connected, then so is its closure. Each point of X is contained in a connected component. Connected components are always closed, but not necessarily open. ` Proof. Let T be the closure of the connected`subset T . Suppose T = T1 T2 with Ti ⊂ T open and closed. Then T = (T ∩ T1 ) (T ∩ T2 ). Hence T equals one of the two, say T = T1 ∩ T . Thus clearly T ⊂ T1 as desired. Pick a point x ∈ X. Consider the set A of connected subsets x ∈ Tα ⊂ X. Note that A is nonempty since {x} ∈ A. There is a partial ordering on A coming from inclusion: α ≤Sα0 ⇔ Tα ⊂ Tα0 . Choose a maximal totally ordered subset A0 ⊂ A, and let ` T = α∈A0 Tα . We claim that T is connected. Namely, suppose that T = T1 T2 is a disjoint union of two open and closed subsets of T . For each α ∈ A0 we have either Tα ⊂ T1 or Tα ⊂ T2 , by connectedness of Tα . Suppose that for some α0 ∈ A0 we have Tα0 6⊂ T1 (say, if not we’re done anyway). Then, since A0 is totally ordered we see immediately that Tα ⊂ T2 for all α ∈ A0 . Hence T = T2 . To get anQexample where connected components are not open, just take an infinite product n∈N {0, 1} with the product topology. This is a totally disconnected space so connected components are singletons, which are not open.  Lemma 5.4.4. Let f : X → Y be a continuous map of topological spaces. Assume that (1) all fibres of f are nonempty and connected, and (2) a set T ⊂ Y is closed if and only if f −1 (T ) is closed. Then f induces a bijection between the sets of connected components of X and Y . Proof. Let T ⊂ Y be a connected component. Note that T is closed, see Lemma 5.4.3. The lemma follows if we show that p−1 (T ) is connected because any connected subset of X maps` into a connected component of Y by Lemma 5.4.2. Suppose that p−1 (T ) = Z1 `Z2 with Z1 , Z2 closed. For any t ∈ T we see that p−1 ({t}) = Z1 ∩ p−1 ({t}) Z2 ∩ p−1 ({t}). By (1) we see p−1 ({t}) is connected ` we conclude that either p−1 ({t}) ⊂ Z1 or p−1 ({t}) ⊂ Z2 . In other words T = T1 T2 with p−1 (Ti ) = Zi . By (2) we conclude that Ti is closed in Y . Hence either T1 = ∅ or T2 = ∅ as desired. 

5.5. IRREDUCIBLE COMPONENTS

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Lemma 5.4.5. Let f : X → Y be a continuous map of topological spaces. Assume that (a) f is open, (b) all fibres of f are nonempty and connected. Then f induces a bijection between the sets of connected components of X and Y . Proof. This is a special case of Lemma 5.4.4.



Lemma 5.4.6. Let f : X → Y be a continuous map of nonempty topological spaces. Assume that (a) Y is connected, (b) f is open and closed, and (c) there is a point y ∈ Y such that the fiber f −1 (y) is a finite set. Then X has at most |f −1 (y)| connected components. Hence any connected component T of X is open and closed, and p(T ) is a nonempty open and closed subset of Y , which is therefore equal to Y . Proof. If the topological space X has at least N connected components for some N ∈ N, we find by induction a decomposition X = X1 q · · · q XN of X as a disjoint union of N nonempty open and closed subsets X1 , . . . , XN of X. As f is open and closed, each f (Xi ) is a nonempty open and closed subset of Y and is hence equal to Y . In particular the intersection Xi ∩ p−1 (y) is nonempty for each 1 ≤ i ≤ N . Hence p−1 (y) has at least N elements.  Definition 5.4.7. A topological space is totally disconnected if the connected components are all singletons. A discrete space is totally disconnected. A totally disconnected space need not be discrete, for example Q ⊂ R is totally disconnected but not discrete. Definition 5.4.8. A topological space X is called locally connected if every point x ∈ X has a fundamental system of connected neighbourhoods. Lemma 5.4.9. Let X be a topological space. If X is locally connected, then (1) any open subset of X is locally connected, and (2) the connected components of X are open. So also the connected components of open subsets of X are open. In particular, every point has a fundamental system of open connected neighbourhoods. Proof. Omitted.

 5.5. Irreducible components

Definition 5.5.1. Let X be a topological space. (1) We say X is irreducible, if X is not empty, and whenever X = Z1 ∪ Z2 with Zi closed, we have X = Z1 or X = Z2 . (2) We say Z ⊂ X is an irreducible component of X if Z is a maximal irreducible subset of X. An irreducible space is obviously connected. Lemma 5.5.2. Let f : X → Y be a continuous map of topological spaces. If E ⊂ X is an irreducible subset, then f (E) ⊂ Y is irreducible as well. Proof. Omitted.



Lemma 5.5.3. Let X be a topological space. If T ⊂ X is irreducible so is its closure in X. Any irreducible component of X is closed. Every point of X is contained in some irreducible component of X.

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Proof. Let T be the closure of the irreducible subset T . If T = Z1 ∪ Z2 with Zi ⊂ T closed, then T = (T ∩ Z1 ) ∪ (T ∩ Z2 ) and hence T equals one of the two, say T = Z1 ∩ T . Thus clearly T ⊂ Z1 as desired. Pick a point x ∈ X. Consider the set A of irreducible subsets x ∈ Tα ⊂ X. Note that A is nonempty since {x} ∈ A. There is a partial ording on A coming from inclusion: α ≤Sα0 ⇔ Tα ⊂ Tα0 . Choose a maximal totally ordered subset A0 ⊂ A, and let T = α∈A0 Tα . We claim that T is irreducible. Namely, suppose that T = Z1 ∪ Z2 is a union of two closed subsets of T . For each α ∈ A0 we have either Tα ⊂ Z1 or Tα ⊂ Z2 , by irreducibility of Tα . Suppose that for some α0 ∈ A0 we have Tα0 6⊂ Z1 (say, if not we’re done anyway). Then, since A0 is totally ordered we see immediately that Tα ⊂ Z2 for all α ∈ A0 . Hence T = Z2 .  A singleton is irreducible. Thus if x ∈ X is a point then the closure {x} is an irreducible closed subset of X. Definition 5.5.4. Let X be a topological space. (1) Let Z ⊂ X be an irreducible closed subset. A generic point of Z is a point ξ ∈ Z such that Z = {ξ}. (2) The space X is called Kolmogorov, if for every x, x0 ∈ X, x 6= x0 there exists a closed subset of X which contains exactly one of the two points. (3) The space X is called sober if every irreducible closed subset has a unique generic point. A space X is Kolmogorov if for x1 , x2 ∈ X we have x1 = x2 if and only if {x1 } = {x2 }. Hence we see that a sober topological space is Kolmogorov. S Lemma 5.5.5. Let X be a topological space. If X has an open covering X = Xi with Xi sober (resp. Kolmogorov), then X is sober (resp. Kolmogorov). Proof. Omitted.



Example 5.5.6. Recall that a topological space X is Hausdorff iff for every distinct pair of points x, y ∈ X there exist disjoint opens U, V ⊂ X such that x ∈ U , y ∈ V . In this case X is irreducible if and only if X is a singleton. Similarly, any subset of X is irreducible if and only if it is a singleton. Hence a Hausdorff space is sober. Lemma 5.5.7. Let f : X → Y be a continuous map of topological spaces. Assume that (a) Y is irreducible, (b) f is open, and (c) there exists a dense collection of points y ∈ Y such that f −1 (y) is irreducible. Then X is irreducible. Proof. Suppose Y = Z1 ∪Z2 with Zi closed. Consider the open sets U1 = Z1 \Z2 = Y \ Z2 and U2 = Z2 \ Z1 = Y \ Z2 . To get a contradiction assume that U1 and U2 are both nonempty. By (b) we see that f (Ui ) is open. By (a) we have X irreducible and hence f (U1 )∩f (U2 ) 6= ∅. By (c) there is a point y which corresponds to a point of this intersection such that the fibre Xy = f −1 (y) is irreducible. Then Xy ∩ U1 and Xy ∩ U2 are nonempty disjoint open subsets of Xy which is a contradiction.  Lemma 5.5.8. Let f : X → Y be a continuous map of topological spaces. Assume that (a) f is open, and (b) for every y ∈ Y the fibre f −1 (y) is irreducible. Then f induces a bijection between irreducible components.

5.6. NOETHERIAN TOPOLOGICAL SPACES

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Proof. We point out that assumption (b) implies that f is surjective (see Definition 5.5.1). Let T ⊂ Y be an irreducible component. Note that T is closed, see Lemma 5.5.3. The lemma follows if we show that p−1 (T ) is irreducible because any irreducible subset of X maps into an irreducible component of Y by Lemma 5.5.2. Note that p−1 (T ) → T satisfies the assumptions of Lemma 5.5.7. Hence we win.  5.6. Noetherian topological spaces Definition 5.6.1. A topological space is called Noetherian if the descending chain condition holds for closed subsets of X. A topological space is called locally Noetherian if every point has a neighbourhood which is Noetherian. Lemma (1) (2) (3)

5.6.2. Let X be a Noetherian topological space. Any subset of X with the induced topology is Noetherian. The space X has finitely many irreducible components. Each irreducible component of X contains a nonempty open of X.

Proof. Let T ⊂ X be a subset of X. Let T1 ⊃ T2 ⊃ . . . be a descending chain of closed subsets of T . Write Ti = T ∩Zi with Zi ⊂ X closed. Consider the descending chain of closed subsets Z1 ⊃ Z1 ∩Z2 ⊃ Z1 ∩Z2 ∩Z3 . . . This stabilizes by assumption and hence the original sequence of Ti stabilizes. Thus T is Noetherian. Let A be the set of closed subsets of X which do not have finitely many irreducible components. Assume that A is not empty to arrive at a contradiction. The set A is partially ordered by inclusion: α ≤ α0 ⇔ Zα ⊂ Zα0 . By the descending chain condition we may find a smallest element of A, say Z. As Z is not a finite union of irreducible components, it is not irreducible. Hence we can write = Z 0 ∪ Z 00Sand S Z 0 0 both are strictly smaller closed subsets. By construction Z =S Zi and Z 00 = Zj00 S are finite unions of their irreducible components. Hence Z = Zi0 ∪ Zj00 is a finite union of irreducible closed subsets. After removing redundant members of this expression, this will be the decomposition of Z into its irreducible components, a contradiction. Let Z ⊂ X be an irreducible component of X. Let Z1 , . . . , Zn be the other irreducible components of X. Consider U = Z \ (Z1 ∪ . . . ∪ Zn ). This is not empty since otherwise the irreducible space Z would be contained in one of the other Zi . Because X = Z ∪ Z1 ∪ . . . Zn (see Lemma 5.5.3), also U = X \ (Z1 ∪ . . . ∪ Zn ) and hence open in X. Thus Z contains a nonempty open of X.  Lemma 5.6.3. Let f : X → Y be a continuous map of topological spaces. (1) If X is Noetherian, then f (X) is Noetherian. (2) If X is locally Noetherian and f open, then f (X) is locally Noetherian. Proof. In case (1), suppose that Z1 ⊃ Z2 ⊃ Z2 ⊃ . . . is a descending chain of closed subsets of f (X) (as usual with the induced topology as a subset of Y ). Then f −1 (Z1 ) ⊃ f −1 (Z2 ) ⊃ f −1 (Z3 ) ⊃ . . . is a descending chain of closed subsets of X. Hence this chain stabilizes. Since f (f −1 (Zi )) = Zi we conclude that Z1 ⊃ Z2 ⊃ Z2 ⊃ . . . stabilizes also. In case (2), let y ∈ f (X). Choose x ∈ X with f (x) = y. By assumption there exists a neighbourhood E ⊂ X of x which is Noetherian. Then f (E) ⊂ f (X) is a neighbourhood which is Noetherian by part (1). 

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Lemma 5.6.4. Let X be a topological space. Let Xi ⊂ X, i = 1, . . . , n be a finite collection of subsets. If each Xi is Noetherian (with the induced topology), then S X i=1,...,n i is Noetherian (with the induced topology). Proof. Omitted.



Example 5.6.5. Any Noetherian topological space has a closed point (combine Lemmas 5.9.6 and 5.9.9). Let X = {1, 2, 3, . . .}. Define a topology on X with opens ∅, {1, 2, . . . , n}, n ≥ 1 and X. Thus X is a locally Noetherian topological space, without any closed points. This space cannot be the underlying topological space of a locally Noetherian scheme, see Properties, Lemma 23.5.8. Lemma 5.6.6. Let X be a locally Noetherian topological space. Then X is locally connected. Proof. Let x ∈ X. Let E be a neighbourhood of x. We have to find a connected neighbourhood of x contained in E. By assumption there exists a neighbourhood E 0 of x which is Noetherian. Then E ∩ E 0 is Noetherian, see Lemma 5.6.2. Let E ∩E 0 = Y1 ∪. . .∪Y Sn be the decomposition into irreducible components, see Lemma 5.6.2. Let E 00 = x∈Yi Yi . This is a connected subset of E ∩ E 0 containing x. It S contains the open E ∩ E 0 \ ( x6∈Yi Yi ) of E ∩ E 0 and hence it is a neighbourhood of x in X. This proves the lemma.  5.7. Krull dimension Definition 5.7.1. Let X be a topological space. (1) A chain of irreducible closed subsets of X is a sequence Z0 ⊂ Z1 ⊂ . . . ⊂ Zn ⊂ X with Zi closed irreducible and Zi 6= Zi+1 for i = 0, . . . , n − 1. (2) The length of a chain Z0 ⊂ Z1 ⊂ . . . ⊂ Zn ⊂ X of irreducible closed subsets of X is the integer n. (3) The dimension or more precisely the Krull dimension dim(X) of X is the element of {∞, 0, 1, 2, 3, . . .} defined by the formula: dim(X) = sup{lengths of chains of irreducible closed subsets} (4) Let x ∈ X. The Krull dimension of X at x is defined as dimx (X) = min{dim(U ), x ∈ U ⊂ X open} the minimum of dim(U ) where U runs over the open neighbourhoods of x in X. Note that if U 0 ⊂ U ⊂ X are open then dim(U 0 ) ≤ dim(U ). Hence if dimx (X) = d then x has a fundamental system of open neighbourhoods U with dim(U ) = dimx (X). Example 5.7.2. The Krull dimension of the usual Euclidean space Rn is 0. Example 5.7.3. Let X = {s, η} with open sets given by {∅, {η}, {s, η}}. In this case a maximal chain of irreducible closed subsets is {s} ⊂ {s, η}. Hence dim(X) = 1. It is easy to generalize this example to get a (n + 1)-element topological space of Krull dimension n. Definition 5.7.4. Let X be a topological space. We say that X is equidimensional if every irreducible component of X has the same dimension.

5.9. QUASI-COMPACT SPACES AND MAPS

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5.8. Codimension and catenary spaces Definition 5.8.1. Let X be a topological space. We say X is catenary if for every pair of irreducible closed subsets T ⊂ T 0 there exist a maximal chain of irreducible closed subsets T = T0 ⊂ T1 ⊂ . . . ⊂ Te = T 0 and every such chain has the same length. Lemma 5.8.2. Let X be a topological space. The following are equivalent: (1) X is catenary, (2) X has an open covering by catenary spaces. Moreover, in this case any locally closed subspace of X is catenary. Proof. Suppose that X is catenary and that U ⊂ X is an open subset. The rule T 7→ T defines a bijective inclusion preserving map between the closed irreducible subsets of U and the closed irreducible subsets of X which meet U . Using this the lemma easily follows. Details omitted.  Definition 5.8.3. Let X be a topological space. Let Y ⊂ X be an irreducible closed subset. The codimension of Y in X is the supremum of the lengths e of chains Y = Y0 ⊂ Y1 ⊂ . . . ⊂ Ye ⊂ X of irreducible closed subsets in X starting with Y . We will denote this codim(Y, X). Lemma 5.8.4. Let X be a topological space. Let Y ⊂ X be an irreducible closed subset. Let U ⊂ X be an open subset such that Y ∩ U is nonempty. Then codim(Y, X) = codim(Y ∩ U, U ) Proof. Follows from the observation made in the proof of Lemma 5.8.2.



Example 5.8.5. Let X = [0, 1] be the unit interval with the following topology: The sets [0, 1], (1 − 1/n, 1] for n ∈ N, and ∅ are open. So the closed sets are ∅, {0}, [0, 1 − 1/n] for n > 1 and [0, 1]. This is clearly a Noetherian topological space. But the irreducible closed subset Y = {0} has infinite codimension codim(Y, X) = ∞. To see this we just remark that all the closed sets [0, 1 − 1/n] are irreducible. Lemma 5.8.6. Let X be a topological space. The following are equivalent: (1) X is catenary, and (2) for pair of irreducible closed subsets Y ⊂ Y 0 we have codim(Y, Y 0 ) < ∞ and for every triple Y ⊂ Y 0 ⊂ Y 00 of irreducible closed subsets we have codim(Y, Y 00 ) = codim(Y, Y 0 ) + codim(Y 0 , Y 00 ). Proof. Omitted.

 5.9. Quasi-compact spaces and maps

The phrase “compact” will be reserved for Hausdorff topological spaces. And many spaces occuring in algebraic geometry are not Hausdorff. Definition 5.9.1. Quasi-compactness. (1) We say that a topological space X is quasi-compact if every open covering of X has a finite refinement.

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(2) We say that a continuous map f : X → Y is quasi-compact if the inverse image f −1 (V ) of every quasi-compact open V ⊂ Y is quasi-compact. (3) We say a subset Z ⊂ X is retrocompact if the inclusion map Z → X is quasi-compact. In many texts on topology a space is called compact if it is quasi-compact and Hausdorff; and in other texts the Hausdorff condition is omitted. To avoid confusion in algebraic geometry we use the term quasi-compact. Note that the notion of quasi-compactness of a map is very different from the notion of a “proper map” in topology, since there one requires the inverse image of any (quasi-)compact subset of the target to be (quasi-)compact, whereas in the definition above we only consider quasi-compact open sets. Lemma 5.9.2. A composition of quasi-compact maps is quasi-compact. Proof. Omitted.



Lemma 5.9.3. A closed subset of a quasi-compact topological space is quasi-compact. Proof. Omitted.



The following is really a reformulation of the quasi-compact property. Lemma 5.9.4. Let X be a quasi-compact topological space. If {Zα }α∈A is a collection of closedTsubsets such that the intersection of each finite subcollection is nonempty, then α∈A Zα is nonempty. Proof. Omitted.



Lemma 5.9.5. Let f : X → Y be a continuous map of topological spaces. (1) If X is quasi-compact, then f (X) is quasi-compact. (2) If f is quasi-compact, then f (X) is retrocompact. S S −1 Proof. If f (X) = Vi is an open covering, then X = f (Vi ) is an open covering. Hence if X is quasi-compact then X = f −1 (Vi1 ) ∪ . . . ∪ f −1 (Vin ) for some i1 , . . . , in ∈ I and hence f (X) = Vi1 ∪ . . . ∪ Vin . This proves (1). Assume f is quasi-compact, and let V ⊂ Y be quasi-compact open. Then f −1 (V ) is quasicompact, hence by (1) we see that f (f −1 (V )) = f (X) ∩ V is quasi-compact. Hence f (X) is retrocompact.  Lemma 5.9.6. Let X be a topological space. Assume that (1) X is nonempty, (2) X is quasi-compact, and (3) X is Kolmogorov. Then X has a closed point. Proof. Consider the set T = {Z ⊂ X | Z = {x} for some x ∈ X} of all closures of singletons in X. It is nonempty since X is nonempty. Make T into a partially ordered set using the relation of inclusion. Suppose Zα , α ∈ A is T a totally ordered subset of T . By Lemma 5.9.4 we see that α∈A Zα 6= ∅. Hence T there exists some x ∈ α∈A Zα and we see that Z = {x} ∈ T is a lower bound for the family. By Zorn’s lemma there exists a minimal element Z ∈ T . As X is Kolmogorov we conclude that Z = {x} for some x and x ∈ X is a closed point. 

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Lemma 5.9.7. Let X be a topological space. Assume (1) X is quasi-compact, (2) X has a basis for the topology consisting of quasi-compact opens, and (3) the intersection of two quasi-compact opens is quasi-compact. For any x ∈ X the connected component of X containing x is the intersection of all open and closed subsets of X containing x. T Proof. Let T be the connected component containing x. Let S = α∈A Zα be the intersection of all open and closed subsets Zα of X containing x. Note that S is closed in X. Note that any finite intersection of Zα ’s is a Zα . Because T is connected and x ∈ T we have T ⊂ S. It suffices to show` that S is connected. If not, then there exists a disjoint union decomposition S = B C with B and C open and closed in S. In particular, B and C are closed in X, and so quasi-compact by Lemma 5.9.3 and assumption (1). By assumption (2) there exist quasi-compact opens U, V ⊂ X with B = S ∩ U and C = S ∩ V (details omitted). Then U ∩ V ∩ S = ∅. Hence T By α U ∩ V ∩ Zα = ∅. By assumption (3) the intersection U ∩ V is quasi-compact. ` Lemma 5.9.4 for some α ∈ A we have U ∩V ∩Zα = ∅. Hence Zα = U ∩Zα V ∩Zα is a decomposition into two open pieces, hence U ∩ Zα and V ∩ Zα are open and closed in X. Thus, if x ∈ B say, then we see that S ⊂ U ∩ Zα and we conclude that C = ∅.  Lemma 5.9.8. Let X be a topological space. Assume (1) X is quasi-compact, (2) X has a basis for the topology consisting of quasi-compact opens, and (3) the intersection of two quasi-compact opens is quasi-compact. For a subset T ⊂ X the following are equivalent: (a) T is an intersection of open and closed subsets of X, and (b) T is closed in X and is a union of connected components of X. Proof. It is clear that (a) implies (b). Assume (b). Let x ∈ X, x 6∈ T . Let x ∈ C ⊂ X be T the connected component of X containing x. By Lemma 5.9.7 we see that C = Vα is the intersection of all open and closed subsets Vα of X which contain C. In particular, any pairwise intersection Vα ∩ Vβ occurs as a VαT. As T is a union of connected components of X we see that C ∩ T = ∅. Hence T ∩ Vα = ∅. Since T is quasi-compact as a closed subset of a quasi-compact space (see Lemma 5.9.3) we deduce that T ∩ Vα = ∅ for some α, see Lemma 5.9.4. For this α we see that Uα = X \ Vα is an open and closed subset of X which contains T and not x. The lemma follows.  Lemma 5.9.9. Let X be a Noetherian topological space. (1) The space X is quasi-compact. (2) Any subset of X is retrocompact. S Proof. Suppose X = Ui is an open covering of X indexed by the set I which does not have a refinement by a finite open covering. Choose i1 , i2 , . . . elements of I inductively in the following way: If X 6= Ui1 ∪ . . . ∪ Uin then choose in+1 such that Uin+1 is not contained in Ui1 ∪ . . . ∪ Uin . Thus we see that X ⊃ (X \ Ui1 ) ⊃ (X \ Ui1 ∪ Ui2 ) ⊃ . . . is a strictly decreasing infinite sequence of closed subsets. This contradicts the fact that X is Noetherian. This proves the first assertion.

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The second assertion is now clear since every subset of X is Noetherian by Lemma 5.6.2.  Lemma 5.9.10. A quasi-compact locally Noetherian space is Noetherian. Proof. The conditions imply immediately that X has a finite covering by Noetherian subsets, and hence is Noetherian by Lemma 5.6.4.  5.10. Constructible sets Definition 5.10.1. Let X be a topological space. Let E ⊂ X be a subset of X. (1) We say E is constructible1 in X if E is a finite union of subsets of the form U ∩ V c where U, V ⊂ X are open and retrocompact in X. (2) We say S E is locally construcible in X if there exists an open covering X = Vi such that each E ∩ Vi is construcible in Vi . Lemma 5.10.2. The collection of constructible sets is closed under finite intersections, finite unions and complements. Proof. Note that if U1 , U2 are open and retrocompact in X then so is U1 ∪ U2 because the union of two quasi-compact subsets of X is quasi-compact. It is also true that U1 ∩ U2 is retrocompact. Namely, suppose U ⊂ X is quasi-compact open, then U2 ∩ U is quasi-compact because U2 is retrocompact in X, and then we conclude U1 ∩ (U2 ∩ U ) is quasi-compact because U1 is retrocompact in X. From this it is formal to show that the complement of a constructible set is constructible, that finite unions of constructibles are constructible, and that finite intersections of constructibles are constructible.  Lemma 5.10.3. Let f : X → Y be a continuous map of topological spaces. If the inverse image of every retrocompact open subset of Y is retrocompact in X, then inverse images of constructible sets are constructible. Proof. This is true because f −1 (U ∩ V c ) = f −1 (U ) ∩ f −1 (V )c , combined with the definition of constructible sets.  Lemma 5.10.4. Let U ⊂ X be open. For a constructible set E ⊂ X the intersection E ∩ U is constructible in U . Proof. Suppose that V ⊂ X is retrocompact open in X. It suffices to show that V ∩ U is retrocompact in U by Lemma 5.10.3. To show this let W ⊂ U be open and quasi-compact. Then W is open and quasi-compact in X. Hence V ∩W = V ∩U ∩W is quasi-compact as V is retrocompact in X.  Lemma 5.10.5. Let X be a topological space. Let E ⊂ X be a subset. Let X = V1 ∪ . . . ∪ Vm be a finite covering by retrocompact opens. Then E is constructible in X if and only if E ∩ Vj is constructible in Vj for each j = 1, . . . , m. Proof. If E is constructible in X, then by Lemma 5.10.4 we see that E ∩ Vj is construcible in Vj for all j. Conversely, suppose that E ∩ Vj is constructible in Vj for each j = 1, . . . , m. Then E is a finite union of sets of the form E 0 = U 0 ∩(Vj \V 0 ) where U 0 , V 0 are open and retrocompact subsets of Vj . Note that U 0 and V 0 are 1In the second edition of EGA I [GD71] this was called a “globally constructible” set and a the terminology “constructible” was used for what we call a locally constructible set.

5.11. CONSTRUCTIBLE SETS AND NOETHERIAN SPACES

143

also open and retrocompact in X (as a composition of quasi-compact maps is quasicompact, see Lemma 5.9.2). Since E 0 = U 0 ∩ (V 0 )c where the complement is in X we win.  Lemma 5.10.6. Let X be a topological space. Suppose that Z ⊂ X is irreducible. Let E ⊂ X be a finite union of locally closed subsets (e.g. E is constructible). The following are equivalent (1) The intersection E ∩ Z contains an open dense subset of Z. (2) The intersection E ∩ Z is dense in Z. If Z has a generic point ξ, then this is also equivalent to (3) We have ξ ∈ E. S Proof. Write E = Ui ∩ Zi as the finite union of intersections of open sets Ui and closed sets Zi . Suppose that E ∩ Z is dense in Z. Note that the closure of E ∩ Z is the union of the closures of the intersections Ui ∩ Zi ∩ Z. Hence we see that Ui ∩ Zi ∩ Z is dense in Z for some i = i0 . As Z is closed we have either Z ∩ Zi = Z or Z ∩ Zi is not dense, hence we conclude Z ⊂ Zi0 . Then Ui0 ∩ Zi0 ∩ Z = Ui0 ∩ Z is an open not empty subset of Z. Because Z is irreducible, it is open dense. The converse is obvious. Suppose that ξ ∈ Z is a generic point. Of course if (1) ⇔ (2) holds, then ξ ∈ E. Conversely, if ξ ∈ E, then ξ ∈ Ui ∩ Zi for some i = i0 . Clearly this implies Z ⊂ Zi0 and hence Ui0 ∩ Zi0 ∩ Z = Ui0 ∩ Z is an open not empty subset of Z. We conclude as before.  5.11. Constructible sets and Noetherian spaces Lemma 5.11.1. Let X be a Noetherian topological space. Constructible sets in X are finite unions of locally closed subsets of X. Proof. This follows immediately from Lemma 5.9.9.



Lemma 5.11.2. Let f : X → Y be a continuous map of Noetherian topological spaces. If E ⊂ Y is constructible in Y , then f −1 (E) is constructible in X. Proof. Follows immediately from Lemma 5.11.1 and the definition of a continuous map.  Lemma 5.11.3. Let X be a Noetherian topological space. Let E ⊂ X be a subset. The following are equivalent (1) E is constructible in X, and (2) for every irreducible closed Z ⊂ X the intersection E ∩ Z either contains a nonempty open of Z or is not dense in Z. Proof. Assume E is constructible and Z ⊂ X irreducible closed. Then E ∩ Z is constructible in Z by Lemma 5.11.2. Hence E ∩ Z is a finite union of nonempty locally closed subsets Ti of Z. Clearly if none of the Ti is open in Z, then E ∩ Z is not dense in Z. In this way we see that (1) implies (2). Conversely, assume (2) holds. Consider the set S of closed subsets Y of X such that E ∩ Y is not constructible in Y . If S = 6 ∅, then it has a smallest element Y as X is Noetherian. Let Y = Y1 ∪ . . . ∪ Yr be the decomposition of Y into its irreducible components, see Lemma 5.6.2. If r > 1, then each Yi ∩ E is constructible in Yi and

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hence a finite union of locally closed subsets of Yi . Thus E ∩ Y is a finite union of locally closed subsets of Y too and we conclude that E ∩ Y is constructible in Y by Lemma 5.11.1. This is a contradication and so r = 1. If r = 1, then Y is irreducible, and by assumption (2) we see that E ∩ Y either (a) contains an open V of Y or (b) is not dense in Y . In case (a) we see, by minimality of Y , that E ∩ (Y \ V ) is a finite union of locally closed subsets of Y \ V . Thus E ∩ Y is a finite union of locally closed subsets of Y and is constructible by Lemma 5.11.1. This is a contradication and so we must be in case (b). In case (b) we see that E ∩ Y = E ∩ Y 0 for some proper closed subset Y 0 ⊂ Y . By minimality of Y we see that E ∩ Y 0 is a finite union of locally closed subsets of Y 0 and we see that E ∩ Y 0 = E ∩ Y is a finite union of locally closed subsets of Y and is constructible by Lemma 5.11.1. This contradication finishes the proof of the lemma.  Lemma 5.11.4. Let X be a Noetherian topological space. Let x ∈ X. Let E ⊂ X be constructible in X. The following are equivalent (1) E is a neighbourhood of x, and (2) for every irreducible closed subset Y of X which contains x the intersection E ∩ Y is dense in Y . Proof. It is clear that (1) implies (2). Assume (2). Consider the set S of closed subsets Y of X containing x such that E ∩ Y is not a neighbourhood of x in Y . If S= 6 ∅, then it has a smallest element Y as X is Noetherian. Let Y = Y1 ∪. . .∪Yr be the decomposition of Y into its irreducible components, see Lemma 5.6.2. If r > 1, then each Yi ∩ E is a neighbourhood of x in Yi by minimality of Y . Thus E ∩ Y is a neighbourhood of x in Y . This is a contradication and so r = 1. If r = 1, then Y is irreducible, and by assumption (2) we see that E ∩ Y is dense in Y . Thus E ∩ Y contains an open V of Y , see Lemma 5.11.3. If x ∈ V then E ∩Y is a neighbourhood of x in Y which is a contradiction. If x 6∈ V , then Y 0 = Y \ V is a proper closed subset of Y containing x. By minimality of Y we see that E ∩ Y 0 contains an open neighbourhood V 0 ⊂ Y 0 of x in Y 0 . But then V 0 ∪ V is an open neighbourhood of x in Y contained in E, a contradiction. This contradication finishes the proof of the lemma.  Lemma 5.11.5. Let X be a Noetherian topological space. Let E ⊂ X be a subset. The following are equivalent (1) E is open in X, and (2) for every irreducible closed subset Y of X the intersection E ∩ Y is either empty or contains a nonempty open of Y . Proof. This follows formally from Lemmas 5.11.3 and 5.11.4.



5.12. Characterizing proper maps We include a section discussing the notion of a proper map in usual topology. It turns out that in topology, the notion of being proper is the same as the notion of being universally closed, in the sense that any base change is a closed morphism (not just taking products with spaces). The reason for doing this is that in algebraic geometry we use this notion of universal closedness as the basis for our definition of properness.

5.12. CHARACTERIZING PROPER MAPS

145

Lemma 5.12.1 (Tube lemma). Let X and Y be topological spaces. Let A ⊂ X and B ⊂ Y be quasi-compact subsets. Let A × B ⊂ W ⊂ X × Y with W open in X × Y . Then there exists opens A ⊂ U ⊂ X and B ⊂ V ⊂ Y such that U × V ⊂ W . Proof. For every a ∈ A and b ∈ B there exist opens U(a,b) of X and V(a,b) of Y such that (a, b) ∈ U(a,b) × V(a,b) ⊂ W . Fix b and we see there exist a finite number a1 , . . . , an such that A ⊂ U(a1 ,b) ∪ . . . ∪ U(an ,b) . Hence A × {b} ⊂ (U(a1 ,b) ∪ . . . ∪ U(an ,b) ) × (V(a1 ,b) ∪ . . . ∪ V(an ,b) ) ⊂ W . Thus for every b ∈ B there exists opens Ub ⊂ X and Vb ⊂ Y such that A × {b} ⊂ Ub × Vb ⊂ W . As above there exist a finite number b1 , . . . , bm such that B ⊂ Vb1 ∪ . . . ∪ Vbm . Then we win because A × B ⊂ (Ub1 ∩ . . . ∩ Ubm ) × (Vb1 ∪ . . . ∪ Vbm ).  The notation in the following definition may be slightly different from what you are used to. Definition 5.12.2. Let f : X → Y be a continuous map between topological spaces. (1) We say that the map f is closed iff the image of every closed subset is closed. (2) We say that the map f is proper2 iff the map Z × X → Z × Y is closed for any topological space Z. (3) We say that the map f is quasi-proper iff the inverse image f −1 (V ) of every quasi-compact V ⊂ Y is quasi-compact. (4) We say that f is universally closed iff the map f 0 : Z ×Y X → Z is closed for any map g : Z → Y . The following lemma is useful later. Lemma 5.12.3. A topological space X is quasi-compact if and only if the projection map Z × X → Z is closed for any topological space Z. Proof. (See also S remark below.) If X is not quasi-compact, there exists an open covering X = i∈I Ui such that no finite number of Ui cover X. Let Z be the subset of the power set P(I) of I consisting of I and all nonempty finite subsets of I. Define a topology on Z with as a basis for the topology the following sets: (1) All subsets of Z \ {I}. (2) The empty set. (3) For every finite subset K of I the set UK := {J ⊂ I | J ∈ Z, K ⊂ J}). It is left to the reader to verify this is the basis for a topology. Consider the subset of Z × X defined by the formula \ M = {(J, x) | J ∈ Z, x ∈ Uic )} i∈J

If (J, x) 6∈ M , then x ∈ Ui for some i ∈ J. Hence U{i} × Ui ⊂ Z × X is an open subset containing (J, x) and not intersecting M . Hence M is closed. The projection of M to Z is Z − {I} which is not closed. Hence Z × X → Z is not closed. Assume X is quasi-compact. Let Z be a topological space. Let M ⊂ Z × X be closed. Let z ∈ Z be a point which is not in pr1 (M ). By the Tube Lemma 5.12.1 there exists an open U ⊂ Z such that U × X is contained in the complement of M . Hence pr1 (M ) is closed.  2This is the terminology used in [Bou71]. Usually this is what is called “universally closed” in the literature. Thus our notion of proper does not involve any separation conditions.

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Remark 5.12.4. Lemma 5.12.3 is a combination of [Bou71, I, p. 75, Lemme 1] and [Bou71, I, p. 76, Corrolaire 1]. Theorem 5.12.5. Let f : X → Y be a continuous map between topological spaces. The following condition is equivalent. (1) (2) (3) (4)

The The The The

map map map map

f f f f

is is is is

quasi-proper and closed. proper. universally closed. closed and f −1 (y) is quasi-compact for any y ∈ Y .

Proof. (See also the remark below.) If the map f satisfies (1), it automatically satisfies (4) because any single point is quasi-compact. Assume map f satisfies (4). We will prove it is universally closed, i.e., (3) holds. Let g : Z → Y be a continuous map of topological spaces and consider the diagram Z ×Y X

g0

f0

 Z

g

/X  /Y

f

During the proof we will use that Z ×Y X → Z × X is a homeomorphism onto its image, i.e., that we may identify Z ×Y X with the corresponding subset of Z × X with the induced topology. The image of f 0 : X ×Y Z → Z is Im(f 0 ) = {z : g(z) ∈ f (X)}. Because f (X) is closed, we see that Im(f 0 ) is a closed subspace of Z. Consider a closed subset P ⊂ X ×Y Z. Let z ∈ Z, z 6∈ f 0 (P ). If z 6∈ Im(f 0 ), then Z \ Im(f 0 ) is an open neighbourhood which avoids f 0 (P ). If z is in Im(f 0 ) then (f 0 )−1 {z} = {z} × f −1 {g(z)} and f −1 {g(z)} is quasi-compact by assumption. Because P is a closed subset of Z ×Y X, we have a closed P 0 of Z × X such that P = P 0 ∩ Z ×Y X. Since (f 0 )−1 {z} is a subset of P c = P 0c ∪ (Z ×Y X)c , we see that (f 0 )−1 {z} is disjoint from (Z ×Y X)c . Hence (f 0 )−1 {z} is contained in P 0c . We may apply the Tube Lemma 5.12.1 to (f 0 )−1 {z} = {z} × f −1 {g(z)} ⊂ (P 0 )c ⊂ Z × X. This gives U × V containing (f 0 )−1 {z} where U and V are open sets in X and Z respectively and U × V has empty intersection with P 0 . Hence z is contained in V and V has empty intersection with the image of P . As a result, the map f is universally closed. The implication (3) ⇒ (2) is trivial. Namely, given any topological space Z consider the projection morphism g : Z × Y → Y . Then it is easy to see that f 0 is the map Z × X → Z × Y , in other words that (Z × Y ) ×Y X = Z × X. (This identification is a purely categorical property having nothing to do with topological spaces per se.) Assume f satisfies (2). We will prove it satisfies (1). Note that f is closed as f can be identified with the map {pt} × X → {pt} × Y which is assumed closed. Choose any quasi-compact subset K ⊂ Y . Let Z be any topological space. Because Z × X → Z × Y is closed we see the map Z × f −1 (K) → Z × K is closed (if T is closed in Z × f −1 (K), write T = Z × f −1 (K) ∩ T 0 for some closed T 0 ⊂ Z × X). Because K is quasi-compact, K × Z → Z is closed by Lemma 5.12.3. Hence the composition Z × f −1 (K) → Z × K → Z is closed and therefore f −1 (K) must be quasi-compact by Lemma 5.12.3 again. 

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Remark 5.12.6. Here are some references to the literature. In [Bou71, I, p. 75, Theorem 1] you can find: (2) ⇔ (4). In [Bou71, I, p. 77, Proposition 6] you can find: (2) ⇒ (1). Of course, trivially we have (1) ⇒ (4). Thus (1), (2) and (4) are equivalent. Fan Zhou claimed and proved that (3) and (4) are equivalent; let me know if you find a reference in the literature. 5.13. Jacobson spaces Definition 5.13.1. Let X be a topological space. Let X0 be the set of closed points of X. We say that X is Jacobson if every closed subset Z ⊂ X is the closure of Z ∩ X0 . Let X be a Jacobson space and let X0 be the set of closed points of X with the induced topology. Clearly, the definition implies that the morphism X0 → X induces a bijection between the closed subsets of X0 and the closed subsets of X. Thus many properties of X are inherted by X0 . For example, the Krull dimensions of X and X0 are the same. Lemma 5.13.2. Let X be a topological space. Let X0 be the set of closed points of X. Suppose that for every irreducible closed subset Z ⊂ X the intersection X0 ∩ Z is dense in Z. Then X is Jacobson. S Proof. Let Z ⊂ X be closed. According to Lemma 5.5.3 we have Z = Zi with Zi irreducible and closed. Thus is X0 ∩ Zi is dense in each Zi , then X0 ∩ Z is dense in Z.  Lemma 5.13.3. Let X be a sober, Noetherian topological space. If X is not Jacobson, then there exists a non-closed point ξ ∈ X such that {ξ} is locally closed. Proof. Assume X is sober, Noetherian and not Jacobson. By Lemma 5.13.2 there exists an irreducible closed subset Z ⊂ X which is not the closure of its closed points. Since X is Noetherian we may assume Z is minimal with this property. Let ξ ∈ Z be the unique generic point (here we use X is sober). Note that the closed points are dense in {z} for any z ∈ Z, z 6= ξ by minimality of Z. Hence the closure of the set of closed points of Z is a closed subset containing all z ∈ Z, z 6= ξ. Hence {ξ} is locally closed as desired.  S Lemma 5.13.4. Let X be a topological space. Let X = Ui be an open covering. Then S X is Jacobson if and only if each Ui is Jacobson. Moreover, in this case X0 = Ui,0 . Proof. Let X be a topological space. Let X0 be the set of closed points of X. Let Ui,0 be the set of closed points of Ui . Then X0 ∩ Ui ⊂ Ui,0 but equality may not hold in general. First, assume that each Ui is Jacobson. We claim that in this case X0 ∩ Ui = Ui,0 . Namely, suppose that x ∈ Ui,0 , i.e., x is closed in Ui . Let {x} be the closure in X. Consider {x} ∩ Uj . If x 6∈ Uj , then {x} ∩ Uj = ∅. If x ∈ Uj , then Ui ∩`Uj ⊂ Uj is an open subset of Uj containing x. Let T 0 = Uj \ Ui ∩ Uj and T = {x} T 0 . Then T , T 0 are closed subsets of Uj and T contains x. As U `j is Jacobson we see that the closed points of Uj are dense in T . Because T = {x} T 0 this can only be the case if x is closed in Uj . Hence {x} ∩ Uj = {x}. We conlude that {x} = {x} as desired.

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Let Z ⊂ X be a closed subset (still assuming each Ui is Jacobson). Since now we know that X0 ∩ Z ∩ Ui = Ui,0 ∩ Z are dense in Z ∩ Ui it follows immediately that X0 ∩ Z is dense in Z. Conversely, assume that X is Jacobson. Let Z ⊂ Ui be closed. Then X0 ∩Z is dense in Z. Hence also X0 ∩ Z is dense in Z, because Z \ Z is closed. As X0 ∩ Ui ⊂ Ui,0 we see that Ui,0 ∩ Z is dense in Z. Thus Ui is Jacobson as desired.  Lemma 5.13.5. Let X be Jacobson. The following types of subsets T ⊂ X are Jacobson: (1) (2) (3) (4) (5) (6)

Open subspaces. Closed subspaces. Locally closed subspaces. Finite unions of locally closed subspaces. Constructible sets. Any subset T ⊂ X which locally on X is a finite union of locally closed subsets.

In each of these cases closed points of T are closed in X. Proof. Let X0 be the set of closed points of X. For any subset T ⊂ X we let (∗) denote the property: (∗) For every closed subset Z ⊂ T the set Z ∩ X0 is dense in Z. Note that always X0 ∩ T ⊂ T0 . Hence property (∗) implies that T is Jacobson. In addition it clearly implies that every closed point of T is closed in X. Let U ⊂ X be an open subset. Suppose Z ⊂ U is closed. Then X0 ∩ Z is dense in Z. Hence X0 ∩ Z is dense in Z, because Z \ Z is closed. Thus (∗) holds. Let Z ⊂ X be a closed subset. Since closed subsets of Z are the same as closed subsets of X contained in Z property (∗) is immediate. Let T ⊂ X be locally closed. Write T = U ∩ Z for some open U ∩ X and some closed Z ⊂ X. Note that closed subsets of T are the same thing as closed subsets of U which happen to be contained in Z. Hence (∗) holds for T because we proved it for U above. Suppose Ti ⊂ X, i = 1, . . . , n are locally closed subsets. Let T = T1 ∪ . . . ∪ Tn . Suppose Z ⊂ T is closed. Then Zi = Z ∩ Ti is closed in Ti . By (∗) for Ti we see that Zi ∩ X0 is dense in Zi . Clearly this implies that X0 ∩ Z is dense in Z, and property (∗) holds for T . The case of constructible subsets is subsumed in the case of finite unions of locally closed subsets, see Definition 5.10.1. The S condition of the last assertion means that there exists an open covering X = Ui such that each T ∩ Ui is a finite union of locally closed subsets of Ui . We conclude that T is Jacobson by Lemma 5.13.4 and the case of a finite union of locally closed subsets dealt with above. It is formal to deduce (∗) for T from S S (∗) for all the inclusions T ∩ Ui ⊂ Ui and the assertions X0 = Ui,0 and T0 = (T ∩ Ui )0 from Lemma 5.13.4.  Lemma 5.13.6. A finite Kolmogorov Jacobson space is discrete.

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149

Proof. By induction on the number of points. The lemma holds if the space is empty. If X is a non-empty finite Kolmogorov space, choose a closed point x ∈ X, see Lemma 5.9.6. Then U = X \ {x} is a finite Jacobson space, see Lemma 5.13.5. By induction U is a finite discrete space, hence all its points are closed. By Lemma 5.13.5 all the points of U are also closed in X and we win.  Lemma 5.13.7. Suppose X is a Jacobson topological space. Let X0 be the set of closed points of X. There is a bijective, inclusion preserving correspondence {constructible subsets of X} ↔ {constructible subsets of X0 } given by E 7→ E ∩ X0 . This correspondence preserves the subset of retrocompact open subsets, as well as complements of these. Proof. Obvious from Lemma 5.13.5 above.



Lemma 5.13.8. Suppose X is a Jacobson topological space. Let X0 be the set of closed points of X. There is a bijective, inclusion preserving correspondence {finite unions loc. closed subsets of X} ↔ {finite unions loc. closed subsets of X0 } given by E 7→ E ∩ X0 . This correspondence preserves the subsets of locally closed, of open and of closed subsets. Proof. Obvious from Lemma 5.13.5 above.



5.14. Specialization Definition 5.14.1. Let X be a toplogical space. (1) If x, x0 ∈ X then we say x is a specialization of x0 , or x0 is a generalization x. of x if x ∈ {x0 }. Notation: x0 (2) A subset T ⊂ X is stable under specialization if for all x0 ∈ T and every specialization x0 x we have x ∈ T . (3) A subset T ⊂ X is stable under generalization if for all x ∈ T and every generalization x0 of x we have x0 ∈ T . Lemma (1) (2) (3)

5.14.2. Let X be a toplogical space. Any closed subset of X is stable under specialization. Any open subset of X is stable under generalization. A subset T ⊂ X is stable under specialization if and only if the complement T c is stable under generalization.

Proof. Omitted.



Definition 5.14.3. Let f : X → Y be a continuous map of topological spaces. (1) We say that specializations lift along f or that f is specializing if given y0 y in Y and any x0 ∈ X with f (x0 ) = y 0 there exists a specialization 0 x x of x0 in X such that f (x) = y. (2) We say that generalizations lift along f or that f is generalizing if given y0 y in Y and any x ∈ X with f (x) = y there exists a generalization x0 x of x in X such that f (x0 ) = y 0 . Lemma 5.14.4. Suppose f : X → Y and g : Y → Z are continuous maps of topological spaces. If specializations lift along both f and g then specializations lift along g ◦ f . Similarly for “generalizations lift along”.

150

Proof. Omitted.

5. TOPOLOGY



Lemma 5.14.5. Let f : X → Y be a continuous map of topological spaces. (1) If specializations lift along f , and if T ⊂ X is stable under specialization, then f (T ) ⊂ Y is stable under specialization. (2) If generalizations lift along f , and if T ⊂ X is stable under generalization, then f (T ) ⊂ Y is stable under generalization. Proof. Omitted.



Lemma 5.14.6. Let f : X → Y be a continuous map of topological spaces. (1) If f is closed then specializations lift along f . (2) If f is open, X is a Noetherian topological space, each irreducible closed subset of X has a generic point, and Y is Kolmogorov then generalizations lift along f . Proof. Assume f is closed. Let y 0 y in Y and any x0 ∈ X with f (x0 ) = y 0 be given. Consider the closed subset T = {x0 } of X. Then f (T ) ⊂ Y is a closed subset, and y 0 ∈ f (T ). Hence also y ∈ f (T ). Hence y = f (x) with x ∈ T , i.e., x0 x. Assume f is open, X Noetherian, every irreducible closed subset of X has a generic point, and Y is Kolmogorov. Let y 0 y in Y and any x ∈ X with f (x) = y be given. Consider T = f −1 ({y 0 }) ⊂ X. Take an open neighbourhood x ∈ U ⊂ X of x. Then f (U ) ⊂ Y is open and y ∈ f (U ). Hence also y 0 ∈ f (U ). In other words, T ∩U 6= ∅. This proves that x ∈ T . Since X is Noetherian, T is Noetherian (Lemma 5.6.2). Hence it has a decomposition T = T1 ∪ . . . ∪ Tn into irreducible components. Then correspondingly T = T1 ∪ . . . ∪ Tn . By the above x ∈ Ti for some i. By assumption there exists a generic point x0 ∈ Ti , and we see that x0 x. As x0 ∈ T 0 0 0 0 we see that f (x ) ∈ {y }. Note that f (Ti ) = f ({x }) ⊂ {f (x )}. If f (x0 ) 6= y 0 , then since Y is Kolmogorov f (x0 ) is not a generic point of the irreducible closed subset {y 0 } and the inclusion {f (x0 )} ⊂ {y 0 } is strict, i.e., y 0 6∈ f (Ti ). This contradicts the fact that f (Ti ) = {y 0 }. Hence f (x0 ) = y 0 and we win.  Lemma 5.14.7. Suppose that s, t : R → U and π : U → X are continuous maps of topological spaces such that (1) π is open, (2) U is sober, (3) s, t have finite fibres, (4) generalizations lift along s, t, (5) (t, s)(R) ⊂ U × U is an equivalence relation on U and X is the quotient of U by this equivalence relation (as a set). Then X is Kolmogorov. Proof. Properties (3) and (5) imply that a point x corresponds to an finite equivalence class {u1 , . . . , un } ⊂ U of the equivalence relation. Suppose that x0 ∈ X is a second point corresponding to the equivalence class {u01 , . . . , u0m } ⊂ U . Suppose that ui u0j for some i, j. Then for any r0 ∈ R with s(r0 ) = u0j by (4) we can find r r0 with s(r) = ui . Hence t(t) t(r0 ). Since {u01 , . . . , u0m } = t(s−1 ({u0j })) we 0 conclude that every element of {u1 , . . . , u0m } is the specialization of an element of {u1 , . . . , un }. Thus {u1 } ∪ . . . ∪ {un } is a union of equivalence classes, hence of the

5.15. SUBMERSIVE MAPS

151

form π −1 (Z) for some subset Z ⊂ X. By (1) we see that Z is closed in X and in fact Z = {x} because π({ui }) ⊂ {x} for each i. In other words, x x0 if and only 0 if some lift of x in U specializes to some lift of x in U , if and only if every lift of x0 in U is a specialization of some lift of x in U . Suppose that both x x0 and x0 x. Say x corresponds to {u1 , . . . , un } and 0 0 x corresponds to {u1 , . . . , u0m } as above. Then, by the resuls of the preceding paragraph, we can find a sequence ...

u0j3

ui3

u0j2

ui2

u0j1

ui1

which must repeat, hence by (2) we conclude that {u1 , . . . , un } = {u01 , . . . , u0m }, i.e., x = x0 . Thus X is Kolmogorov.  Lemma 5.14.8. Let f : X → Y be a morphism of topological spaces. Suppose that Y is a sober topological space, and f is surjective. If either specializations or generalizations lift along f , then dim(X) ≥ dim(Y ). Proof. Assume specializations lift along f . Let Z0 ⊂ Z1 ⊂ . . . Ze ⊂ Y be a chain of irreducible closed subsets of X. Let ξe ∈ X be a point mapping to the generic point of Ze . By assumption there exists a specialization ξe ξe−1 in X such that ξe−1 maps to the generic point of Ze−1 . Continuing in this manner we find a sequence of specializations ξe ξe−1 ... ξ0 with ξi mapping to the generic point of Zi . This clearly implies the sequence of irreducible closed subsets {ξ0 } ⊂ {ξ1 } ⊂ . . . {ξe } is a chain of length e in X. The case when generalizations lift along f is similar.  Lemma 5.14.9. Let X be a Noetherian sober topological space. Let E ⊂ X be a subset of X. (1) If E is constructible and stable under specialization, then E is closed. (2) If E is constructible and stable under generalization, then E is open. Proof. Let E be constructible and stable under generalization. Let Y ⊂ X be an irreducible closed subset with generic point ξ ∈ Y . If E ∩ Y is nonempty, then it contains ξ (by stability under generalization) and hence is dense in Y , hence it contains a nonempty open of Y , see Lemma 5.11.3. Thus E is open by Lemma 5.11.5. This proves (2). To prove (1) apply (2) to the complement of E in X.  5.15. Submersive maps Definition 5.15.1. Let f : X → Y be a continuous map of topological spaces. We say f is submersive3 if f is surjective and for any T ⊂ Y we have T is open or closed if and only if f −1 (T ) is so. Another way to express the second condition is that Y has the quotient topology relative to the map X → Y . Here is an example where this holds. Lemma 5.15.2. Let f : X → Y be surjective, open, continuous map of topological spaces. Let T ⊂ Y be a subset. Then (1) f −1 (T ) = f −1 (T ), 3This is very different from the notion of a submersion between differential manifolds!

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(2) T ⊂ Y is closed if and only f −1 (T ) is closed, (3) T ⊂ Y is open if and only f −1 (T ) is open, and (4) T ⊂ Y is locally closed if and only f −1 (T ) is locally closed. In particular we see that f is submersive. Proof. It is clear that f −1 (T ) ⊂ f −1 (T ). If x ∈ X, and x 6∈ f −1 (T ), then there exists an open neighbourhood x ∈ U ⊂ X with U ∩ f −1 (T ) = ∅. Since f is open we see that f (U ) is an open neighbourhood of f (x) not meeting T . Hence x 6∈ f −1 (T ). This proves (1). Part (2) is an easy consequences of this. Part (3) is obvious from the fact that f is open. For (4), if f −1 (T ) is locally closed, then f −1 (T ) ⊂ f −1 (T ) = f −1 (T ) is open, and hence by (3) applied to the map f −1 (T ) → T we see that T is open in T , i.e., T is locally closed.  5.16. Dimension functions It scarcely makes sense to consider dimension functions unless the space considered is sober (Definition 5.5.4). Thus the definition below can be improved by considering the sober topological space associated to X. Since the underlying topological space of a scheme is sober we do not bother with this improvement. Definition 5.16.1. Let X be a topological space. (1) Let x, y ∈ X, x 6= y. Suppose x y, that is y is a specialization of x. We say y is an immediate specialization of x if there is no z ∈ X \ {x, y} with x z and z y. (2) A map δ : X → Z is called a dimension function4 if (a) whenever x y and x 6= y we have δ(x) > δ(y), and (b) for every immediate specialization x y in X we have δ(x) = δ(y) + 1. It is clear that if δ is a dimension function, then so is δ + t for any t ∈ Z. Here is a fun lemma. Lemma 5.16.2. Let X be a topological space. If X is sober and has a dimension function, then X is catenary. Moreover, for any x y we have   δ(x) − δ(y) = codim {y}, {x} . Proof. Suppose Y ⊂ Y 0 ⊂ X are irreducible closed subsets. Let ξ ∈ Y , ξ 0 ∈ Y 0 be their generic points. Then we see immediately from the definitions that codim(Y, Y 0 ) ≤ δ(ξ) − δ(ξ 0 ) < ∞. In fact the first inequality is an equality. Namely, suppose Y = Y0 ⊂ Y1 ⊂ . . . ⊂ Ye = Y 0 is any maximal chain of irreducible closed subsets. Let ξi ∈ Yi denote the generic point. Then we see that ξi ξi+1 is an immediate specialization. Hence we see that e = δ(ξ) − δ(ξ 0 ) as desired. This also proves the last statement of the lemma.  Lemma 5.16.3. Let X be a topological space. Let δ, δ 0 be two dimension functions on X. If X is locally Noetherian and sober then δ − δ 0 is locally constant on X. 4This is likely nonstandard notation. This notion is usually introduced only for (locally) Noetherian schemes, in which case condition (a) is implied by (b).

5.17. NOWHERE DENSE SETS

153

Proof. Let x ∈ X be a point. We will show that δ − δ 0 is constant in a neighbourhood of x. We may replace X by an open neighbourhood of x in X which is Noetherian. Hence we may assume X is Noetherian and sober. Let Z1 , . . . , Zr be the irreducible components of X passing through x. (There are finitely many as X is Noetherian, see Lemma 5.6.2.) Let ξi ∈ Zi be the generic point. Note Z1 ∪ . . . ∪ Zr is a neighbourhood of x in X (not necessarily closed). We claim that δ − δ 0 is constant on Z1 ∪ . . . ∪ Zr . Namely, if y ∈ Zi , then δ(x) − δ(y) = δ(x) − δ(ξi ) + δ(ξi ) − δ(y) = −codim({x}, Zi ) + codim({y}, Zi ) by Lemma 5.16.2. Similarly for δ 0 . Whence the result.



Lemma 5.16.4. Let X be locally Noetherian, sober and catenary. Then any point has an open neighbourhood U ⊂ X which has a dimension function. Proof. We will use repeatedly that an open subspace of a catenary space is catenary, see Lemma 5.8.2 and that a Noetherian topological space has finitely many irreducible components, see Lemma 5.6.2. In the proof of Lemma 5.16.3 we saw how to construct such a function. Namely, we first replace X by a Noetherian open neighbourhood of x. Next, we let Z1 , . . . , Zr ⊂ X be the irreducible components of X. Let [ Zi ∩ Zj = Zijk be the decomposition into irreducible components. We replace X by [  [ X\ Zi ∪ Zijk x6∈Zi

x6∈Zijk

so that we may assume x ∈ Zi for all i and x ∈ Zijk for all i, j, k. For y ∈ X choose any i such that y ∈ Zi and set δ(y) = −codim({x}, Zi ) + codim({y}, Zi ). We claim this is a dimension function. First we show that it is well defined, i.e., independent of the choice of i. Namely, suppose that y ∈ Zijk for some i, j, k. Then we have (using Lemma 5.8.6) δ(y) = −codim({x}, Zi ) + codim({y}, Zi ) = −codim({x}, Zijk ) − codim(Zijk , Zi ) + codim({y}, Zijk ) + codim(Zijk , Zi ) = −codim({x}, Zijk ) + codim({y}, Zijk ) which is symmetric in i and j. We omit the proof that it is a dimension function.



Remark 5.16.5. Combining Lemmas 5.16.3 and 5.16.4 we see that on a catenary, locally Noetherian, sober topological space the obstruction to having a dimension function is an element of H 1 (X, Z). 5.17. Nowhere dense sets Definition 5.17.1. Let X be a topological space. (1) Given a subset T ⊂ X the interior of T is the largest open subset of X contained in T . (2) A subset T ⊂ X is called nowhere dense if the closure of T has empty interior.

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Lemma 5.17.2. Let X be a topological space. The union of a finite number of nowhere dense sets is a nowhere dense set. Proof. Omitted.



Lemma 5.17.3. Let X be a topological space. Let U ⊂ X be an open. Let T ⊂ U be a subset. If T is nowhere dense in U , then T is nowhere dense in X. Proof. Assume T is nowhere dense in U . Suppose that x ∈ X is an interior point of the closure T of T in X. Say x ∈ V ⊂ T with V ⊂ X open in X. Note that T ∩ U is the closure of T in U . Hence the interior of T ∩ U being empty implies V ∩ U = ∅. Thus x cannot be in the closure of U , a fortiori cannot be in the closure of T , a contradiction.  S Lemma 5.17.4. Let X be a topological space. Let X = Ui be an open covering. Let T ⊂ X be a subset. If T ∩ Ui is nowhere dense in Ui for all i, then T is nowhere dense in X. Proof. Omitted. (Hint: closure commutes with intersecting with opens.)



Lemma 5.17.5. Let f : X → Y be a continuous map of topological spaces. Let T ⊂ X be a subset. If f identifies X with a closed subset of Y and T is nowhere dense in X, then also f (T ) is nowhere dense in Y . Proof. Omitted.



Lemma 5.17.6. Let f : X → Y be a continuous map of topological spaces. Let T ⊂ Y be a subset. If f is open and T is a closed nowhere dense subset of Y , then also f −1 (T ) is a closed nowhere dense subset of X. If f is surjective and open, then T is closed nowhere dense if and only if f −1 (T ) is closed nowhere dense. Proof. Omitted. (Hint: In the first case the interior of f −1 (T ) maps into the interior of T , and in the second case the interior of f −1 (T ) maps onto the interior of T .)  5.18. Miscellany Recall that a neighbourhood of a point need not be open. Definition 5.18.1. A topological space X is called locally quasi-compact5 if every point has a fundamental system of quasi-compact neighbourhoods. The following lemma applies to the underlying topological space associated to a quasi-separated scheme. Lemma 5.18.2. Let X be a topological space which (1) has a basis of the topology consisting of quasi-compact opens, and (2) has the property that the intersection of any two quasi-compact opens is quasi-compact. Then (1) X is locally quasi-compact, 5This may not be standard notation. Alternative notions used in the literature are: (1) Every point has some quasi-compact neighbourhood, and (2) Every point has a closed quasi-compact neighbourhood. A scheme has the property that every point has a fundamental system of open quasi-compact neighbourhoods.

5.19. OTHER CHAPTERS

155

(2) a quasi-compact open U ⊂ X is retrocompact, (3) any quasi-compact open U ⊂ X has a cofinal system of open coverings S U : U = j∈J Uj with J finite and all Uj and Uj ∩ Uj 0 quasi-compact, (4) add more here. Proof. Omitted.



Definition 5.18.3. Let X be a topological space. We say x ∈ X is an isolated point of X if {x} is open in X. 5.19. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology

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Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 6

Sheaves on Spaces 6.1. Introduction Basic properties of sheaves on topological spaces will be explained in this document. A reference is [God73]. This will be superceded by the discussion of sheaves over sites later in the documents. But perhaps it makes sense to briefly define some of the notions here. 6.2. Basic notions The following notions are considered basic and will not be defined, and or proved. This does not mean they are all necessarily easy or well known. S (1) Let X be a topological space. The phrase: “Let U = i∈I Ui be an open covering” means the following: I is a set, and for each i ∈ I we are given an open subset Ui ⊂ X. Furthermore U is the union of the Ui . It is allowed to have I = ∅ in which case there are no Ui and U = ∅. It is also allowed, in case I 6= ∅ to have any or all of the Ui be empty. (2) etc, etc. 6.3. Presheaves Definition 6.3.1. Let X be a topological space. (1) A presheaf F of sets on X is a rule which assigns to each open U ⊂ X a set F(U ) and to each inclusion V ⊂ U a map ρU V : F(U ) → F(V ) such V U that ρU = id and whenever W ⊂ V ⊂ U we have ρU F (U ) U W = ρW ◦ ρV . (2) A morphism ϕ : F → G of presheaves of sets on X is a rule which assigns to each open U ⊂ X a map of sets ϕ : F(U ) → G(U ) compatible with restriction maps, i.e., whenever V ⊂ U ⊂ X are open the diagram F(U )

ϕ

ρU V

 F(V )

/ G(U ) ρU V

ϕ

 / G(V )

commutes. (3) The category of presheaves of sets on X will be denoted PSh(X). The elements of the set F(U ) are called the sections of F over U . For every V ⊂ U the map ρU V : F(U ) → F(V ) is called the restricton map. We will use the notation s|V := ρU (s) if s ∈ F(U ). This notation is consistent with the notion of restriction V of functions from topology because if W ⊂ V ⊂ U and s is a section of F over 157

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U then s|W = (s|V )|W by the property of the restriction maps expressed in the definition above. Another notation that is often used is to indicate sections over an open U by the symbol Γ(U, −) or by H 0 (U, −). In other words, the following equalities are tautological Γ(U, F) = F(U ) = H 0 (U, F). In this chapter we will not use this notation, but in others we will. Definition 6.3.2. Let X be a topological space. Let A be a set. The constant presheaf with value A is the presheaf that assigns the set A to every open U ⊂ X, and such that all restriction mappings are idA . 6.4. Abelian presheaves In this section we briefly point out some features of the category of presheaves that allow one to define presheaves of abelian groups. Example 6.4.1. Let X be a topological space X. Consider a rule F that associates to every open subset a singleton set. Since every set has a unique map into a singleton set, there exist unique restriction maps ρU V . The resulting structure is a presheaf of sets. It is a final object in the category of presheaves of sets, by the property of singleton sets mentioned above. Hence it is also unique up to unique isomorphism. We will sometimes write ∗ for this presheaf. Lemma 6.4.2. Let X be a topological space. The category of presheaves of sets on X has products (see Categories, Definition 4.13.5). Moreover, the set of sections of the product F × G over an open U is the product of the sets of sections of F and G over U . Proof. Namely, suppose F and G are presheaves of sets on the topological space X. Consider the rule U 7→ F(U ) × G(U ), denoted F × G. If V ⊂ U ⊂ X are open then define the restriction mapping (F × G)(U ) −→ (F × G)(V ) by mapping (s, t) 7→ (s|V , t|V ). Then it is immediately clear that F ×G is a presheaf. Also, there are projection maps p : F × G → F and q : F × G → G. We leave it to the reader to show that for any third presheaf H we have Mor(H, F × G) = Mor(H, F) × Mor(H, G).  Recall that if (A, + : A × A → A, − : A → A, 0 ∈ A) is an abelian group, then the zero and the negation maps are uniquely determined by the addition law. In other words, it makes sense to say “let (A, +) be an abelian group”. Lemma 6.4.3. Let X be a topological space. Let F be a presheaf of sets. Consider the following types of structure on F: (1) For every open U the structure of an abelian group on F(U ) such that all restriction maps are abelian group homomorphisms. (2) A map of presheaves + : F × F → F, a map of presheaves − : F → F and a map 0 : ∗ → F (see Example 6.4.1) satisfying all the axioms of +, −, 0 in a usual abelian group.

6.5. PRESHEAVES OF ALGEBRAIC STRUCTURES

159

(3) A map of presheaves + : F × F → F, a map of presheaves − : F → F and a map 0 : ∗ → F such that for each open U ⊂ X the quadruple (F(U ), +, −, 0) is an abelian group, (4) A map of presheaves + : F × F → F such that for every open U ⊂ X the map + : F(U ) × F(U ) → F(U ) defines the structure of an abelian group. There are natural bijections between the collections of types of data (1) - (4) above. Proof. Omitted.



The lemma says that to give an abelian group object F in the category of presheaves is the same as giving a presheaf of sets F such that all the sets F(U ) are endowed with the structure of an abelian group and such that all the restriction mappings are group homomorphisms. For most algebra structures we will take this approach to (pre)sheaves of such objects, i.e., we will define a (pre)sheaf of such objects to be a (pre)sheaf F of sets all of whose sets of sections F(U ) are endowed with this structure compatibly with the restriction mappings. Definition 6.4.4. Let X be a topological space. (1) A presheaf of abelian groups on X or an abelian presheaf over X is a presheaf of sets F such that for each open U ⊂ X the set F(U ) is endowed with the structure of an abelian group, and such that all restriction maps ρU V are homomorphisms of abelian groups, see Lemma 6.4.3 above. (2) A morphism of abelian presheaves over X ϕ : F → G is a morphism of presheaves of sets which induces a homomorphism of abelian groups F(U ) → G(U ) for every open U ⊂ X. (3) The category of presheaves of abelian groups on X is denoted PAb(X). Example 6.4.5. Let X be a topological space. For each x ∈ X suppose given an abelian group Mx . For U ⊂ X open we set M F(U ) = Mx . x∈U Pn We denote a typical element in this abelian group by i=1 mxi , where xi ∈ U and mxi ∈ Mxi . (Of course we may always choose our representation such that x1 , . . . , xn are pairwise distinct.) We define for V ⊂ UP⊂ X open a restriction n mapping i=1 mxi to the element P F(U ) → F(V ) by mapping an element s = s|V = xi ∈V mxi . We leave it to the reader to verify that this is a presheaf of abelian groups. 6.5. Presheaves of algebraic structures Let us clarify the definition of presheaves of algebraic structures. Suppose that C is a category and that F : C → Sets is a faithful functor. Typically F is a “forgetful” functor. For an object M ∈ Ob(C) we often call F (M ) the underlying set of the object M . If M → M 0 is a morphism in C we call F (M ) → F (M 0 ) the underlying map of sets. In fact, we will often not distinguish between an object and its underlying set, and similarly for morphisms. So we will say a map of sets F (M ) → F (M 0 ) is a morphism of algebraic structures, if it is equal to F (f ) for some morphism f : M → M 0 in C. In analogy with Definition 6.4.4 above a “presheaf of objects of C” could be defined by the following data:

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(1) a presheaf of sets F, and (2) for every open U ⊂ X a choice of an object A(U ) ∈ Ob(C) subject to the following conditions (using the phraseology above) (1) for every open U ⊂ X the set F(U ) is the underlying set of A(U ), and (2) for every V ⊂ U ⊂ X open the map of sets ρU V : F(U ) → F(V ) is a morphism of algebraic structures. In other words, for every V ⊂ U open in X the restriction mappings ρU V is the image F (αVU ) for some unique morphism αVU : A(U ) → A(V ) in the category C. The uniqueness is forced by the condition that F is faithful; it also implies that U V αW = αW ◦ αVU whenever W ⊂ V ⊂ U are open in X. The system (A(−), αVU ) is what we will define as a presheaf with values in C on X, compare Sites, Definition 9.2.2. We recover our presheaf of sets (F, ρU V ) via the rules F(U ) = F (A(U )) and U U ρV = F (αV ). Definition 6.5.1. Let X be a topological space. Let C be a category. (1) A presheaf F on X with values in C is given by a rule which assigns to every open U ⊂ X an object F(U ) of C and to each inclusion V ⊂ U a morphism ρU V : F(U ) → F(V ) in C such that whenever W ⊂ V ⊂ U we U V = ρ have ρU W ◦ ρV . W (2) A morphism ϕ : F → G of presheaves with value in C is given by a morphism ϕ : F(U ) → G(U ) in C compatible with restriction morphisms. Definition 6.5.2. Let X be a topological space. Let C be a category. Let F : C → Sets be a faithful functor. Let F be a presheaf on X with values in C. The presheaf of sets U 7→ F (F(U )) is called the underlying presheaf of sets of F. It is customary to use the same letter F to denote the underlying presheaf of sets, and this makes sense according to our discussion preceding Definition 6.5.1. In particular, the phrase “let s ∈ F(U )” or “let s be a section of F over U ” signifies that s ∈ F (F(U )). This notation and these definitions apply in particular to: Presheaves of (not necessarily abelian) groups, rings, modules over a fixed ring, vector spaces over a fixed field, etc and morphisms between these. 6.6. Presheaves of modules Suppose that O is a presheaf of rings on X. We would like to define the notion of a presheaf of O-modules over X. In analogy with Definition 6.4.4 we are tempted to define this as a sheaf of sets F such that for every open U ⊂ X the set F(U ) is endowed with the structure of an O(U )-module compatible with restriction mappings (of F and O). However, it is customary (and equivalent) to define it as in the following definition. Definition 6.6.1. Let X be a topological space, and let O be a presheaf of rings on X. (1) A presheaf of O-modules is given by an abelian presheaf F together with a map of presheaves of sets O × F −→ F such that for every open U ⊂ X the map O(U ) × F(U ) → F(U ) defines the structure of an O(U )-module structure on the abelian group F(U ).

6.7. SHEAVES

161

(2) A morphism ϕ : F → G of presheaves of O-modules is a morphism of abelian presheaves ϕ : F → G such that the diagram O×F id×ϕ

 O×G

/F  /G

ϕ

commutes. (3) The set of O-module morphisms as above is denoted HomO (F, G). (4) The category of presheaves of O-modules is denoted PMod(O). Suppose that O1 → O2 is a morphism of presheaves of rings on X. In this case, if F is a presheaf of O2 -modules then we can think of F as a presheaf of O1 -modules by using the composition O1 × F → O2 × F → F. We sometimes denote this by FO1 to indicate the restriction of rings. We call this the restriction of F. We obtain the restriction functor PMod(O2 ) −→ PMod(O1 ) On the other hand, given a presheaf of O1 -modules G we can construct a presheaf of O2 -modules O2 ⊗p,O1 G by the rule (O2 ⊗p,O1 G) (U ) = O2 (U ) ⊗O1 (U ) G(U ) The index p stands for “presheaf” and not “point”. This presheaf is called the tensor product presheaf. We obtain the change of rings functor PMod(O1 ) −→ PMod(O2 ) Lemma 6.6.2. With X, O1 , O2 , F and G as above there exists a canonical bijection HomO1 (G, FO1 ) = HomO2 (O2 ⊗p,O1 G, F) In other words, the restriction and change of rings functors are adjoint to each other. Proof. This follows from the fact that for a ring map A → B the restriction functor and the change of ring functor are adjoint to each other.  6.7. Sheaves In this section we explain the sheaf condition. Definition 6.7.1. Let X be a topological space. (1) A sheaf F of sets on X is a presheaf of sets which satsifies the followS ing additional property: Given any open covering U = i∈I Ui and any collection of sections si ∈ F(Ui ), i ∈ I such that ∀i, j ∈ I si |Ui ∩Uj = sj |Ui ∩Uj there exists a unique section s ∈ F(U ) such that si = s|Ui for all i ∈ I. (2) A morphism of sheaves of sets is simply a morphism of presheaves of sets. (3) The category of sheaves of sets on X is denoted Sh(X).

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6. SHEAVES ON SPACES

Remark 6.7.2. There is always a bit of confusion as to whether it is necessary to say something about the set of sections of a sheaf over the empty set ∅ ⊂ X. It is necessary, and we already did if you read the definition right. Namely, note that the empty set is covered by the empty open covering, and hence the “collection of section si ” from the definition above actually form an element of the empty product which is the final object of the category the sheaf has values in. In other words, if you read the definition right you automatically deduce that F(∅) = a final object, which in the case of a sheaf of sets is a singleton. If you do not like this argument, then you can just require that F(∅) = {∗}. In particular, this condition will then ensure that if U, V ⊂ X are open and disjoint then F(U ∪ V ) = F(U ) × F(V ). (Because the fibre product over a final object is a product.) Example 6.7.3. Let X, Y be topological spaces. Consider the rule F wich associates to the open U ⊂ X the set F(U ) = {f : U → Y | f is continuous} with the obvious S restriction mappings. We claim that F is a sheaf. To see this suppose that U = i∈I Ui is an open covering, and fi ∈ F(Ui ), i ∈ I with fi |Ui ∩Uj = fj |Ui ∩Uj for all i, j ∈ I. In this case define f : U → Y by setting f (u) equal to the value of fi (u) for any i ∈ I such that u ∈ Ui . This is well defined by assumption. Moreover, f : U → Y is a map such that its restriction to Ui agrees with the continuous map Ui . Hence clearly f is continuous! We can use the result of the example to define constant sheaves. Namely, suppose that A is a set. Endow A with the discrete topology. Let U ⊂ X be an open subset. Then we have {f : U → A | f continuous} = {f : U → A | f locally constant}. Thus the rule which assigns to an open all locally constant maps into A is a sheaf. Definition 6.7.4. Let X be a topological space. Let A be a set. The constant sheaf with value A denoted A, or AX is the sheaf that assigns to an open U ⊂ X the set of all locally constant maps U → A with restriction mappings given by restrictions of functions. Example 6.7.5. Let X be a topological space. Let (Ax )x∈X be a family of sets Ax indexed by points x ∈ X. We are going to construct a sheaf of sets Π from this data. For U ⊂ X open set Y Π(U ) = Ax . x∈U

For V ⊂ U ⊂ X open define a restriction mapping by the following rule: An element s = (ax )x∈U ∈ Π(U ) restricts to s|V = (ax )x∈V . It is obvious S that this defines a presheaf of sets. We claim this is a sheaf. Namely, let U = Ui be an open covering. Suppose that si ∈ Π(Ui ) are such that si and sj agree over Ui ∩ Uj . Write si = (ai,x )x∈Ui . The compatibility condition implies that ai,x = aj,x in the set Ax whenever x ∈ Ui ∩ Uj . Hence there exists a unique element s = (ax )x∈U in Q Π(U ) = x∈U Ax with the property that ax = ai,x whenever x ∈ Ui for some i. Of course this element s has the property that s|Ui = si for all i.

6.9. SHEAVES OF ALGEBRAIC STRUCTURES

163

Example 6.7.6. Let X be a topological space. Suppose L for each x ∈ X we are given an abelian group Mx . Consider the presheaf F : U 7→ x∈U Mx defined in Example 6.4.5. This is not a sheaf in general. For example, if X is an infiniteQset with the discrete topology, then the sheaf condition L would imply L that F(X) = x∈X F({x}) but by definition we have F(X) = M = x x∈X x∈X F({x}). And an infinite direct sum is in general different from an infinite direct product. However, if X is a topological space such that every open of X is quasi-compact, then F is a sheaf. This is left as an exercise to the reader. 6.8. Abelian sheaves Definition 6.8.1. Let X be a topological space. (1) An abelian sheaf on X or sheaf of abelian groups on X is an abelian presheaf on X such that the underlying presheaf of sets is a sheaf. (2) The category of sheaves of abelian groups is denoted Ab(X). Let X be a topological space. In the case of an S abelian presheaf F the sheaf condition with regards to an open covering U = Ui is often expressed by saying that the complex of abelian groups Y Y 0 → F(U ) → F(Ui ) → F(Ui0 ∩ Ui1 ) i

(i0 ,i1 )

is exact. The first map is the usual one, whereas the second maps the element (si )i∈I to the element Y (si0 |Ui0 ∩Ui1 − si1 |Ui0 ∩Ui1 )(i0 ,i1 ) ∈ F(Ui0 ∩ Ui1 ) (i0 ,i1 )

6.9. Sheaves of algebraic structures Let us clarify the definition of sheaves of certain types of structures. First, let us reformulate the sheaf condition. Namely, suppose that F is a presheaf of sets on the topological space X. The sheaf condition can be reformulated as follows. Let S U = i∈I Ui be an open covering. Consider the diagram /Q / Q F(Ui ) F(U ) / (i0 ,i1 )∈I×I F(Ui0 ∩ Ui1 ) i∈I Q Here the left map is defined by the rule s 7→ i∈I s|Ui . The two maps on the right are the maps Y Y Y Y si 7→ si0 |Ui0 ∩Ui1 resp. si 7→ si1 |Ui0 ∩Ui1 . i

(i0 ,i1 )

i

(i0 ,i1 )

The sheaf condition exactly says that the left arrow is the equalizer of the right two. This generalizes immediately to the case of presheaves with values in a category as long as the category has products. Definition 6.9.1. Let X be a topological space. Let C be a category with products. A presheaf F with values in C on X is a sheaf if for every open covering the diagram /Q / Q F(Ui ) F(U ) / (i0 ,i1 )∈I×I F(Ui0 ∩ Ui1 ) i∈I is an equalizer diagram in the category C.

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Suppose that C is a category and that F : C → Sets is a faithful functor. A good example to keep in mind is the case where C is the category of abelian groups and F is the forgetful functor. Consider a presheaf F with values in C on X. We would like to reformulate the condition above in terms of the underlying presheaf of sets (Definition 6.5.2). Note that the underlying presheaf of sets is a sheaf of sets if and only if all the diagrams /Q / Q F (F(Ui )) F (F(U )) / (i0 ,i1 )∈I×I F (F(Ui0 ∩ Ui1 )) i∈I of sets – after applying the forgetful functor F – are equalizer diagrams! Thus we would like C to have products and equalizers and we would like F to commute with them. This is equivalent to the condition that C has limits and that F commutes with them, see Categories, Lemma 4.13.10. But this is not yet good enough (see Example 6.9.4); we also need F to reflect isomorphisms. This property means that given a morphism f : A → A0 in C, then f is an isomorphism if (and only if) F (f ) is a bijection. Lemma 6.9.2. Suppose the category C and the functor F : C → Sets have the following properties: (1) F is faithful, (2) C has limits and F commutes with them, and (3) the functor F reflects isomorphisms. Let X be a topological space. Let F be a presheaf with values in C. Then F is a sheaf if and only if the underlying presheaf of sets is a sheaf. Proof. Assume that F is a sheaf. Then F(U ) is the equalizer of the diagram above and by assumption we see F (F(U )) is the equalizer of the corresponding diagram of sets. Hence F (F) is a sheaf of sets. Assume that F (F) is a sheaf. Let E ∈ Ob(C) be the equalizer of the two parrallel arrows in Definition 6.9.1. We get a canonical morphism F(U ) → E, simply because F is a presheaf. By assumption, the induced map F (F(U )) → F (E) is an isomorphism, because F (E) is the equalizer of the corresponding diagram of sets. Hence we see F(U ) → E is an isomorphism by condition (3) of the lemma.  The lemma in particular applies to sheaves of groups, rings, algebras over a fixed ring, modules over a fixed ring, vector spaces over a fixed field, etc. In other words, these are presheaves of groups, rings, modules over a fixed ring, vector spaces over a fixed field, etc such that the underlying presheaf of sets is a sheaf. Example 6.9.3. Let X be a topological space. For each open U ⊂ X consider the R-algebra C 0 (U ) = {f : U → R | f is continuous}. There are obvious restriction mappings that turn this into a presheaf of R-algebras over X. By Example 6.7.3 it is a sheaf of sets. Hence by the Lemma 6.9.2 it is a sheaf of R-algebras over X. Example 6.9.4. Consider the category of topological spaces Top. There is a natural faithful functor Top → Sets which commutes with products and equalizers. But it does not reflect isomorphisms. And, in fact it turns out that the analogue of Lemma 6.9.2 is wrong. Namely, suppose X = N with the discrete topology. Let Ai , for i ∈ N be a discrete topological space. For any subset U ⊂ N define F(U ) = Q i∈U Ai with the discrete topology. Then this is a presheaf of topological spaces whose underlying presheaf of sets is a sheaf, see Example 6.7.5. However, if each Ai

6.11. STALKS

165

has at least two elements, then this is not a sheaf of topological spaces according to Definition Q6.9.1. The reader may check that putting the product topology on each F(U ) = i∈U Ai does lead to a sheaf of topological spaces over X. 6.10. Sheaves of modules Definition 6.10.1. Let X be a topological space. Let O be a sheaf of rings on X. (1) A sheaf of O-modules is a presheaf of O-modules F, see Definition 6.6.1, such that the underlying presheaf of abelian groups F is a sheaf. (2) A morphism of sheaves of O-modules is a morphism of presheaves of Omodules. (3) Given sheaves of O-modules F and G we denote HomO (F, G) the set of morphism of sheaves of O-modules. (4) The category of sheaves of O-modules is denoted Mod(O). This definition kind of makes sense even if O is just a presheaf of rings, allthough we do not know any examples where this is useful, and we will avoid using the terminology “sheaves of O-modules” in case O is not a sheaf of rings. 6.11. Stalks Let X be a topological space. Let x ∈ X be a point. Let F be a presheaf of sets on X. The stalk of F at x is the set Fx = colimx∈U F(U ) where the colimit is over the set of open neighbourhoods U of x in X. The set of open neighbourhoods is (partially) ordered by (reverse) inclusion: We say U ≥ U 0 ⇔ U ⊂ U 0 . The transition maps in the system are given by the restriction maps of F. See Categories, Section 4.19 for notation and terminology regarding (co)limits over systems. Note that the colimit is a directed colimit. Thus it is easy to describe Fx . Namely, Fx = {(U, s) | x ∈ U, s ∈ F(U )}/ ∼ with equivalence relation given by (U, s) ∼ (U 0 , s0 ) if and only if s|U ∩U 0 = s0 |U ∩U 0 . By abuse of notation we will often denote (U, s), sx , or even s the corresponding element in Fx . Also we will say s = s0 in Fx for two local sections of F defined in an open neighbourhod of x to denote that they have the same image in Fx . An obvious consequence of this definition is that for any open U ⊂ X there is a canonical map Y F(U ) −→ Fx x∈U Q defined by s 7→ x∈U (U, s). Think about it! Lemma 6.11.1. Let F be a sheaf of sets on the topological space X. For every open U ⊂ X the map Y F(U ) −→ Fx x∈U

is injective. Proof. Suppose that s, s0 ∈ F(U ) map to the same element in every stalk Fx for all x ∈ U . This means that for every x S ∈ U , there exists an open V x ⊂ U , x ∈ V x 0 such that s|V x = s |V x . But then U = x∈U V x is an open covering. Thus by the uniqueness in the sheaf condition we see that s = s0 . 

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Definition 6.11.2. Let X be a topological space. A Q presheaf of sets F on X is separated if for every open U ⊂ X the map F(U ) → x∈U Fx is injective. Another observation is that the construction of the stalk Fx is functorial in the presheaf F. In other words, it gives a functor PSh(X) −→ Sets, F 7−→ Fx . This functor is called the stalk functor. Namely, if ϕ : F → G is a morphism of presheaves, then we define ϕx : Fx → Gx by the rule (U, s) 7→ (U, ϕ(s)). To see that this works we have to check that if (U, s) = (U 0 , s0 ) in Fx then also (U, ϕ(s)) = (U 0 , ϕ(s0 )) in Gx . This is clear since ϕ is compatible with the restriction mappings. Example 6.11.3. Let X be a topological space. Let A be a set. Denote temporarily Ap the constant presheaf with value A (p for presheaf – not for point). There is a canonical map of presheaves Ap → A into the constant sheaf with value A. For evey point we have canonical bijections A = (Ap )x = Ax , where the second map is induced by functoriality from the map Ap → A. Example 6.11.4. Suppose X = Rn with the Euclidean topology. Consider the ∞ ∞ presheaf of C ∞ functions on X, denoted CR n . In other words, CRn (U ) is the set ∞ of C -functions f : U → R. As in Example 6.7.3 it is easy to show that this is a sheaf. In fact it is a sheaf of R-vector spaces. Next, let x ∈ X = Rn be a point. How do we think of an element in the stalk ∞ ∞ CR -function f whose domain contains x. And n ,x ? Such an element is given by a C a pair of such functions f , g determine the same element of the stalk if they agree ∞ in a neighbourhood of x. In other words, an element if CR n ,x is the same thing as ∞ what is sometimes called a germ of a C -function at x. Example 6.11.5. Let X beQa topological space. Let Ax be a set for each x ∈ X. Consider the sheaf F : U 7→ x∈U Ax of Example 6.7.5. We would just like to point out here that the stalk Fx of F at x is in general not equal to the set Ax . Of course there is a map Fx → Ax , but that is in general the best you can say. For example, if each neighbourhood of x has infinitely many points, and each Ax0 has exactly two elements, then Fx has infinitely many elements. (Left to the reader.) On the other hand, if every neighbourhood of x contains a point y such that Ay = ∅, then Fx = ∅. 6.12. Stalks of abelian presheaves We first deal with the case of abelian groups as a model for the general case. Lemma 6.12.1. Let X be a topological space. Let F be a presheaf of abelian groups on X. There exists a unique structure of an abelian group on Fx such that for every U ⊂ X open, x ∈ U the map F(U ) → Fx is a group homomorphism. Moreover, Fx = colimx∈U F(U ) holds in the category of abelian groups. Proof. We define addition of a pair of elements (U, s) and (V, t) as the pair (U ∩ V, s|U ∩V + t|U ∩V ). The rest is easy to check. 

6.14. STALKS OF PRESHEAVES OF MODULES

167

What is crucial in the proof above is that the partially ordered set of open neighbourhoods is a directed system (compare Categories, Definition 4.19.2). Namely, the coproduct of two abelian groups A, B is the direct`sum A ⊕ B, whereas the coproduct in the category of sets is the disjoint union A B, showing that colimits in the category of abelian groups do not agree with colimits in the category of sets in general. 6.13. Stalks of presheaves of algebraic structures The proof of Lemma 6.12.1 will work for any type of algebraic structure such that directed colimits commute with the forgetful functor. Lemma 6.13.1. Let C be a category. Let F : C → Sets be a functor. Assume that (1) F is faithful, and (2) directed colimits exist in C and F commutes with them. Let X be a topological space. Let x ∈ X. Let F be a presheaf with values in C. Then Fx = colimx∈U F(U ) exists in C. Its underlying set is equal to the stalk of the underlying presheaf of sets of F. Furthermore, the construction F 7→ Fx is a functor from the category of presheaves with values in C to C. Proof. Omitted.



By the very definition, all the morphisms F(U ) → Fx are morphisms in the category C which (after applying the forgetful functor F ) turn into the corresponding maps for the underlying sheaf of sets. As usual we will not distinguish between the morphism in C and the underlying map of sets, which is permitted since F is faithful. This lemma applies in particular to: Presheaves of (not necessarily abelian) groups, rings, modules over a fixed ring, vector spaces over a fixed field. 6.14. Stalks of presheaves of modules Lemma 6.14.1. Let X be a topological space. Let O be a presheaf of rings on X Let F be a presheaf O-modules. Let x ∈ X. The canonical map Ox × Fx → Fx coming from the multiplication map O × F → F defines a Ox -module structure on the abelian group Fx . Proof. Omitted.



Lemma 6.14.2. Let X be a topological space. Let O → O0 be a morphism of presheaves of rings on X Let F be a presheaf O-modules. Let x ∈ X. We have Fx ⊗Ox Ox0 = (F ⊗p,O O0 )x as Ox0 -modules. Proof. Omitted.



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6.15. Algebraic structures In this section we mildly formalize the notions we have encountered in the sections above. Definition 6.15.1. A type of algebraic structure is given by a category C and a functor F : C → Sets with the following properties (1) (2) (3) (4)

F is faithful, C has limits and F commutes with limits, C has filtered colimits and F commutes with them, and F reflects isomorphisms.

We make this definition to point out the properties we will use in a number of arguments below. But we will not actually study this notion in any great detail, since we are prohibited from studying “big” categories by convention, except for those listed in Categories, Remark 4.2.2. Among those the following have the required properties. Lemma 6.15.2. The following categories, endowed with the obvious forgetful functor, define types of algebraic structures: (1) (2) (3) (4) (5) (6) (7)

The The The The The The The

category category category category category category category

of of of of of of of

pointed sets. abelian groups. groups. monoids. rings. R-modules for a fixed ring R. Lie algebras over a fixed field.

Proof. Omitted.



From now on we will think of a (pre)sheaf of algebraic structures and their stalks, in terms of the underlying (pre)sheaf of sets. This is allowable by Lemmas 6.9.2 and 6.13.1. In the rest of this section we point out some results on algebraic structures that will be useful in the future. Lemma 6.15.3. Let (C, F ) be a type of algebraic structure. C has a final object 0 and = {∗}. Q F (0) Q C has products and F ( Ai ) = F (Ai ). C has fibre products and F (A ×B C) = F (A) ×F (B) F (C). C has equalizers, and if E → A is the equalizer of a, b : A → B, then F (E) → F (A) is the equalizer of F (a), F (b) : F (A) → F (B). (5) A → B is a monomorphism if and only if F (A) → F (B) is injective. (6) if F (a) : F (A) → F (B) is surjective, then a is an epimorphism. (7) given A1 → A2 → A3 → . . ., then colim Ai exists and F (colim Ai ) = colim F (Ai ), and more generally for any filtered colimit.

(1) (2) (3) (4)

Proof. Omitted. The only interesting statement is (5) which follows because A → B is a monomorphism if and only if A → A ×B A is an isomorphism, and then applying the fact that F reflects isomorphisms. 

6.16. EXACTNESS AND POINTS

169

Lemma 6.15.4. Let (C, F ) be a type of algebraic structure. Suppose that A, B, C ∈ Ob(C). Let f : A → B and g : C → B be morphisms of C. If F (g) is injective, and Im(F (f )) ⊂ Im(F (g)), then f factors as f = g ◦ t for some morphism t : A → C. Proof. Consider A ×B C. The assumptions imply that F (A ×B C) = F (A) ×F (B) F (C) = F (A). Hence A = A ×B C because F reflects isomorphisms. The result follows.  Example 6.15.5. The lemma will be applied often to the following situation. Suppose that we have a diagram A

/B

C

 /D

in C. Suppose C → D is injective on underlying sets, and suppose that the composition A → B → D has image on underlying sets in the image of C → D. Then we get a commutative diagram /B A  C

 /D

in C. Example 6.15.6. Let F : C → Sets be a type of algebraic structures. Let X be a topological space. Suppose that for every x ∈ X we are given an object Ax ∈ ob(C). Q Consider the presheaf Π with values in C on X defined by the rule Π(U ) = x∈U Ax (with obvious Q restriction mappings). Note that the associated presheaf of sets U 7→ F (Π(U )) = x∈U F (Ax ) is a sheaf by Example 6.7.5. Hence Π is a sheaf of algebraic structures of type (C, F ). This gives many examples of sheaves of abelian groups, groups, rings, etc. 6.16. Exactness and points In any category we have the notion of epimorphism, monomorphism, isomorphism, etc. Lemma 6.16.1. Let X be a topological space. Let ϕ : F → G be a morphism of sheaves of sets on X. (1) The map ϕ is a monomorphism in the category of sheaves if and only if for all x ∈ X the map ϕx : Fx → Gx is injective. (2) The map ϕ is an epimorphism in the category of sheaves if and only if for all x ∈ X the map ϕx : Fx → Gx is surjective. (3) The map ϕ is a isomorphism in the category of sheaves if and only if for all x ∈ X the map ϕx : Fx → Gx is bijective. Proof. Omitted.



It follows that in the category of sheaves of sets the notions epimorphism and monomorphism can be described as follows. Definition 6.16.2. Let X be a topological space.

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(1) A presheaf F is called a subpresheaf of a presheaf G if F(U ) ⊂ G(U ) for all open U ⊂ X such that the restriction maps of G induce the restriction maps of F. If F and G are sheaves, then F is called a subsheaf of G. We sometimes indicate this by the notation F ⊂ G. (2) A morphism of presheaves of sets ϕ : F → G on X is called injective if and only if F(U ) → G(U ) is injective for all U open in X. (3) A morphism of presheaves of sets ϕ : F → G on X is called surjective if and only if F(U ) → G(U ) is surjective for all U open in X. (4) A morphism of sheaves of sets ϕ : F → G on X is called injective if and only if F(U ) → G(U ) is injective for all U open in X. (5) A morphism of sheaves of sets ϕ : F → G on X is called surjective if and only if for every open S U of X and every section s of F(U ) there exists an open covering U = Ui such that s|Ui is in the image of F(Ui ) → G(U ) for all i. Lemma 6.16.3. Let X be a topological space. (1) Epimorphisms (resp. monomorphisms) in the category of presheaves are exactly the surjective (resp. injective) maps of presheaves. (2) Epimorphisms (resp. monomorphisms) in the category of sheaves are exactly the surjective (resp. injective) maps of sheaves, and are exactly those maps with are surjective (resp. injective) on all the stalks. (3) The sheafification of a surjective (resp. injective) morphism of presheaves of sets is surjective (resp. injective). Proof. Omitted.



Lemma 6.16.4. let X be a topological space. Let (C, F ) be a type of algebraic structure. Suppose that F, G are sheaves on X with values in C. Let ϕ : F → G be a map of the underlying sheaves of sets. If for all points x ∈ X the map Fx → Gx is a morphism of algebraic structures, then ϕ is a morphism of sheaves of algebraic structures. Proof. Let U be an open subset of X. Consider the diagram of (underlying) sets /Q F(U ) x∈U Fx  G(U ) /

Q 

x∈U

Gx

By assumption, and previous results, all but the left vertical arrow are morphisms of algebraic structures. In addition the bottom horizontal arrow is injective, see Lemma 6.11.1. Hence we conclude by Lemma 6.15.4, see also Example 6.15.5  Short exact sequences of abelian sheaves, etc will be discussed in the chapter on sheaves of modules. See Modules, Section 15.3. 6.17. Sheafification In this section we explain how to get the sheafification of a presheaf on a topological space. We will use stalks to describe the sheafification in this case. This is different from the general procedure described in Sites, Section 9.10, and perhaps somewhat easier to understand.

6.17. SHEAFIFICATION

171

The basic construction is the following. Let F be a presheaf of sets F on a topological space X. For every open U ⊂ X we define Y F # (U ) = {(su ) ∈ Fu such that (∗)} u∈U

where (∗) is the property: (∗) For every u ∈ U , there exists an open neighbourhood u ∈ V ⊂ U , and a section σ ∈ F(V ) such that for all v ∈ V we have sv = (V, σ) in Fv . Note that (∗) is a condition for each u ∈ U , and that given u ∈ U the truth of this condition depends only on the values sv for v in any open neighbourhood of u. Thus it is clear that, if V ⊂ U ⊂ X are open, the projection maps Y Y Fu −→ Fv u∈U

#

v∈V

#

maps elements of F (U ) into F (V ). In other words, we get the structure of a presheaf of sets on F # . Q Furthermore, the map F(U ) → u∈U Fu described in Section 6.11 clearly has image in F # (U ). In addition, if V ⊂ U ⊂ X are open then we have the following commutative diagram /Q / F # (U ) F(U ) u∈U Fu  F(V )

 / F # (V ) /

Q

 v∈V

Fv

where the vertical maps are induced from the restriction mappings. Thus we see that there is a canonical morphism of presheaves F → F # . Q In Example 6.7.5 we saw that the rule Π(F) : U 7→ u∈U Fu is a sheaf, with obvious restriction mappings. And by construction F # is a subpresheaf of this. In other words, we have morphisms of presheaves F → F # → Π(F). In addition the rule that associates to F the sequence above is clearly functorial in the presheaf F. This notation will be used in the proofs of the lemmas below. Lemma 6.17.1. The presheaf F # is a sheaf. Proof. It is probably better for the reader to find their own explanation of this than to read the proof here. In fact the lemma is true for the same reason as why the presheaf of continuous function is a sheaf, see Example 6.7.3 (and this analogy can be made precise using the “espace ´etal´e”). S Anyway, let U = Ui be an open covering. Suppose that si = (si,u )u∈Ui ∈ F # (Ui ) such that si andQ sj agree over Ui ∩ Uj . Because Π(F) is a sheaf, we find an element s = (su )u∈U in u∈U Fu restricting to si on Ui . We have to check property (∗). Pick u ∈ U . Then u ∈ Ui for some i. Hence by (∗) for si , there exists a V open, u ∈ V ⊂ Ui and a σ ∈ F(V ) such that si,v = (V, σ) in Fv for all v ∈ V . Since si,v = sv we get (∗) for s.  Lemma 6.17.2. Let X be a topological space. Let F be a presheaf of sets on X. Let x ∈ X. Then Fx = Fx# .

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Proof. The map Fx → Fx# is injective, since already the map Fx → Π(F)x is injective. Namely, there is a canonical map Π(F)x → Fx which is a left inverse to the map Fx → Π(F)x , see Example 6.11.5. To show that it is surjective, suppose that s ∈ Fx# . We can find an open neighbourhood U of x such that s is the equivalence class of (U, s) with s ∈ F # (U ). By definition, this means there exists an open neighbourhood V ⊂ U of x and a section σ ∈ F(V ) such that s|V is the image of σ in Π(F)(V ). Clearly the class of (V, σ) defines an element of Fx mapping  to s. Lemma 6.17.3. Let F be a presheaf of sets on X. Any map F → G into a sheaf of sets factors uniquely as F → F # → G. Proof. Clearly, there is a commutative diagram F

/ F#

/ Π(F)

 G

 / G#

 / Π(G)

So it suffices to prove that G = G # . To see this it suffices to prove, for every point x ∈ X the map Gx → Gx# is bijective, by Lemma 6.16.1. And this is Lemma 6.17.2 above.  This lemma really says that there is an adjoint pair of functors: i : Sh(X) → PSh(X) (inclusion) and # : PSh(X) → Sh(X) (sheafification). The formula is that MorPSh(X) (F, i(G)) = MorSh(X) (F # , G) which says that sheafification is a left adjoint of the inclusion functor. See Categories, Section 4.22. Example 6.17.4. See Example 6.11.3 for notation. The map Ap → A induces a map A# p → A. It is easy to see that this is an isomorphism. In words: The sheafification of the constant presheaf with value A is the constant sheaf with value A. Lemma 6.17.5. Let X be a topological space. A presheaf F is separated (see Definition 6.11.2) if and only if the canonical map F → F # is injective. Proof. This is clear from the construction of F # in this section.



6.18. Sheafification of abelian presheaves The following strange looking lemma is likely unnecessary, but very convenient to deal with sheafification of presheaves of algebraic structures. Lemma 6.18.1. Let X be a topological space. Let F be a presheaf of sets on X. Let U ⊂ X be open. There is a canonical fibre product diagram F # (U )

Q



x∈U

Fx

where the maps are the following:

/ Π(F)(U )

/Q

x∈U

 Π(F)x

6.19. SHEAFIFICATION OF PRESHEAVES OF ALGEBRAIC STRUCTURES

173

(1) The left vertical map has components F # (U ) → Fx# = Fx where the equality is Lemma 6.17.2. (2) The top horizontal map comes from the map of presheaves F → Π(F) described in Section 6.17. (3) The right vertical map has obvious component maps Π(F)(U ) → Π(F)x . (4) The bottom horizontal map has components Fx → Π(F)x which come from the map of presheaves F → Π(F) described in Section 6.17. Proof. It is clear that the diagram commutes. We have to show it is a fibre product diagram. The bottom horizontal arrow is injective since all the maps Fx → Π(F)x are injective (see beginning proof of Lemma 6.17.2). A section s ∈ Π(F)(U ) is in F # if and only if (∗) holds. But (∗) says that around every point the section s comes from a section of F. By definition of the stalk functors, this is equivalent to saying that the value of s in every stalk Π(F)x comes from an element of the stalk Fx . Hence the lemma.  Lemma 6.18.2. Let X be a topological space. Let F be an abelian presheaf on X. Then there exists a unique structure of abelian sheaf on F # such that F → F # is a morphism of abelian presheaves. Moreover, the following adjointness property holds MorPAb(X) (F, i(G)) = MorAb(X) (F # , G). Proof. Recall the sheaf of sets Π(F) defined in Section 6.17. All the stalks Fx are abelian groups, see Lemma 6.12.1. Hence Π(F) is a sheaf of abelian groups by Example 6.15.6. Also, it is clear that the map F → Π(F) is a morphism of abelian presheaves. If we show that condition (∗) of Section 6.17 defines a subgroup of Π(F)(U ) for all open subsets U ⊂ X, then F # canonically inherits the structure of abelian sheaf. This is quite easy to do by hand, and we leave it to the reader to find a good simple argument. The argument we use here, which generalizes to presheaves of algebraic structures is the following: Lemma 6.18.1 show that F # (U ) is the fibre product of a diagram of abelian groups. Thus F # is an abelian subgroup as desired. Note that at this point Fx# is an abelian group by Lemma 6.12.1 and that Fx → Fx# is a bijection (Lemma 6.17.2) and a homomorphism of abelian groups. Hence Fx → Fx# is an isomorphism of abelian groups. This will be used below without further mention. To prove the adjointness property we use the adjointness property of sheafification of presheaves of sets. For example if ψ : F → i(G) is morphism of presheaves then we obtain a morphism of sheaves ψ 0 : F # → G. What we have to do is to check that this is a morphism of abelian sheaves. We may do this for example by noting that it is true on stalks, by Lemma 6.17.2, and then using Lemma 6.16.4 above.  6.19. Sheafification of presheaves of algebraic structures Lemma 6.19.1. Let X be a topological space. Let (C, F ) be a type of algebraic structure. Let F be a presheaf with values in C on X. Then there exists a sheaf F # with values in C and a morphism F → F # of presheaves with values in C with the following properties: (1) The map F → F # identifies the underlying sheaf of sets of F # with the sheafification of the underlying presheaf of sets of F.

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(2) For any morphism F → G, where G is a sheaf with values in C there exists a unique factorization F → F # → G. Proof. The proof is the same as the proof of Lemma 6.18.2, with repeated application of Lemma 6.15.4 (see also Example 6.15.5). The main idea however, is to define F # (U ) as the fibre product in C of the diagram Π(F)(U )

Q

x∈U

Fx

/

Q

x∈U

 Π(F)x

compare Lemma 6.18.1.



6.20. Sheafification of presheaves of modules Lemma 6.20.1. Let X be a topological space. Let O be a presheaf of rings on X Let F be a presheaf O-modules. Let O# be the sheafification of O. Let F # be the sheafification of F as a presheaf of abelian groups. There exists a map of sheaves of sets O# × F # −→ F # which makes the diagram /F O×F  O# × F #

 / F#

commute and which makes F # into a sheaf of O# -modules. In addition, if G is a sheaf of O# -modules, then any morphism of presheaves of O-modules F → G (into the restriction of G to a O-module) factors uniquely as F → F # → G where F # → G is a morphism of O# -modules. Proof. Omitted.



This actually means that the functor i : Mod(O# ) → PMod(O) (combining restriction and including sheaves into presheaves) and the sheafification functor of the lemma # : PMod(O) → Mod(O# ) are adjoint. In a formula MorPMod(O) (F, iG) = MorMod(O# ) (F # , G) Let X be a topological space. Let O1 → O2 be a morphism of sheaves of rings on X. In Section 6.6 we defined a restriction functor and a change of rings functor on presheaves of modules associated to this situation. If F is a sheaf of O2 -modules then the restriction FO1 of F is clearly a sheaf of O1 -modules. We obtain the restriction functor Mod(O2 ) −→ Mod(O1 ) On the other hand, given a sheaf of O1 -modules G the presheaf of O2 -modules O2 ⊗p,O1 G is in general not a sheaf. Hence we define the tensor product sheaf O2 ⊗O1 G by the formula O2 ⊗O1 G = (O2 ⊗p,O1 G)#

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175

as the sheafification of our construction for presheaves. We obtain the change of rings functor Mod(O1 ) −→ Mod(O2 ) Lemma 6.20.2. With X, O1 , O2 , F and G as above there exists a canonical bijection HomO1 (G, FO1 ) = HomO2 (O2 ⊗O1 G, F) In other words, the restriction and change of rings functors are adjoint to each other. Proof. This follows from Lemma 6.6.2 and the fact that HomO2 (O2 ⊗O1 G, F) = HomO2 (O2 ⊗p,O1 G, F) because F is a sheaf.  Lemma 6.20.3. Let X be a topological space. Let O → O0 be a morphism of sheaves of rings on X Let F be a sheaf O-modules. Let x ∈ X. We have Fx ⊗Ox Ox0 = (F ⊗O O0 )x as Ox0 -modules. Proof. Follows directly from Lemma 6.14.2 and the fact that taking stalks commutes with sheafification.  6.21. Continuous maps and sheaves Let f : X → Y be a continuous map of topological spaces. We will define the pushforward and pullback functors for presheaves and sheaves. Let F be a presheaf of sets on X. We define the pushforward of F by the rule f∗ F(V ) = F(f −1 (V )) for any open V ⊂ Y . Given V1 ⊂ V2 ⊂ Y open the restriction map is given by the commutativity of the diagram f∗ F(V2 )

F(f −1 (V2 ))

 f∗ F(V1 )



restriction for F

F(f −1 (V1 ))

It is clear that this defines a presheaf of sets. The construction is clearly functorial in the presheaf F and hence we obtain a functor f∗ : PSh(X) −→ PSh(Y ). Lemma 6.21.1. Let f : X → Y be a continuous map. Let F be a sheaf of sets on X. Then f∗ F is a sheaf on Y . S Proof. This immediately follows from the fact that if V = Vj is an open covering S in Y , then f −1 (V ) = f −1 (Vj ) is an open covering in X.  As a consequence we obtain a functor f∗ : Sh(X) −→ Sh(Y ). This is compatible with composition in the following strong sense.

176

6. SHEAVES ON SPACES

Lemma 6.21.2. Let f : X → Y and g : Y → Z be continuous maps of topological spaces. The functors (g ◦ f )∗ and g∗ ◦ f∗ are equal (on both presheaves and sheaves of sets). Proof. This is because (g ◦ f )∗ F(W ) = F((g ◦ f )−1 W ) and (g∗ ◦ f∗ )F(W ) = F(f −1 g −1 W ) and (g ◦ f )−1 W = f −1 g −1 W .  Let G be a presheaf of sets on Y . The pullback presheaf fp G of a given presheaf G is defined as the left adjoint of the pushforward f∗ on presheaves. In other words it should be a presheaf fp G on X such that MorPSh(X) (fp G, F) = MorPSh(Y ) (G, f∗ F). By the Yoneda lemma this determines the pullback uniquely. It turns out that it actually exists. Lemma 6.21.3. Let f : X → Y be a continuous map. There exists a functor fp : PSh(Y ) → PSh(X) which is right adjoint to f∗ . For a presheaf G it is determined by the rule fp G(U ) = colimf (U )⊂V G(V ) where the colimit is over the collection of open neighbourhoods V of f (U ) in Y . The colimits are over directed partially ordered sets. (The restriction mappings of fp G are explained in the proof.) Proof. The colimit is over the partially ordered set consisting of open subset V ⊂ Y which contain f (U ) with ordering by reverse inclusion. This is a directed partially ordered set, since if V, V 0 are in it then so is V ∩ V 0 . Furthermore, if U1 ⊂ U2 , then every open neighbourhood of f (U2 ) is an open neighbourhood of f (U1 ). Hence the system defining fp G(U2 ) is a subsystem of the one defining fp G(U1 ) and we obtain a restiction map (for example by applying the generalities in Categories, Lemma 4.13.7). Note that the construction of the colimit is clearly functorial in G, and similarly for the restriction mappings. Hence we have defined fp as a functor. A small useful remark is that there exists a canonical map G(U ) → fp G(f −1 (U )), because the system of open neighbourhoods of f (f −1 (U )) contains the element U . This is compatible with restriction mappings. In other words, there is a canonical map iG : G → f∗ fp G. Let F be a presheaf of sets on X. Suppose that ψ : fp G → F is a map of presheaves of sets. The corresponding map G → f∗ F is the map f∗ ψ ◦ iG : G → f∗ fp G → f∗ F. Another small useful remark is that there exists a canonical map cF : fp f∗ F → F. Namely, let U ⊂ X open. For every open neighbourhood V ⊃ f (U ) in Y there exists a map f∗ F(V ) = F(f −1 (V )) → F(U ), namely the restriction map on F. And this is certainly compatible wrt restriction mappings between values of F on f −1 of varying opens containing f (U ). Thus we obtain a canonical map fp f∗ F(U ) → F(U ). Another trivial verification show that these maps are compatible with restrictions and define a map cF of presheaves of sets. Suppose that ϕ : G → f∗ F is a map of presheaves of sets. Consider fp ϕ : fp G → fp f∗ F. Postcomposing with cF gives the desired map cF ◦ fp ϕ : fp G → F. We omit the verification that this construction is inverse to the construction in the other direction given above. 

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177

Lemma 6.21.4. Let f : X → Y be a continuous map. Let x ∈ X. Let G be a presheaf of sets on Y . There is a canonical bijection of stalks (fp G)x = Gf (x) . Proof. This you can see as follows (fp G)x

=

colimx∈U fp G(U )

=

colimx∈U colimf (U )⊂V G(V )

=

colimf (x)∈V G(V )

= Gf (x) Here we have used Categories, Lemma 4.13.9, and the fact that any V open in Y containing f (x) occurs in the third description above. Details omitted.  Let G be a sheaf of sets on Y . The pullback sheaf f −1 G is defined by the formula f −1 G = (fp G)# . Sheafification is a left adjoint to the inclusion of sheaves in presheaves, and fp is a left adjoint to f∗ on presheaves. As a formal consequence we obtain that f −1 is a left adjoint of pushforward on sheaves. In other words, MorSh(X) (f −1 G, F) = MorSh(Y ) (G, f∗ F). The formal argument is given in the setting of abelian sheaves in the next section. Lemma 6.21.5. Let x ∈ X. Let G be a sheaf of sets on Y . There is a canonical bijection of stalks (f −1 G)x = Gf (x) . Proof. This is a combination of Lemmas 6.17.2 and 6.21.4.



Lemma 6.21.6. Let f : X → Y and g : Y → Z be continuous maps of topological spaces. The functors (g ◦ f )−1 and f −1 ◦ g −1 are canonically isomorphic. Similarly (g ◦ f )p ∼ = fp ◦ gp on presheaves. Proof. To see this use that adjoint functors are unique up to unique isomorphism, and Lemma 6.21.2.  Definition 6.21.7. Let f : X → Y be a continuous map. Let F be a sheaf of sets on X and let G be a sheaf of sets on Y . An f -map ξ : G → F is a collection of maps ξV : G(V ) → F(f −1 (V )) indexed by open subsets V ⊂ Y such that G(V )

ξV

restriction of G

 G(V 0 )

ξV 0

/ F(f −1 V ) 

restriction of F

/ F(f −1 V 0 )

commutes for all V 0 ⊂ V ⊂ Y open. Lemma 6.21.8. Let f : X → Y be a continuous map. Let F be a sheaf of sets on X and let G be a sheaf of sets on Y . There are canonical bijections between the following three sets: (1) The set of maps G → f∗ F. (2) The set of maps f −1 G → F. (3) The set of f -maps ξ : G → F. Proof. We leave the easy verification to the reader.



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6. SHEAVES ON SPACES

It is sometimes convenient to think about f -maps instead of maps between sheaves either on X or on Y . We define composition of f -maps as follows. Definition 6.21.9. Suppose that f : X → Y and g : Y → Z are continuous maps of topological spaces. Suppose that F is a sheaf on X, G is a sheaf on Y , and H is a sheaf on Z. Let ϕ : G → F be an f -map. Let ψ : H → G be an g-map. The composition of ϕ and ψ is the (g ◦ f )-map ϕ ◦ ψ defined by the commutativity of the diagrams / F(f −1 g −1 W ) H(W ) 7 (ϕ◦ψ)W % G(g −1 W )

ψW

ϕg−1 W

We leave it to the reader to verify that this works. Another way to think about this is to think of ϕ ◦ ψ as the composition ψ

g∗ ϕ

H− → g∗ G −−→ g∗ f∗ F = (g ◦ f )∗ F Now, doesn’t it seem that thinking about f -maps is somehow easier? Finally, given a continuous map f : X → Y , and an f -map ϕ : G → F there is a natural map on stalks ϕx : Gf (x) −→ Fx for all x ∈ X. The image of a representative (V, s) of an element in Gf (x) is mapped to the element in Fx with representative (f −1 V, ϕV (s)). We leave it to the reader to see that this is well defined. Another way to state it is that it is the unique map such that all diagrams / Fx F(f −1 V ) O O ϕx

ϕV

G(V )

/ Gf (x)

(for x ∈ V ⊂ Y open) commute. Lemma 6.21.10. Suppose that f : X → Y and g : Y → Z are continuous maps of topological spaces. Suppose that F is a sheaf on X, G is a sheaf on Y , and H is a sheaf on Z. Let ϕ : G → F be an f -map. Let ψ : H → G be an g-map. Let x ∈ X be a point. The map on stalks (ϕ ◦ ψ)x : Hg(f (x)) → Fx is the composition ψf (x)

ϕx

Hg(f (x)) −−−→ Gf (x) −−→ Fx Proof. Immediate from Definition 6.21.9 and the definition of the map on stalks above.  6.22. Continuous maps and abelian sheaves Let f : X → Y be a continuous map. We claim there are functors f∗ : PAb(X) −→ f∗ : Ab(X) −→ fp : PAb(Y ) −→ f

−1

: Ab(Y ) −→

PAb(Y ) Ab(Y ) PAb(X) Ab(X)

6.22. CONTINUOUS MAPS AND ABELIAN SHEAVES

179

with similar properties to their counterparts in Section 6.21. To see this we argue in the following way. Each of the functors will be constructed in the same way as the corresponding functor in Section 6.21. This works because all the colimits in that section are directed colimits (but we will work through it below). First off, given an abelian presheaf F on X and an abelian presheaf G on Y we define f∗ F(V )

= F(f −1 (V ))

fp G(U )

=

colimf (U )⊂V G(V )

as abelian groups. The restriction mappings are the same as the restriction mappings for presheaves of sets (and they are all homomorphisms of abelian groups). The assignments F 7→ f∗ F and G → fp G are functors on the categories of presheaves of abelian groups. This is clear, as (for example) a map of abelian presheaves G1 → G2 gives rise to a map of directed systems {G1 (V )}f (U )⊂V → {G2 (V )}f (U )⊂V all of whose maps are homomorphisms and hence gives rise to a homomorphism of abelian groups fp G1 (U ) → fp G2 (U ). The functors f∗ and fp are adjoint on the category of presheaves of abelian groups, i.e., we have MorPAb(X) (fp G, F) = MorPAb(Y ) (G, f∗ F). To prove this, note that the map iG : G → f∗ fp G from the proof of Lemma 6.21.3 is a map of abelian presheaves. Hence if ψ : fp G → F is a map of abelian presheaves, then the corresponding map G → f∗ F is the map f∗ ψ ◦ iG : G → f∗ fp G → f∗ F is also a map of abelian presheaves. For the other direction we point out that the map cF : fp f∗ F → F from the proof of Lemma 6.21.3 is a map of abelian presheaves as well (since it is made out of restriction mappings of F which are all homomorphisms). Hence given a map of abelian presheaves ϕ : G → f∗ F the map cF ◦fp ϕ : fp G → F is a map of abelian presheaves as well. Since these constructions ψ 7→ f∗ ψ and ϕ 7→ cF ◦ fp ϕ are inverse to each other as constructions on maps of presheaves of sets we see they are also inverse to each other on maps of abelian presheaves. If F is an abelian sheaf on Y , then f∗ F is an abelian sheaf on X. This is true because of the definition of an abelian sheaf and because this is true for sheaves of sets, see Lemma 6.21.1. This defines the functor f∗ on the category of abelian sheaves. We define f −1 G = (fp G)# as before. Adjointness of f∗ and f −1 follows formally as in the case of presheaves of sets. Here is the argument: MorAb(X) (f −1 G, F)

=

MorPAb(X) (fp G, F)

=

MorPAb(Y ) (G, f∗ F)

=

MorAb(Y ) (G, f∗ F)

Lemma 6.22.1. Let f : X → Y be a continuous map. (1) Let G be an abelian presheaf on Y . Let x ∈ X. The bijection Gf (x) → (fp G)x of Lemma 6.21.4 is an isomorphism of abelian groups. (2) Let G be an abelian sheaf on Y . Let x ∈ X. The bijection Gf (x) → (f −1 G)x of Lemma 6.21.5 is an isomorphism of abelian groups.

180

6. SHEAVES ON SPACES

Proof. Omitted.



Given a continuous map f : X → Y and sheaves of abelian groups F on X, G on Y , the notion of an f -map G → F of sheaves of abelian groups makes sense. We can just define it exactly as in Definition 6.21.7 (replacing maps of sets with homomorphisms of abelian groups) or we can simply say that it is the same as a map of abelian sheaves G → f∗ F. We will use this notion freely in the following. The group of f -maps between G and F will be in canonical bijection with the groups MorAb(X) (f −1 G, F) and MorAb(Y ) (G, f∗ F). Composition of f -maps is defined in exactly the same manner as in the case of f maps of sheaves of sets. In addition, given an f -map G → F as above, the induced maps on stalks ϕx : Gf (x) −→ Fx are abelian group homomorphisms.

6.23. Continuous maps and sheaves of algebraic structures Let (C, F ) be a type of algebraic structure. For a topological space X let us introduce the notation: (1) PSh(X, C) will be the category of presheaves with values in C. (2) Sh(X, C) will be the category of sheaves with values in C. Let f : X → Y be a continuous map of topological spaces. The same arguments as in the previous section show there are functors f∗ : PSh(X, C) −→ f∗ : Sh(X, C) −→ fp : PSh(Y, C) −→ f

−1

: Sh(Y, C) −→

PSh(Y, C) Sh(Y, C) PSh(X, C) Sh(X, C)

constructed in the same manner and with the same properties as the functors constructed for abelian (pre)sheaves. In particular there are commutative diagrams PSh(X, C)

f∗

/ PSh(Y, C)

f∗

 / PSh(Y )

 Sh(X)

/ PSh(X, C)

Sh(Y, C)

F

 PSh(X)

PSh(Y, C)

F

fp

F

 PSh(Y )

F

fp

 / PSh(X)

Sh(X, C)

f∗

/ Sh(Y, C)

f∗

 / Sh(Y )

F

F

f −1

F

 Sh(Y )

/ Sh(X, C) F

f −1

 / Sh(X)

6.24. CONTINUOUS MAPS AND SHEAVES OF MODULES

181

The main formulas to keep in mind are the following f∗ F(V )

= F(f −1 (V ))

fp G(U )

=

colimf (U )⊂V G(V )

=

(fp G)#

f

−1

G

(fp G)x (f

−1

G)x

= Gf (x) = Gf (x)

Each of these formulas has the property that they hold in the category C and that upon taking underlying sets we get the corresponding formula for presheaves of sets. In addition we have the adjointness properties MorPSh(X,C) (fp G, F) MorSh(X,C) (f

−1

G, F)

=

MorPSh(Y,C) (G, f∗ F)

=

MorSh(Y,C) (G, f∗ F).

To prove these, the main step is to construct the maps iG : G −→ f∗ fp G and cF : fp f∗ F −→ F which occur in the proof of Lemma 6.21.3 as morphisms of presheaves with values in C. This may be safely left to the reader since the constructions are exactly the same as in the case of presheaves of sets. Given a continuous map f : X → Y and sheaves of algebraic structures F on X, G on Y , the notion of an f -map G → F of sheaves of algebraic structures makes sense. We can just define it exactly as in Definition 6.21.7 (replacing maps of sets with morphisms in C) or we can simply say that it is the same as a map of sheaves of algebraic structures G → f∗ F. We will use this notion freely in the following. The set of f -maps between G and F will be in canonical bijection with the sets MorSh(X,C) (f −1 G, F) and MorSh(Y,C) (G, f∗ F). Composition of f -maps is defined in exactly the same manner as in the case of f maps of sheaves of sets. In addition, given an f -map G → F as above, the induced maps on stalks ϕx : Gf (x) −→ Fx are homomorphisms of algebraic structures. Lemma 6.23.1. Let f : X → Y be a continuous map of topological spaces. Suppose given sheaves of algebraic structures F on X, G on Y . Let ϕ : G → F be an f -map of underlying sheaves of sets. If for every V ⊂ Y open the map of sets ϕV : G(V ) → F(f −1 V ) is the effect of a morphism in C on underlying sets, then ϕ comes from a unique f -morphism between sheaves of algebraic structures. Proof. Omitted.



6.24. Continuous maps and sheaves of modules The case of sheaves of modules is more complicated. The reason is that the natural setting for defining the pullback and pushforward functors, is the setting of ringed spaces, which we will define below. First we state a few obvious lemmas.

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6. SHEAVES ON SPACES

Lemma 6.24.1. Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on X. Let F be a presheaf of O-modules. There is a natural map of underlying presheaves of sets f∗ O × f∗ F −→ f∗ F which turns f∗ F into a presheaf of f∗ O-modules. This construction is functorial in F. Proof. Let V ⊂ Y is open. We define the map of the lemma to be the map f∗ O(V ) × f∗ F(V ) = O(f −1 V ) × F(f −1 V ) → F(f −1 V ) = f∗ F(V ). Here the arrow in the middle is the multiplication map on X. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of f∗ O-module on f∗ F.  Lemma 6.24.2. Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on Y . Let G be a presheaf of O-modules. There is a natural map of underlying presheaves of sets fp O × fp G −→ fp G which turns fp G into a presheaf of fp O-modules. This construction is functorial in G. Proof. Let U ⊂ X is open. We define the map of the lemma to be the map fp O(U ) × fp G(U )

=

colimf (U )⊂V O(V ) × colimf (U )⊂V G(V )

=

colimf (U )⊂V (O(V ) × G(V ))

→ colimf (U )⊂V G(V ) =

fp G(U ).

Here the arrow in the middle is the multiplication map on Y . The second equality holds because directed colimits commute with finite limits, see Categories, Lemma 4.17.2. We leave it to the reader to see this is compatible with restriction mappings and defines a structure of fp O-module on fp G.  Let f : X → Y be a continuous map. Let OX be a presheaf of rings on X and let OY be a presheaf of rings on Y . So at the moment we have defined functors f∗ : PMod(OX ) −→

PMod(f∗ OX )

fp : PMod(OY ) −→

PMod(fp OY )

These satisfy some compatibilities as follows. Lemma 6.24.3. Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on Y . Let G be a presheaf of O-modules. Let F be a presheaf of fp O-modules. Then MorPMod(fp O) (fp G, F) = MorPMod(O) (G, f∗ F). Here we use Lemmas 6.24.2 and 6.24.1, and we think of f∗ F as an O-module via the map iO : O → f∗ fp O (defined first in the proof of Lemma 6.21.3).

6.24. CONTINUOUS MAPS AND SHEAVES OF MODULES

183

Proof. Note that we have MorPAb(X) (fp G, F) = MorPAb(Y ) (G, f∗ F). according to Section 6.22. So what we have to prove is that under this correspondence, the subsets of module maps correspond. In addition, the correspondence is determined by the rule     ψ : fp G → F 7−→ f∗ ψ ◦ iG : G → f∗ fp G → f∗ F Hence, using the functoriality of the pushforward we see that it suffices to prove that the map iG : G → f∗ fp G is compatible with module structure, which we leave to the reader.  Lemma 6.24.4. Let f : X → Y be a continuous map of topological spaces. Let O be a presheaf of rings on X. Let F be a presheaf of O-modules. Let G be a presheaf of f∗ O-modules. Then MorPMod(O) (O ⊗p,fp f∗ O fp G, F) = MorPMod(f∗ O) (G, f∗ F). Here we use Lemmas 6.24.2 and 6.24.1, and we use the map cO : fp f∗ O → O in the definition of the tensor product. Proof. This follows from the equalities MorPMod(O) (O ⊗p,fp f∗ O fp G, F)

=

MorPMod(fp f∗ O) (fp G, Ffp f∗ O )

=

MorPMod(f∗ O) (G, f∗ F).

which is a combination of Lemmas 6.6.2 and 6.24.3.



Lemma 6.24.5. Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on X. Let F be a sheaf of O-modules. The pushforward f∗ F, as defined in Lemma 6.24.1 is a sheaf of f∗ O-modules. Proof. Obvious from the definition and Lemma 6.21.1.



Lemma 6.24.6. Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on Y . Let G be a sheaf of O-modules. There is a natural map of underlying presheaves of sets f −1 O × f −1 G −→ f −1 G which turns f −1 G into a sheaf of f −1 O-modules. Proof. Recall that f −1 is defined as the composition of the functor fp and sheafification. Thus the lemma is a combination of Lemma 6.24.2 and Lemma 6.20.1.  Let f : X → Y be a continuous map. Let OX be a sheaf of rings on X and let OY be a sheaf of rings on Y . So now we have defined functors

f

f∗ −1

: Mod(OX ) −→

Mod(f∗ OX )

: Mod(OY ) −→

Mod(f −1 OY )

These satisfy some compatibilities as follows.

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Lemma 6.24.7. Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on Y . Let G be a sheaf of O-modules. Let F be a sheaf of f −1 O-modules. Then MorMod(f −1 O) (f −1 G, F) = MorMod(O) (G, f∗ F). Here we use Lemmas 6.24.6 and 6.24.5, and we think of f∗ F as an O-module by restriction via O → f∗ f −1 O. Proof. Argue by the equalities MorMod(f −1 O) (f −1 G, F)

=

MorMod(fp O) (fp G, F)

=

MorMod(O) (G, f∗ F).

where the second is Lemmas 6.24.3 and the first is by Lemma 6.20.1.



Lemma 6.24.8. Let f : X → Y be a continuous map of topological spaces. Let O be a sheaf of rings on X. Let F be a sheaf of O-modules. Let G be a sheaf of f∗ O-modules. Then MorMod(O) (O ⊗f −1 f∗ O f −1 G, F) = MorMod(f∗ O) (G, f∗ F). Here we use Lemmas 6.24.6 and 6.24.5, and we use the canonical map f −1 f∗ O → O in the definition of the tensor product. Proof. This follows from the equalities MorMod(O) (O ⊗f −1 f∗ O f −1 G, F)

=

MorMod(f −1 f∗ O) (f −1 G, Ff −1 f∗ O )

=

MorMod(f∗ O) (G, f∗ F).

which are a combination of Lemma 6.20.2 and 6.24.7.



Let f : X → Y be a continuous map. Let OX be a (pre)sheaf of rings on X and let OY be a (pre)sheaf of rings on Y . So at the moment we have defined functors f∗ : PMod(OX ) −→ f∗ : Mod(OX ) −→ fp : PMod(OY ) −→ f

−1

: Mod(OY ) −→

PMod(f∗ OX ) Mod(f∗ OX ) PMod(fp OY ) Mod(f −1 OY )

Clearly, usually the pair of functors (f∗ , f −1 ) on sheaves of modules are not adjoint, because their target categories do not match. Namely, as we saw above, it works only if by some miracle the sheaves of rings OX , OY satisfy the relations OX = f −1 OY and OY = f∗ OX . This is almost never true in practice. We interrupt the discussion to define the correct notion of morphism for which a suitable adjoint pair of functors on sheaves of modules exists. 6.25. Ringed spaces Let X be a topological space and let OX be a sheaf of rings on X. We are supposed to think of the sheaf of rings OX as a sheaf of functions on X. And if f : X → Y is a “suitable” map, then by composition a function on Y turns into a function on X. Thus there should be a natural f -map from OY to OX See Definition 6.21.7, and the remarks in previous sections for terminology. For a precise example, see Example 6.25.2 below. Here is the relevant abstract definition.

6.26. MORPHISMS OF RINGED SPACES AND MODULES

185

Definition 6.25.1. A ringed space is a pair (X, OX ) consisting of a topological space X and a sheaf of rings OX on X. A morphism of ringed spaces (X, OX ) → (Y, OY ) is a pair consisting of a continuous map f : X → Y and an f -map of sheaves of rings f ] : OY → OX . Example 6.25.2. Let f : X → Y be a continuous map of topological spaces. 0 Consider the sheaves of continuous real valued functions CX on X and CY0 on Y , see ] 0 Example 6.9.3. We claim that there is a natural f -map f : CY0 → CX associated to f . Namely, we simply define it by the rule CY0 (V ) −→ h 7−→

0 CX (f −1 V )

h◦f

Stricly speaking we should write f ] (h) = h◦f |f −1 (V ) . It is clear that this is a family of maps as in Definition 6.21.7 and compatible with the R-algebra structures. Hence it is an f -map of sheaves of R-algebras, see Lemma 6.23.1. Of course there are lots of other situations where there is a canonical morphism of ringed spaces associated to a geometrical type of morphism. For example, if M , N are C ∞ -manifolds and f : M → N is a infinitely differentiable map, then f induces a ∞ ∞ ). The construction (which ) → (N, CN canonical morphism of ringed spaces (M, CM is identical to the above) is left to the reader. It may not be completely obvious how to compose morphisms of ringed spaces hence we spell it out here. Definition 6.25.3. Let (f, f ] ) : (X, OX ) → (Y, OY ) and (g, g ] ) : (Y, OY ) → (Z, OZ ) be morphisms of ringed spaces. Then we define the composition of morphisms of ringed spaces by the rule (g, g ] ) ◦ (f, f ] ) = (g ◦ f, f ] ◦ g ] ). Here we use composition of f -maps defined in Definition 6.21.9. 6.26. Morphisms of ringed spaces and modules We have now introduced enough notation so that we are able to define the pullback and pushforward of modules along a morphism of ringed spaces. Definition 6.26.1. Let (f, f ] ) : (X, OX ) → (Y, OY ) be a morphism of ringed spaces. (1) Let F be a sheaf of OX -modules. We define the pushforward of F as the sheaf of OY -modules which as a sheaf of abelian groups equals f∗ F and with module structure given by the restriction via f ] : OY → f∗ OX of the module structure given in Lemma 6.24.5. (2) Let G be a sheaf of OY -modules. We define the pullback f ∗ G to be the sheaf of OX -modules defined by the formula f ∗ F = OX ⊗f −1 OY f −1 F where the ring map f −1 OY → OX is the map corresponding to f ] , and where the module structure is given by Lemma 6.24.6.

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Thus we have defined functors f∗ : Mod(OX ) −→ ∗

f : Mod(OY ) −→

Mod(OY ) Mod(OX )

The final result on these functors is that they are indeed adjoint as expected. Lemma 6.26.2. Let (f, f ] ) : (X, OX ) → (Y, OY ) be a morphism of ringed spaces. Let F be a sheaf of OX -modules. Let G be a sheaf of OY -modules. There is a canonical bijection HomOX (f ∗ G, F) = HomOY (G, f∗ F). In other words: the functor f ∗ is the left adjoint to f∗ . Proof. This follows from the work we did before: HomOX (f ∗ G, F)

=

MorMod(OX ) (OX ⊗f −1 OY f −1 G, F)

=

MorMod(f −1 OY ) (f −1 G, Ff −1 OY )

=

HomOY (G, f∗ F).

Here we use Lemmas 6.20.2 and 6.24.7.



Lemma 6.26.3. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. The functors (g ◦ f )∗ and g∗ ◦ f∗ are equal. There is a canonical isomorphism of functors (g ◦ f )∗ ∼ = f ∗ ◦ g∗ . Proof. The result on pushforwards is a consequence of Lemma 6.21.2 and our definitions. The result on pullbacks follows from this by the same argument as in the proof of Lemma 6.21.6.  Given a morphism of ringed spaces (f, f ] ) : (X, OX ) → (Y, OY ), and a sheaf of OX -modules F, a sheaf of OY -modules G on Y , the notion of an f -map ϕ : G → F of sheaves of modules makes sense. We can just define it as an f -map ϕ : G → F of abelian sheaves such that for all open V ⊂ Y the map G(V ) −→ F(f −1 V ) is an OY (V )-module map. Here we think of F(f −1 V ) as an OY (V )-module via the map fV] : OY (V ) → OX (f −1 V ). The set of f -maps between G and F will be in canonical bijection with the sets MorMod(OX ) (f ∗ G, F) and MorMod(OY ) (G, f∗ F). See above. Composition of f -maps is defined in exactly the same manner as in the case of f -maps of sheaves of sets. In addition, given an f -map G → F as above, and x ∈ X the induced map on stalks ϕx : Gf (x) −→ Fx is an OY,f (x) -module map where the OY,f (x) -module structure on Fx comes from the OX,x -module structure via the map fx] : OY,f (x) → OX,x . Here is a related lemma. Lemma 6.26.4. Let (f, f ] ) : (X, OX ) → (Y, OY ) be a morphism of ringed spaces. Let G be a sheaf of OY -modules. Let x ∈ X. Then f ∗ Gx = Ff (x) ⊗OY,f (x) ,fx] OX,x as OX,x -modules.

6.27. SKYSCRAPER SHEAVES AND STALKS

187

Proof. This follows from Lemma 6.20.3 and the identification of the stalks of pullback sheaves at x with the corresponding stalks at f (x). See the formulae in Section 6.23 for example.  6.27. Skyscraper sheaves and stalks Definition 6.27.1. Let X be a topological space. (1) Let x ∈ X be a point. Denote ix : {x} → X the inclusion map. Let A be a set and think of A as a sheaf on the one point space {x}. We call ix,∗ A the skyscraper sheaf at x with value A. (2) If in (1) above A is an abelian group then we think of ix,∗ A as a sheaf of abelian groups on X. (3) If in (1) above A is an algebraic structure then we think of ix,∗ A as a sheaf of algeberaic structures. (4) If (X, OX ) is a ringed space, then we think of ix : {x} → X as a morphism of ringed spaces ({x}, OX,x ) → (X, OX ) and if A is a OX,x -module, then we think of ix,∗ A as a sheaf of OX -modules. (5) We say a sheaf of sets F is a skyscraper sheaf if there exists an point x of X and a set A such that F ∼ = ix,∗ A. (6) We say a sheaf of abelian groups F is a skyscraper sheaf if there exists an point x of X and an abelian group A such that F ∼ = ix,∗ A as sheaves of abelian groups. (7) We say a sheaf of algebraic structures F is a skyscraper sheaf if there exists an point x of X and an algebraic structure A such that F ∼ = ix,∗ A as sheaves of algebraic structures. (8) If (X, OX ) is a ringed space and F is a sheaf of OX -modules, then we say F is a skyscraper sheaf if there exists a point x ∈ X and a OX,x -module A such that F ∼ = ix,∗ A as sheaves of OX -modules. Lemma 6.27.2. Let X be a topological space, x ∈ X a point, and A a set. For any point x0 ∈ X the stalk of the skyscraper sheaf at x with value A at x0 is  A if x0 ∈ {x} (ix,∗ A)x0 = {∗} if x0 6∈ {x} A similar description holds for the case of abelian groups, algebraic structures and sheaves of modules. Proof. Omitted.



Lemma 6.27.3. Let X be a topological space, and let x ∈ X a point. The functors F 7→ Fx and A 7→ ix,∗ A are adjoint. In a formula MorSets (Fx , A) = MorSh(X) (F, ix,∗ A). A similar satement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have HomOX,x (Fx , A) = HomOX (F, ix,∗ A). Proof. Omitted. Hint: The stalk functor can be seen as the pullback functor for the morphism ix : {x} → X. Then the adjointness follows from adjointness of i−1 x and ix,∗ (resp. i∗x and ix,∗ in the case of sheaves of modules. 

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6.28. Limits and colimits of presheaves Let X be a topological space. Let I → PSh(X), i 7→ Fi be a diagram. (1) Both limi Fi and colimi Fi exist. (2) For any open U ⊂ X we have (limi Fi )(U ) = limi Fi (U ) and (colimi Fi )(U ) = colimi Fi (U ). (3) Let x ∈ X be a point. In general the stalk of limi Fi at x is not equal to the limit of the stalks. But if the diagram category is finite then it is the case. In other words, the stalk functor is left exact (see Categories, Definition 4.21.1). (4) Let x ∈ X. We always have (colimi Fi )x = colimi Fi,x . The proofs are all easy. 6.29. Limits and colimits of sheaves Let X be a topological space. Let I → Sh(X), i 7→ Fi be a diagram. (1) Both limi Fi and colimi Fi exist. (2) The inclusion functor i : Sh(X) → PSh(X) commutes with limits. In other words, we may compute the limit in the category of sheaves as the limit in the category of presheaves. In particular, for any open U ⊂ X we have (limi Fi )(U ) = limi Fi (U ). (3) The inclusion functor i : Sh(X) → PSh(X) does not commute with colimits in general (not even with finite colimits – think surjections). The colimit is computed as the sheafification of the colimit in the category of presheaves:  # colimi Fi = U 7→ colimi Fi (U ) . (4) Let x ∈ X be a point. In general the stalk of limi Fi at x is not equal to the limit of the stalks. But if the diagram category is finite then it is the case. In other words, the stalk functor is left exact. (5) Let x ∈ X. We always have (colimi Fi )x = colimi Fi,x . (6) The sheafification functor # : PSh(X) → Sh(X) commutes with all colimits, and with finite limits. But it does not commute with all limits. The proofs are all easy. Here is an example of what is true for directed colimits of sheaves. Lemma 6.29.1. Let X be a topological space. Let I be a directed partially ordered set. Let (Fi , ϕii0 ) be a system of sheaves of sets over I, see Categories, Section 4.19. Let U ⊂ X be an open subset. Consider the canonical map Ψ : colimi Fi (U ) −→ (colimi Fi ) (U ) (1) If all the transition maps are injective then Ψ is injective for any open U .

6.30. BASES AND SHEAVES

189

(2) If U is quasi-compact, then Ψ is injective. (3) If U is quasi-compact and all the transition maps are injective then Ψ is an isomorphism. S (4) If U has a cofinal system of open coverings U : U = j∈J Uj with J finite and Uj ∩ Uj 0 quasi-compact for all j, j 0 ∈ J, then Ψ is bijective. Proof. Assume all the transition maps are injective. In this case the presheaf F 0 : V 7→ colimi Fi (V ) is separated (see Definition 6.11.2). By the discussion above we have (F 0 )# = colimi Fi . By Lemma 6.17.5 we see that F 0 → (F 0 )# is injective. This proves (1). Assume U is quasi-compact. Suppose that s ∈ Fi (U ) and s0 ∈ Fi0 (U ) give rise to elements on the left hand side which have the same image under S Ψ. Since U is quasi-compact this means there exists a finite open covering U = j=1,...,m Uj and for each j an index ij ∈ I, ij ≥ i, ij ≥ i0 such that ϕiij (s) = ϕi0 ij (s0 ). Let i00 ∈ I be ≥ than all of the ij . We conclude that ϕii00 (s) and ϕi0 i00 (s) agree on the opens Uj for all j and hence that ϕii00 (s) = ϕi0 i00 (s). This proves (2). Assume U is quasi-compact and all transition maps injective. Let s be an element of theS target of Ψ. Since U is quasi-compact there exists a finite open covering U = j=1,...,m Uj , for each j an index ij ∈ I and sj ∈ Fij (Uj ) such that s|Uj comes from sj for all j. Pick i ∈ I which is ≥ than all of the ij . By (1) the sections ϕij i (sj ) agree over the overlaps Uj ∩ Uj 0 . Hence they glue to a section s0 ∈ Fi (U ) which maps to s under Ψ. This proves (3). Assume the hypothesis of (4). Let s be anSelement of the target of Ψ. By assumption there exists a finite open covering U = j=1,...,m Uj , with Uj ∩ Uj 0 quasi-compact for all j, j 0 ∈ J and for each j an index ij ∈ I and sj ∈ Fij (Uj ) such that s|Uj is the image of sj for all j. Since Uj ∩ Uj 0 is quasi-compact we can apply (2) and we see that there exists an ijj 0 ∈ I, ijj 0 ≥ ij , ijj 0 ≥ ij 0 such that ϕij ijj0 (sj ) and ϕij0 ijj0 (sj 0 ) agree over Uj ∩ Uj 0 . Choose an index i ∈ I wich is bigger or equal than all the ijj 0 . Then we see that the sections ϕij i (sj ) of Fi glue to a section of Fi over U . This section is mapped to the element s as desired.  Example 6.29.2. Let X = {s1 , s2 , ξ1 , ξ2 , ξ3 , . . .} as a set. Declare a subset U ⊂ X to be open if s1 ∈ U or s2 ∈ U implies U contains all of the ξi . Let Un = {ξn , ξn+1 , . . .}, and let jn : Un → X be the inclusion map. Set Fn = jn,∗ Z. There are transition maps Fn → Fn+1 . Let F = colim Fn . Note that Fn,ξm = 0 if m < n because {ξm } is an open subset of X which misses Un . Hence we see that Fξn = 0 for all n. On the other hand the stalk Fsi , i = 1, 2 is the colimit Y M = colimn Z m≥n

which is not zero. We conclude that the sheaf F is the direct sum of the skyscraper sheaves with value M at the closed points s1 and s2 . Hence Γ(X, F) = M ⊕ M . On the other hand, the reader can verify that colimn Γ(X, Fn ) = M . Hence some condition is necessary in part (4) of Lemma 6.29.1 above. 6.30. Bases and sheaves Sometimes there exists a basis for the topology consisting of opens that are easier to work with than general opens. For convenience we give here some definitions and simple lemmas in order to facilitate working with (pre)sheaves in such a situation.

190

6. SHEAVES ON SPACES

Definition 6.30.1. Let X be a topological space. Let B be a basis for the topology on X. (1) A presheaf F of sets on B is a rule which assigns to each U ∈ B a set F(U ) and to each inclusion V ⊂ U of elements of B a map ρU V : F(U ) → F(V ) V U such that whenever W ⊂ V ⊂ U in B we have ρU W = ρW ◦ ρV . (2) A morphism ϕ : F → G of presheaves of sets on B is a rule which assigns to each element U ∈ B a map of sets ϕ : F(U ) → G(U ) compatible with restriction maps. As in the case of usual presheaves we use the terminology of sections, restrictions of sections, etc. In particular, we may define the stalk of F at a point x ∈ X by the colimit Fx = colimU ∈B,x∈U F(U ). As in the case of the stalk of a presheaf on X this limit is directed. The reason is that the collection of U ∈ B, x ∈ U is a fundamental system of open neighbourhoods of x. It is easy to make examples to show that the notion of a presheaf on X is very different from the notion of a presheaf on a basis for the topology on X. This does not happen in the case of sheaves. A much more useful notion therefore, is the following. Definition 6.30.2. Let X be a topological space. Let B be a basis for the topology on X. (1) A sheaf F of sets on B is a presheaf of sets on B which satisfies the following additional property: Given any U ∈ B, S and any covering U = S i∈I Ui with Ui ∈ B, and any coverings Ui ∩ Uj = k∈Iij Uijk with Uijk ∈ B the sheaf condition holds: (∗∗) For any collection of sections si ∈ F(Ui ), i ∈ I such that ∀i, j ∈ I, ∀k ∈ Iij si |Uijk = sj |Uijk there exists a unique section s ∈ F(U ) such that si = s|Ui for all i ∈ I. (2) A morphism of sheaves of sets on B is simply a morphism of presheaves of sets. First we explain that it suffices to check the sheaf condition (∗∗) on a cofinal system of coverings. In the situation of the definition, suppose U ∈ B. Let us temporarily denote CovB (U ) the set of all coverings of U by elements of B. Note that CovB (U ) is partially ordered by refinement. A subset C ⊂ CovB (U ) is a cofinal system, if for every U ∈ CovB (U ) there exists a covering V ∈ C which refines U. Lemma 6.30.3. With notation as above. For each U S∈ B, let C(U ) ⊂ CovB (U ) be a cofinal system. S For each U ∈ B, and each U : U = Ui in C(U ), let coverings Uij : Ui ∩ Uj = Uijk , Uijk ∈ B be given. Let F be a presheaf of sets on B. The following are equivalent (1) The presheaf F is a sheaf on B. S (2) For every U ∈ B and every covering U : U = Ui in C(U ) the sheaf condition (∗∗) holds (for the given coverings Uij ).

6.30. BASES AND SHEAVES

191

Proof. We S have to show that (2) implies (1). Suppose that U ∈ B, and that U : U = i∈I Ui is an arbitrary covering by elements of B. Because the system S C(U ) is cofinal we can find an element V : U = j∈J Vj in C(U ) which refines U. This means there exists a map α : J → I such that Vj ⊂ Uα(i) . Note that if s, s0 ∈ F(U ) are sections such that s|Ui = s0 |Ui , then s|Vj = (s|Uα(j) )|Vj = (s0 |Uα(j) )|Vj = s0 |Vj for all j. Hence by the uniqueness in (∗∗) for the covering V we conclude that s = s0 . Thus we have proved the uniqueness part of (∗∗) for our arbitrary covering U. S Suppose furthermore that Ui ∩ Ui0 = k∈Iii0 Uii0 k are arbitrary coverings by Uii0 k ∈ B. Let us try to prove the existence part of (∗∗) for the system (U, Uij ). Thus let si ∈ F(Ui ) and suppose we have si |Uijk = si0 |Uii0 k 0

for all i, i , k. Set tj = sα(i) |Vj , where V and α are as above. There is one small kink in the argument here. Namely, let Vjj 0 : Vj ∩ Vj 0 = S 0 l∈Jjj 0 Vjj l be the covering given to us by the statement of the lemma. It is not a priori clear that tj |Vjj0 l = tj 0 |Vjj0 l for all j, j 0 , l. To see this, note that we do have tj |W = tj 0 |W for all W ∈ B, W ⊂ Vjj 0 l ∩ Uα(j)α(j 0 )k for all k ∈ Iα(j)α(j 0 ) , by our assumption on the family of elements si . And since Vj ∩ Vj 0 ⊂ Uα(j) ∩ Uα(j 0 ) we see that tj |Vjj0 l and tj 0 |Vjj0 l agree on the members of a covering of Vjj 0 l by elements of B. Hence by the uniqueness part proved above we finally deduce the desired equality of tj |Vjj0 l and tj 0 |Vjj0 l . Then we get the existence of an element t ∈ F(U ) by property (∗∗) for (V, Vjj 0 ). Again there is a small snag. We know that t restricts to tj on Vj but we do not yet know that t restricts to si on Ui . To conclude this note that the sets Ui ∩ Vj , j ∈ J cover Ui . Hence also the sets Uiα(j)k ∩ Vj , j ∈ J, k ∈ Iiα(j) cover Ui . We leave it to the reader to see that t and si restrict to the same section of F on any W ∈ B which is contained in one of the open sets Uiα(j)k ∩ Vj , j ∈ J, k ∈ Iiα(j) . Hence by the uniqueness part seen above we win.  Lemma 6.30.4. Let X be a topological space. Let B be a basis for the topology on X. Assume that for every pair U, U 0 ∈ B we have U ∩ U 0 ∈ B. For each U ∈ B, let C(U ) ⊂ CovB (U ) be a cofinal system. Let F be a presheaf of sets on B. The following are equivalent (1) The presheaf F is a sheaf on B. S (2) For every U ∈ B and every covering U : U = Ui in C(U ) and for every family of sections si ∈ F(Ui ) such that si |Ui ∩Uj = sj |Ui ∩Uj there exists a unique section s ∈ F(U ) which restricts to si on Ui . Proof. This is a reformulation of Lemma 6.30.3 above in the special case where the coverings Uij each consist of a single element. But also this case is much easier and is an easy exercise to do directly. 

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Lemma 6.30.5. Let X be a topological space. Let B be a basis for the topology on X. Let U ∈ B. Let F be a sheaf of sets on B. The map Y F(U ) → Fx x∈U

identifies F(U ) with the elements (sx )x∈U with the property (∗) For any x ∈ U there exists a V ∈ B, x ∈ V and a section σ ∈ F(V ) such that for all y ∈ V we have sy = (V, σ) in Fy . Q Proof. First note that the map F(U ) → x∈U Fx is injective by the uniqueness in the sheaf condition of Definition 6.30.2. Let (sx ) be any element on the S right hand side which satisfies (∗). Clearly this means we can find a covering U = Ui , Ui ∈ B such that (sx )x∈Ui comes from certain σi ∈ F(Ui ). For every y ∈ Ui ∩Uj the sections σi and σj agree in the stalk Fy . Hence there exists an element Vijy ∈ B, y ∈ Vijy such that σi |Vijy = σj |Vijy . Thus the sheaf condition (∗∗) of Definition 6.30.2 applies to the system of σi and we obtain a section s ∈ F(U ) with the desired property.  Let X be a topological space. Let B be a basis for the topology on X. There is a natural restriction functor from the category of sheaves of sets on X to the category of sheaves of sets on B. It turns out that this is an equivalence of categories. In down to earth terms this means the following. Lemma 6.30.6. Let X be a topological space. Let B be a basis for the topology on X. Let F be a sheaf of sets on B. There exists a unique sheaf of sets F ext on X such that F ext (U ) = F(U ) for all U ∈ B compatibly with the restriction mappings. Proof. We first construct a presheaf F ext with the desired property. Namely, for an arbitrary open U ⊂ X we define F ext (U ) as the set of elements (sx )x∈U such that (∗) of Lemma 6.30.5 holds. It is clear that there are restriction mappings that turn F ext into a presheaf of sets. Also, by Lemma 6.30.5 we see that F(U ) = F ext (U ) whenever U is an element of the basis B. To see F ext is a sheaf one may argue as in the proof of Lemma 6.17.1.  Note that we have Fx = Fxext in the situation of the lemma. This is so because the collection of elements of B containing x forms a fundamental system of open neighbourhoods of x. Lemma 6.30.7. Let X be a topological space. Let B be a basis for the topology on X. Denote Sh(B) the category of sheaves on B. There is an equivalence of categories Sh(X) −→ Sh(B) which assigns to a sheaf on X its restriction to the members of B. Proof. The inverse functor in given in Lemma 6.30.6 above. Checking the obvious functorialities is left to the reader.  This ends the discussion of sheaves of sets on a basis B. Let (C, F ) be a type of algebraic structure. At the end of this section we would like to point out that the constructions above work for sheaves with values in C. Let us briefly define the relevant notions.

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Definition 6.30.8. Let X be a topological space. Let B be a basis for the topology on X. Let (C, F ) be a type of algebraic structure. (1) A presheaf F with values in C on B is a rule which assigns to each U ∈ B an object F(U ) of C and to each inclusion V ⊂ U of elements of B a morphism ρU V : F(U ) → F(V ) in C such that whenever W ⊂ V ⊂ U in B V U we have ρU W = ρW ◦ ρV . (2) A morphism ϕ : F → G of presheaves with values in C on B is a rule which assigns to each element U ∈ B a morphism of algebraic structures ϕ : F(U ) → G(U ) compatible with restriction maps. (3) Given a presheaf F with values in C on B we say that U 7→ F (F(U )) is the underlying presheaf of sets. (4) A sheaf F with values in C on B is a presheaf with values in C on B whose underlying presheaf of sets is a sheaf. At this point we can define the stalk at x ∈ X of a presheaf with values in C on B as the directed colimit Fx = colimU ∈B,x∈U F(U ). It exists as an object of C because of our assumptions on C. Also, we see that the underlying set of Fx is the stalk of the underlying presheaf of sets on B. Note that Lemmas 6.30.3, 6.30.4 and 6.30.5 refer to the sheaf property which we have defined in terms of the associated presheaf of sets. Hence they generalize without change to the notion of a presheaf with values in C. The analogue of Lemma 6.30.6 need some care. Here it is. Lemma 6.30.9. Let X be a topological space. Let (C, F ) be a type of algebraic structure. Let B be a basis for the topology on X. Let F be a sheaf with values in C on B. There exists a unique sheaf F ext with values in C on X such that F ext (U ) = F(U ) for all U ∈ B compatibly with the restriction mappings. Proof. By the conditions imposed on the pair (C, F ) it suffices to come up with a presheaf F ext which does the correct thing on the level of underlying presheaves of sets. Thus our first task is to construct a suitable object F ext (U ) for all open U ⊂ X. We could do this by imitating Lemma 6.18.1 in the setting of presheaves on B. However, a slightly different method (but basically equivalent) is the following: Define it as the directed colimit F ext (U ) := colimU F IB(U) S over all coverings U : U = i∈I Ui by Ui ∈ B of the fibre product /Q F IB(U) x∈U Fx  F(U i) i∈I

Q

/Q

i∈I

Q

x∈Ui

Fx

By the usual arguments, see Lemma 6.15.4 and Example 6.15.5 it suffices to show that this construction on underlying sets is the same as the definition using (∗∗) above. Details left to the reader.  Note that we have Fx = Fxext

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as objects in C in the situation of the lemma. This is so because the collection of elements of B containing x forms a fundamental system of open neighbourhoods of x. Lemma 6.30.10. Let X be a topological space. Let B be a basis for the topology on X. Let (C, F ) be a type of algebraic structure. Denote Sh(B, C) the category of sheaves with values in C on B. There is an equivalence of categories Sh(X, C) −→ Sh(B, C) which assigns to a sheaf on X its restriction to the members of B. Proof. The inverse functor in given in Lemma 6.30.9 above. Checking the obvious functorialities is left to the reader.  Finally we come to the case of (pre)sheaves of modules on a basis. We will use the easy fact that the category of presheaves of sets on a basis has products and that they are described by taking products of values on elements of the bases. Definition 6.30.11. Let X be a topological space. Let B be a basis for the topology on X. Let O be a presheaf of rings on B. (1) A presheaf of O-modules F on B is a presheaf of abelian groups on B together with a morphism of presheaves of sets O × F → F such that for all U ∈ B the map O(U ) × F(U ) → F(U ) turns the group F(U ) into an O(U )-module. (2) A morphism ϕ : F → G of presheaves of O-modules on B is a morphism of abelian presheaves on B which induces an O(U )-module homomorphism F(U ) → G(U ) for every U ∈ B. (3) Suppose that O is a sheaf of rings on B. A sheaf F of O-modules on B is a presheaf of O-modules on B whose underlying presheaf of abelain groups is a sheaf. We can define the stalk at x ∈ X of a presheaf of O-modules on B as the directed colimit Fx = colimU ∈B,x∈U F(U ). It is a Ox -module. Note that Lemmas 6.30.3, 6.30.4 and 6.30.5 refer to the sheaf property which we have defined in terms of the associated presheaf of sets. Hence they generalize without change to the notion of a presheaf of O-modules. The analogue of Lemma 6.30.6 is as follows. Lemma 6.30.12. Let X be a topological space. Let O be a sheaf of rings on B. Let B be a basis for the topology on X. Let F be a sheaf with values in C on B. Let Oext be the sheaf of rings on X extending O and let F ext be the abelian sheaf on X extending F, see Lemma 6.30.9. There exists a canonical map Oext × F ext −→ F ext which agrees with the given map over elements of B and which endows F ext with the structure of an Oext -module.

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Proof. It suffices to construct the multiplication map on the level of presheaves of sets. Perhaps the easiest way to see this is to prove directly that if (fx )x∈U , fx ∈ Ox and (mx )x∈U , mx ∈ Fx satisfy (∗), then the element (fx mx )x∈U also satisfies (∗). Then we get the desired result, because in the proof of Lemma 6.30.6 we construct the extension in terms of families of elements of stalks satisfying (∗).  Note that we have Fx = Fxext as Ox -modules in the situation of the lemma. This is so because the collection of elements of B containing x forms a fundamental system of open neighbourhoods of x, or simply because it is true on the underlying sets. Lemma 6.30.13. Let X be a topological space. Let B be a basis for the topology on X. Let O be a sheaf of rings on X. Denote Mod(O|B ) the category of sheaves of O|B -modules on B. There is an equivalence of categories Mod(O) −→ Mod(O|B ) which assigns to a sheaf of O-modules on X its restriction to the members of B. Proof. The inverse functor in given in Lemma 6.30.12 above. Checking the obvious functorialities is left to the reader.  Finally, we address the question of the relationship of this with continuous maps. This is now very easy thanks to the work above. First we do the case where there is a basis on the target given. Lemma 6.30.14. Let f : X → Y be a continuous map of topological spaces. Let (C, F ) be a type of algebraic structures. Let F be a sheaf with values in C on X. Let G be a sheaf with values in C on Y . Let B be a basis for the topology on Y . Suppose given for every V ∈ B a morphism ϕV : G(V ) −→ F(f −1 V ) of C compatible with restriction mappings. Then there is a unique f -map (see Definition 6.21.7 and discussion of f -maps in Section 6.23) ϕ : G → F recovering ϕV for V ∈ B. Proof. This is trivial because the collection of maps amounts to a morphism between the restrictions of G and f∗ F to B. By Lemma 6.30.10 this is the same as giving a morphism from G to f∗ F, which by Lemma 6.21.8 is the same as an f -map. See also Lemma 6.23.1 and the discussion preceding it for how to deal with the case of sheaves of algebraic structures.  Here is the analogue for ringed spaces. Lemma 6.30.15. Let (f, f ] ) : (X, OX ) → (Y, OY ) be a morphism of ringed spaces. Let F be a sheaf of OX -modules. Let G be a sheaf of OY -modules. Let B be a basis for the topology on Y . Suppose given for every V ∈ B a OY (V )-module map ϕV : G(V ) −→ F(f −1 V ) (where F(f −1 V ) has a module structure using fV] : OY (V ) → OX (f −1 V )) compatible with restriction mappings. Then there is a unique f -map (see discussion of f -maps in Section 6.26) ϕ : G → F recovering ϕV for V ∈ B.

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Proof. Same as the proof of the corresponding lemma for sheaves of algebraic structures above.  Lemma 6.30.16. Let f : X → Y be a continuous map of topological spaces. Let (C, F ) be a type of algebraic structures. Let F be a sheaf with values in C on X. Let G be a sheaf with values in C on Y . Let BY be a basis for the topology on Y . Let BX be a basis for the topology on X. Suppose given for every V ∈ BY , and U ∈ BX such that f (U ) ⊂ V a morphism ϕU V : G(V ) −→ F(U ) of C compatible with restriction mappings. Then there is a unique f -map (see Definition 6.21.7 and the discussion of f -maps in Section 6.23) ϕ : G → F recovering ϕU V as the composition ϕV

restr.

G(V ) −−→ F(f −1 (V )) −−−→ F(U ) for every pair (U, V ) as above. Proof. Let us first proves this for sheaves of sets. Fix V ⊂ Y open. Pick s ∈ G(V ). We are going to construct an element ϕV (s) ∈ F(f −1 V ). We can define a value ϕ(s)x in the stalk Fx for every x ∈ f −1 V by picking a U ∈ BX with x ∈ U ⊂ f −1 V and setting ϕ(s)x equal to the equivalence class of (U, ϕU V (s)) in the stalk. Clearly, the family (ϕ(s)x )x∈f −1 V satisfies condition (∗) because the maps ϕU V for varying U are compatible with restrictions in the sheaf F. Thus, by the proof of Lemma 6.30.6 we see that (ϕ(s)x )x∈f −1 V corresponds to a unique element ϕV (s) of F(f −1 V ). Thus we have defined a set map ϕV : G(V ) → F(f −1 V ). The compatibility between ϕV and ϕU V follows from Lemma 6.30.5. We leave it to the reader to show that the construction of ϕV is compatible with restriction mappings as we vary v ∈ BY . Thus we may apply Lemma 6.30.14 above to “glue” them to the desired f -map. Finally, we note that the map of sheaves of sets so constructed satisfies the property that the map on stalks Gf (x) −→ Fx is the colimit of the system of maps ϕU V as V ∈ BY varies over those elements that contain f (x) and U ∈ BX varies over those elements that contain x. In particular, if G and F are the underlying sheaves of sets of sheaves of algebraic structures, then we see that the maps on stalks is a morphism of algebraic structures. Hence we conclude that the associated map of sheaves of underlying sets f −1 G → F satisfies the assumptions of Lemma 6.23.1. We conclude that f −1 G → F is a morphism of sheaves with values in C. And by adjointness this means that ϕ is an f -map of sheaves of algebraic structures.  Lemma 6.30.17. Let (f, f ] ) : (X, OX ) → (Y, OY ) be a morphism of ringed spaces. Let F be a sheaf of OX -modules. Let G be a sheaf of OY -modules. Let BY be a basis for the topology on Y . Let BX be a basis for the topology on X. Suppose given for every V ∈ BY , and U ∈ BX such that f (U ) ⊂ V a OY (V )-module map ϕU V : G(V ) −→ F(U ) compatible with restriction mappings. Here the OY (V )-module structure on F(U ) comes from the OX (U )-module structure via the map fV] : OY (V ) → OX (f −1 V ) →

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OX (U ). Then there is a unique f -map of sheaves of modules (see Definition 6.21.7 and the discussion of f -maps in Section 6.26) ϕ : G → F recovering ϕU V as the composition ϕV

restrc.

G(V ) −−→ F(f −1 (V )) −−−−→ F(U ) for every pair (U, V ) as above. Proof. Similar to the above and omitted.



6.31. Open immersions and (pre)sheaves Let X be a topological space. Let j : U → X be the inclusion of an open subset U into X. In Section 6.21 we have defined functors j∗ and j −1 such that j∗ is right adjoint to j −1 . It turns out that for an open immersion there is a left adjoint for j −1 , which we will denote j! . First we point out that j −1 has a particularly simple description in the case of an open immersion. Lemma 6.31.1. Let X be a topological space. Let j : U → X be the inclusion of an open subset U into X. (1) Let G be a presheaf of sets on X. The presheaf jp G (see Section 6.21) is given by the rule V 7→ G(V ) for V ⊂ U open. (2) Let G be a sheaf of sets on X. The sheaf j −1 G is given by the rule V 7→ G(V ) for V ⊂ U open. (3) For any point u ∈ U and any sheaf G on X we have a canonical identification of stalks j −1 Gu = (G|U )u = Gu . (4) On the category of presheaves of U we have jp j∗ = id. (5) On the category of sheaves of U we have j −1 j∗ = id. The same description holds for (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures, and (pre)sheaves of modules. Proof. The colimit in the definition of jp G(V ) is over collection of all W ⊂ X open such that V ⊂ W ordered by reverse inclusion. Hence this has a largest element, namely V . This proves (1). And (2) follows because the assignment V 7→ G(V ) for V ⊂ U open is clearly a sheaf if G is a sheaf. Assertion (3) follows from (2) since the collection of open neighbourhoods of u which are contained in U is cofinal in the collection of all open neighbourhoods of u in X. Parts (4) and (5) follow by computing j −1 j∗ F(V ) = j∗ F(V ) = F(V ). The exact same arguments work for (pre)sheaves of abelian groups and (pre)sheaves of algebraic structures.  Definition 6.31.2. Let X be a topological space. Let j : U → X be the inclusion of an open subset. (1) Let G be a presheaf of sets, abelian groups or algebraic structures on X. The presheaf jp G described in Lemma 6.31.1 is called the restriction of G to U and denoted G|U . (2) Let G be a sheaf of sets on X, abelian groups or algebraic structures on X. The sheaf j −1 G is called the restriction of G to U and denoted G|U . (3) If (X, O) is a ringed space, then the pair (U, O|U ) is called the open subspace of (X, O) associated to U .

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(4) If G is a presheaf of O-modules then G|U together with the multiplication map O|U × G|U → G|U (see Lemma 6.24.6) is called the restriction of G to U . We leave a definition of the restriction of presheaves of modules to the reader. Ok, so in this section we will discuss a left adjoint to the restriction functor. Here is the definition in the case of (pre)sheaves of sets. Definition 6.31.3. Let X be a topological space. Let j : U → X be the inclusion of an open subset. (1) Let F be a presheaf of sets on U . We define the extension of F by the empty set jp! F to be the presheaf of sets on X defined by the rule  ∅ if V 6⊂ U jp! F(V ) = F(V ) if V ⊂ U with obvious restriction mappings. (2) Let F be a sheaf of sets on U . We define the extension of F by the empty set j! F to be the sheafification of the presheaf jp! F. Lemma 6.31.4. Let X be a topological space. Let j : U → X be the inclusion of an open subset. (1) The functor jp! is a left adjoint to the restriction functor jp (see Lemma 6.31.1). (2) The functor j! is a left adjoint to restriction, in a formula MorSh(X) (j! F, G) = MorSh(U ) (F, j −1 G) = MorSh(U ) (F, G|U ) bifunctorially in F and G. (3) Let F be a sheaf of sets on U . The stalks of the sheaf j! F are described as follows  ∅ if x 6∈ U j! Fx = Fx if x ∈ U (4) On the category of presheaves of U we have jp jp! = id. (5) On the category of sheaves of U we have j −1 j! = id. Proof. To map jp! F into G it is enough to map F(V ) → G(V ) whenever V ⊂ U compatibly with restriction mappings. And by Lemma 6.31.1 the same description holds for maps F → G|U . The adjointness of j! and restriction follows from this and the properties of sheafification. The identification of stalks is obvious from the definition of the extension by the empty set and the definition of a stalk. Statements (4) and (5) follow by computing the value of the sheaf on any open of U .  Note that if F is a sheaf of abelian groups on U , then in general j! F as defined above, is not a sheaf of abelian groups, for example because some of its stalks are empty (hence not abelian groups for sure). Thus we need to modify the definition of j! depending on the type of sheafs we consider. The reason for choosing the empty set in the definition of the extension by the empty set, is that it is the initial object in the category of sets. Thus in the case of abelian groups we use 0 (and more generally for sheaves with values in any abelian category). Definition 6.31.5. Let X be a topological space. Let j : U → X be the inclusion of an open subset.

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(1) Let F be an abelian presheaf on U . We define the extension jp! F of F by 0 to be the abelian presheaf on X defined by the rule  0 if V 6⊂ U jp! F(V ) = F(V ) if V ⊂ U with obvious restriction mappings. (2) Let F be an abelian sheaf on U . We define the extension j! F of F by 0 to be the sheafification of the abelian presheaf jp! F. (3) Let C be a category having an initial object e. Let F be a presheaf on U with values in C. We define the extension jp! F of F by e to be the presheaf on X with values in C defined by the rule  e if V 6⊂ U jp! F(V ) = F(V ) if V ⊂ U with obvious restriction mappings. (4) Let (C, F ) be a type of algebraic structure such that C has an initial object e. Let F be a sheaf of algebraic structures on U (of the give type). We define the extension j! F of F by e to be the sheafification of the presheaf jp! F defined above. (5) Let O be a presheaf of rings on X. Let F be a presheaf of O|U -modules. In this case we define the extension by 0 to be the presheaf of O-modules which is equal to jp! F as an abelian presheaf endowed with the multiplication map O × jp! F → jp! F. (6) Let O be a sheaf of rings on X. Let F be a sheaf of O|U -modules. In this case we define the extension by 0 to be the O-module which is equal to j! F as an abelian sheaf endowed with the multiplication map O × j! F → j! F. It is true that one can define j! in the setting of sheaves of algebraic structures (see below). However, it depends on the type of algebraic structures involved what the resulting object is. For example, if O is a sheaf of rings on U , then j!,rings O 6= j!,abelian O since the initial object in the category of rings is Z and the initial object in the category of abelian groups is 0. In particular the functor j! does not commute with taking underlying sheaves of sets, in contrast to what we have seen sofar! We separate out the case of (pre)sheaves of abelian groups, (pre)sheaves of algebraic structures and (pre)sheaves of modules as usual. Lemma 6.31.6. Let X be a topological space. Let j : U → X be the inclusion of an open subset. Consider the functors of restriction and extension by 0 for abelian (pre)sheaves. (1) The functor jp! is a left adjoint to the restriction functor jp (see Lemma 6.31.1). (2) The functor j! is a left adjoint to restriction, in a formula MorAb(X) (j! F, G) = MorAb(U ) (F, j −1 G) = MorAb(U ) (F, G|U ) bifunctorially in F and G. (3) Let F be an abelian sheaf on U . The stalks of the sheaf j! F are described as follows  0 if x 6∈ U j! Fx = Fx if x ∈ U (4) On the category of abelian presheaves of U we have jp jp! = id.

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(5) On the category of abelian sheaves of U we have j −1 j! = id. Proof. Omitted.



Lemma 6.31.7. Let X be a topological space. Let j : U → X be the inclusion of an open subset. Let (C, F ) be a type of algebraic structure such that C has an initial object e. Consider the functors of restriction and extension by e for (pre)sheaves of algebraic structure defined above. (1) The functor jp! is a left adjoint to the restriction functor jp (see Lemma 6.31.1). (2) The functor j! is a left adjoint to restriction, in a formula MorSh(X,C) (j! F, G) = MorSh(U,C) (F, j −1 G) = MorSh(U,C) (F, G|U ) bifunctorially in F and G. (3) Let F be a sheaf on U . The stalks of the sheaf j! F are described as follows  e if x 6∈ U j! Fx = Fx if x ∈ U (4) On the category of presheaves of algebraic structures on U we have jp jp! = id. (5) On the category of sheaves of algebraic structures on U we have j −1 j! = id. Proof. Omitted.



Lemma 6.31.8. Let (X, O) be a ringed space. Let j : (U, O|U ) → (X, O) be an open subspace. Consider the functors of restriction and extension by 0 for (pre)sheaves of modules defined above. (1) The functor jp! is a left adjoint to restriction, in a formula MorPMod(O) (jp! F, G) = MorPMod(O|U ) (F, G|U ) bifunctorially in F and G. (2) The functor j! is a left adjoint to restriction, in a formula MorMod(O) (j! F, G) = MorMod(O|U ) (F, G|U ) bifunctorially in F and G. (3) Let F be a sheaf of O-modules on U . The stalks of the sheaf j! F are described as follows  0 if x 6∈ U j! Fx = Fx if x ∈ U (4) On the category of sheaves of O|U -modules on U we have j −1 j! = id. Proof. Omitted.



Note that by the lemmas above, both the functors j∗ and j! are fully faithful embeddings of the category of sheaves on U into the category of sheaves on X. It is only true for the functor j! that one can easily describe the essential image of this functor. Lemma 6.31.9. Let X be a topological space. Let j : U → X be the inclusion of an open subset. The functor j! : Sh(U ) −→ Sh(X)

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is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = ∅ for all x ∈ X \ U . Proof. Fully faithfullness follows formally from j −1 j! = id. We have seen that any sheaf in the image of the functor has the property on the stalks mentioned in the lemma. Conversely, suppose that G has the indicated property. Then it is easy to check that j! j −1 G → G is an isomorphism on all stalks and hence an isomorphism.



Lemma 6.31.10. Let X be a topological space. Let j : U → X be the inclusion of an open subset. The functor j! : Ab(U ) −→ Ab(X) is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = 0 for all x ∈ X \ U . Proof. Omitted.



Lemma 6.31.11. Let X be a topological space. Let j : U → X be the inclusion of an open subset. Let (C, F ) be a type of algebraic structure such that C has an initial object e. The functor j! : Sh(U, C) −→ Sh(X, C) is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = e for all x ∈ X \ U . Proof. Omitted.



Lemma 6.31.12. Let (X, O) be a ringed space. Let j : (U, O|U ) → (X, O) be an open subspace. The functor j! : Mod(O|U ) −→ Mod(O) is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = 0 for all x ∈ X \ U . Proof. Omitted.



Remark 6.31.13. Let j : U → X be an open immersion of topological spaces as above. Let x ∈ X, x 6∈ U . Let F be a sheaf of sets on U . Then Fx = ∅ by Lemma 6.31.4. Hence j! does not transform a final object of Sh(U ) into a final object of Sh(X) unless U = X. According to our conventions in Categories, Section 4.21 this means that the functor j! is not left exact as a functor between the categories of sheaves of sets. It will be shown later that j! on abelian sheaves is exact, see Modules, Lemma 15.3.5. 6.32. Closed immersions and (pre)sheaves Let X be a topological space. Let i : Z → X be the inclusion of a closed subset Z into X. In Section 6.21 we have defined functors i∗ and i−1 such that i∗ is right adjoint to i−1 .

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Lemma 6.32.1. Let X be a topological space. Let i : Z → X be the inclusion of a closed subset Z into X. Let F be a sheaf of sets on Z. The stalks of i∗ F are described as follows  {∗} if x 6∈ Z i∗ Fx = Fx if x ∈ Z where {∗} denotes a singleton set. Moreover, i−1 i∗ = id on the category of sheaves of sets on Z. Moreover, the same holds for abelian sheaves on Z, resp. sheaves of algebraic structures on Z where {∗} has to be replaced by 0, resp. a final object of the category of algebraic structures. Proof. If x 6∈ Z, then there exist arbitrarily small open neighbourhoods U of x which do not meet Z. Because F is a sheaf we have F(i−1 (U )) = {∗} for any such U , see Remark 6.7.2. This proves the first case. The second case comes from the fact that for z ∈ Z any open neighbourhood of z is of the form Z ∩ U for some open U of X. For the statement that i−1 i∗ = id consider the canonical map i−1 i∗ F → F. This is an isomorphism on stalks (see above) and hence an isomorphism. For sheaves of abelian groups, and sheaves of algebraic structures you argue in the same manner.  Lemma 6.32.2. Let X be a topological space. Let i : Z → X be the inclusion of a closed subset. The functor i∗ : Sh(Z) −→ Sh(X) is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = {∗} for all x ∈ X \ Z. Proof. Fully faithfullness follows formally from i−1 i∗ = id. We have seen that any sheaf in the image of the functor has the property on the stalks mentioned in the lemma. Conversely, suppose that G has the indicated property. Then it is easy to check that G → i∗ i−1 G is an isomorphism on all stalks and hence an isomorphism.



Lemma 6.32.3. Let X be a topological space. Let i : Z → X be the inclusion of a closed subset. The functor i∗ : Ab(Z) −→ Ab(X) is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = 0 for all x ∈ X \ Z. Proof. Omitted.



Lemma 6.32.4. Let X be a topological space. Let i : Z → X be the inclusion of a closed subset. Let (C, F ) be a type of algebraic structure with final object 0. The functor i∗ : Sh(Z, C) −→ Sh(X, C) is fully faithful. Its essential image consists exactly of those sheaves G such that Gx = 0 for all x ∈ X \ Z. Proof. Omitted.



6.33. GLUEING SHEAVES

203

Remark 6.32.5. Let i : Z → X be a closed immersion of topological spaces as above. Let x ∈ X, x 6∈ Z. Let F be a sheaf of sets on Z. Then (i∗ F)x = {∗} by Lemma 6.32.1. Hence if F = ∗q∗, where ∗ is the singleton sheaf, then i∗ Fx = {∗} = 6 i∗ (∗)x q i∗ (∗)x because the latter is a two point set. According to our conventions in Categories, Section 4.21 this means that the functor i∗ is not right exact as a functor between the categories of sheaves of sets. In particular, it cannot have a right adjoint, see Categories, Lemma 4.22.4. On the other hand, we will see later (see Modules, Lemma 15.6.3) that i∗ on abelian sheaves is exact, and does have a right adjoint, namely the functor that associates to an abelian sheaf on X the sheaf of sections supported in Z. Remark 6.32.6. We have not discussed the relationship between closed immersions and ringed spaces. This is because the notion of a closed immersion of ringed spaces is best discussed in the setting of quasi-coherent sheaves, see Modules, Section 15.13. 6.33. Glueing sheaves In this section we glue sheaves defined on the members of a covering of X. We first deal with maps. S Lemma 6.33.1. Let X be a topological space. Let X = Ui be an open covering. Let F, G be sheaves of sets on X. Given a collection ϕi : F|Ui −→ G|Ui of maps of sheaves such that for all i, j ∈ I the maps ϕi , ϕj restrict to the same map F|Ui ∩Uj → G|Ui ∩Uj then there exists a unique map of sheaves ϕ : F −→ G whose restriction to each Ui agrees with ϕi . Proof. Omitted.



The previous lemma implies that given two sheaves F, G on the topological space X the rule U 7−→ MorSh(U ) (F|U , G|U ) defines a sheaf. This is a kind of internal hom sheaf. It is seldom used in the setting of sheaves of sets, and more usually in the setting of sheaves of modules, see Modules, Section 15.19. S Let X be a topological space. Let X = i∈I Ui be an open covering. For each i ∈ I let Fi be a sheaf of sets on Ui . For each pair i, j ∈ I, let ϕij : Fi |Ui ∩Uj −→ Fj |Ui ∩Uj be an isomorphism of sheaves of sets. Assume in addition that for every triple of indices i, j, k ∈ I the following diagram is commutative Fi |Ui ∩Uj ∩Uk

/ Fk |U ∩U ∩U 7 i j k

ϕik ϕij

'

Fj |Ui ∩Uj ∩Uk

ϕjk

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We will call such a collectionS of data (Fi , ϕij ) a glueing data for sheaves of sets with respect to the covering X = Ui . S Lemma 6.33.2. Let X be a topological space. Let X = i∈I Ui be an open covering. Given any glueing data (Fi , ϕij ) for sheaves of sets with respect to the covering S X = Ui there exists a sheaf of sets F on X together with isomorphisms ϕi : F|Ui → Fi such that the diagrams F|Ui ∩Uj id

ϕi

/ Fi |Ui ∩Uj ϕij



F|Ui ∩Uj

ϕj



/ Fj |U ∩U i j

are commutative. Proof. Actually we can write a formula for the set of sections of F over an open W ⊂ X. Namely, we define F(W ) = {(si )i∈I | si ∈ Fi (W ∩ Ui ), ϕij (si |W ∩Ui ∩Uj ) = sj |W ∩Ui ∩Uj }. Restriction mappings for W 0 ⊂ W are defined by the restricting each of the si to W 0 ∩ Ui . The sheaf condition for F follows immediately from the sheaf condition for each of the Fi . We still have to prove that F|Ui maps isomorphically to Fi . Let W ⊂ Ui . In this case the condition in the definition of F(W ) implies that sj = ϕij (si |W ∩Uj ). And the commutativity of the diagrams in the definition of a glueing data assures that we may start with any section s ∈ Fi (W ) and obtain a compatible collection by setting si = s and sj = ϕij (si |W ∩Uj ). Thus the lemma follows.  S Lemma 6.33.3. Let X be a topological space. Let X = Ui be an open covering. Let (Fi , ϕij ) be a glueing data of sheaves of abelian groups, resp. sheaves of algebraic structures, resp. sheaves of O-modules for some sheaf of rings O on X. Then the construction in the proof of Lemma 6.33.2 above leads to a sheaf of abelian groups, resp. sheaf of algebraic structures, resp. sheaf of O-modules. Proof. This is true because in the construction the set of sections F(W ) over an open W is given as the equalizer of the maps Q /Q / i,j∈I Fi (W ∩ Ui ∩ Uj ) i∈I Fi (W ∩ Ui ) And in each of the cases envisioned this equalizer gives an object in the relevant category whose underlying set is the object considered in the cited lemma.  S Lemma 6.33.4. Let X be a topological space. Let X = i∈I Ui be an open covering. The functor which associates to a sheaf of sets F the following collection of glueing data (F|Ui , (F|Ui )|Ui ∩Uj → (F|Uj )|Ui ∩Uj ) S with respect to the covering X = Ui defines an equivalence of categories between Sh(X) and the category of glueing data. A similar statement holds for abelian sheaves, resp. sheaves of algebraic structures, resp. sheaves of O-modules. Proof. The functor is fully faithful by Lemma 6.33.1 and essentially surjective (via an explicitly given quasi-inverse functor) by Lemma 6.33.2. 

6.34. OTHER CHAPTERS

205

This lemma means that if the sheaf F was constructed from the glueing data (Fi , ϕij ) and if G is a sheaf on X, then a morphism f : F → G is given by a collection of morphisms of sheaves fi : Fi −→ G|Ui compatible with the glueing maps ϕij . Similarly, to give a morphism of sheaves g : G → F is the same as giving a collection of morphisms of sheaves gi : G|Ui −→ Fi compatible with the glueing maps ϕij . 6.34. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes

(37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70)

´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style

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(71) Obsolete (72) GNU Free Documentation License

(73) Auto Generated Index

CHAPTER 7

Commutative Algebra 7.1. Introduction Basic commutative algebra will be explained in this document. A reference is [Mat70]. 7.2. Conventions A ring is commutative with 1. The zero ring is a ring. In fact it is the only ring that does not have a prime ideal. The Kronecker symbol δij will be used. If R → S is a ring map and q a prime of S, then we use the notation “p = R ∩ q” to indicate the prime which is the inverse image of q under R → S even if R is not a subring of S and even if R → S is not injective. 7.3. Basic notions The following notions are considered basic and will not be defined, and or proved. This does not mean they are all necessarily easy or well known. (1) R is a ring, (2) x ∈ R is nilpotent, (3) x ∈ R is a zerodivisor, (4) x ∈ R is a unit, (5) e ∈ R is an idempotent, (6) an idempotent e ∈ R is called trivial if e = 1 or e = 0, (7) ϕ : R1 → R2 is a ring homomorphism, (8) ϕ : R1 → R2 is of finite presentation, or R2 is a finitely presented R1 algebra, see Definition 7.6.1, (9) ϕ : R1 → R2 is of finite type, or R2 is a finitely type R1 -algebra, see Definition 7.6.1, (10) ϕ : R1 → R2 is finite, or R2 is a finite R1 -algebra, (11) R is a (integral) domain, (12) R is reduced, (13) R is Noetherian, (14) R is a principal ideal domain or a PID, (15) R is a Euclidean domain, (16) R is a unique factorization domain or a UFD, (17) R is a discrete valuation ring or a dvr, (18) K is a field, (19) K ⊂ L is a field extension, (20) K ⊂ L is an algebraic field extension, (21) {ti }i∈I is a transcendence basis for L over K, (22) the transcendence degree trdeg(L/K) of L over K, 207

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(23) the field k is algebraically closed, (24) if K ⊂ L is algebraic, and K → k a field map, then there exists a map L → k extending the map on K, (25) I ⊂ R is an ideal, (26) I ⊂ R is radical, √ (27) if I is an ideal then we have its radical I, (28) I ⊂ R is nilpotent means that I n = 0 for some n ∈ N, (29) I ⊂ R is locally nilpotent means that every element of I is nilpotent, (30) p ⊂ R is a prime ideal, (31) if p ⊂ R is prime and if I, J ⊂ R are ideal, and if IJ ⊂ p, then I ⊂ p or J ⊂ p. (32) m ⊂ R is a maximal ideal, (33) any nonzero ring has a maximal ideal, T (34) the Jacobson radical of R is rad(R) = m⊂R m the intersection of all the maximal ideals of R, (35) the ideal (T ) generated by a subset T ⊂ R, (36) the quotient ring R/I, (37) an ideal I in the ring R is prime if and only if R/I is a domain, (38) an ideal I in the ring R is maximal if and only if the ring R/I is a field, (39) if ϕ : R1 → R2 is a ring homomorphism, and if I ⊂ R2 is an ideal, then ϕ−1 (I) is an ideal of R1 , (40) if ϕ : R1 → R2 is a ring homomorphism, and if I ⊂ R1 is an ideal, then ϕ(I) · R2 (sometimes denoted I · R2 , or IR2 ) is the ideal of R2 generated by ϕ(I), (41) if ϕ : R1 → R2 is a ring homomorphism, and if p ⊂ R2 is a prime ideal, then ϕ−1 (p) is a prime ideal of R1 , (42) M is an R-module, (43) for m ∈ M the annihilator I = {f ∈ R | f m = 0} of m in R, (44) N ⊂ M is an R-submodule, (45) M is an Noetherian R-module, (46) M is a finite R-module, (47) M is a finitely generated R-module, (48) M is a finitely presented R-module, (49) M is a free R-module, (50) if 0 → K → L → M → 0 is a short exact sequence of R-modules and K, M are free, then L is free, (51) if N ⊂ M ⊂ L are R-modules, then L/M = (L/N )/(M/N ), (52) S is a multiplicative subset of R, (53) the localization R → S −1 R of R, (54) if R is a ring and S is a multiplicative subset of R then S −1 R is the zero ring if and only if S contains 0, (55) if R is a ring and if the multiplicative subset S consists completely of nonzerodivisors, then R → S −1 R is injective, (56) if ϕ : R1 → R2 is a ring homomorphism, and S is a multiplicative subsets of R1 , then ϕ(S) is a multiplicative subset of R2 , (57) if S, S 0 are multiplicative subsets of R, and if SS 0 denotes the set of products SS 0 = {r ∈ R | ∃s ∈ S, ∃s0 ∈ S 0 , r = ss0 } then SS 0 is a multiplicative subset of R,

7.4. SNAKE LEMMA

209

(58) if S, S 0 are multiplicative subsets of R, and if S denotes the image of S −1 in (S 0 )−1 R, then (SS 0 )−1 R = S ((S 0 )−1 R), (59) the localization S −1 M of the R-module M , (60) the functor M 7→ S −1 M preserves injective maps, surjective maps, and exactness, (61) if S, S 0 are multiplicative subsets of R, and if M is an R-module, then (SS 0 )−1 M = S −1 ((S 0 )−1 M ), (62) if R is a ring, I and ideal of R and S a multiplicative subset of R, then −1 S −1 I is an ideal of S −1 R, and we have S −1 R/S −1 I = S (R/I), where S is the image of S in R/I, (63) if R is a ring, and S a multiplicative subset of R, then any ideal I 0 of S −1 R is of the form S −1 I, where one can take I to be the inverse image of I 0 in R, (64) if R is a ring, M an R-module, and S a multiplicative subset of R, then any submodule N 0 of S −1 M is of the form S −1 N for some submodule N ⊂ M , where one can take N to be the inverse image of N 0 in M , (65) if S = {1, f, f 2 , . . .} then Rf = S −1 R and Mf = S −1 M , (66) if S = R \ p = {x ∈ R | x 6∈ p} for some prime ideal p, then it is customary to denote Rp = S −1 R and Mp = S −1 M , (67) a local ring is a ring with exactly one maximal ideal, (68) a semi-local ring is a ring with finitely many maximal ideals, (69) if p is a prime in R, then Rp is a local ring with maximal ideal pRp , (70) the residue field, denoted κ(p), of the prime p in the ring R is the quotient Rp /pRp = (R \ p)−1 R/p, (71) given R and M1 , M2 the tensor product M1 ⊗R M2 , (72) etc. 7.4. Snake lemma The snake lemma and its variants are discussed in the setting of abelian categories in Homology, Section 10.3. Lemma 7.4.1. Suppose given a commutative diagram X

0

 /U

α

/Y  /V

/Z β

/0

γ

 /W

of abelian groups with exact rows, then there is a canonical exact sequence Ker(α) → Ker(β) → Ker(γ) → Coker(α) → Coker(β) → Coker(γ) Moreover, if X → Y is injective, then the first map is injective, and if V → W is surjective, then the last map is surjective. Proof. The map ∂ : Ker(γ) → Coker(α) is defined as follows. Take z ∈ Ker(γ). Choose y ∈ Y mapping to z. Then β(y) ∈ V maps to zero in W . Hence β(y) is the image of some u ∈ U . Set ∂z = u the class of u in the cokernel of α. Proof of exactness is omitted. 

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7.5. Finite modules and finitely presented modules Just some basic notation and lemmas. Definition 7.5.1. Let R be a ring. Let M be an R-module (1) We say M is a finite R-module, or a finitely generated R-module if there exist n ∈ N and x1 , . . . , xn ∈ M such that every element of M is a R-linear combination of the xi . Equivalently, this means there exists a surjection R⊕n → M for some n ∈ N. (2) We say M is a finitely presented R-module or an R-module of finite presentation if there exist integers n, m ∈ N and an exact sequence R⊕m −→ R⊕n −→ M −→ 0 Informally this means that M is finitely generated and that the module of relations among these generators is finitely generated as well. A choice of an exact sequence as in the definition is called a presentation of M . Lemma 7.5.2. Let R be a ring. Let α : R⊕n → M and β : N → M be module maps. If Im(α) ⊂ Im(β), then there exists an R-module map γ : R⊕n → N such that α = β ◦ γ. Proof. Let ei = (0, . . . , 0, 1, 0, . . . , 0) be the ith basis vector of R⊕n . Let xi ∈ N be an element with α(ei ) = β(xi ) which exists by assumption. Set γ(a1 , . . . , an ) = P ai xi . By construction α = β ◦ γ.  Lemma 7.5.3. Let M be an R-module of finite presentation. For any surjection α : R⊕n → M the kernel of α is a finitely generated R-module. Proof. Choose a presentation R⊕l → R⊕m → M → 0 Let K = Ker(α). By Lemma 7.5.2 there exists a map R⊕m → R⊕n such that the solid diagram / R⊕m /M /0 R⊕l id

   /K / R⊕n α / M /0 0 commutes. This produces the dotted arrow. By the snake lemma (Lemma 7.4.1) we see that we get an isomorphism Coker(R⊕l → K) ∼ = Coker(R⊕m → R⊕n ) In particular we conclude that Coker(R⊕l → K) is a finite R-module. Hence there are finitely many elements of K which together with the images of the basis vectors of R⊕l generate K, i.e., K is finitely generated.  Lemma 7.5.4. Let R be a ring. Let 0 → M1 → M2 → M3 → 0 be a short exact sequence of R-modules. (1) If M1 and M3 are finite R-modules, then M2 is a finite R-module. (2) If M1 and M3 are finitely presented R-modules, then M2 is a finitely presented R-module.

7.6. RING MAPS OF FINITE TYPE AND OF FINITE PRESENTATION

(3) If M2 (4) If M2 M3 is (5) If M3 M1 is

211

is a finite R-module, then M3 is a finite R-module. is a finitely presented R-module and M1 is a finite R-module, then a finitely presented R-module. is a finitely presented R-module and M2 is a finite R-module, then a finite R-module.

Proof. We prove part (5). Assume M3 is finitely presented and M2 finite. Let α : R⊕n → M2 be a surjection. Then we can find k1 , . . . , km ∈ R⊕n which generate the kernel of the composition R⊕n → M2 → M3 . Then α(k1 ), . . . , α(km ) generate M1 as a submodule of M2 . The proofs of the other parts are omitted.  Lemma 7.5.5. Let R be a ring, and let M be a finite R-module. There exists a filtration by R-submodules 0 = M0 ⊂ M1 ⊂ . . . ⊂ Mn = M such that each quotient Mi /Mi−1 is isomorphic to R/Ii for some ideal Ii of R. Proof. By induction on the number of generators of M . Let x1 , . . . , xr ∈ M be a minimal number of generators. Let M 0 = Rx1 ⊂ M . Then M/M 0 has r − 1 generators and the induction hypothesis applies. And clearly M 0 ∼ = R/I1 with I1 = {f ∈ R | f x1 = 0}.  Lemma 7.5.6. Let R → S be a ring map. Let M be an S-module. If M is finite as an R-module, then M is finite as an S-module. Proof. In fact, any R-generating set of M is also an S-generating set of M , since the R-module structure is induced by the image of R in S.  7.6. Ring maps of finite type and of finite presentation Definition 7.6.1. Let R → S be a ring map. (1) We say R → S is of finite type, or that S is a finite type R-algebra if there exists an n ∈ N and an surjection of R-algebras R[x1 , . . . , xn ] → S. (2) We say R → S is of finite presentation if there exist integers n, m ∈ N and polynomials f1 , . . . , fm ∈ R[x1 , . . . , xn ] and an isomorphism of R-algebras R[x1 , . . . , xn ]/(f1 , . . . , fm ) ∼ = S. Informally this means that S is finitely generated as an R-algebra and that the ideal of relations among the generators is finitely generated. A choice of a surjection R[x1 , . . . , xn ] → S as in the definition is sometimes called a presentation of S. Lemma 7.6.2. The notions finite type and finite presentation have the following permanence properties. (1) A composition of ring maps of finite type is of finite type. (2) A composition of ring maps of finite presentation is of finite presentation. (3) Given R → S 0 → S with R → S of finite type, then S 0 → S is of finite type. (4) Given R → S 0 → S, with R → S of finite presentation, and R → S 0 of finite type, then S 0 → S is of finite presentation. Proof. We only prove the last assertion. Write S = R[x1 , . . . , xn ]/(f1 , . . . , fm ) and S 0 = R[y1 , . . . , ya ]/I. Say that the class y¯i of yi maps to hi mod (f1 , . . . , fm ) in S. Then it is clear that S 0 = S[x1 , . . . , xn ]/(f1 , . . . , fm , h1 − y¯1 , . . . , hm − y¯m ). 

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Lemma 7.6.3. Let R → S be a ring map of finite presentation. For any surjection α : R[x1 , . . . , xn ] → S the kernel of α is a finitely generated ideal in R[x1 , . . . , xn ]. Proof. Write S = R[y1 , . . . , ym ]/(f1 , . . . , fk ). Choose gi ∈ R[y1 , . . . , ym ] which are lifts of α(xi ). Then we see that S = R[xi , yj ]/(fj , xi − gi ). Choose hj ∈ R[x1 , . . . , xn ] such that α(hj ) corresponds to yj mod (f1 , . . . , fk ). Consider the map ψ : R[xi , yj ] → R[xi ], xi 7→ xi , yj 7→ hj . Then the kernel of α is the image of (fj , xi − gi ) under ψ and we win.  Lemma 7.6.4. Let R → S be a ring map. Let M be an S-module. Assume R → S is of finite type and M is finitely presented as an R-module. Then M is finitely presented as an S-module. Proof. This is similar to the proof of part (4) of Lemma 7.6.2. We may assume S = R[x1 , . . . , xn ]/J.PChoose y1 , . . . , ym ∈ M which generate M as an R-module aij yj = 0, i = 1, . . . , t which generate the kernel of R⊕n → and choose relations M . For any i = 1, . . . , n and j = 1, . . . , m write X xi yj = aijk yk for some aijk P ∈ R. Consider the S-module N generated P by y1 , . . . , ym subject to the relations aij yj = 0, i = 1, . . . , t and xi yj = aijk yk , i = 1, . . . , n and j = 1, . . . , m. Then N has a presentation S ⊕nm+t −→ S ⊕m −→ M −→ 0 By construction there is a surjective map ϕ : N → M . To finish the proof we show P ϕ is injective. Suppose z = bj yj ∈ N for some bj ∈ S. We may think of bj as a polynomial P in x1 , . . . , xn with coefficients in R. By applying the relations of the form xi yj = aijkP yk we can inductively lower the degree of the polynomials. Hence we see that z = cj yj for some cj ∈ R. Hence if ϕ(z) = 0 then the vector (c1 , . . . , cm ) is an R-linear combination of the vectors (ai1 , . . . , aim ) and we conclude that z = 0 as desired.  7.7. Finite ring maps Definition 7.7.1. Let ϕ : R → S be a ring map. We say ϕ : R → S is finite if if S is finite as an R-module. Lemma 7.7.2. Let R → S be a finite ring map. Let M be an S-module. Then M is finite as an R-module if and only if M is finite as an S-module. Proof. One of the implications follows from Lemma 7.5.6. To see the other assume that M is finite as an S-module. Pick x1 , . . . , xn ∈ S which generate S as an Rmodule. Pick y1 , . . . , ym ∈ M which generate M as an S-module. Then xj yj generate M as an R-module.  Lemma 7.7.3. Suppose that R → S and S → T are finite ring maps. Then R → T is finite. Proof. If ti generate T as an S-module and sj generate S as an R-module, then ti sj generate T as an R-module. (Also follows from Lemma 7.7.2.)  Lemma 7.7.4. Let R → S be a finite and finitely presented ring map. Let M be an S-module. Then M is finitely presented as an R-module if and only if M is finitely presented as an S-module.

7.8. COLIMITS

213

Proof. One of the implications follows from Lemma 7.6.4. To see the other assume that M is finitely presented as an S-module. Pick a presentation S ⊕m −→ S ⊕n −→ M −→ 0 As S is finite as an R-module, the kernel of S ⊕n → M is a finite R-module. Thus from Lemma 7.5.4 we see that it suffices to prove that S is finitely presented as an R-module. P Pick x1 , . . . , xn ∈ S which generate S as an R-module. Write xi xj = aijk xk for some aijk ∈ R. Let J = Ker(R[X1 , . . . , Xn ] → S) where R[XP , . . . , X ] → S is the 1 n R-algebra map determined by Xi 7→ xi . Let gij = Xi Xj − aijk Xk which is an element of J. Let I = (gij ) so that I ⊂ J. By Lemma 7.6.3 there exist finitely many g1 , . . . , gN ∈ J such that J = (g1 , . . . , gN ). For every index l ∈ {1, . . . , N } we can write gl = hl mod I for some hl ∈ J which has degree ≤ 1 in X1 , . . . , Xn . (Details omitted; hint: use the gij get P rid of the monomial of highest degree in gl and use induction.) Write hl = al0 + ali Xi for some ali ∈ R. Then S has the following presentation R⊕N −→ R⊕n+1 −→ M −→ 0 as an R-module where the first arrow maps the lth basis P vector to (al0 , al1 , . . . , aln ) and the second arrow maps (a0 , a1 , . . . , an ) to a0 + ai xi .  7.8. Colimits Some of the material in this section overlaps with the general discussion on colimits in Categories, Sections 4.13 – 4.19. Definition 7.8.1. A partially ordered set is a set I together with a relation ≤ which is associative (if i ≤ j and j ≤ k then i ≤ k) and reflexive (i ≤ i for all i ∈ I). A directed set (I, ≤) is a partially ordered set (I, ≤) such that I is not empty and such that ∀i, j ∈ I, there exists k ∈ I with i ≤ k, j ≤ k. It is customary to drop the ≤ from the notation when talking about a partially ordered set. This is the same as the notion defined in Categories, Section 4.19. Definition 7.8.2. Let (I, ≤) be a partially ordered set. A system (Mi , µij ) of R-modules over I consists of a family of R-modules {Mi }i∈I indexed by I and a family of R-module maps {µij : Mi → Mj }i≤j such that for all i ≤ j ≤ k (7.8.2.1)

µii = idMi

(7.8.2.2)

µik = µjk ◦ µij

We say (Mi , µij ) is a directed system if I is a directed set. This is the same as the notion defined in Categories, Definition 4.19.1 and Section 4.19. We refer to Categories, Definition 4.13.2 for the definition of a colimit of a diagram/system in any category. Lemma 7.8.3. Let (Mi , µij ) be a system of R-modules over the partially L ordered set I. The colimit of the system (Mi , µij ) is the quotient R-module ( i∈I Mi )/Q where Q is the R-submodule generated by all elements ιi (xi ) − ιj (µij (xi ))

214

7. COMMUTATIVE ALGEBRA

L where ιi : Mi → i is the natural inclusion. We denote the colimit M = i∈I ML colimi Mi . We denote π : i∈I Mi → M the projection map and φi = π ◦ ιi : Mi → M. Proof. This lemma is a special case of Categories, Lemma 4.13.11 but we will also prove it directly in this case. Namely, note that φi = φj ◦ µij in the above construction. To show the pair (M, φi ) is the colimit we have to show it satisfies the universal property: for any other such pair (Y, ψi ) with ψi : Mi → Y , ψi = ψj ◦ µij , there is a unique R-module homomorphism g : M → Y such that the following diagram commutes: µij / Mj Mi φj

φi

ψi

M

} ψj

g

   Y And this is clear because we can define g by taking the map ψi on the summand L Mi in the direct sum Mi .  Lemma 7.8.4. Let (Mi , µij ) be a system of R-modules over the partially ordered set I. Assume that I is directed. The colimit of the system (Mi , µij ) is canonically isomorphic to the module M defined as follows: (1) as a set let a  M= Mi / ∼ i∈I

where for m ∈ Mi and m0 ∈ Mi0 we have m ∼ m0 ⇔ µij (m) = µi0 j (m0 ) for some j ≥ i, i0 (2) as an abelian group for m ∈ Mi and m0 ∈ Mi0 we define the sum of the classes of m and m0 in M to be the class of µij (m) + µi0 j (m0 ) where j ∈ I is any index with i ≤ j and i0 ≤ j, and (3) as an R-module define for m ∈ Mi and x ∈ R the product of x and the class of m in M to be the class of xm in M . The canonical maps φi : Mi → M are induced by the canonical maps Mi → ` M . i i∈I Proof. Omitted. Compare with Categories, Section 4.17.



Lemma 7.8.5. Let (Mi , µij ) be a directed system. Let M = colim Mi with µi : Mi → M , then µi (xi ) = 0 for xi ∈ Mi if and only if there exists j  i such that µij (xi ) = 0. Proof. This is clear from the description of the directed colimit in Lemma 7.8.4.  Example 7.8.6. Consider the partially ordered set I = {a, b, c} with a ≺ b and a ≺ c and no other strict inequalities. A system (Ma , Mb , Mc , µab , µac ) over I consists of three R-modules Ma , Mb , Mc and two R-module homomorphisms µab : Ma → Mb and µac : Ma → Mc . The colimit of the system is just M := colimi∈I Mi = Coker(Ma → Mb ⊕ Mc )

7.8. COLIMITS

215

where the map is µab ⊕ −µac . Thus the kernel of the canonical map Ma → M is Ker(µab ) + Ker(µac ). And the kernel of the canonical map Mb → M is the image of Ker(µac ) under the map µab . Hence clearly the result of Lemma 7.8.5 is false for general systems. Definition 7.8.7. Let (Mi , µij ), (Ni , νij ) be systems of R-modules over the same partially ordered set I. A homomorphism of systems Φ from (Mi , µij ) to (Ni , νij ) is by definition a family of R-module homomorphisms φi : Mi → Ni such that φj ◦ µij = νij ◦ φi for all i ≤ j. This is the same notion as a transformation of functors between the associated diagrams M : I → ModR and N : I → ModR , in the language of categories. The following lemma is a special case of Categories, Lemma 4.13.7. Lemma 7.8.8. Let (Mi , µij ), (Ni , νij ) be systems of R-modules over the same partially ordered set. A morphism of systems Φ = (φi ) from (Mi , µij ) to (Ni , νij ) induces a unique homomorphism colim φi : colim Mi −→ colim Ni such that

/ colim Mi

Mi

colim φi

φi

 / colim Ni

 Ni

commutes for all i ∈ I. Proof. Write M = colim Mi and N = colim Ni and φ = colim φi (as yet to be constructed). We will use the explicit description of M and N in Lemma 7.8.3 without further mention. The condition of the lemma is equivalent to the condition that L /M i∈I Mi L

φi

L 

i∈I

φ

 /N

Ni

commutes. Hence it is clear that if φ exists, then it is unique. To see that φ exists, L it suffices to show that the kernel of the upper horizontal arrow is mapped by φi to the kernel of the lower horizontal arrow. To see this, let j ≤ k and xj ∈ Mj . Then M ( φi )(xj − µjk (xj )) = φj (xj ) − φk (µjk (xj )) = φj (xj ) − νjk (φi (xj )) which is in the kernel of the lower horizontal arrow as required.



Lemma 7.8.9. Let I be a directed partially ordered set. Let (Li , λij ), (Mi , µij ), and (Ni , νij ) be systems of R-modules over I. Let ϕi : Li → Mi and ψi : Mi → Ni be morphisms of systems over I. Assume that for all i ∈ I the sequence of R-modules Li

ϕi

/ Mi

ψi

/ Ni

is a complex with homology Hi . Then the R-modules Hi form a system over I, the sequence of R-modules colimi Li

ϕ

/ colimi Mi

ψ

/ colimi Ni

216

7. COMMUTATIVE ALGEBRA

is a complex as well, and denoting H its homology we have H = colimi Hi . Proof. We are going to repeatedly use the description of colimits over I as in Lemma 7.8.4 without further mention. Let h ∈ H. Since H = ker(ϕ)/Im(ψ) we see that h is the class mod Im(ψ) of an element [m] in Ker(ψ) ⊂ colimi Mi . Choose an i such that [m] comes from an element m ∈ Mi . Choose a j ≥ i such that νij (ψi (m)) = 0 which is possible since [m] ∈ Ker(ψ). After replacing i by j and m by µij (m) we see that we may assume m ∈ Ker(ψi ). This shows that the map colimi Hi → H is surjective. Suppose that hi ∈ Hi has image zero on H. Since Hi = Ker(ψi )/Im(ϕi ) we may represent hi by an element m ∈ Ker(ψi ) ⊂ Mi . The assumption on the vanishing of hi in H means that the class of m in colimi Mi lies in the image of ϕ. Hence there exists an j ≥ i and a l ∈ Lj such that ϕj (l) = µij (m). Clearly this shows that the image of hi in Hj is zero. This proves the injectivity of colimi Hi → H.  Example 7.8.10. Taking colimits is not exact in general. Consider the partially ordered set I = {a, b, c} with a ≺ b and a ≺ c and no other strict inequalities, as in Example 7.8.6. Consider the map of systems (0, Z, Z, 0, 0) → (Z, Z, Z, 1, 1). From the description of the colimit in Example 7.8.6 we see that the associated map of colimits is not injective, even though the map of systems is injective on each object. Hence the result of Lemma 7.8.9 is false for general systems. Lemma 7.8.11. Let I be an index category satisfying the assumptions of Categories, Lemma 4.17.3. Then taking colimits of diagrams of abelian groups over I is exact (i.e., the analogue of Lemma 7.8.9 holds in this situation). ` Proof. By Categories, Lemma 4.17.3 we may write I = j∈J Ij with each Ij a filtered category, and J possibly empty. By Categories, Lemma 4.19.3 taking colimits over the index categories Ij is the same as taking the colimit over some directed partially ordered set. Hence Lemma 7.8.9 applies to these colimits. This reduces the problem to showing that coproducts in the category of R-modules over the set J are exact. In other words, exact sequences Lj → Mj → Nj of R modules we have to show that M M M Lj −→ Mj −→ Nj j∈J

j∈J

j∈J

is exact. This can be verified by hand, and holds even if J is empty.



For purposes of reference, we define what it means to have a relation between elements of a module. Definition 7.8.12. Let R be a ring. Let M be an R-module. Let n ≥ 0 and xi ∈ M for i = 1, . . . , n. A relation P between x1 , . . . , xn in M is a sequence of elements f1 , . . . , fn ∈ R such that i=1,...,n fi xi = 0. Lemma 7.8.13. Let R be a ring and let M be an R-module. Then M is the colimit of a directed system (Mi , µij ) of R-modules with all Mi finitely presented R-modules. Proof. Consider any finite subset S ⊂ M and any finite collection of relations E among the elements of S. So each s ∈ S corresponds to xs ∈ M and each e ∈ E

7.9. LOCALIZATION

217

P consists of a vector of elements fe,s ∈ R such that fe,s xs = 0. Let MS,E be the cokernel of the map X R#E −→ R#S , (ge )e∈E 7−→ ( ge fe,s )s∈S . There are canonical maps MS,E → M . If S ⊂ S 0 and if the elements of E correspond, via this map, to relations in E 0 , then there is an obvious map MS,E → MS 0 ,E 0 commuting with the maps to M . Let I be the set of pairs (S, E) with ordering by inclusion as above. It is clear that the colimit of this directed system is M .  7.9. Localization Definition 7.9.1. Let R be a ring, S a subset of R. We say S is a multiplicative subset of R is 1 ∈ S and S is closed under multiplication, i.e., s, s0 ∈ S ⇒ ss0 ∈ S. Given a ring A and a multiplicative subset S, we define a relation on A × S as follows: (x, s) ∼ (y, t) ⇐⇒ ∃u ∈ S, such that (xt − ys)u = 0 It is easily checked that this is an equivalence relation. Let x/s (or xs ) be the equivalence class of (x, s) and S −1 A be the set of all equivalence classes. Define addition and multiplication in S −1 A as follows: (7.9.1.1) (7.9.1.2)

x/s + y/t = (xt + ys)/st x/s · y/t = xy/st

One can check that S −1 A becomes a ring under these operations. Definition 7.9.2. This ring is called the localization of A with respect to S. We have a natural ring map from A to its localization S −1 A, A −→ S −1 A,

x 7−→ x/1

which is sometimes called the localization map. In general the localization map is not injective, unless S contains no zerodivisors. For, if x/1 = 0, then there is a u ∈ S such that xu = 0 in A and hence x = 0 since there are no zerodivisors in S. The localization of a ring has the following universal property. Proposition 7.9.3. Let f : A → B be a ring map that sends every element in S to a unit of B. Then there is a unique homomorphism g : S −1 A → B such that the following diagram commutes. A

/B
2 there exists an inverse σn : S/I n → P/J n of Ψn .

522

7. COMMUTATIVE ALGEBRA

Namely, as S is formally smooth over R (by Proposition 7.128.13) we see that in the solid diagram / P/J n S σn−1

"  P/J n−1

of R-algebras we can fill in the dotted arrow by some R-algebra map τ : S → P/J n making the diagram commute. This induces an R-algebra map τ : S/I n → P/J n which is equal to σn−1 modulo J n . By construction the map Ψn is surjective and now τ ◦ Ψn is an R-algebra endomorphism of P/J n which maps xi to xi + δi,n with δi,n ∈ J n−1 /J n . It follows that Ψn is an isomorphism and hence it has an inverse σn . This proves the lemma.  7.130. Smooth algebras over fields Warning: The following two lemmas do not hold over nonperfect fields in general. Lemma 7.130.1. Let k be an algebraically closed field. Let S be a finite type k-algebra. Let m ⊂ S be a maximal ideal. Then dimκ(m) ΩS/k ⊗S κ(m) = dimκ(m) m/m2 . Proof. Since k is algebraically closed we have κ(m) = k. We may choose a presentation 0 → I → k[x1 , . . . , xn ] → S → 0 such that all xi end up in m. Write I = (f1 , . . . , fm ). Note that each fi is contained in (x1 , . . . , xn ), i.e., each fi has zero constant term. Hence we may write X fj = aij xi + h.o.t. By Lemma 7.123.9 there is an exact sequence M M S · fj → S · dxi → ΩS/k → 0. Tensoring with κ(m) = k we get an exact sequence M M k · fj → k · dxi → ΩS/k ⊗ κ(m) → 0. The matrix of the map is given by the partial derivatives of the fj evaluated at 0. In other words by the matrix (aij ). Similarly there is a short exact sequence (f1 , . . . , fm )/(x1 f1 , . . . , xn fm ) → (x1 , . . . , xn )/(x1 , . . . , xn )2 → m/m2 → 0. Note that the first map is given by expanding the fj in terms of the xi , i.e., by the same matrix (aij ). Hence the two numbers are the same.  Lemma 7.130.2. Let k be an algebraically closed field. Let S be a finite type k-algebra. Let m ⊂ S be a maximal ideal. The following are equivalent: (1) (2) (3) (4)

The ring Sm is a regular local ring. We have dimκ(m) ΩS/k ⊗S κ(m) ≤ dim(Sm ). We have dimκ(m) ΩS/k ⊗S κ(m) = dim(Sm ). There exists a g ∈ S, g 6∈ m such that Sg is smooth over k. In other words S/k is smooth at m.

7.130. SMOOTH ALGEBRAS OVER FIELDS

523

Proof. Note that (1), (2) and (3) are equivalent by Lemma 7.130.1 and Definition 7.103.6. Assume that S is smooth at q. By Lemma 7.127.10 we see that Sg is standard smooth over k for a suitable g ∈ S, g 6∈ m. Hence by Lemma 7.127.7 we see that ΩSg /k is free of rank dim(Sg ). Hence by Lemma 7.130.1 we see that dim(Sm ) = dim(m/m2 ) in other words Sm is regular. Conversely, suppose that Sm is regular. Let d = dim(Sm ) = dim m/m2 . Choose a presentation S = k[x1 , . . . , xn ]/I such that xi maps to an element of m for all i. In other words, m00 = (x1 , . . . , xn ) is the corresponding maximal ideal of k[x1 , . . . , xn ]. Note that we have a short exact sequence I/m00 I → m00 /(m00 )2 → m/(m)2 → 0 Pick c = n − d elements f1 , . . . , fd ∈ I such that their images in m00 /(m00 )2 span the kernel of the map to m/(m)2 . This is clearly possible. Denote J = (f1 , . . . , fc ). So J ⊂ I. Denote S 0 = k[x1 , . . . , xn ]/J so there is a surjection S 0 → S. Denote m0 = m00 S 0 the corresponding maximal ideal of S 0 . Hence we have k[x1 , . . . , xn ] O

/ S0 O

/S O

m00

/ m0

/m

By our choice of J the exact sequence J/m00 J → m00 /(m00 )2 → m0 /(m0 )2 → 0 0 shows that dim(m0 /(m0 )2 ) = d. Since Sm 0 surjects onto Sm we see that dim(Sm0 ) ≥ 0 d. Hence by the discussion preceding Definition 7.58.9 we conclude that Sm 0 is 0 regular of dimension d as well. Because S was cut out by c = n − d equations we conclude that there exists a g 0 ∈ S 0 , g 0 6∈ m0 such that Sg0 0 is a global complete 0 intersection over k, see Lemma 7.125.4. Also the map Sm 0 → Sm is a surjection of Noetherian local domains of the same dimension and hence an isomorphism. By Lemma 7.118.6 we see that Sg0 00 ∼ = Sg00 for some g 00 ∈ S 0 , g 00 6∈ m0 . All in all we conclude that after replacing S by a principal localization we may assume that S is a global complete intersection.

At this point we may write S = k[x1 , . . . , xn ]/(f1 , . . . , fc ) with dim S = n − c. Recall that the naive cotangent complex of this algebra is given by M M S · fj → S · dxi see Lemma 7.127.15. By this same lemma in order to show that S is smooth at m we have to show that one of the c × c minors gI of the matrix “A” giving the map above does not vanish at m. By Lemma 7.130.1 the matrix A mod m has rank c. Thus we win.  Lemma 7.130.3. Let k be any field. Let S be a finite type k-algebra. Let X = Spec(S). Let q ⊂ S be a prime corresponding to x ∈ X. The following are equivalent: (1) The k-algebra S is smooth at q over k. (2) We have dimκ(q) ΩS/k ⊗S κ(q) ≤ dimx X. (3) We have dimκ(q) ΩS/k ⊗S κ(q) = dimx X.

524

7. COMMUTATIVE ALGEBRA

Moreover, in this case the local ring Sq is regular. Proof. If S is smooth at q over k, then there exists a g ∈ S, g 6∈ q such that Sg is standard smooth over k, see Lemma 7.127.10. A standard smooth algebra over k has a module of differentials which is free of rank equal to the dimension, see Lemma 7.127.7. Thus we see that (1) implies (3). To finish the proof of the lemma it suffices to show that (2) implies (1) and that it implies that Sq is regular. Assume (2). By Nakayama’s Lemma 7.18.1 we see that ΩS/k,q can be generated by ≤ dimx X elements. We may replace S by Sg for some g ∈ S, g 6∈ q such that ΩS/k is generated by at most dimx X elements. Let K ⊃ k be an algebraically closed field extension such that there exists a k-algebra map ψ : κ(q) → K. Consider SK = K ⊗k S. Let m ⊂ SK be the maximal ideal corresponding to the surjection / K ⊗k κ(q) idK ⊗ψ/ K.

SK = K ⊗k S

Note that m ∩ S = q, in other words m lies over q. By Lemma 7.108.6 the dimension of XK = Spec(SK ) at the point corresponding to m is dimx X. By Lemma 7.106.6 this is equal to dim((SK )m ). By Lemma 7.123.12 the module of differentials of SK over K is the base change of ΩS/k , hence also generated by at most dimx X = dim((SK )m ) elements. By Lemma 7.130.2 we see that SK is smooth at m over K. By Lemma 7.127.17 this implies that S is smooth at q over k. This proves (1). Moreover, we know by Lemma 7.130.2 that the local ring (SK )m is regular. Since Sq → (SK )m is flat we conclude from Lemma 7.103.8 that Sq is regular.  The following lemma can be significantly generalized (in several different ways). Lemma 7.130.4. Let k be a field. Let R be a Noetherian local ring containing k. Assume that the residue field κ = R/m is a finitely generated separable extension of k. Then the map d : m/m2 −→ ΩR/k ⊗R κ(m) is injective. Proof. We may replace R by R/m2 . Hence we may assume that m2 = 0. By assumption we may write κ = k(x1 , . . . , xr , y) where x1 , . . . , xr is a transcendence basis of κ over k and y is separable algebraic P over k(x1 , . . . , xr ). Say its minimal equation is P (y) = 0 with P (T ) = T d + i 1 minimal, then 0 = θ(f n ) = nf n−1 contradicting the minimality of n. We conclude that f is not nilpotent. Suppose f a = 0. If f is a unit then a = 0 and we win. Assume f is not a unit. Then 0 = θ(f a) = f θ(a) + a by the Leibniz rule and hence a ∈ (f ). By induction suppose we have shown f a = 0 ⇒ a ∈ (f n ). Then writing a = f n b we get 0 = n θ(f n+1 b) = (n + 1)f n b + f n+1 θ(b). Hence a = fT b = −f n+1 θ(b)/(n + 1) ∈ (f n+1 ). Since in the Noetherian local ring S we have (f n ) = 0, see Lemma 7.48.6 we win.  The following is probably quite useless in applications. Lemma 7.130.7. Let k be a field of characteristic 0. Let S be a finite type kalgebra. Let q ⊂ S be a prime. The following are equivalent: (1) The algebra S is smooth at q over k. (2) The Sq -module ΩS/k,q is (finite) free. (3) The ring Sq is regular. Proof. In characteristic zero any field extension is separable and hence the equivalence of (1) and (3) follows from Lemma 7.130.5. Also (1) implies (2) by definition of smooth algebras. Assume that ΩS/k,q is free over Sq . We are going to use the notation and observations made in the proof of Lemma 7.130.5. So R = Sq with maximal ideal m and residue field κ. Our goal is to prove R is regular. If m/m2 = 0, then m = 0 and R ∼ = κ. Hence R is regular and we win. If m/m2 6= 0, then choose any f ∈ m whose image in m/m2 is not zero. By Lemma 7.130.4 we see that df has nonzero image in ΩR/k /mΩR/k . By assumption ΩR/k = ΩS/k,q is finite free and hence by Nakayama’s Lemma 7.18.1 we see that

526

7. COMMUTATIVE ALGEBRA

df generates a direct summand. We apply Lemma 7.130.6 to deduce that f is a nonzerodivisor in R. Furthermore, by Lemma 7.123.9 we get an exact sequence (f )/(f 2 ) → ΩR/k ⊗R R/f R → Ω(R/f R)/k → 0 This implies that Ω(R/f R)/k is finite free as well. Hence by induction we see that R/f R is a regular local ring. Since f ∈ m was a nonzerodivisor we conclude that R is regular, see Lemma 7.99.7.  Example 7.130.8. Lemma 7.130.7 does not hold in characteristic p > 0. The standard examples are the ring maps Fp −→ Fp [x]/(xp ) whose module of differentials is free but is clearly not smooth, and the ring map (p > 2) Fp (t) → Fp (t)[x, y]/(xp + y 2 + α) which is not smooth at the prime q = (y, xp − α) but is regular. Using the material above we can characterize smoothness at the generic point in terms of field extensions. Lemma 7.130.9. Let R → S be an injective finite type ring map with R and S domains. Then R → S is smooth at q = (0) if and only if f.f.(R) ⊂ f.f.(S) is a separable extension of fields. Proof. Assume R → S is smooth at (0). We may replace S by Sg for some nonzero g ∈ S and assume that R → S is smooth. Set K = f.f.(R). Then K → S ⊗R K is smooth (Lemma 7.127.4). Moreover, for any field extension K ⊂ K 0 the ring map K 0 → S ⊗R K 0 is smooth as well. Hence S ⊗R K 0 is a regular ring by Lemma 7.130.3, in particular reduced. It follows that S ⊗R K is a geometrically reduced over K. Hence f.f.(S) is geometricaly reduced over K, see Lemma 7.41.3. Hence f.f.(S)/K is separable by Lemma 7.42.1. Conversely, assume that f.f.(R) ⊂ f.f.(S) is separable. We may assume R → S is of finite presentation, see Lemma 7.28.1. It suffices to prove that K → S ⊗R K is smooth at (0), see Lemma 7.127.17. This follows from Lemma 7.130.5, the fact that a field is a regular ring, and the assumption that f.f.(R) → f.f.(S) is separable.  7.131. Smooth ring maps in the Noetherian case Definition 7.131.1. Let ϕ : B 0 → B be a ring map. We say ϕ is a small extension if B 0 and B are local Artinian rings, ϕ is surjective and I = Ker(ϕ) has length 1 as a B 0 -module. Clearly this means that I 2 = 0 and that I = (x) for some x ∈ B 0 such that m0 x = 0 where m0 ⊂ B 0 is the maximal ideal. Lemma 7.131.2. Let R → S be a ring map. Let q be a prime ideal of S lying over p ⊂ R. Assume R is Noetherian and R → S of finite type. The following are equivalent: (1) R → S is smooth at q,

7.131. SMOOTH RING MAPS IN THE NOETHERIAN CASE

527

(2) for every surjection of local R-algebras (B 0 , m0 ) → (B, m) with Ker(B 0 → B) having square zero and every solid commutative diagram SO

/B O

R

/ B0

such that q = S ∩ m there exists a dotted arrow making the diagram commute, (3) same as in (2) but with B 0 → B ranging over small extensions, and (4) same as in (2) but with B 0 → B ranging over small extensions such that in addition S → R induces an isomorphism κ(q) ∼ = κ(m). Proof. Assume (1). This means there exists a g ∈ S, g 6∈ q such that R → Sg is smooth. By Proposition 7.128.13 we know that R → Sg is formally smooth. Note that given any diagram as in (2) the map S → B factors automatically through Sq and a fortiori through Sg . The formal smoothness of Sg over R gives us a morphism Sg → B 0 fitting into a similar diagram with Sg at the upper left corner. Composing with S → Sg gives the desired arrow. In other words, we have shown that (1) implies (2). Clearly (2) implies (3) and (3) implies (4). Assume (4). We are going to show that (1) holds, thereby finishing the proof of the lemma. Choose a presentation S = R[x1 , . . . , xn ]/(f1 , . . . , fm ). This is possible as S is of finite type over R and therefore of finite presentation (see Lemma 7.29.4). Set I = (f1 , . . . , fm ). Consider the naive cotangent complex Mm d : I/I 2 −→ Sdxj j=1

of this presentation (see Section 7.124). It suffices to show that when we localize this complex at q then the map becomes a split injection, see Lemma 7.127.12. Denote S 0 = R[x1 , . . . , xn ]/I 2 . By Lemma 7.123.11 we have Mm S ⊗S 0 ΩS 0 /R = S ⊗R[x1 ,...,xn ] ΩR[x1 ,...,xn ]/R = Sdxj . j=1

Thus the map d : I/I 2 −→ S ⊗S 0 ΩS 0 /R is the same as the map in the naive cotangent complex above. In particular the truth of the assertion we are trying to prove depends only on the three rings R → S 0 → S. Let q0 ⊂ R[x1 , . . . , xn ] be the prime ideal corresponding to q. Since localization commutes with taking modules of differentials (Lemma 7.123.8) we see that it suffices to show that the map d : Iq0 /Iq20 −→ Sq ⊗Sq0 0 ΩS 0 0 /R

(7.131.2.1) coming from R →

q

Sq0 0

→ Sq is a split injection.

Let N ∈ N be an integer. Consider the ring 0 BN = Sq0 0 /(q0 )N Sq0 0 = (S 0 /(q0 )N S 0 )q0 0 0 0 and its quotient BN = BN /IBN . Note that BN ∼ = Sq /qN Sq . Observe that BN is an Artinian local ring since it is the quotient of a local Noetherian ring by a

528

7. COMMUTATIVE ALGEBRA

0 power of its maximal ideal. Consider a filtration of the kernel IN of BN → BN by 0 BN -submodules 0 ⊂ JN,1 ⊂ JN,2 ⊂ . . . ⊂ JN,n(N ) = IN 0 such that each successive quotient JN,i /JN,i−1 has length 1. (As BN is Artinian such a filtration exists.) This gives a sequence of small extensions 0 0 0 0 0 BN → BN /JN,1 → BN /JN,2 → . . . → BN /JN,n(N ) = BN /IN = BN = Sq /qN Sq

Applying condition (4) successively to these small extensions starting with the map S → BN we see there exists a commutative diagram SO

/ BN O

R

/ B0 N

0 0 0 Clearly the ring map S → BN factors as S → Sq → BN where Sq → BN is a local 0 to the homomorphism of local rings. Moreover, since the maximal ideal of BN 0 factors through Sq /(q)N Sq = BN . N th power is zero we conclude that Sq → BN In other words we have shown that for all N ∈ N the surjection of R-algebras 0 BN → BN has a splitting.

Consider the presentation 0 ΩB 0 /R → ΩB /R → 0 IN → BN ⊗BN N N

0 coming from the surjection BN → BN with kernel IN (see Lemma 7.123.9). By the 0 → BN has a right inverse. Hence by Lemma 7.123.10 above the R-algebra map BN we see that the sequence above is split exact! Thus for every N the map 0 ΩB 0 /R IN −→ BN ⊗BN N

is a split injection. The rest of the proof is gotten by unwinding what this means exactly. Note that IN = Iq0 /(Iq20 + (q0 )N ∩ Iq0 ) By Artin-Rees (Lemma 7.48.4) we find a c ≥ 0 such that Sq /qN −c Sq ⊗Sq IN = Sq /qN −c Sq ⊗Sq Iq0 /Iq20 for all N ≥ c (these tensor product are just a fancy way of dividing by qN −c ). We may of course assume c ≥ 1. By Lemma 7.123.11 we see that 0 0 N −c 0 0 /R = S 0 /(q ) Sq0 0 /(q0 )N −c Sq0 0 ⊗Sq0 0 ΩBN Sq0 ⊗Sq0 0 ΩS 0 0 /R q q

we can further tensor this by BN = Sq /q Sq /q

N −c

N

to see that

N −c 0 /R = Sq /q Sq ⊗Sq0 0 ΩBN Sq ⊗Sq0 0 ΩS 0 0 /R . q

Since a split injection remains a split injection after tensoring with anything we see that 0 ΩB 0 /R ) Sq /qN −c Sq ⊗Sq (7.131.2.1) = Sq /qN −c Sq ⊗Sq (IN −→ BN ⊗BN N

is a split injection for all N ≥ c. By Lemma 7.69.1 we see that (7.131.2.1) is a split injection. This finishes the proof. 

7.132. OVERVIEW OF RESULTS ON SMOOTH RING MAPS

529

7.132. Overview of results on smooth ring maps Here is a list of results on smooth ring maps that we proved in the preceding sections. For more precise statements and definitions please consult the references given.

(1) A ring map R → S is smooth if it is of finite presentation and the naive cotangent complex of S/R is quasi-isomorphic to a finite projective Smodule in degree 0, see Definition 7.127.1. (2) If S is smooth over R, then ΩS/R is a finite projective S-module, see discussion following Definition 7.127.1. (3) The property of being smooth is local on S, see Lemma 7.127.13. (4) The property of being smooth is stable under base change, see Lemma 7.127.4. (5) The property of being smooth is stable under composition, see Lemma 7.127.14. (6) A smooth ring map is syntomic, in particular flat, see Lemma 7.127.10. (7) A finitely presented, flat ring map with smooth fibre rings is smooth, see Lemma 7.127.16. (8) A finitely presented ring map R → S is smooth if and only if it is formally smooth, see Proposition 7.128.13. (9) If R → S is a finite type ring map with R Noetherian then to check that R → S is smooth it suffices to check the lifting property of formal smoothness along small extensions of Artinian local rings, see Lemma 7.131.2. (10) A smooth ring map R → S is the base change of a smooth ring map R0 → S0 with R0 of finite type over Z, see Lemma 7.128.14. (11) Formation of the set of points where a ring map is smooth commutes with flat base change, see Lemma 7.127.17. (12) If S is of finite type over an algebraically closed field k, and m ⊂ S a maximal ideal, then the following are equivalent (a) S is smooth over k in a neighbourhood of m, (b) Sm is a regular local ring, (c) dim(Sm ) = dimκ(m) ΩS/k ⊗S κ(m). see Lemma 7.130.2. (13) If S is of finite type over a field k, and q ⊂ S a prime ideal, then the following are equivalent (a) S is smooth over k in a neighbourhood of q, (b) dimq (S/k) = dimκ(q) ΩS/k ⊗S κ(q). see Lemma 7.130.3. (14) If S is smooth over a field, then all its local rings are regular, see Lemma 7.130.3. (15) If S is of finite type over a field k, q ⊂ S a prime ideal, the field extension k ⊂ κ(q) is separable and Sq is regular, then S is smooth over k at q, see Lemma 7.130.5. (16) If S is of finite type over a field k, if k has characteristic 0, if q ⊂ S a prime ideal, and if ΩS/k,q is free, then S is smooth over k at q, see Lemma 7.130.7.

530

7. COMMUTATIVE ALGEBRA

Some of these results were proved using the notion of a standard smooth ring map, see Definition 7.127.6. This is the analogue of what a relative global complete intersection map is for the case of syntomic morphisms. It is also the easiest way to make examples. ´ 7.133. Etale ring maps An ´etale ring map is a smooth ring map whose relative dimension is equal to zero. This is the same as the following slightly more direct definition. Definition 7.133.1. Let R → S be a ring map. We say R → S is ´etale if it is of finite presentation and the naive cotangent complex N LS/R is quasi-isomorphic to zero. Given a prime q of S we say that R → S is ´etale at q if there exists a g ∈ S, g 6∈ q such that R → Sg is ´etale. In particular we see that ΩS/R = 0 if S is ´etale over R. If R → S is smooth, then R → S is ´etale if and only if ΩS/R = 0. From our results on smooth ring maps we automatically get a whole host of results for ´etale maps. We summarize these in Lemma 7.133.3 below. But before we do so we prove that any ´etale ring map is standard smooth. Lemma 7.133.2. Any ´etale ring map is standard smooth. More precisely, if R → S is ´etale, then there exists a presentation S = R[x1 , . . . , xn ]/(f1 , . . . , fn ) such that the image of det(∂fj /∂xi ) is invertible in S. Proof. Let R → S be ´etale. Choose a presentation S = R[x1 , . . . , xn ]/I. As R → S is ´etale we know that M d : I/I 2 −→ Sdxi i=1,...,n

2

is an isomorphism, in particular I/I is a free S-module. Thus by Lemma 7.126.6 we may assume (after possibly changing the presentation), that I = (f1 , . . . , fc ) such that the classes fi mod I 2 form a basis of I/I 2 . It follows immediately from the fact that the displayed map above is an isomorphism that c = n and that det(∂fj /∂xi ) is invertible in S.  Lemma 7.133.3. Results on ´etale ring maps. (1) (2) (3) (4)

(5)

(6) (7) (8)

If R → Rf is ´etale for any ring R and any f ∈ R. Compositions of ´etale ring maps are ´etale. A base change of an ´etale ring map is ´etale. The property of being ´etale is local: Given a ring map R → S and elements g1 , . . . , gm ∈ S which generate the unit ideal such that R → Sgj is ´etale for j = 1, . . . , m then R → S is ´etale. Given R → S of finite presentation, and a flat ring map R → R0 , set S 0 = R0 ⊗R S. The set of primes where R → S 0 is ´etale is the inverse image via Spec(S 0 ) → Spec(S) of the set of primes where R → S is ´etale. An ´etale ring map is syntomic, in particular flat. If S is finite type over a field k, then S is ´etale over k if and only if ΩS/k = 0. Any ´etale ring map R → S is the base change of an ´etale ring map R0 → S0 with R0 of finite type over Z.

´ 7.133. ETALE RING MAPS

531

(9) Let A = colim Ai be a filtered colimit of rings. Let A → B be an ´etale ring map. Then there exists an ´etale ring map Ai → Bi for some i such that B∼ = A ⊗Ai Bi . (10) Let A be a ring. Let S be a multiplicative subset of A. Let S −1 A → B 0 be ´etale. Then there exists an ´etale ring map A → B such that B 0 ∼ = S −1 B. Proof. In each case we use the corresponding result for smooth ring maps with a small argument added to show that ΩS/R is zero. Proof of (1). The ring map R → Rf is smooth and ΩRf /R = 0. Proof of (2). The composition A → C of smooth maps A → B and B → C is smooth, see Lemma 7.127.14. By Lemma 7.123.7 we see that ΩC/A is zero as both ΩC/B and ΩB/A are zero. Proof of (3). Let R → S be ´etale and R → R0 be arbitrary. Then R0 → S 0 = R0 ⊗R S is smooth, see Lemma 7.127.4. Since ΩS 0 /R0 = S 0 ⊗S ΩS/R by Lemma 7.123.12 we conclude that ΩS 0 /R0 = 0. Hence R0 → S 0 is ´etale. Proof of (4). Assume the hypotheses of (4). By Lemma 7.127.13 we see that R → S is smooth. We are also given that ΩSgi /R = (ΩS/R )gi = 0 for all i. Then ΩS/R = 0, see Lemma 7.22.2. Proof of (5). The result for smooth maps is Lemma 7.127.17. In the proof of that lemma we used that N LS/R ⊗S S 0 is homotopy equivalent to N LS 0 /R0 . This reduces us to showing that if M is a finitely presented S-module the set of primes q0 of S 0 such that (M ⊗S S 0 )q0 = 0 is the inverse image of the set of primes q of S such that Mq = 0. This is true (proof omitted). Proof of (6). Follows directly from the corresponding result for smooth ring maps (Lemma 7.127.10). Proof of (7). Follows from Lemma 7.130.3 and the definitions. Proof of (8). Lemma 7.128.14 gives the result for smooth ring maps. The resulting smooth ring map R0 → S0 satisfies the hypotheses of Lemma 7.122.8, and hence we may replace S0 by the factor of relative dimension 0 over R0 . Proof of (9). Follows from (8) since R0 → A will factor through Ai for some i. Proof of (10). Follows from (9), (1), and (2) since S −1 A is a filtered colimit of principal localizations of A.  Next we work out in more detail what it means to be ´etale over a field. Lemma 7.133.4. Let k be a field. A ring map k → S is ´etale if and only if S is isomorphic as a k-algebra to a finite product of finite separable extensions of k. Proof. If k → k 0 is a finite separable field extension then we can write k 0 = k(α) ∼ = k[x]/(f ). Here f is the minimal polynomial of the element α. Since k 0 is separable over k we have gcd(f, f 0 ) = 1. This implies that d : k 0 ·f → k 0 ·dx is an isomorphism. Hence k → k 0 is ´etale. Conversely, suppose that k → S is ´etale. Let k be an algebraic closure of k. Then S ⊗k k is ´etale over k. Suppose we have the result over k. Then S ⊗k k is reduced and hence S is reduced. Also, S Q ⊗k k is finite over k and hence S is finite over k. Hence S is a finite product S = ki of fields, see Lemma 7.50.2 and Proposition 7.58.6. The result over k means S ⊗k k is isomorphic to a finite product of copies

532

7. COMMUTATIVE ALGEBRA

of k, which implies that each k ⊂ ki is finite separable, see for example Lemmas 7.42.1 and 7.42.3. Thus we have reduced to the case k = k. In this case Lemma 7.130.2 (combined with ΩS/k = 0) we see that Sm ∼ = k for all maximal ideals m ⊂ S. This implies the result because S is the product of the localizations at its maximal ideals by Lemma 7.50.2 and Proposition 7.58.6 again.  Lemma 7.133.5. Let R → S be a ring map. Let q ⊂ S be a prime lying over p in R. If S/R is ´etale at q then (1) we have pSq = qSq is the maximal ideal of the local ring Sq , and (2) the field extension κ(p) ⊂ κ(q) is finite separable. Proof. First we may replace S by Sg for some g ∈ S, g 6∈ q and assume that R → S is ´etale. Then the lemma follows from Lemma 7.133.4 by unwinding the fact that S ⊗R κ(p) is ´etale over κ(p).  Lemma 7.133.6. An ´etale ring map is quasi-finite. Proof. Let R → S be an ´etale ring map. By definition R → S is of finite type. For any prime p ⊂ R the fibre ring S ⊗R κ(p) is ´etale over κ(p) and hence a finite products of fields finite separable over κ(p), in particular finite over κ(p). Thus R → S is quasi-finite by Lemma 7.114.4.  Lemma 7.133.7. Let R → S be a ring map. Let q be a prime of S lying over a prime p of R. If (1) R → S is of finite presentation, (2) Rp → Sq is flat (3) pSq is the maximal ideal of the local ring Sq , and (4) the field extension κ(p) ⊂ κ(q) is finite separable, then R → S is ´etale at q. Proof. Apply Lemma 7.114.2 to find a g ∈ S, g 6∈ q such that q is the only prime of Sg lying over p. We may and do replace S by Sg . Then S ⊗R κ(p) has a unique prime, hence is a local ring, hence is equal to Sq /pSq ∼ = κ(q). By Lemma 7.127.16 there exists a g ∈ S, g 6∈ q such that R → Sg is smooth. Replace S by Sg again we may assume that R → S is smooth. By Lemma 7.127.10 we may even assume that R → S is standard smooth, say S = R[x1 , . . . , xn ]/(f1 , . . . , fc ). Since S ⊗R κ(p) = κ(q) has dimension 0 we conclude that n = c, i.e., if R → S is ´etale.  Here is a completely new phenomenon. Lemma 7.133.8. Let R → S and R → S 0 be ´etale. Then any R-algebra map S 0 → S is ´etale. Proof. First of all we note that S 0 → S is of finite presentation by Lemma 7.6.2. Let q ⊂ S be a prime ideal lying over the primes q0 ⊂ S 0 and p ⊂ R. By Lemma 7.133.5 the ring map Sq /pSq → Sq0 0 /pSq0 0 is a map finite separable extensions of κ(p). In particular it is flat. Hence by Lemma 7.120.8 we see that Sq0 0 → Sq is flat. Thus S 0 → S is flat. Moreover, the above also shows that q0 Sq is the maximal ideal of Sq and that the residue field extension of Sq0 0 → Sq is finite separable. Hence from Lemma 7.133.7 above we conclude that S 0 → S is ´etale at q. Since being ´etale is local (see Lemma 7.133.3) we win. 

´ 7.133. ETALE RING MAPS

533

Lemma 7.133.9. Let ϕ : R → S be a ring map. If R → S is surjective, flat and finitely presented then there exist an idempotent e ∈ R such that S = Re . Proof. Since Spec(S) → Spec(R) is a homeomorphism onto a closed subset (see Lemma 7.16.7) and is open (see Proposition 7.37.8) we see that the image is D(e) for some idempotent e ∈ R (see Lemma 7.19.3). Thus Re → S induces a bijection on spectra. Now this map induces an isomorphism on all local rings for example by Lemmas 7.73.4 and 7.18.1. Then it follows that Re → S is also injective, for example see Lemma 7.22.1.  Lemma 7.133.10. Let R be a ring and let I ⊂ R be an ideal. Let R/I → S be an ´etale ring map. Then there exists an ´etale ring map R → S such that S ∼ = S/IS as R/I-algebras. Proof. By Lemma 7.133.2 we can write S = (R/I)[x1 , . . . , xn ]/(f 1 , . . . , f n ) as in ∂f i )i,j=1,...,n invertible in S. Just take some lifts fi Definition 7.127.6 with ∆ = det( ∂x j ∂fi )i,j=1,...,c and set S = R[x1 , . . . , xn , xn+1 ]/(f1 , . . . , fc , xn+1 ∆−1) where ∆ = det( ∂x j as in Example 7.127.8. This proves the lemma. 

Lemma 7.133.11. Consider a commutative diagram 0

/J O

/ B0 O

/B O

/0

0

/I

/ A0

/A

/0

with exact rows where B 0 → B and A0 → A are surjective ring maps whose kernels are ideals of square zero. If A → B is ´etale, and J = I ⊗A B, then A0 → B 0 is ´etale. Proof. By Lemma 7.133.10 there exists an ´etale ring map A0 → C such that C/IC = B. Then A0 → C is formally smooth (by Proposition 7.128.13) hence we get an A0 -algebra map ϕ : C → B 0 . Since A0 → C is flat we have I ⊗A B = I ⊗A C/IC = IC. Hence the assumption that J = I ⊗A B implies that ϕ induces an isomorphism IC → J and an isomorphism C/IC → B 0 /IB 0 , whence ϕ is an isomorphism.  Example 7.133.12. Let n, m ≥ 1 be integers. Consider the ring map R = Z[a1 , . . . , an+m ] −→

S = Z[b1 , . . . , bn , c1 , . . . , cm ]

a1

7−→

b1 + c1

a2

7−→

b2 + b1 c1 + c2

...

...

...

an+m

7−→

bn cm

of Example 7.126.7. Write symbolically S = R[b1 , . . . , cm ]/({ak (bi , cj ) − ak }k=1,...,n+m )

534

7. COMMUTATIVE ALGEBRA

where for example a1 (bi , cj ) = b1 + c1 .  1 c1 . . . 0 1 c1  . . . . . . . . .   0 ... 0   1 b1 . . .  0 1 b1  . . . . . . . . . 0 ... 0

The matrix of partial derivatives is  cm 0 . . . 0 . . . cm . . . 0   . . . . . . . . . . . .  1 c1 . . . c m   bn 0 ... 0   . . . bn . . . 0   . . . . . . . . . . . . 1 b1 . . . bn

The determinant ∆ of this matrix is better known as the resultant of the polynomials g = xn + b1 xn−1 + . . . + bn and h = xm + c1 xm−1 + . . . + cm , and the matrix above is known as the Sylvester matrix associated to g, h. In a formula ∆ = Resx (g, h). The Sylvester matrix is the tranpose of the matrix of the linear map S[x]<m ⊕ S[x] 0 the ring extension n

R = Z[1/p] ⊂ R0 = Z[1/p][x]/(xp − 1) has the following property: For d < pn there exist elements α0 , . . . , αd−1 ∈ R0 such that Y (αi − αj ) 0≤i<j0 R/pi associated prime of the module R/p(n) is p. Hence the set of associate primes of R/(R ∩ xR) is a subset of {pi } and there are no inclusion relations among them. This proves (3). 

7.142. FORMAL SMOOTHNESS OF FIELDS

571

7.142. Formal smoothness of fields In this section we show that field extensions are formally smooth if and only if they are separable. Lemma 7.142.1. Let K be a field of characteristic p > 0. Let a ∈ K. Then da = 0 in ΩK/Fp if and only if a is a pth power. Proof. By Lemma 7.123.4 we see that there exists a subfield Fp ⊂ L ⊂ K such that Fp ⊂ L is a finitely generated field extension and such that da is zero in ΩL/Fp . Hence we may assume that K is a finitely generated field extension of Fp . Choose a transcendence basis x1 , . . . , xr ∈ K such that K is finite separable over Fp (x1 , . . . , xr ). We remark that the result holds for the purely transcendental subfield Fp (x1 , . . . , xr ) ⊂ K. Namely, Mr Fp (x1 , . . . , xr )dxi ΩFp (x1 ,...,xr )/Fp = i=1

and any rational function all of whose partial derivatives are zero is a pth power. Moreover, we also have Mr ΩK/Fp = Kdxi i=1

since Fp (x1 , . . . , xr ) ⊂ K is finite separable (computation omitted). Suppose a ∈ K is an element such that da = 0 in the module of differentials. By our choice of xi we see that the minimal polynomial P (T ) ∈ k(x1 , . . . , xr )[T ] of a is separable. Write Xd P (T ) = T d + ai T d−i i=1

and hence 0 = dP (a) =

Xd i=1

ad−i dai

in ΩK/Fp . By the description of ΩK/Fp above and the fact that P was the minimal polynomial of a, we see that this implies dai = 0. Hence ai = bpi for each i. Therefore by Lemma 7.39.8 we see that a is a pth power.  Lemma 7.142.2. Let k be a field of characteristic p > 0. Let a1 , . . . , an ∈ k be elements such that da1 , . . . , dan are linearly independent in Ωk/Fp . Then the field 1/p

1/p

extension k(a1 , . . . , an ) has degree pn over k. Proof. By induction on n. If n = 1 the result is Lemma 7.142.1. For the induction 1/p 1/p step, suppose that k(a1 , . . . , an−1 ) has degree pn−1 over k. We have to show that 1/p 1/p an does not map to a pth power in k(a1 , . . . , an−1 ). If it does then we can write X p in−1 /p i1 /p an = λI a1 . . . an−1 I=(i1 ,...,in−1 ), 0≤ij ≤p−1 X in−1 = λpI ai11 . . . an−1 I=(i1 ,...,in−1 ), 0≤ij ≤p−1

Applying d we see that dan is linearly dependent on dai , i < n. This is a contradiction.  Lemma 7.142.3. Let k be a field of characteristic p > 0. The following are equivalent: (1) the field extension K/k is separable (see Definition 7.40.1), and (2) the map K ⊗k Ωk/Fp → ΩK/Fp is injective.

572

7. COMMUTATIVE ALGEBRA

Proof. Write K as a directed colimit K = colimi Ki of finitely generated field extensions k ⊂ Ki . By definition K is separable if and only if each Ki is separable over k, and by Lemma 7.123.4 we see that K ⊗k Ωk/Fp → ΩK/Fp is injective if and only if each Ki ⊗k Ωk/Fp → ΩKi /Fp is injective. Hence we may assume that K/k is a finitely generated field extension. Assume k ⊂ K is a finitely generated field extension which is separable. Choose x1 , . . . , xr+1 ∈ K as in Lemma 7.40.3. In this case there exists an irreducible polynomial G(X1 , . . . , Xr+1 ) ∈ k[X1 , . . . , Xr+1 ] such that G(x1 , . . . , xr+1 ) = 0 and such that ∂G/∂Xr+1 is not identically zero. Moreover K is the field of fractions of the domain. S = K[X1 , . . . , Xr+1 ]/(G). Write X ir+1 G= aI X I , X I = X1i1 . . . Xr+1 . Using the presentation of S above we see that L S ⊗k Ωk ⊕ i=1,...,r+1 SdXi P ΩS/Fp = P I h X daI + ∂G/∂Xi dXi i Since ΩK/Fp is the localization of the S-module ΩS/Fp (see Lemma 7.123.8) we conclude that L K ⊗k Ωk ⊕ i=1,...,r+1 KdXi P P ΩK/Fp = h X I daI + ∂G/∂Xi dXi i Now, since the polynomial ∂G/∂Xr+1 is not identically zero we conclude that the map K ⊗k Ωk/Fp → ΩS/Fp is injective as desired. Assume k ⊂ K is a finitely generated field extension and that K ⊗k Ωk/Fp → ΩK/Fp is injective. (This part of the proof is the same as the argument proving Lemma 7.42.1.) Let x1 , . . . , xr be a transcendence basis of K over k such that the degree of inseparability of the finite extension k(x1 , . . . , xr ) ⊂ K is minimal. If K is separable over k(x1 , . . . , xr ) then we win. Assume this is not the case to get a contradiction. Then there exists an element α ∈ K which is not separable over k(x1 , . . . , xr ). Let P (T ) ∈ k(x1 , . . . , xr )[T ] be its minimal polynomial. Because α is not separable actually P is a polynomial in T p . Clear denominators to get an irreducible polynomial X G(X1 , . . . , Xr , T ) = aI,i X I T i ∈ k[X1 , . . . , Xr , T ] such that G(x1 , . . . , xr , α) = 0 in L. Note that this means k[X1 , . . . , Xr , T ]/(G) ⊂ L. We may assume that for some pair (I0 , i0 ) the coefficient aI0 ,i0 = 1. We claim that dG/dXi is not identically zero for at least one i. Namely, if this is not the case, then G is actually a polynomial in X1p , . . . , Xrp , T p . Then this means that X xI αi daI,i (I,i)6=(I0 ,i0 )

is zero in ΩK/Fp . Note that there is no k-linear relation among the elements {xI αi | aI,i 6= 0 and (I, i) 6= (I0 , i0 )} of K. Hence the assumption that K ⊗k Ωk/Fp → ΩK/Fp is injective this implies that daI,i = 0 in Ωk/Fp for all (I, i). By Lemma 7.142.1 we see that each aI,i is a pth power, which implies that G is a pth power contradicting the irreducibility of G. Thus, after renumbering, we may assume that dG/dX1 is not zero. Then we see that x1 is separably algebraic over k(x2 , . . . , xr , α), and that x2 , . . . , xr , α is a transcendence basis of L over k. This means that the degree of inseparability of

7.142. FORMAL SMOOTHNESS OF FIELDS

573

the finite extension k(x2 , . . . , xr , α) ⊂ L is less than the degree of inseparability of the finite extension k(x1 , . . . , xr ) ⊂ L, which is a contradiction.  Lemma 7.142.4. Let k ⊂ K be an extension of fields. If K is formally smooth over k, then K is a separable extension of k. Proof. Assume K is formally smooth over k. By Lemma 7.128.9 we see that K ⊗k Ωk/Z → ΩK/Z is injective. Hence K is separable over k by Lemma 7.142.3 above.  Lemma 7.142.5. Let k ⊂ K be an extension of fields. Then K is formally smooth over k if and only if H1 (LK/k ) = 0. Proof. This follows from Proposition 7.128.8 and the fact that a vector spaces is free (hence projective).  Lemma (1) (2) (3)

7.142.6. Let k ⊂ K be an extension of fields. If K is purely transcendental over k, then K is formally smooth over k. If K is separable algebraic over k, then K is formally smooth over k. If K is separable over k, then K is formally smooth over k.

Proof. For (1) write K = k(xj ; j ∈ J). Suppose that A is a k-algebra, and I ⊂ A is an ideal of square zero. Let ϕ : K → A/I be a k-algebra map. Let aj ∈ A be an element such that aj mod I = ϕ(xj ). Then it is easy to see that there is a unique k-algebra map K → A which maps xj to aj and which reduces to ϕ mod I. Hence k ⊂ K is formally smooth. In case (2) we see that k ⊂ K is a colimit of ´etale ring extensions. An ´etale ring map is formally ´etale (Lemma 7.138.2). Hence this case follows from Lemma 7.138.3 and the trivial observation that a formally ´etale ring map is formally smooth. In case (3), write K = colim Ki as the filtered colimit of its finitely generated sub k-extensions. By Definition 7.40.1 each Ki is separable algebraic over a purely transcendental extension of k. Hence Ki /k is formally smooth by cases (1) and (2) and Lemma 7.128.3. Thus H1 (LKi /k ) = 0 by Lemma 7.142.5. Hence H1 (LK/k ) = 0 by Lemma 7.124.8. Hence K/k is formally smooth by Lemma 7.142.5 again.  Lemma 7.142.7. Let k be a field. (1) If the characteristic of k is zero, then any extension field of k is formally smooth over k. (2) If the characteristic of k is p > 0, then k ⊂ K is formally smooth if and only if it is a separable field extension. Proof. Combine Lemmas 7.142.4 and 7.142.6.



Here we put together all the different characterizations of separable field extensions. Proposition 7.142.8. Let k ⊂ K be a field extension. If the characteristic of k is zero then (1) K is separable over k, (2) K is geometrically reduced over k, (3) K is formally smooth over k, (4) H1 (LK/k ) = 0, and (5) the map K ⊗k Ωk/Z → ΩK/Z is injective.

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If the characteristic of k is p > 0, then the following are equivalent: (1) K is separable over k, (2) the ring K ⊗k k 1/p is reduced, (3) K is geometrically reduced over k, (4) the map K ⊗k Ωk/Fp → ΩK/Fp is injective, (5) H1 (LK/k ) = 0, and (6) K is formally smooth over k. Proof. This is a combination of Lemmas 7.42.1, 7.142.7 7.142.4, and 7.142.3.



Here is yet another characterization of finitely generated separable field extensions. Lemma 7.142.9. Let k ⊂ K be a finitely generated field extension. Then K is separable over k if and only if K is the localization of a smooth k-algebra. Proof. Choose a finite type k-algebra R which is a domain whose fraction field is K. Lemma 7.130.9 says that k → R is smooth at (0) if and only if K/k is separable. This proves the lemma.  Lemma 7.142.10. Let k ⊂ K be a field extension. Then K is a filtered colimit of global complete intersection algebras over k. If K/k is separable, then K is a filtered colimit of smooth algebras over k. Proof. Suppose that E ⊂ K is a finite subset. It suffices to show that there exists a k subalgebra A ⊂ K which contains E and which is a global complete intersection (resp. smooth) over k. The separable/smooth case follows from Lemma 7.142.9. In general let L ⊂ K be the subfield generated by E. Pick a transcendence basis x1 , . . . , xd ∈ L over k. The extension k(x1 , . . . , xd ) ⊂ L is finite. Say L = k(x1 , . . . , xd )[y1 , . . . , yr ]. Pick inductively polynomials Pi ∈ k(x1 , . . . , xd )[Y1 , . . . , Yr ] such that Pi = Pi (Y1 , . . . , Yi ) is monic in Yi over k(x1 , . . . , xd )[Y1 , . . . , Yi−1 ] and maps to the minimum polynomial of yi in k(x1 , . . . , xd )[y1 , . . . , yi−1 ][Yi ]. Then it is clear that P1 , . . . , Pr is a regular sequence in k(x1 , . . . , xr )[Y1 , . . . , Yr ] and that L = k(x1 , . . . , xr )[Y1 , . . . , Yr ]/(P1 , . . . , Pr ). If h ∈ k[x1 , . . . , xd ] is a polynomial such that Pi ∈ k[x1 , . . . , xd , 1/h, Y1 , . . . , Yr ], then we see that P1 , . . . , Pr is a regular sequence in k[x1 , . . . , xd , 1/h, Y1 , . . . , Yr ] and A = k[x1 , . . . , xd , 1/h, Y1 , . . . , Yr ]/(P1 , . . . , Pr ) is a global complete intersection. After adjusting our choice of h we may assume E ⊂ A and we win.  7.143. Constructing flat ring maps The following lemma is occasionally useful. Lemma 7.143.1. Let (R, m, k) be a local ring. Let k ⊂ K be a field extension. There exists a local ring R0 , a flat local ring map R → R0 such that m0 = mR0 and the residue field extension k = R/m ⊂ R0 /m0 is isomorphic to k ⊂ K. Proof. Suppose that k ⊂ k 0 = k(α) is a monogenic extension of fields. Then k 0 is the residue field of a flat local extension R ⊂ R0 as in the lemma. Namely, if α is transcendental over k, then we let R0 be the localization at the prime P of R[x] d−i d mR[x]. If α is algebraic with minimal polynomial T + λ T , then we let i P R0 = R[T ]/(T d + λi T d−i ). Consider the collection of triples (k 0 , R → R0 , φ), where k ⊂ k 0 ⊂ K is a subfield, R → R0 is a local ring map as in the lemma, and φ : R0 → k 0 induces an isomorphism

7.144. THE COHEN STRUCTURE THEOREM

575

R0 /mR0 ∼ = k 0 of k-extensions. These form a “big” category C with morphisms (k1 , R1 , φ1 ) → (k2 , R2 , φ2 ) given by ring maps ψ : R1 → R2 such that R1

φ1

/ k1

/K

/ k2

/K

ψ

 R2

φ2

commutes. This implies that k1 ⊂ k2 . Suppose that I is a directed partially ordered set, and ((Ri , ki , φi ), ψii0 ) is a system over I, see Categories, Section 4.19. In this case we can consider R0 = colimi∈I Ri S This is a local ring with maximal ideal mR0 , and residue field k 0 = i∈I ki . Moreover, the ring map R → R0 is flat as it is a colimit of flat maps (and tensor products commute with directed colimits). Hence we see that (R0 , k 0 , φ0 ) is an “upper bound” for the system. An almost trivial application of Zorn’s Lemma would finish the proof if C was a set, but it isn’t. (Actually, you can make this work by finding a reasonable bound on the cardinals of the local rings occuring.) To get around this problem we choose a total ordering on K. For x ∈ K we let K(x) be the subfield of K generated by all elements of K which are ≤ x. By transfinite induction on x ∈ K we will produce ring maps R ⊂ R(x) as in the lemma with residue field extension k ⊂ K(x). Moreover, by construction we will have that R(x) will contain R(y) for all y ≤ x. Namely, if x has a predecessor x0 , then K(x) = K(x0 )[x] and hence we can let R(x0 ) ⊂ R(x) be the local ring extension constructed in the first paragraph of the proof. If x does 0 not have a predecessor, then we first set R0 (x) = colimx0 <x R(x third S ) as in the 0 0 paragraph of the proof. The residue field of R (x) is K (x) = x0 <x K(x0 ). Since K(x) = K 0 (x)[x] we see that we can use the construction of the first paragraph of the proof to produce R0 (x) ⊂ R(x). This finishes the proof of the lemma.  Lemma 7.143.2. Let R be a ring. Let p ⊂ R be a prime and let κ(p) ⊂ L be a finite extension of fields. Then there exists a finite free ring map R → S such that q = pS is prime and κ(p) ⊂ κ(q) is isomorphic to the given extension κ(p) ⊂ L. Proof. By induction of the degree of κ(p) ⊂ L. If the degree is 1, then we take R = S. In general, if there exists a sub extension κ(p) ⊂ L0 ⊂ L then we win by induction on the degree (by first constructing R ⊂ S 0 corresponding to L0 /κ(p) and then construction S 0 ⊂ S corresponding to L/L0 ). Thus weP may assume that L ⊃ κ(p) is generated by a single element α ∈ L. Let X d + i 0. In this section we mostly focus on Noetherian complete local rings. Lemma 7.144.2. Let R be a Noetherian complete local ring. Any quotient of R is also a Noetherian complete local ring. Given a finite ring map R → S, then S is a product of Noetherian complete local rings. Proof. The ring S is Noetherian by Lemma 7.29.1. As an R-module S is complete by Lemma 7.91.2. Hence S is the product of the completions at its maximal ideals by Lemma 7.91.17.  Lemma 7.144.3. Let (R, m) be a complete local ring. If m is a finitely generated ideal then R is Noetherian. Proof. See Lemma 7.91.9.



Definition 7.144.4. Let (R, m) be a complete local ring. A subring Λ ⊂ R is called a coefficient ring if the following conditions hold: (1) Λ is a complete local ring with maximal ideal Λ ∩ m, (2) the residue field of Λ maps isomorphically to the residue field of R, and (3) Λ ∩ m = pΛ, where p is the characteristic of the residue field of R. Let us make some remarks on this definition. We split the discussion into the following cases: (1) The local ring R contains a field. This happens if either Q ⊂ R, or pR = 0 where p is the characteristic of R/m. In this case a coefficient ring Λ is a field contained in R which maps isomorphically to R/m. (2) The characteristic of R/m is p > 0 but no power of p is zero in R. In this case Λ is a complete discrete valuation ring with uniformizer p and residue field R/m. (3) The characteristic of R/m is p > 0, and for some n > 1 we have pn−1 6= 0, pn = 0 in R. In this case Λ is an Artinian local ring whose maximal ideal is generated by p and which has residue field R/m. The complete discrete valuation rings with uniformizer p above play a special role and we baptize them as follows. Definition 7.144.5. A Cohen ring is a complete discrete valuation ring with uniformizer p a prime number. Lemma 7.144.6. Let p be a prime number. Let k be a field of characteristic p. There exists a Cohen ring Λ with Λ/pΛ ∼ = k. 7This includes the condition that T mn = (0); in some texts this may be indicated by saying

that R is complete and separated. Warning: It can happen that the completion limn R/mn of a local ring is non-complete, see Examples, Lemma 66.2.1. This does not happen when m is finitely generated, see Lemma 7.91.7 in which case the completion is Noetherian, see Lemma 7.91.9.

7.144. THE COHEN STRUCTURE THEOREM

577

Proof. First note that the p-adic integers Zp form a Cohen ring for Fp . Let k be an arbitrary field of characteristic p. Let Zp → R be a flat local ring map such that mR = pR and R/pR = k, see Lemma 7.143.1. Then clearly R is a discrete valuation ring. Hence its completion is a Cohen ring for k.  Lemma 7.144.7. Let p > 0 be a prime. Let Λ be a Cohen ring with residue field of characteristic p. For every n ≥ 1 the ring map Z/pn Z → Λ/pn Λ is formally smooth. Proof. If n = 1, this follows from Proposition 7.142.8. For general n we argue by induction on n. Namely, if Z/pn Z → Λ/pn Λ is formally smooth, then we can apply Lemma 7.128.12 to the ring map Z/pn+1 Z → Λ/pn+1 Λ and the ideal I = (pn ) ⊂ Z/pn+1 Z.  Theorem 7.144.8 (Cohen structure theorem). Let (R, m) be a complete local ring. (1) R has a coefficient ring (see Definition 7.144.4), (2) if m is a finitely generated ideal, then R is isomorphic to a quotient Λ[[x1 , . . . , xn ]]/I where Λ is either a field or a Cohen ring. Proof. Let us prove a coefficient ring exists. First we prove this in case the characteristic of the residue field κ is zero. Namely, in this case we will prove by induction on n > 0 that there exists a section ϕn : κ −→ R/mn to the canonical map R/mn → κ = R/m. This is trivial for n = 1. If n > 1, let ϕn−1 be given. The field extension Q ⊂ κ is formally smooth by Proposition 7.142.8. Hence we can find the dotted arrow in the following diagram R/mn−1 o O

R/mn 9 O

ϕn−1

κo

Q

This proves the induction step. Putting these maps together limn ϕn : κ −→ R = limn R/mn gives a map whose image is the desired coefficient ring. Next, we prove the existence of a coefficient ring in the case where the characteristic of the residue field κ is p > 0. Namely, choose a Cohen ring Λ with κ = Λ/pΛ, see Lemma 7.144.6. In this case we will prove by induction on n > 0 that there exists a map ϕn : Λ/pn Λ −→ R/mn whose composition with the reduction map R/mn → κ produces the given isomorphism Λ/pΛ = κ. This is trivial for n = 1. If n > 1, let ϕn−1 be given. The ring

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map Z/pn Z → Λ/pn Λ is formally smooth by Lemma 7.144.7. Hence we can find the dotted arrow in the following diagram R/mn−1 o O

R/mn 9 O

ϕn−1

Λ/pn Λ o

Z/pn Z

This proves the induction step. Putting these maps together limn ϕn : Λ = limn Λ/pn Λ −→ R = limn R/mn gives a map whose image is the desired coefficient ring. The final statement of the theorem is now clear. Namely, if y1 , . . . , yn are generators of the ideal m, then we can use the map Λ → R just constructed to get a map Λ[[x1 , . . . , xn ]] −→ R,

xi 7−→ yi .

n

This map is surjective on each R/m and hence is surjective as R is complete. Some details omitted.  Remark 7.144.9. If k is a field then the power series ring k[[X1 , . . . , Xd ]] is a Noetherian complete local regular ring of dimension d. If Λ is a Cohen ring then Λ[[X1 , . . . , Xd ]] is a complete local Noetherian regular ring of dimension d+1. Hence the Cohen structure theorem implies that any Noetherian complete local ring is a quotient of a regular local ring. In particular we see that a Noetherian complete local ring is universally catenary, see Lemma 7.98.6 and Lemma 7.99.3. Lemma 7.144.10. Let (R, m) be a Noetherian complete local domain. Then there exists a R0 ⊂ R with the following properties (1) R0 is a regular complete local ring, (2) R0 ⊂ R is finite and induces an isomorphism on residue fields, (3) R0 is either isomorphic to k[[X1 , . . . , Xd ]] where k is a field or Λ[[X1 , . . . , Xd ]] where Λ is a Cohen ring. Proof. Let Λ be a coefficient ring of R. Since R is a domain we see that either Λ is a field or Λ is a Cohen ring. Case I: Λ = k is a field. Let d = dim(R). Choose x1 , . . . , xd ∈ m which generate an ideal of definition I ⊂ R. (See Section 7.58.) By Lemma 7.91.12 we see that R is Iadically complete as well. Consider the map R0 = k[[X1 , . . . , Xd ]] → R which maps Xi to xi . Note that R0 is complete with respect to the ideal I0 = (X1 , . . . , Xd ), and that R/I0 R ∼ = R/IR is finite over k = R0 /I0 (because dim(R/I) = 0, see Section 7.58.) Hence we conclude that R0 → R is finite by Lemma 7.91.15. Since dim(R) = dim(R0 ) this implies that R0 → R is injective (see Lemma 7.104.3), and the lemma is proved. Case II: Λ is a Cohen ring. Let d + 1 = dim(R). Let p > 0 be the characteristic of the residue field k. As R is a domain we see that p is a nonzerodivisor in R. Hence dim(R/pR) = d, see Lemma 7.58.11. Choose x1 , . . . , xd ∈ R which generate an ideal of definition in R/pR. Then I = (p, x1 , . . . , xd ) is an ideal of definition of R. By Lemma 7.91.12 we see that R is I-adically complete as well. Consider the map R0 = Λ[[X1 , . . . , Xd ]] → R which maps Xi to xi . Note that R0 is complete with respect to the ideal I0 = (p, X1 , . . . , Xd ), and that R/I0 R ∼ = R/IR is finite

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over k = R0 /I0 (because dim(R/I) = 0, see Section 7.58.) Hence we conclude that R0 → R is finite by Lemma 7.91.15. Since dim(R) = dim(R0 ) this implies that R0 → R is injective (see Lemma 7.104.3), and the lemma is proved.  7.145. Nagata and Japanese rings In this section we discuss finiteness of integral closure. It turns out that this is closely related to the relationship between a local ring and its completion. Definition 7.145.1. Let R be a a domain with field of fractions K. (1) We say R is N-1 if the integral closure of R in K is a finite R-module. (2) We say R is N-2, or Japanese if for any finite extension K ⊂ L of fields the integral closure of R in L is finite over R. The main interest in these notions is for Noetherian rings, but here is a nonNoetherian example. Example 7.145.2. Let k be a field. The domain R = k[x1 , x2 , x3 , . . .] is Japanese, but not Noetherian. The reason is the following. Suppose that R ⊂ L and the field L is a finite extension of the fraction field of R. Then there exists an integer n such that L comes from a finite extension k(x1 , . . . , xn ) ⊂ L0 by adjoining the (transcendental) elements xn+1 , xn+2 , etc. Let S0 be the integral closure of k[x1 , . . . , xn ] in L0 . By Proposition 7.145.31 below it is true that S0 is finite over k[x1 , . . . , xn ]. Moreover, the integral closure of R in L is S = S0 [xn+1 , xn+2 , . . .] (use Lemma 7.34.8) and hence finite over R. The same argument works for R = Z[x1 , x2 , x3 , . . .]. Lemma 7.145.3. Let R be a domain. If R is N-1 then so is any localization of R. Same for N-2. Proof. These statements hold because taking integral closure commutes with localization, see Lemma 7.33.9.  Lemma 7.145.4. Let R be a domain. Let f1 , . . . , fn ∈ R generate the unit ideal. If each domain Rfi is N-1 then so is R. Same for N-2. Proof. Assume Rfi is N-2 (or N-1). Let L be a finite extension of the fraction field of R (equal to the fraction field in the N-1 case). Let S be the integral closure of R in L. By Lemma 7.33.9 we see that Sfi is the integral closure of Rfi in L. Hence Sfi is finite over Rfi by assumption. Thus S is finite over R by Lemma 7.22.2.  Lemma 7.145.5. Let R be a domain. Let R ⊂ S be a quasi-finite extension of domains (for example finite). Assume R is N-2 and Noetherian. Then S is N-2. Proof. Let K = f.f.(R) ⊂ L = f.f.(S). Note that this is a finite field extension (for example by Lemma 7.114.2 (2) applied to the fibre S ⊗R K, and the definition of a quasi-finite ring map). Let S 0 be the integral closure of R in S. Then S 0 is contained in the integral closure of R in L which is finite over R by assumption. As R is Noetherian this implies S 0 is finite over R. By Lemma 7.115.15 there exist elements g1 , . . . , gn ∈ S 0 such that Sg0 i ∼ = Sgi and such that g1 , . . . , gn generate the unit ideal in S. Hence it suffices to show that S 0 is N-2 by Lemmas 7.145.3 and 7.145.4. Thus we have reduced to the case where S is finite over R. Assume R ⊂ S with hypotheses as in the lemma and moreover that S is finite over R. Let M be a finite field extension of the fraction field of S. Then M is also a

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finite field extension of f.f (R) and we conclude that the integral closure T of R in M is finite over R. By Lemma 7.33.14 we see that T is also the integral closure of S in M and we win by Lemma 7.33.13.  Lemma 7.145.6. Let R be a Noetherian domain. If R[z, z −1 ] is N-1, then so is R. Proof. Let R0 be the integral closure of R in its field of fractions K. Let S 0 be the integral closure of R[z, z −1 ] in its field of fractions. Clearly R0 ⊂ S 0 . Since K[z, z −1 ] −1 0 is a normal domain we see that S 0 ⊂ K[z, P z ].j Suppose that f1 , . . . , fn ∈ S 0 −1 generate S as R[z, z ]-module. Say fi = aij z (finite sum), with aij ∈ K. For any x ∈ R0 we can write X x= hi fi −1 0 with the finite R-submodule P hi ∈ R[z, z ]. Thus we see that R is contained in Raij ⊂ K. Since R is Noetherian we conclude that R0 is a finite R-module. 

Lemma 7.145.7. Let R be a Noetherian domain, and let R ⊂ S be a finite extension of domains. If S is N-1, then so is R. If S is N-2, then so is R. Proof. Omitted. (Hint: Integral closures of R in extension fields are contained in integral closures of S in extension fields.)  Lemma 7.145.8. Let R be a Noetherian normal domain with fraction field K. Let K ⊂ L be a finite separable field extension. Then the integral closure of R in L is finite over R. Proof. Consider the trace pairing L × L −→ K,

(x, y) 7−→ hx, yi := TrL/K (xy).

Since L/K is separable this is nondegenerate (exercise in Galois theory). Moreover, if x ∈ L is integral over R, then TrL/K (x) is integral over R also, and since R is normal we see TrL/K (x) ∈ R. Pick x1 , . . . , xn ∈ L which are integral over R and which form a K-basis of L. Then the integral closure S ⊂ L is contained in the R-module M = {y ∈ L | hxi , yi ∈ R, i = 1, . . . , n} By linear algebra we see that M ∼ = R⊕n as an R-module. Hence S ⊂ R⊕n is a finitely generated R-module as R is Noetherian.  Example 7.145.9. Lemma 7.145.8 does not work if the ring is not Noetherian. For example consider the action of G = {+1, −1} on A = C[x1 , x2 , x3 , . . .] where −1 acts by mapping xi to −xi . The invariant ring R = AG is the C-algebra generated by all xi xj . Hence R ⊂ A is not finite. But R is a normal domain with fraction field K = LG the G-invariants in the fraction field L of A. And clearly A is the integral closure of R in L. Lemma 7.145.10. A Noetherian domain of characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese). Proof. This is clear from Lemma 7.145.8 since every field extension in characteristic zero is separable. 

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Lemma 7.145.11. Let R be a Noetherian domain with fraction field K of characteristic p > 0. Then R is Japanese if and only if for every finite purely inseparable extension K ⊂ L the integral closure of R in L is finite over R. Proof. Assume the integral closure of R in every finite purely inseparable field extension of K is finite. Let K ⊂ L be any finite extension. We have to show the integral closure of R in L is finite over R. Choose a finite normal field extension K ⊂ M containing L. As R is Noetherian it suffices to show that the integral closure of R in M is finite over R. By Lemma 7.39.4 there exists a subextension K ⊂ Minsep ⊂ M such that Minsep /K is purely inseparable, and M/Minsep is separable. By assumption the integral closure R0 of R in Minsep is finite over R. By Lemma 7.145.8 the integral closure R00 of R0 in M is finite over R0 . Then R00 is finite over R by Lemma 7.7.3. Since R00 is also the integral closure of R in M (see Lemma 7.33.14) we win.  Lemma 7.145.12. Let R be a Noetherian domain. If R is N-1 then R[x] is N-1. If R is N-2 then R[x] is N-2. Proof. Assume R is N-1. Let R0 be the integral closure of R which is finite over R. Hence also R0 [x] is finite over R[x]. The ring R0 [x] is normal (see Lemma 7.34.8), hence N-1. This proves the first assertion. For the second assertion, by Lemma 7.145.7 it suffices to show that R0 [x] is N2. In other words we may and do assume that R is a normal N-2 domain. In characteristic zero we are done by Lemma 7.145.10. In characteristic p > 0 we have to show that the integral closure of R[x] is finite in any finite purely inseparable extension of f.f.(R[x]) = K(x) ⊂ L with K = f.f.(R). Clearly there exists a finite purely inseparable field extension K ⊂ L0 and q = pe such that L ⊂ L0 (x1/q ). As R[x] is Noetherian it suffices to show that the integral closure of R[x] in L0 (x1/q ) is finite over R[x]. And this integral closure is equal to R0 [x1/q ] with R ⊂ R0 ⊂ L0 the integral closure of R in L0 . Since R is N-2 we see that R0 is finite over R and hence R0 [x1/q ] is finite over R[x].  Lemma 7.145.13 (Tate). Let R be a ring. Let x ∈ R. Assume (1) R is a normal Noetherian domain, (2) R/xR is a Japanese domain, (3) R ∼ = limn R/xn R is complete with respect to x. Then R is Japanese. Proof. We may assume x 6= 0 since otherwise the lemma is trivial. Let K be the fraction field of R. If the characteristic of K is zero the lemma follows from (1), see Lemma 7.145.10. Hence we may assume that the characteristic of K is p > 0, and we may apply Lemma 7.145.11. Thus given K ⊂ L be a finite purely inseparable field extension we have to show that the integral closure S of R in L is finite over R. Let q be a power of p such that Lq ⊂ K. By enlarging L if necessary we may assume there exists an element y ∈ L such that y q = x. Since R → S induces a homeomorphism of spectra (see Lemma 7.44.2) there is a unique prime ideal q ⊂ S lying over the prime ideal p = xR. It is clear that q = {f ∈ S | f q ∈ p} = yS

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since y q = x. Hence Rp and Sq are discrete valuation rings, see Lemma 7.111.6. By Lemma 7.111.8 we see that κ(p) ⊂ κ(q) is a finite field extension. Hence the integral closure S 0 ⊂ κ(q) of R/xR is finite over R/xR by assumption (2). Since S/yS ⊂ S 0 this implies that S/yS is finite over R. Note that S/y n S has a finite filtration whose subquotients are the modules y i S/y i+1 S ∼ = S/yS. Hence we see that each T S/y n S is finite over R. In particular S/xS is finite over R. Also, it is clear xn S = (0) since an element in the intersection has qth power contained T that n in x R = (0) (Lemma 7.48.6). Thus we may apply Lemma 7.91.15 to conclude that S is finite over R, and we win.  Lemma 7.145.14. Let R be a ring. If R is Noetherian, a domain, and N-2, then so is R[[x]]. Proof. Apply Lemma 7.145.13 to the element x ∈ R[[x]].



Definition 7.145.15. Let R be a ring. (1) We say R is universally Japanese if for any finite type ring map R → S with S a domain we have that S is Japanese (i.e., N-2). (2) We say that R is a Nagata ring if R is Noetherian and for every prime ideal p the ring R/p is Japanese. It is clear that a Noetherian universally Japanese ring is a Nagata ring. It is our goal to show that a Nagata ring is universally Japanese. This is not obvious at all, and requires some work. But first, here is a useful lemma. Lemma 7.145.16. Let R be a Nagata ring. Let R → S be essentially of finite type with S reduced. Then the integral closure of R in S is finite over R. Proof. As S is essentially of finite type over R it is Noetherian and has finitely manyQminimal primes q1 , . . . , qm , see Lemma 7.29.6. Since S is reduced we have S ⊂ Sqi and each Sqi = Ki is a field, see Lemmas 7.23.2 and 7.24.3. It suffices to show that the integral closure Q A0i of R in each Ki is finite over R. This is true because R is Noetherian and A ⊂ A0i . Let pi ⊂ R be the prime of R corresponding to qi . As S is essentially of finite type over R we see that Ki = Sqi = κ(qi ) is a finitely generated field extension of κ(pi ). Hence the algebraic closure Li of κ(pi ) in ⊂ Ki is finite over κ(pi ), see Lemma 7.39.7. It is clear that A0i is the integral closure of R/pi in Li , and hence we win by definition of a Nagata ring.  Lemma 7.145.17. Let R be a ring. To check that R is universally Japanese it suffices to show: If R → S is of finite type, and S a domain then S is N-1. Proof. Namely, assume the condition of the lemma. Let R → S be a finite type ring map with S a domain. Let f.f.(S) ⊂ L be a finite extension of its fraction field. Then there exists a finite ring extension S ⊂ S 0 ⊂ L with f.f.(S 0 ) = L. By assumption S 0 is N-1, and hence the integral closure S 00 of S 0 in L is finite over S 0 . Thus S 00 is finite over S (Lemma 7.7.3) and S 00 is the integral closure of S in L (Lemma 7.33.14). We conclude that R is universally Japanese.  Lemma 7.145.18. If R is universally Japanese then any algebra essentially of finite type over R is universally Japanese. Proof. The case of an algebra of finite type over R is immediate from the definition. The general case follows on applying Lemma 7.145.3. 

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Lemma 7.145.19. Let R be a Nagata ring. If R → S is a quasi-finite ring map (for example finite) then S is a Nagata ring also. Proof. First note that S is Noetherian as R is Noetherian and a quasi-finite ring map is of finite type. Let q ⊂ S be a prime ideal, and set p = R ∩ q. Then R/p ⊂ S/q is quasi-finite and hence we conclude that S/q is N-2 by Lemma 7.145.5 as desired.  Lemma 7.145.20. A localization of a Nagata ring is a Nagata ring. Proof. Clear from Lemma 7.145.3.



Lemma 7.145.21. Let R be a ring. Let f1 , . . . , fn ∈ R generate the unit ideal. (1) If each Rfi is universally Japanese then so is R. (2) If each Rfi is Nagata then so is R. Proof. Let ϕ : R → S be a finite type ring map so that S is a domain. Then ϕ(f1 ), . . . , ϕ(fn ) generate the unit ideal in S. Hence if each Sfi = Sϕ(fi ) is N-1 then so is S, see Lemma 7.145.4. This proves (1). If each Rfi is Nagata, then each Rfi is Noetherian and hence R is Noetherian, see Lemma 7.22.2. And if p ⊂ R is a prime, then we see each Rfi /pRfi = (R/p)fi is Japanese and hence we conclude R/p is Japanese by Lemma 7.145.4. This proves (2).  Lemma 7.145.22. A Noetherian complete local ring is a Nagata ring. Proof. Let R be a complete local Noetherian ring. Let p ⊂ R be a prime. Then R/p is also a complete local Noetherian ring, see Lemma 7.144.2. Hence it suffices to show that a Noetherian complete local domain R is N-2. By Lemmas 7.145.5 and 7.144.10 we reduce to the case R = k[[X1 , . . . , Xd ]] where k is a field or R = Λ[[X1 , . . . , Xd ]] where Λ is a Cohen ring. In the case k[[X1 , . . . , Xd ]] we reduce to the statement that a field is N-2 by Lemma 7.145.14. This is clear. In the case Λ[[X1 , . . . , Xd ]] we reduce to the statement that a Cohen ring Λ is N-2. Applying Lemma 7.145.13 once more with x = p ∈ Λ we reduce yet again to the case of a field. Thus we win.  Definition 7.145.23. Let (R, m) be a Noetherian local ring. We say R is analytically unramified if its completion R∧ = limn R/mn is reduced. A prime ideal p ⊂ R is said to be analytically unramified if R/p is analytically unramified. At this point we know the following are true for any Noetherian local ring R: The map R → R∧ is a faithfully flat local ring homomorphism (Lemma 7.91.4). The completion R∧ is Noetherian (Lemma 7.91.9) and complete (Lemma 7.91.8). Hence the completion R∧ is a Nagata ring (Lemma 7.145.22). Moreover, we have seen in Section 7.144 that R∧ is a quotient of a regular local ring (Theorem 7.144.8), and hence universally catenary (Remark 7.144.9). Lemma 7.145.24. Let (R, m) be a Noetherian local ring. (1) If R is analytically unramified, then R is reduced. (2) If R is analytically unramified, then each minimal prime of R is analytically unramified. (3) If R is reduced with minimal primes q1 , . . . , qt , and each qi is analytically unramified, then R is analytically unramified.

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(4) If R is analytically unramified, then the integral closure of R in its total ring of fractions Q(R) is finite over R. (5) If R is a domain and analytically unramified, then R is N-1. Proof. In this proof we will use the remarks immediately following Definition 7.145.23. As R → R∧ is a faithfully flat local ring homomorphism it is injective and (1) follows. Let q be a minimal prime of R, and assume R is analytically unramified. Then q is an associated prime of R (see Proposition 7.61.6). Hence there exists an f ∈ R such that {x ∈ R | f x = 0} = q. Note that (R/q)∧ = R∧ /q∧ , and that {x ∈ R∧ | f x = 0} = q∧ , because completion is exact (Lemma 7.91.3). If x ∈ R∧ is such that x2 ∈ q∧ , then f x2 = 0 hence (f x)2 = 0 hence f x = 0 hence x ∈ q∧ . Thus q is analytically unramified and (2) holds. Assume R is reduced with minimal primes q1 , . . . , qt , and each qi is analytically unramified. Then R → R/q1 × . . . × R/qt is injective. Since completion is exact (see Lemma 7.91.3) we see that R∧ ⊂ (R/q1 )∧ × . . . × (R/qt )∧ . Hence (3) is clear. Assume R is analytically unramified. Let p1 , . . . , ps be the minimal primes of R∧ . Then we see that Q(R∧ ) = Rp1 × . . . × Rps with each Rpi a field as R∧ is reduced (see Lemma 7.23.2). Hence the integral closure S of R∧ in Q(R∧ ) is equal to S = S1 × . . . × Ss with Si the integral closure of R/pi in its fraction field. In particular S is finite over R∧ . Denote R0 the integral closure of R in Q(R). As R → R∧ is flat we see that R0 ⊗R R∧ ⊂ Q(R) ⊗R R∧ ⊂ Q(R∧ ). Moreover R0 ⊗R R∧ is integral over R∧ (Lemma 7.33.11). Hence R0 ⊗R R∧ ⊂ S is a R∧ -submodule. As R∧ is Noetherian it is a finite R∧ module. Thus we may find f1 , . . . , fn ∈ R0 such that R0 ⊗R R∧ is generated by the elements fi ⊗ 1 as a R∧ -module. By faithful flatness we see that R0 is generated by f1 , . . . , fn as an R-module. This proves (4). Part (5) is a special case of part (4).



Lemma 7.145.25. Let R be a Noetherian local ring. Let p ⊂ R be a prime. Assume (1) Rp is a discrete valuation ring, and (2) p is analytically unramified. Then for any associated prime q of R∧ /pR∧ the local ring (R∧ )q is a discrete valuation ring. Proof. Assumption (2) says that R∧ /pR∧ is a reduced ring. Hence an associated prime q ⊂ R∧ of R∧ /pR∧ is the same thing as a minimal prime over pR∧ . In particular we see that the maximal ideal of (R∧ )q is p(R∧ )q . Choose x ∈ R such that xRp = pRp . By the above we see that x ∈ (R∧ )q generates the maximal ideal. As R → R∧ is faithfully flat we see that x is a nonzerodivisor in (R∧ )q . Hence we win.  Lemma (1) (2) (3)

7.145.26. Let (R, m) be a Noetherian local domain. Let x ∈ m. Assume x 6= 0, R/xR has no embedded primes, and for each associated prime p ⊂ R of R/xR we have

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(a) the local ring Rp is regular, and (b) p is analytically unramified. Then R is analytically unramified. Proof. Let p1 , . . . , pt be the associated primes of the R-module R/xR. Since R/xR has no embedded primes we see that each pi has height 1, and is a minimal prime over (x). For each i, let qi1 , . . . , qisi be the associated primes of the R∧ -module R∧ /pi R∧ . By Lemma 7.145.25. we see that (R∧ )qij is regular. By Lemma 7.63.3 we see that [ AssR∧ (R∧ /xR∧ ) = AssR∧ (R∧ /pR∧ ) = {qij }. p∈AssR (R/xR)

∧

2

∧

Let y ∈ R with y = 0. As (R )qij is regular, and hence a domain (Lemma 7.99.2) we see that y maps to zero in (R∧ )qij . Hence y maps to zero in R∧ /xR∧ by Lemma 7.61.18. Hence y = xy 0 . Since x is a nonzerodivisor (as R → R∧ is flat) we see that T n ∧ 0 2 (y ) = 0. Hence we conclude that y ∈ x R = (0) (Lemma 7.48.6).  Lemma 7.145.27. Let (R, m) be a local ring. If R is Noetherian, a domain, and Nagata, then R is analytically unramified. Proof. By induction on dim(R). The case dim(R) = 0 is trivial. Hence we assume dim(R) = d and that the lemma holds for all Noetherian Nagata domains of dimension < d. Let R ⊂ S be the integral closure of R in the field of fractions of R. By assumption S is a finite R-module. By Lemma 7.145.19 we see that S is Nagata. By Lemma 7.104.4 we see dim(R) = dim(S). Let m1 , . . . , mt be the maximal ideals of S. Each of these lies over the maximal ideal m of R. Moreover (m1 ∩ . . . ∩ mt )n ⊂ mS for sufficiently large n as S/mS is Artinian. By Lemma 7.91.3 R∧ → S ∧ is an injective map, and by the Q Chinese Remainder Lemma 7.14.4 combined with Lemma 7.91.12 we have S ∧ = Si∧ where Si∧ is the completion of S with respect to the maximal ideal mi . Hence it suffices to show that Smi is analytically unramified. In other words, we have reduced to the case where R is a Noetherian normal Nagata domain. Assume R is a Noetherian, normal, local Nagata domain. Pick a nonzero x ∈ m in the maximal ideal. We are going to apply Lemma 7.145.26. We have to check properties (1), (2), (3)(a) and (3)(b). Property (1) is clear. We have that R/xR has no embedded primes by Lemma 7.141.6. Thus property (2) holds. The same lemma also tells us each associated prime p of R/xR has height 1. Hence Rp is a 1dimensional normal domain hence regular (Lemma 7.111.6). Thus (3)(a) holds. Finally (3)(b) holds by induction hypothesis, since R/p is Nagata (by Lemma 7.145.19 or directly from the definition). Thus we conclude R is analytically unramified.  Lemma 7.145.28. Let R be a Noetherian domain. If there exists an f ∈ R such that Rf is normal then U = {p ∈ Spec(R) | Rp is normal} is open in Spec(R).

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Proof. It is clear that the standard open D(f ) is contained in U . By Serre’s criterion Lemma 7.141.4 we see that p 6∈ U implies that for some q ⊂ p we have either (1) Case I: depth(Rq ) < 2 and dim(Rq ) ≥ 2, and (2) Case II: Rq is not regular and dim(Rq ) = 1. This in particular also means that Rq is not normal, and hence f ∈ q. In case I we see that depth(Rq ) = depth(Rq /f Rq ) + 1. Hence such a prime q is the same thing as an embedded associated prime of R/f R. In case II q is an associated prime of R/f R of height 1. Thus there is a finite set E of such primes q (see Lemma 7.61.5) and [ Spec(R) \ U = V (q) q∈E

as desired.



Lemma 7.145.29. Let R be a Noetherian domain. Assume (1) there exists a nonzero f ∈ R such that Rf is normal, and (2) for every maximal ideal m ⊂ R the local ring Rm is N-1. Then R is N-1. Proof. Set K = f.f.(R). Suppose that R ⊂ R0 ⊂ K is a finite extension of R contained in K. Note that Rf = Rf0 since Rf is already normal. Hence by Lemma 7.145.28 the set of primes p0 ∈ Spec(R0 ) with Rp0 0 non-normal is closed in Spec(R0 ). Since Spec(R0 ) → Spec(R) is closed the image of this set is closed in Spec(R). For such a ring R0 denote ZR0 ⊂ Spec(R) this image. 0 be the integral closure of the local ring Pick a maximal ideal m ⊂ R. Let Rm ⊂ Rm in K. By assumption this is a finite ring extension. By Lemma 7.33.9 we can find 0 finitely many elements r1 , . . . , rn ∈ K integral over R such that Rm is generated by 0 r1 , . . . , rn over Rm . Let R = R[x1 , . . . , xn ] ⊂ K. With this choice it is clear that m 6∈ ZR0 .

As Spec(R) is quasi-compact, the above shows that we can find a finite collection T R ⊂ Ri0 ⊂ K such that ZRi0 = ∅. Let R0 be the subring of K generated by all of these. It is finite over R. Also ZR0 = ∅. Namely, every prime p0 lies over a prime p0i such that (Ri0 )p0i is normal. This implies that Rp0 0 = (Ri0 )p0i is normal too. Hence R0 is normal, in other words R0 is the integral closure of R in K.  The following proposition says in particular that an algebra of finite type over a Nagata ring is a Nagata ring. Proposition 7.145.30 (Nagata). Let R be a ring. The following are equivalent: (1) R is a Nagata ring, (2) any finite type R-algebra is Nagata, and (3) R is universally Japanese and Noetherian. Proof. It is clear that a Noetherian universally Japanese ring is universally Nagata (i.e., condition (2) holds). Let R be a Nagata ring. We will show that any finitely generated R-algebra S is Nagata. This will prove the proposition. Step 1. There exists a sequence of ring maps R = R0 → R1 → R2 → . . . → Rn = S such that each Ri → Ri+1 is generated by a single element. Hence by induction it suffices to prove S is Nagata if S ∼ = R[x]/I.

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Step 2. Let q ⊂ S be a prime of S, and let p ⊂ R be the corresponding prime of R. We have to show that S/q is N-2. Hence we have reduced to the proving the following: (*) Given a Nagata domain R and a monogenic extension R ⊂ S of domains then S is N-2. Step 3. Let R be a Nagata domain and R ⊂ S a monogenic extension of domains. Let R ⊂ R0 be the integral closure of R in its fraction field. Let S 0 be the subring of f.f.(S) generated by R0 and S. As R0 is finite over R (by the Nagata property) also S 0 is finite over S. Since S is Noetherian it suffices to prove that S 0 is N-2 (Lemma 7.145.7). Hence we have reduced to proving the following: (**) Given a normal Nagata domain R and a monogenic extension R ⊂ S of domains then S is N-2. Step 4: Let R be a normal Nagata domain and let R ⊂ S be a monogenic extension of domains. Suppose the extension of fraction fields f.f.(R) ⊂ f.f.(S) is purely transcendental. In this case S = R[x]. By Lemma 7.145.12 we see that S is N2. Hence we have reduced to proving the following: (**) Given a normal Nagata domain R and a monogenic extension R ⊂ S of domains inducing a finite extension of fraction fields then S is N-2. Step 5. Let R be a normal Nagata domain and let R ⊂ S be a monogenic extension of domains inducing a finite extension of fraction fields K = f.f.(R) ⊂ f.f.(S) = L. Choose an element x ∈ S which generates S as an R-algebra. Let L ⊂ M be a finite extension of fields. Let R0 be the integral closure of R in M . Then the integral closure S 0 of S in M is equal to the integral closure of R0 [x] in M . Also f.f.(R0 ) = M , and R ⊂ R0 is finite (by the Nagata property of R). This implies that R0 is a Nagata ring (Lemma 7.145.19). To show that S 0 is finite over S is the same as showing that S 0 is finite over R0 [x]. Replace R by R0 and S by S 0 to reduce to the following statement: (***) Given a normal Nagata domain R with fraction field K, and x ∈ K, the ring S ⊂ K generated by R and x is N-1. Step 6. Let R be a normal Nagata domain with fraction field K. Let x = b/a ∈ K. We have to show that the ring S ⊂ K generated by R and x is N-1. Note that Sa ∼ = Ra is normal. Hence by Lemma 7.145.29 it suffices to show that Sm is N-1 for every maximal ideal m of S. With assumptions as in the preceding paragraph, pick such a maximal ideal and set n = R ∩ m. The residue field extension κ(n) ⊂ κ(m) is finite (Theorem 7.31.1) and generated by the image of x. Hence there exists a monic polynomial f (X) = P X d + i=1,...,d ai X d−i with f (x) ∈ m. Let K ⊂ K 00 be a finite extension of fields such that f (X) splits completely in K 00 [X]. Let R0 be the integral closure of R in K 00 . Let S 0 ⊂ K 0 be the subring generated by R0 and x. As R is Nagata we see R0 is finite over R and Nagata (Lemma 7.145.19). Moreover, S 0 is finite over S. If for 0 0 every maximal ideal m0 of S 0 the local ring Sm 0 is N-1, then Sm is N-1 by Lemma 7.145.29, which in turn implies that Sm is N-1 by Lemma 7.145.7. After replacing R by R0 and S by S 0 , and m by any of the maximal ideals m0 lying over Q m we reach the situation where the polynomial f above split completely: f (X) = i=1,...,d (X − ai ) with ai ∈ R. Since f (x) ∈ m we see that x − ai ∈ m for some i. Finally, after replacing x by x − ai we may assume that x ∈ m.

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To recapitulate: R is a normal Nagata domain with fraction field K, x ∈ K and S is the subring of K generated by x and R, finally m ⊂ S is a maximal ideal with x ∈ m. We have to show Sm is N-1. We will show that Lemma 7.145.26 applies to the local ring Sm and the element x. This will imply that Sm is analytically unramified, whereupon we see that it is N-1 by Lemma 7.145.24. We have to check properties (1), (2), (3)(a) and (3)(b). Property (1) is trivial. Let I = Ker(R[X] → S) where X 7→ x. We claim that I is generated by all linear forms aX + b such that ax = b in K. Clearly all these linear forms are in I. If g = ad X d + . . . a1 X + a0 ∈ I, then we see that ad x is integral over R (Lemma 7.115.1) and hence b := ad x ∈ R as R is normal. Then g − (ad X − b)X d−1 ∈ I and we win by induction on the degree. As a consequence we see that S/xS = R[X]/(X, I) = R/J where J = {b ∈ R | ax = b for some a ∈ R} = xR ∩ R By Lemma 7.141.6 we see that S/xS = R/J has no embedded primes as an Rmodule, hence as an R/J-module, hence as an S/xS-module, hence as an S-module. This proves property (2). Take such an associated prime q ⊂ S with the property q ⊂ m (so that it is an associated prime of Sm /xSm – it does not matter for the arguments). Then q is minimal over xS and hence has height 1. By the sequence of equalities above we see that p = R ∩ q is an associated prime of R/J, and so has height 1 (see Lemma 7.141.6). Thus Rp is a discrete valuation ring and therefore Rp ⊂ Sq is an equality. This shows that Sq is regular. This proves property (3)(a). Finally, (S/q)m is a localization of S/q, which is a quotient of S/xS = R/J. Hence (S/q)m is a localization of a quotient of the Nagata ring R, hence Nagata (Lemmas 7.145.19 and 7.145.20) and hence analytically unramified (Lemma 7.145.27). This shows (3)(b) holds and we are done.  Proposition 7.145.31. The following types of rings are Nagata and in particular universally Japanese: (1) (2) (3) (4) (5)

fields, Noetherian complete local rings, Z, Dedekind domains with fraction field of characteristic zero, finite type ring extensions of any of the above.

Proof. The Noetherian complete local ring case is Lemma 7.145.22. In the other cases you just check if R/p is N-2 for every prime ideal p of the ring. This is clear whenever R/p is a field, i.e., p is maximal. Hence for the Dedeking ring case we only need to check it when p = (0). But since we assume the fraction field has characteristic zero Lemma 7.145.10 kicks in.  7.146. Ascending properties In this section we start proving some algebraic facts concerning the “ascent” of properties of rings.

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Lemma 7.146.1. Suppose that R → S is a flat and local ring homomorphism of Noetherian local rings. Then depthS (S) = depthR (R) + depthS (S/mR S). Proof. Denote n the right hand side of the equality of the lemma. First assume that n is zero. Then depth(R) = 0 (resp. depth(S/mR S) = 0) which means there is a z ∈ R (resp. y ∈ S/mR S) whose annihilator is mR (resp. mS /mR ). Let y ∈ S be a lift of y. It follows from the fact that R → S is flat that the annihilator of z in S is mR S, and hence the annihilator of zy is mS . Thus depthS (S) = 0 as well. Assume n > 0. If depth(S/mR S) > 0, then choose an f ∈ mS which maps to a nonzerodivisor in S/mR S. According to Lemma 7.92.2 the element f ∈ S is a nonzerodivisor and S/f S is flat over R. Hence by induction on n we have depth(S/f S) = depth(R) + depth(S/(f, mR )). Since f and f are nonzerodivisors we see that depth(S) = depth(S/f S) + 1 and depth(S/mR S) = depth(S/(f, mR )) + 1, see Lemma 7.68.10. Hence we see that the equality holds for R → S as well. If n > 0, but depth(S/mR S) = 0, then choose f ∈ mR a nonzerodivisor. As R → S is flat it is also the case that f is a nonzerodivisor on S. By induction on n again we have depth(S/f S) = depth(R/f R) + depth(S/mR ). By a similar argument as above we conclude that the equality holds for R → S as well.  Here is a more general statement involving modules, see [DG67, IV, Proposition 6.3.1]. Lemma 7.146.2. We have depthS (M ⊗R N ) = depthR (M ) + depthS/mR S (N/mR N ) where R → S is a local homomorphism of local Noetherian rings, M is a finite R-module, and N is a finite S-module flat over R. Proof. Omitted, but similar to the proof of Lemma 7.146.1.



Lemma 7.146.3. Let R → S be a local homomorphism of local Noetherian rings. Assume (1) S/mR S is Cohen-Macaulay, and (2) R → S is flat. Then S is Cohen-Macaulay if and only if R is Cohen-Macaulay. Proof. This follows from the definitions combined with Lemmas 7.146.1 and 7.104.7.  Lemma 7.146.4. Let ϕ : R → S be a ring map. Assume (1) R is Noetherian, (2) S is Noetherian, (3) ϕ is flat, (4) the fibre rings S ⊗R κ(p) are Cohen-Macaulay, and (5) R has property (Sk ). Then S has property (Sk ).

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Proof. Let q be a prime of S lying over a prime p of R. By Lemma 7.146.1 we have depth(Sq ) = depth(Sq /pSq ) + depth(Rp ). On the other hand, we have dim(Sq ) ≤ dim(Rp ) + dim(Sq /pSq ). by Lemma 7.104.6. (Actually equality holds, by Lemma 7.104.7 but strictly speaking we do not need this.) Finally, as the fibre rings of the map are assumed CohenMacaulay we see that depth(Sq /pSq ) = dim(Sq /pSq ). Thus the lemma follows by the following string of inequalities depth(Sq )

=

dim(Sq /pSq ) + depth(Rp )

≥ dim(Sq /pSq ) + min(k, dim(Rp )) =

min(dim(Sq /pSq ) + k, dim(Sq /pSq ) + dim(Rp ))

≥ min(k, dim(Sq )) as desired.



Lemma 7.146.5. Let ϕ : R → S be a ring map. Assume (1) R is Noetherian, (2) S is Noetherian (3) ϕ is flat, (4) the fibre rings S ⊗R κ(p) are regular, and (5) R has property (Rk ). Then S has property (Rk ). Proof. Let q be a prime of S lying over a prime p of R. Assume that dim(Sq ) ≤ k. Since dim(Sq ) = dim(Rp ) + dim(Sq /pSq ) by Lemma 7.104.7 we see that dim(Rp ) ≤ k. Hence Rp is regular by assumption. It follows that Sq is regular by Lemma 7.104.8.  Lemma 7.146.6. Let ϕ : R → S be a ring map. Assume (1) ϕ is smooth, (2) R is reduced. Then S is reduced. Proof. First assume R is Noetherian. In this case being reduced is the same as having properties (S1 ) and (R0 ), see Lemma 7.141.3. Note that S is noetherian, and R → S is flat with regular fibres (see the list of results on smooth ring maps in Section 7.132). Hence we may apply Lemmas 7.146.4 and 7.146.5 and we see that S is (S1 ) and (R0 ), in other words reduced by Lemma 7.141.3 again. In the general case we may find a finitely generated Z-subalgebra R0 ⊂ R and a smooth ring map R0 → S0 such that S ∼ = R ⊗R0 S0 , see remark (10) in Section 7.132. Now, if x ∈ S is an element with x2 = 0, then we can enlarge R0 and assume that x comes from an element x0 ∈ S0 . After enlarging R0 once more we may assume that x20 = 0 in S0 . However, since R0 ⊂ R is reduced we see that S0 is reduced and hence x0 = 0 as desired.  Lemma 7.146.7. Let ϕ : R → S be a ring map. Assume (1) ϕ is smooth,

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(2) R is normal. Then S is normal. Proof. First assume R is Noetherian. In this case being reduced is the same as having properties (S2 ) and (R1 ), see Lemma 7.141.4. Note that S is noetherian, and R → S is flat with regular fibres (see the list of results on smooth ring maps in Section 7.132). Hence we may apply Lemmas 7.146.4 and 7.146.5 and we see that S is (S2 ) and (R1 ), in other words reduced by Lemma 7.141.4 again. The general case. First note that R is reduced and hence S is reduced by Lemma 7.146.6. Let q be a prime of S and let p be the corresponding prime of R. Note that Rp is a normal domain. We have to show that Sq is a normal domain. To do this we may replace R by Rp and S by Sp . Hence we may assume that R is a normal domain. Assume R → S smooth, and R a normal domain. We may find a finitely generated Z-subalgebra R0 ⊂ R and a smooth ring map R0 → S0 such that S ∼ = R ⊗R0 S0 , see remark (10) in Section 7.132. As R0 is a Nagata domain (see Proposition 7.145.31) we see that its integral closure R00 is finite over R0 . Moreover, as R is a normal domain it is clear that R00 ⊂ R. Hence we may replace R0 by R00 and S0 by R00 ⊗R0 S0 and assume that R0 is a normal Noetherian domain. By the first paragraph of the proof we conclude that S0 isSa normal ring (it need not be a domain of course). In this way we see that R = Rλ is the union of normal Noetherian domains and correspondingly S = colim Rλ ⊗R0 S0 is the colimit of normal rings. This implies that S is a normal ring. Some details omitted.  Lemma 7.146.8. Let ϕ : R → S be a ring map. Assume (1) ϕ is smooth, (2) R is a regular ring. Then S is regular. Proof. This follows from Lemma 7.146.5 applied for all (Rk ) using Lemma 7.130.3 to see that the hypotheses are satisfied.  7.147. Descending properties In this section we start proving some algebraic facts concerning the “descent” of properties of rings. It turns out that it is often “easier” to descend properties than it is to ascend them. In other words, the assumption on the ring map R → S are often weaker than the assumptions in the corresponding lemma of the preceding section. However, we warn the reader that the results on descent are often useless unless the corresponding ascent can also be shown! Here is a typical result which illustrates this phenomenon. Lemma 7.147.1. Let R → S be a ring map. Assume that (1) R → S is faithfully flat, and (2) S is Noetherian. Then R is Noetherian. Proof. Let I0 ⊂ I1 ⊂ I2 ⊂ . . . be a growing sequence of ideals of R. By assumption we have In S = In+1 S = In+2 S = . . . for some n. Since R → S is flat we have Ik S = Ik ⊗R S. Hence, as R → S is faithfully flat we see that In S = In+1 S = In+2 S = . . . implies that In = In+1 = In+2 = . . . as desired. 

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Lemma 7.147.2. Let R → S be a ring map. Assume that (1) R → S is faithfully flat, and (2) S is reduced. Then R is reduced. Proof. This is clear as R → S is injective.



Lemma 7.147.3. Let R → S be a ring map. Assume that (1) R → S is faithfully flat, and (2) S is a normal ring. Then R is a normal ring. Proof. Since S is reduced it follows that R is reduced. Let p be a prime of R. We have to show that Rp is a normal domain. Since Sp is faithfully over Rp too we may assume that R is local with maximal ideal m. Let q be a prime of S lying over m. Then we see that R → Sq is faithfully flat (Lemma 7.36.16). Hence we may assume S is local as well. In particular S is a normal domain. Since R → S is faithfully flat and S is a normal domain we see that R is a domain. Next, suppose that a/b is integral over R with a, b ∈ R. Then a/b ∈ S as S is normal. Hence a ∈ bS. This means that a : R → R/bR becomes the zero map after base change to S. By faithful flatness we see that a ∈ bR, so a/b ∈ R. Hence R is normal.  Lemma 7.147.4. Let R → S be a ring map. Assume that (1) R → S is faithfully flat, and (2) S is a regular ring. Then R is a regular ring. Proof. We see that R is Noetherian by Lemma 7.147.1. Let p ⊂ R be a prime. Choose a prime q ⊂ S lying over p. Then Lemma 7.103.8 applies to Rp → Sq and we conclude that Rp is regular. Since p was arbitrary we see R is regular.  Lemma 7.147.5. Let R → S be a ring map. Assume that (1) R → S is faithfully flat of finite presentation, and (2) S is Noetherian and has property (Sk ). Then R is Noetherian and has property (Sk ). Proof. We have already seen that (1) and (2) imply that R is Noetherian, see Lemma 7.147.1. Let p ⊂ R be a prime ideal. Choose a prime q ⊂ S lying over p which corresponds to a minimal prime of the fibre ring S ⊗R κ(p). Then A = Rp → Sq = B is a flat local ring homomorphism of Noetherian local rings with mA B an ideal of definition of B. Hence dim(A) = dim(B) (Lemma 7.104.7) and depth(A) = depth(B) (Lemma 7.146.1). Hence since B has (Sk ) we see that A has (Sk ).  Lemma 7.147.6. Let R → S be a ring map. Assume that (1) R → S is faithfully flat and of finite presentation, and (2) S is Noetherian and has property (Rk ). Then R is Noetherian and has property (Rk ).

7.147. DESCENDING PROPERTIES

593

Proof. We have already seen that (1) and (2) imply that R is Noetherian, see Lemma 7.147.1. Let p ⊂ R be a prime ideal and assume dim(Rp ) ≤ k. Choose a prime q ⊂ S lying over p which corresponds to a minimal prime of the fibre ring S⊗R κ(p). Then A = Rp → Sq = B is a flat local ring homomorphism of Noetherian local rings with mA B an ideal of definition of B. Hence dim(A) = dim(B) (Lemma 7.104.7). As S has (Rk ) we conclude that B is a regular local ring. By Lemma 7.103.8 we conclude that A is regular.  Lemma 7.147.7. Let R → S be a ring map. Assume that (1) R → S is smooth and surjective on spectra, and (2) S is a Nagata ring. Then R is a Nagata ring. Proof. Recall that a Nagata ring is the same thing as a Noetherian universally Japanese ring (Proposition 7.145.30). We have already seen that R is Noetherian in Lemma 7.147.1. Let R → A be a finite type ring map into a domain. According to Lemma 7.145.17 it suffices to check that A is N-1. It is clear that B = A ⊗R S is a finite type S-algebra and hence Nagata (Proposition 7.145.30). Since A → B is smooth (Lemma 7.127.4) we see that B is reduced (Lemma 7.146.6). Since B is Noetherian it has only a finite number of minimal primes q1 , . . . , qt (see Lemma 7.29.6). As A → B is flat each of these lies over (0) ⊂ A (by going down, see Lemma 7.36.17) The total ring of fractions Q(B) is the product of the Li = f.f.(B/qi ) (Lemmas 7.23.2 and 7.24.3). Moreover, the integral closure B 0 of B in Q(B) is the product of the integral closures Bi0 of the B/qi in the factors Li (compare with 0 Lemma 7.34.14). Since B is universally Q 0 Japanese the ring extensions B/qi ⊂ Bi are 0 finite and we conclude that B = Bi is finite over B. Since A → B is flat we see that any nonzerodivisor on A maps to a nonzerodivisor on B. The corresponding map Q(A) ⊗A B = (A \ {0})−1 A ⊗A B = (A \ {0})−1 B → Q(B) is injective (we used Lemma 7.11.15). Via this map A0 maps into B 0 . This induces a map A0 ⊗A B −→ B 0 which is injective (by the above and the flatness of A → B). Since B 0 is a finite B-module and B is Noetherian we see that A0 ⊗A B is a finite B-module. Hence there exist finitely many elements xi ∈ A0 such that the elements xi ⊗ 1 generate A0 ⊗A B as a B-module. Finally, by faithful flatness of A → B we conclude that the xi also generated A0 as an A-module, and we win.  Remark 7.147.8. The property of being “universally catenary” does not descend; not even along ´etale ring maps. In Examples, Section 66.9 there is a construction of a finite ring map A → B with A local Noetherian and not universally catenary, B semi-local with two maximal ideals m, n with Bm and Bn regular of dimension 2 and 1 respectively, and the same residue fields as that of A. Moreover, mA generates the maximal ideal in both Bm and Bn (so A → B is unramified as well as finite). By Lemma 7.139.10 there exists a local ´etale ring map A → A0 such that B ⊗A A0 = B1 × B2 decomposes with A0 → Bi surjective. This shows that A0 has two minimal primes qi with A0 /qi ∼ = Bi . Since Bi is regular local (since it is ´etale over either Bm or Bn ) we conclude that A0 is universally catenary.

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7.148. Geometrically normal algebras In this section we put some applications of ascent and descent of properties of rings. Lemma 7.148.1. Let k be a field. Let A be a k-algebra. The following properties of A are equivalent: (1) k 0 ⊗k A is a normal ring for every field extension k ⊂ k 0 , (2) k 0 ⊗k A is a normal ring for every finitely generated field extension k ⊂ k 0 , and (3) k 0 ⊗k A is a normal ring for every finite purely inseparable extension k ⊂ k0 . where normal ring is as defined in Definition 7.34.10. Proof. It is clear that (1) ⇒ (2) ⇒ (3). Assume (2) and let k ⊂ k 0 be any field extension. Then we can write k 0 = colimi ki as a directed colimit of finitely generated field extensions. Hence we see that k 0 ⊗k A = colimi ki ⊗k A is a directed colimit of normal rings. Thus we see that k 0 ⊗k A is a normal ring by Lemma 7.34.15. Hence (1) holds. Assume (3) and let k ⊂ K be a finitely generated field extension. By Lemma 7.43.3 we can find a diagram / K0 KO O k

/ k0

where k ⊂ k 0 , K ⊂ K 0 are finite purely inseparable field extensions such that k 0 ⊂ K 0 is separable. By Lemma 7.142.9 there exists a smooth k 0 -algebra B such that K 0 is the fraction field of B. Now we can argue as follows: Step 1: k 0 ⊗k A is a normal ring because we assumed (3). Step 2: B ⊗k0 k 0 ⊗k A is a normal ring as k 0 ⊗k A → B ⊗k0 k 0 ⊗k A is smooth (Lemma 7.127.4) and ascent of normality along smooth maps (Lemma 7.146.7). Step 3. K 0 ⊗k0 k 0 ⊗k A = K 0 ⊗k A is a normal ring as it is a localization of a normal ring (Lemma 7.34.12). Step 4. Finally K ⊗k A is a normal ring by descent of normality along the faithfully flat ring map K ⊗k A → K 0 ⊗k A (Lemma 7.147.3). This proves the lemma.  Definition 7.148.2. Let k be a field. A k-algebra R is called geometrically normal over k if the equivalent conditions of Lemma 7.148.1 hold. Lemma 7.148.3. Let k be a field. A localization of a geometrically normal kalgebra is geometrically normal. Proof. This is clear as being a normal ring is checked at the localizations at prime ideals.  Lemma 7.148.4. Let k be a field. Let A, B be k-algebras. Assume A is geometrically normal over k and B is a normal ring. Then A ⊗k B is a normal ring. Proof. Let r be a prime ideal of A ⊗k B. Denote p, resp. q the corresponding prime of A, resp. B. Then (A ⊗k B)r is a localization of Ap ⊗k Bq . Hence it suffices to prove the result for the ring Ap ⊗k Bq , see Lemma 7.34.12 and Lemma 7.148.3. Thus we may assume A and B are domains.

7.149. GEOMETRICALLY REGULAR ALGEBRAS

595

Assume that A and B are domains with fractions fields K and L. Note that B is the filtered colimit of its finite type normal k-sub algebras (as k is a Nagata ring, see Proposition 7.145.31, and hence the integral closure of a finite type k-sub algebra is still a finite type k-sub algebra by Proposition 7.145.30). By Lemma 7.34.15 we reduce to the case that B is of finite type over k. Assume that A and B are domains with fractions fields K and L and B of finite type over k. In this case the ring K ⊗k B is of finite type over K, hence Noetherian (Lemma 7.29.1). In particular K ⊗k B has finitely many minimal primes (Lemma 7.29.6). Since A → A ⊗k B is flat, this implies that A ⊗k B has finitely many minimal primes (by going down for flat ring maps – Lemma 7.36.17 – these primes all lie over (0) ⊂ A). Thus it suffices to prove that A ⊗k B is integrally closed in its total ring of fractions (Lemma 7.34.14). We claim that K ⊗k B and A ⊗k L are both normal rings. If this is true then any element x of Q(A ⊗k B) which is integral over A ⊗k B is (by Lemma 7.34.11) contained in K ⊗k B ∩ A ⊗k L = A ⊗k B and we’re done. Since A ⊗K L is a normal ring by assumption, it suffices to prove that K ⊗k B is normal. As A is geometrically normal over k we see K is geometrically normal S over k (Lemma 7.148.3) hence K is geometrically reduced over k. Hence K = Ki is the union of finitely generated field extensions of k which are geometrically reduced (Lemma 7.41.2). Each Ki is the localization of a smooth k-algebra (Lemma 7.142.9). So Ki ⊗k B is the localization of a smooth B-algebra hence normal (Lemma 7.146.7). Thus K ⊗k B is a normal ring (Lemma 7.34.15) and we win.  7.149. Geometrically regular algebras Let k be a field. Let A be a Noetherian k-algebra. Let k ⊂ K be a finitely generated field extension. Then the ring K ⊗k A is Noetherian as well, see Lemma 7.29.7. Thus the following lemma makes sense. Lemma 7.149.1. Let k be a field. Let A be a k-algebra. Assume A is Noetherian. The following properties of A are equivalent: (1) k 0 ⊗k A is regular for every finitely generated field extension k ⊂ k 0 , and (2) k 0 ⊗k A is regular for every finite purely inseparable extension k ⊂ k 0 . Here regular ring is as in Definition 7.103.6. Proof. The lemma makes sense by the remarks preceding the lemma. It is clear that (1) ⇒ (2). Assume (2) and let k ⊂ K be a finitely generated field extension. By Lemma 7.43.3 we can find a diagram / K0 KO O / k0 k where k ⊂ k , K ⊂ K are finite purely inseparable field extensions such that k 0 ⊂ K 0 is separable. By Lemma 7.142.9 there exists a smooth k 0 -algebra B such that K 0 is the fraction field of B. Now we can argue as follows: Step 1: k 0 ⊗k A is a regular ring because we assumed (2). Step 2: B ⊗k0 k 0 ⊗k A is a regular ring as k 0 ⊗k A → B ⊗k0 k 0 ⊗k A is smooth (Lemma 7.127.4) and ascent of regularity along 0

0

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smooth maps (Lemma 7.146.8). Step 3. K 0 ⊗k0 k 0 ⊗k A = K 0 ⊗k A is a regular ring as it is a localization of a regular ring (immediate from the definition). Step 4. Finally K ⊗k A is a regular ring by descent of regularity along the faithfully flat ring map K ⊗k A → K 0 ⊗k A (Lemma 7.147.4). This proves the lemma.  Definition 7.149.2. Let k be a field. Let R be a Noetherian k-algebra. The k-algebra R is called geometrically regular over k if the equivalent conditions of Lemma 7.149.1 hold. It is clear from the definition that K ⊗k R is a geometrically regular algebra over K for any finitely generated field extension K of k. We will see later (More on Algebra, Proposition 12.28.1) that it suffices to check R ⊗k k 0 is regular whenever k ⊂ k 0 ⊂ k 1/p (finite). Lemma 7.149.3. Let k be a field. Let A → B be a faithfully flat k-algebra map. If B is geometrically regular over k, so is A. Proof. Assume B is geometrically regular over k. Let k ⊂ k 0 be a finite, purely inseparable extension. Then A ⊗k k 0 → B ⊗k k 0 is faithfully flat as a base change of A → B (by Lemmas 7.28.3 and 7.36.6) and B ⊗k k 0 is regular by our assumption on B over k. Then A ⊗k k 0 is regular by Lemma 7.147.4.  Lemma 7.149.4. Let k be a field. Let A → B be a smooth ring map of k-algebras. If A is geometrically regular over k, then B is geometrically regular over k. Proof. Let k ⊂ k 0 be a finitely generated field extension. Then A ⊗k k 0 → B ⊗k k 0 is a smooth ring map (Lemma 7.127.4) and A ⊗k k 0 is regular. Hence B ⊗k k 0 is regular by Lemma 7.146.8.  Lemma 7.149.5. Let k be a field. Let A be an algebra over k. Let k = colim ki be a directed colimit of subfields. If A is geometrically regular over each ki , then A is geometrically regular over k. Proof. Let k ⊂ k 0 be a finite purely inseparable field extension. We can get k 0 by adjoining finitely many variables to k and imposing finitely many polynomial relations. Hence we see that there exists an i and a finite purely inseparable field extension ki ⊂ ki0 such that ki = k ⊗ki ki0 . Thus A ⊗k k 0 = A ⊗ki ki0 and the lemma is clear.  Lemma 7.149.6. Let k ⊂ k 0 be a separable algebraic field extension. Let A be a an algebra over k 0 . Then A is geometrically regular over k if and only if it is geometrically regular over k 0 . Proof. Let k ⊂ L be a finite purely inseparable field extension. Then L0 = k 0 ⊗k L is a field (see material in Section 7.39) and A ⊗k L = A ⊗k0 L0 . Hence if A is geometrically regular over k 0 , then A is geometrically regular over k. Assume A is geometrically regular over k. Since k 0 is the filtered colimit of finite extensions of k we may assume by Lemma 7.149.5 that k 0 /k is finite separable. Consider the ring maps k 0 → A ⊗k k 0 → A. Note that A ⊗k k 0 is geometrically regular over k 0 as a base change of A to k 0 . Note that A⊗k k 0 → A is the base change of k 0 ⊗k k 0 → k 0 by the map k 0 → A. Since k 0 /k is an ´etale extension of rings, we see that k 0 ⊗k k 0 → k 0 is ´etale (Lemma 7.133.3). Hence A is geometrically regular over k 0 by Lemma 7.149.4. 

7.151. COLIMITS AND MAPS OF FINITE PRESENTATION, II

597

7.150. Geometrically Cohen-Macaulay algebras This section is a bit of a misnomer, since Cohen-Macaulay algebras are automatically geometrically Cohen-Macaulay. Namely, see Lemma 7.122.6 and Lemma 7.150.2 below. Lemma 7.150.1. Let k be a field and let k ⊂ K and k ⊂ L be two field extensions such that one of them is a field extension of finite type. Then K ⊗k L is a Noetherian Cohen-Macaulay ring. Proof. The ring K ⊗k L is Noetherian by Lemma 7.29.7. Say K is a finite extension of the purely transcendental extension k(t1 , . . . , tr ). Then k(t1 , . . . , tr )⊗k L → K ⊗k L is a finite free ring map. By Lemma 7.104.9 it suffices to show that k(t1 , . . . , tr )⊗k L is Cohen-Macaulay. This is clear because it is a localization of the polynomial ring L[t1 , . . . , tr ]. (See for example Lemma 7.97.7 for the fact that a polynomial ring is Cohen-Macaulay.)  Lemma 7.150.2. Let k be a field. Let S be a Noetherian k-algebra. Let k ⊂ K be a finitely generated field extension, and set SK = K ⊗k S. Let q ⊂ S be a prime of S. Let qK ⊂ SK be a prime of SK lying over q. Then Sq is Cohen-Macaulay if and only if (SK )qK is Cohen-Macaulay. Proof. By Lemma 7.29.7 the ring SK is Noetherian. Hence Sq → (SK )qK is a flat local homomorphism of Noetherian local rings. Note that the fibre (SK )q /q(SK )q ∼ = (κ(q) ⊗k K)q0 K

K

is the localization of the Cohen-Macaulay (Lemma 7.150.1) ring κ(q) ⊗k K at a suitable prime ideal q0 . Hence the lemma follows from Lemma 7.146.3.  7.151. Colimits and maps of finite presentation, II This section is a continuation of Section 7.119. We start with an application of the openness of flatness. It says that we can approximate flat modules by flat modules which is useful. Lemma 7.151.1. Let R → S be a ring map. Let M be an S-module. Assume that (1) R → S is of finite presentation, (2) M is a finitely presented S-module, and (3) M is flat over R. In this case we have the following: (1) There exists a finite type Z-algebra R0 and a finite type ring map R0 → S0 and a finite S0 -module M0 such that M0 is flat over R0 , together with a ring maps R0 → R and S0 → S and an S0 -module map M0 → M such that S ∼ = R ⊗R0 S0 and M = S ⊗S0 M0 . (2) If R = colimλ∈Λ Rλ is written as a directed colimit, then there exists a λ and a ring map Rλ → Sλ of finite presentation, and an Sλ -module Mλ of finite presentation such that Mλ is flat over Rλ and such that S = R ⊗Rλ Sλ and M = S ⊗Sλ Mλ . (3) If (R → S, M ) = colimλ∈Λ (Rλ → Sλ , Mλ ) is written as a directed colimit such that (a) Rµ ⊗Rλ Sλ → Sµ and Sµ ⊗Sλ Mλ → Mµ are isomorphisms for µ ≥ λ,

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(b) Rλ → Sλ is of finite presentation, (c) Mλ is a finitely presented Sλ -module, then for all sufficiently large λ the module Mλ is flat over Rλ . Proof. We first write (R → S, M ) as the directed colimit of a system (Rλ → Sλ , Mλ ) as in as in Lemma 7.119.15. Let q ⊂ S be a prime. Let p ⊂ R, qλ ⊂ Sλ , and pλ ⊂ Rλ the corresponding primes. As seen in the proof of Theorem 7.121.4 ((Rλ )pλ , (Sλ )qλ , (Mλ )qλ ) is a system as in Lemma 7.119.11, and hence by Lemma 7.120.3 we see that for some λq ∈ Λ for all λ ≥ λq the module Mλ is flat over Rλ at the prime qλ . By Theorem 7.121.4 we get an open subset Uλ ⊂ Spec(Sλ ) such that Mλ flat over Rλ at all the primes of Uλ . Denote Vλ ⊂ Spec(S) the inverse image of Uλ under the map Spec(S) → Spec(Sλ ). The argument above shows that for every q ∈ Spec(S) there exists a λq such that q ∈ Vλ for all λ ≥ λq . Since Spec(S) is quasi-compact we see this implies there exists a single λ0 ∈ Λ such that Vλ0 = Spec(S). The complement Spec(Sλ0 ) \ Uλ0 is V (I) for some ideal I ⊂ Sλ0 . As Vλ0P = Spec(S) we see that IS = S. Choose f1 , . . . , fr ∈ I and s1 , . . . , sn ∈ S such that fi si = 1. SinceP colim Sλ = S, after increasing λ0 we may assume there exist si,λ0 ∈ Sλ0 such that fi si,λ0 = 1. Hence for this λ0 we have Uλ0 = Spec(Sλ0 ). This proves (1). Proof of (2). Let (R0 → S0 , M0 ) be as in (1) and suppose that R = colim Rλ . Since R0 is a finite type Z algebra, there exists a λ and a map R0 → Rλ such that R0 → Rλ → R is the given map R0 → R (see Lemma 7.119.2). Then, part (2) follows by taking Sλ = Rλ ⊗R0 S0 and Mλ = Sλ ⊗S0 M0 . Finally, we come to the proof of (3). Let (Rλ → Sλ , Mλ ) be as in (3). Choose (R0 → S0 , M0 ) and R0 → R as in (1). As in the proof of (2), there exists a λ0 and a ring map R0 → Rλ0 such that R0 → Rλ0 → R is the given map R0 → R. Since S0 is of finite presentation over R0 and since S = colim Sλ we see that for some λ1 ≥ λ0 we get an R0 -algebra map S0 → Sλ1 such that the composition S0 → Sλ1 → S is the given map S0 → S (see Lemma 7.119.2). For all λ ≥ λ1 this gives maps Ψλ : Rλ ⊗R0 S0 −→ Rλ ⊗Rλ1 Sλ1 ∼ = Sλ the last isomorphism by assumption. By construction colimλ Ψλ is an isomorphism. Hence Ψλ is an isomorphism for all λ large enough by Lemma 7.119.6. In the same vein, there exists a λ2 ≥ λ1 and an S0 -module map M0 → Mλ2 such that M0 → Mλ2 → M is the given map M0 → M (see Lemma 7.119.3). For λ ≥ λ2 there is an induced map Sλ ⊗S M0 −→ Sλ ⊗S Mλ ∼ = Mλ 0

2

λ2

and for λ large enough this map is an isomorphism by Lemma 7.119.4. This implies (3) because M0 is flat over R0 .  Lemma 7.151.2. Let R → A → B be ring maps. Assume A → B faithfully flat of finite presentation. Then there exists a commutative diagram / B0 / A0 R

R

 /A

 /B

7.151. COLIMITS AND MAPS OF FINITE PRESENTATION, II

599

with R → A0 of finite presentation, A0 → B0 faithfully flat of finite presentation and B = A ⊗A0 B0 . Proof. We first prove the lemma with R replaced Z. By Lemma 7.151.1 there exists a diagram /A AO 0 O B0

/B

where A0 is of finite type over Z, B0 is flat of finite presentation over A0 such that B = A ⊗A0 B0 . As A0 → B0 is flat of finite presentation we see that the image of Spec(B0 ) → Spec(A0 ) is open, see Proposition 7.37.8. Hence the complement of the image is V (I0 ) for some ideal I0 ⊂ A0 . As A → B is faithfully flat the map Spec(B) → Spec(A) is surjective, see Lemma 7.36.15. Now we use that the base change P of the image is the image of the base change. Hence I0 A = A. Pick a relation fi ri = 1, with ri ∈ A, fi ∈ I0 . Then after enlarging A0 to contain the elements ri (and correspondingly enlarging B0 ) we see that A0 → B0 is surjective on spectra also, i.e., faithfully flat. Thus the lemma holds in case R = Z. In the general case, take the solution A00 → B00 just obtained and set A0 = A00 ⊗Z R, B0 = B00 ⊗Z R.  Lemma 7.151.3. Let A = colimi∈I Ai be a directed colimit of rings. Let 0 ∈ I and ϕ0 : B0 → C0 a map of A0 -algebras such that A ⊗A0 B0 −→ A ⊗A0 C0 is finite. If C0 is of finite type over A0 , then for some i ≥ 0 the map Ai ⊗A0 B0 −→ Ai ⊗A0 C0 is finite. Proof. Let x1 , . . . , xm be generators for C0 over A0 . Pick monic polynomials Pj ∈ A ⊗A0 B0 [T ] such that Pj (1 ⊗ xj ) = 0 in A ⊗A0 C0 . For some i ≥ 0 we can find Pj,i ∈ Ai ⊗A0 B0 [T ] mapping to Pj . Since ⊗ commutes with colimits we see that Pj,i (1 ⊗ xj ) is zero in Ai ⊗A0 C0 after possibly increasing i. Then this i works.  Lemma 7.151.4. Let A = colimi∈I Ai be a directed colimit of rings. Let 0 ∈ I and ϕ0 : B0 → C0 a map of A0 -algebras such that A ⊗A0 B0 −→ A ⊗A0 C0 is surjective. If C0 is of finite type over A0 , then for some i ≥ 0 the map Ai ⊗A0 B0 −→ Ai ⊗A0 C0 is surjective. Proof. Let x1 , . . . , xm be generators for C0 over A0 . Pick bj ∈ A ⊗A0 B0 mapping to 1 ⊗ xj in A ⊗A0 C0 . For some i ≥ 0 we can find bj,i ∈ Ai ⊗A0 B0 mapping to bj . Then this i works.  Lemma 7.151.5. Let A = colimi∈I Ai be a directed colimit of rings. Let 0 ∈ I and ϕ0 : B0 → C0 a map of A0 -algebras such that A ⊗A0 B0 −→ A ⊗A0 C0 is ´etale. If B0 is of finite type over A0 and C0 is of finite presentation over A0 , then for some i ≥ 0 the map Ai ⊗A0 B0 −→ Ai ⊗A0 C0 is ´etale. Proof. Note that B0 → C0 is of finite presentation, see Lemma 7.6.2. Write C0 = B0 [x1 , . . . , xn ]/(f1,0 , . . . , fm,0 ). Write Bi = Ai ⊗A0 B0 and Ci = Ai ⊗A0 C0 . Note that Ci = Bi [x1 , . . . , xn ]/(f1,i , . . . , fm,i ) where fj,i is the image of fj,0 in the polynomial ring over Bi . Write B = A ⊗A0 B0 and C = A ⊗A0 C0 . Note that

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C = B[x1 , . . . , xn ]/(f1 , . . . , fm ) where fj is the image of fj,0 in the polynomial ring over B. The assumption is that the map M d : (f1 , . . . , fm )/(f1 , . . . , fm )2 −→ Cdxk is an isomorphism. Thus for sufficiently large i we can find elements ξk,i ∈ (f1,i , . . . , fm,i )/(f1,i , . . . , fm,i )2 L with Ci dxk . Moreover, on increasing i if necessary, we see that P dξk,i = dxk in (∂fj,i /∂xk )ξk,i = fj,i mod (f1,i , . . . , fm,i )2 since this is true in the limit. Then this i works.  The following lemma is an application of the results above which doesn’t seem to fit well anywhere else. Lemma 7.151.6. Let R → S be a faithfully flat ring map of finite presentation. Then there exists a commutative diagram / S0 >

S_ R

where R → S 0 is quasi-finite, faithfully flat and of finite presentation. Proof. As a first step we reduce this lemma to the case where R is of finite type over Z. By Lemma 7.151.2 there exists a diagram SO 0

/S O

R0

/R

where R0 is of finite type over Z, and S0 is faithfully flat of finite presentation over R0 such that S = R ⊗R0 S0 . If we prove the lemma for the ring map R0 → S0 , then the lemma follows for R → S by base change, as the base change of a quasifinite ring map is quasi-finite, see Lemma 7.114.8. (Of course we also use that base changes of flat maps are flat and base changes of maps of finite presentation are of finite presentation.) Assume R → S is a faithfully flat ring map of finite presentation and that R is Noetherian (which we may assume by the preceding paragraph). Let W ⊂ Spec(S) be the open set of Lemma 7.122.4. As R → S is faithfully flat the map Spec(S) → Spec(R) is surjective, see Lemma 7.36.15. By Lemma 7.122.5 the map W → Spec(R) is also surjective. Hence by replacing S with a product Sg1 ×. . .×Sgm we may assume W = Spec(S); here we use that Spec(R) is quasi-compact (Lemma 7.16.10), and that the map Spec(S) → Spec(R) is open (Proposition 7.37.8). Suppose that p ⊂ R is a prime. Choose a prime q ⊂ S lying over p which corresponds to a maximal ideal of the fibre ring S ⊗R κ(p). The Noetherian local ring S q = Sq /pSq is Cohen-Macaulay, say of dimension d. We may choose f1 , . . . , fd in the maximal ideal of Sq which map to a regular sequence in S q . Choose a common denominator g ∈ S, g 6∈ q of f1 , . . . , fd , and consider the R-algebra S 0 = Sg /(f1 , . . . , fd ).

7.152. OTHER CHAPTERS

601

By construction there is a prime ideal q0 ⊂ S 0 lying over p and corresponding to q (via Sg → Sg0 ). Also by construction the ring map R → S 0 is quasi-finite at q as the local ring Sq0 0 /pSq0 0 = Sq /(f1 , . . . , fd ) + pSq = S q /(f 1 , . . . , f d ) has dimension zero, see Lemma 7.114.2. Also by construction R → S 0 is of finite presentation. Finally, by Lemma 7.92.3 the local ring map Rp → Sq0 0 is flat (this is where we use that R is Noetherian). Hence, by openness of flatness (Theorem 7.121.4), and openness of quasi-finiteness (Lemma 7.115.14) we may after replacing g by gg 0 for a suitable g 0 ∈ S, g 0 6∈ q assume that R → S 0 is flat and quasifinite. The image Spec(S 0 ) → Spec(R) is open and contains p. In other words we have shown a ring S 0 as in the statement of the lemma exists (except possibly the faithfulness part) whose image contains any given prime. Using one more time the quasi-compactness of Spec(R) we see that a finite product of such rings does the job.  7.152. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent

(32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61)

Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms

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(62) Properties of Algebraic Stacks (63) Morphisms of Algebraic Stacks (64) Cohomology of Algebraic Stacks (65) Introducing Algebraic Stacks (66) Examples (67) Exercises

(68) (69) (70) (71) (72)

Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License (73) Auto Generated Index

CHAPTER 8

Brauer groups 8.1. Introduction A reference are the lectures by Serre in the Seminaire Cartan, see [Ser55a]. Serre in turn refers to [Deu68] and [ANT44]. We changed some of the proofs, in particular we used a fun argument of Rieffel to prove Wedderburn’s theorem. Very likely this change is not an improvement and we strongly encourage the reader to read the original exposition by Serre. 8.2. Noncommutative algebras Let k be a field. In this chapter an algebra A over k is a possibly noncommutative ring A together with a ring map k → A such that k maps into the center of A and such that 1 maps to an identity element of A. An A-module is a right A-module such that the identity of A acts as the identity. Definition 8.2.1. Let A be a k-algebra. We say A is finite if dimk (A) < ∞. In this case we write [A : k] = dimk (A). Definition 8.2.2. A skew field is a possibly noncommutative ring with an identity element 1, with 1 6= 0, such that in which every nonzero element has a multiplicative inverse. A skew field is a k-algebra for some k (e.g., for the prime field contained in it). We will use below that any module over a skew field is free because a maximal linearly independent set of vectors forms a basis and exists by Zorn’s lemma. Definition 8.2.3. Let A be a k-algebra. We say an A-module M is simple if it is nonzero and the only A-submodules are 0 and M . We say A is simple if the only two-sided ideals of A are 0 and A. Definition 8.2.4. A k-algebra A is central if the center of A is the image of k → A. Definition 8.2.5. Given a k-algebra A we denote Aop the k-algebra we get by reversing the order of multiplication in A. This is called the opposite algebra. 8.3. Wedderburn’s theorem The following cute argument can be found in a paper of Rieffel, see [Rie65]. The proof could not be simpler (quote from Carl Faith’s review). Lemma 8.3.1. Let A be a possibly noncommutative ring with 1 which contains no nontrivial two-sided ideal. Let M be a nonzero right ideal in A, and view M as a right A-module. Then A coincides with the bicommutant of M . 603

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Proof. Let A0 = EndA (M ), and let A00 = EndA0 (M ) (the bicommutant of M ). Let R : A → A00 be the natural homomorphism R(a)(m) = ma. Then R is injective, since R(1) = idM and A contains no nontrivial two-sided ideal. We claim that R(M ) is a right ideal in A00 . Namely, R(m)a00 = R(ma00 ) for a00 ∈ A00 and m in M , because left multiplication of M by any element n of M represents an element of A0 , and so (nm)a00 = n(ma00 ), that is, (R(m)a00 )(n) = R(ma00 )(n) for all n in M . Finally, the product ideal AM is a two-sided ideal, and so A = AM . Thus R(A) = R(A)R(M ), so that R(A) is a right ideal in A00 . But R(A) contains the identity element of A00 , and so R(A) = A00 .  Lemma 8.3.2. Let A be a k-algebra. If A is finite, then (1) (2) (3) (4)

A has a simple module, any nonzero module contains a simple submodule, a simple module over A has finite dimension over k, and if M is a simple A-module, then EndA (M ) is a skew field.

Proof. Of course (1) follows from (2) since A is a nonzero A-module. For (2), any submodule of minimal (finite) dimension as a k-vector space will be simple. There exists a finite dimensional one because a cyclic submodule is one. If M is simple, then mA ⊂ M is a sub-module, hence we see (3). Any nonzero element of EndA (M ) is an isomorphism, hence (4) holds.  Theorem 8.3.3. Let A be a simple finite k-algebra. Then A is a matrix algebra over a finite k-algebra K which is a skew field. Proof. We may choose a simple submodule M ⊂ A and then the k-algebra K = EndA (M ) is a skew field, see Lemma 8.3.2. By Lemma 8.3.1 we see that A = EndK (M ). Since K is a skew field and M is finitely generated (since dimk (M ) < ∞) we see that M is finite free as a left K-module. It follows immediately that A∼  = Mat(n × n, K op ). 8.4. Lemmas on algebras Let A be a k-algebra. Let B ⊂ A be a subalgebra. The centralizer of B in A is the subalgebra C = {y ∈ A | xy = yx for all x ∈ B}. It is a k-algebra. Lemma 8.4.1. Let A, A0 be k-algebras. Let B ⊂ A, B 0 ⊂ A0 be subalgebras with centralizers C, C 0 . Then the centralizer of B ⊗k B 0 in A ⊗k A0 is C ⊗k C 0 . Proof. Denote C 00 ⊂ A ⊗k A0 the centralizer of B ⊗k B 0 . It is clear that C ⊗k C 0 ⊂ C 00 . Conversely, every element of C 00 commutes with B ⊗ 1 hence is contained in C ⊗k A0 . Similarly C 00 ⊂ A ⊗k C 0 . Thus C 00 ⊂ C ⊗k A0 ∩ A ⊗k C 0 = C ⊗k C 0 .  Lemma 8.4.2. Let A be a finite simple k-algebra. Then the center k 0 of A is a finite field extension of k. Proof. Write A = Mat(n × n, K) for some skew field K finite over k, see Theorem 8.3.3. By Lemma 8.4.1 the center of A is k ⊗k k 0 where k 0 ⊂ K is the center of K. Since the center of a skew field is a field, we win. 

8.4. LEMMAS ON ALGEBRAS

605

Lemma 8.4.3. Let V be a k vector space. Let K be a central k-algebra which is a skew field. Let W ⊂ V ⊗k K be a two-sided K-sub vector space. Then W is generated as a left K-vector space by W ∩ (V ⊗ 1). Proof. Let V 0 ⊂ V be the k-sub vector space generated by v ∈ V such that v ⊗ 1 ∈ W . Then V 0 ⊗k K ⊂ W and we have W/V 0 ⊗k K ⊂ V /V 0 ⊗k K. If v ∈ V /V 0 is a nonzero vector such that v ⊗ 1 is contained in W/V 0 ⊗k K, then we see that v ⊗ 1 ∈ W where v ∈ V lifts v. This contradicts our construction of V 0 . Hence we may replace V by V /V 0 and W by W/V 0 ⊗k K and it suffices to prove that W ∩ (V ⊗ 1) is nonzero if W is nonzero. To see this let w ∈ W be a nonzero element which can be written as W = P −1 and assume i=1,...,n vi ⊗ ki with n minimal. We may right multiply with k1 that k1 = 1. If n = 1, then we win because v1 ⊗ 1 ∈ W . If n > 1, then we see that for any c ∈ K X cv − vc = vi ⊗ (cki − ki c) ∈ W i=2,...,n

and hence cki − ki c = 0 by minimality of n. This implies that ki is in the center P of K which is k by assumption. Hence v = (v1 + ki vi ) ⊗ 1 contradicting the minimality of n.  Lemma 8.4.4. Let A be a k-algebra. Let K be a central k-algebra which is a skew field. Then any two-sided ideal I ⊂ A⊗k K is of the form J ⊗k K for some two-sided ideal J ⊂ A. In particular, if A is simple, then so is A ⊗k K. Proof. Set J = {a ∈ A | a⊗1 ∈ I}. This is a two-sided ideal of A. And I = J ⊗k K by Lemma 8.4.3.  Lemma 8.4.5. Let R be a possibly noncommutative ring. Let n ≥ 1 be an integer. Let Rn = Mat(n × n, R). (1) The functors M 7→ M ⊕n and N 7→ N e11 define quasi-inverse equivalences of categories ModR ↔ ModRn . (2) A two-sided ideal of Rn is of the form IRn for some two-sided ideal I of R. (3) The center of Rn is equal to the center of R. L Proof. Part (1) proves itself. If J ⊂ Rn is a two-sided ideal, then J = eii Jejj and all of the summands eii Jejj are equal to each other and are a two-sided ideal I of R. This proves (2). Part (3) is clear.  Lemma (1) (2) (3)

8.4.6. Let A be a finite simple k-algebra. There exists exactly one simple A-module M up to isomorphism. Any finite A-module is a direct sum of copies of a simple module. Two finite A-modules are isomorphic if and only if they have the same dimension over k. (4) If A = Mat(n × n, K) with K a finite skew field extension of k, then M = K ⊕n is a simple A-module and EndA (M ) = K op . (5) If M is a simple A-module, then L = EndA (M ) is a skew field finite over k acting on the left on M , we have A = EndL (M ), and the centers of A and L agree. Also [A : k][L : k] = dimk (M )2 .

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(6) For a finite A-module N the algebra B = EndA (N ) is a matrix algebra over the skew field L of (3). Moreover EndB (N ) = A. Proof. By Theorem 8.3.3 we can write A = Mat(n×n, K) for some finite skew field extension K of k. By Lemma 8.4.5 the category of modules over A is equivalent to the category of modules over K. Thus (1), (2), and (3) hold because every module over K is free. Part (4) holds because the equivalence transforms the K-module K to M = K ⊕n . Using M = K ⊕n in (5) we see that L = K op . The statement about the center of L = K op follows from Lemma 8.4.5. The statement about EndL (M ) follows from the explicit form of M . The formula of dimensions is clear. Part (6) follows as N is isomorphic to a direct sum of copies of a simple module.  Lemma 8.4.7. Let A, A0 be two simple k-algebras one of which is finite and central over k. Then A ⊗k A0 is simple. Proof. Suppose that A0 is finite and central over k. Write A0 = Mat(n × n, K 0 ), see Theorem 8.3.3. Then the center of K 0 is k and we conclude that A ⊗k K 0 is simple by Lemma 8.4.4. Hence A ⊗k A0 = Mat(n × n, A ⊗k K 0 ) is simple by Lemma 8.4.5.  Lemma 8.4.8. The tensor product of finite central simple algebras over k is finite, central, and simple. Proof. Combine Lemmas 8.4.1 and 8.4.7.



Lemma 8.4.9. Let A be a finite central simple algebra over k. Let k ⊂ k 0 be a field extension. Then A0 = A ⊗k k 0 is a finite central simple algebra over k 0 . Proof. Combine Lemmas 8.4.1 and 8.4.7.



Lemma 8.4.10. Let A be a finite central simple algebra over k. Then A ⊗k Aop ∼ = Mat(n × n, k) where n = [A : k]. Proof. By Lemma 8.4.8 the algebra A ⊗k Aop is simple. Hence the map A ⊗k Aop −→ Endk (A),

a ⊗ a0 7−→ (x 7→ axa0 )

is injective. Since both sides of the arrow have the same dimension we win.



8.5. The Brauer group of a field Let k be a field. Consider two finite central simple algebras A and B over k. We say A and B are similar if there exist n, m > 0 such that Mat(n×n, A) ∼ = Mat(m×m, B) as k-algebras. Lemma 8.5.1. Similarity. (1) Similarity defines an equivalence relation on the set of isomorphism classes of finite central simple algebras over k. (2) Every similarity class contains a unique (up to isomorphism) finite central skew field extension of k. (3) If A = Mat(n × n, K) and B = Mat(m × m, K 0 ) for some finite central skew fields K, K 0 over k then A and B are similar if and only if K ∼ = K0 as k-algebras.

8.6. SKOLEM-NOETHER

607

Proof. Note that by Wedderburn’s theorem (Theorem 8.3.3) we can always write a finite central simple algebra as a matrix algebra over a finite central skew field. Hence it suffices to prove the third assertion. To see this it suffices to show that if A = Mat(n × n, K) ∼ = Mat(m × m, K 0 ) = B then K ∼ = K 0 . To see this note that for a simple module M of A we have EndA (M ) = K op , see Lemma 8.4.6. Hence  A∼ = (K 0 )op and we win. = B implies K op ∼ Given two finite central simple k-algebras A, B the tensor product A ⊗k B is another, see Lemma 8.4.8. Moreover if A is similar to A0 , then A ⊗k B is similar to A0 ⊗k B because tensor products and taking matrix algebras commute. Hence tensor product defines an operation on equivalence classes of finite central simple algebras which is clearly associative and commutative. Finally, Lemma 8.4.10 shows that A ⊗k Aop is isomorphic to a matrix algebra, i.e., that A ⊗k Aop is in the similarity class of k. Thus we obtain an abelian group. Definition 8.5.2. Let k be a field. The Brauer group of k is the abelian group of similarity classes of finite central simple k-algebras defined above. Notation Br(k). For any map of fields k → k 0 we obtain a group homomorphism Br(k) −→ Br(k 0 ),

A 7−→ A ⊗k k 0

see Lemma 8.4.9. In other words, Br(−) is a functor from the category of fields to the category of abelian groups. Observe that the Brauer group of a field is zero if and only if every finite central skew field extension k ⊂ K is trivial. Lemma 8.5.3. The Brauer group of an algebraically closed field is zero. Proof. Let k ⊂ K be a finite central skew field extension. For any element x ∈ K the subring k[x] ⊂ K is a commutative finite integral k-sub algebra, hence a field, see Algebra, Lemma 7.33.17. Since k is algebraically closed we conclude that k[x] = k. Since x was arbitrary we conclude k = K.  Lemma 8.5.4. Let A be a finite central simple algebra over a field k. Then [A : k] is a square. Proof. This is true because A ⊗k k is a matrix algebra over k by Lemma 8.5.3.  8.6. Skolem-Noether Theorem 8.6.1. Let A be a finite central simple k-algebra. Let B be a simple k-algebra. Let f, g : B → A be two k-algebra homomorphisms. Then there exists an invertible element x ∈ A such that f (b) = xg(b)x−1 for all b ∈ B. Proof. Choose a simple A-module M . Set L = EndA (M ). Then L is a skew field with center k which acts on the left on M , see Lemmas 8.3.2 and 8.4.6. Then M has two B ⊗k Lop -module structures defined by m ·1 (b ⊗ l) = lmf (b) and m ·2 (b ⊗ l) = lmg(b). Since B ⊗k Lop is a finite simple k-algebra by Lemma 8.4.7 we see that these module structures are isomorphic by Lemma 8.4.6. Hence we find ϕ : M → M intertwining these operations. In particular ϕ is in the commutant of L which implies that ϕ is multiplication by some x ∈ A, see Lemma 8.4.6. Working out the definitions we see that x is a solution to our problem.  Lemma 8.6.2. Let A be a finite simple k-algebra. Any automorphism of A is inner. In particular, any automorphism of Mat(n × n, k) is inner.

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Proof. Note that A is a finite central simple algebra over the center of A which is a finite field extension of k, see Lemma 8.4.2. Hence the Skolem-Noether theorem (Theorem 8.6.1) applies.  8.7. The centralizer theorem Theorem 8.7.1. Let A be a finite central simple algebra over k, and let B be a simple subalgebra of A. Then (1) the centralizer C of B in A is simple, (2) [A : k] = [B : k][C : k], and (3) the centralizer of C in A is B. Proof. Throughout this proof we use the results of Lemma 8.4.6 freely. Choose a simple A-module M . Set L = EndA (M ). Then L is a skew field with center k which acts on the left on M and A = EndL (M ). Then M is a right B ⊗k Lop -module and C = EndB⊗k Lop (M ). Since the algebra B ⊗k Lop is simple by Lemma 8.4.7 we see that C is simple (by Lemma 8.4.6 again). Write B ⊗k Lop = Mat(m × m, K) for some skew field K finite over k. Then C = Mat(n × n, K op ) if M is isomorphic to a direct sum of n copies of the simple B ⊗k Lop -module K ⊕m (the lemma again). Thus we have dimk (M ) = nm[K : k], [B : k][L : k] = m2 [K : k], [C : k] = n2 [K : k], and [A : k][L : k] = dimk (M )2 (by the lemma again). We conclude that (2) holds. Part (3) follows because of (2) applied to C ⊂ A shows that [B : k] = [C 0 : k] where C 0 is the centralizer of C in A (and the obvious fact that B ⊂ C 0 ).  Lemma 8.7.2. Let A be a finite central simple algebra over k, and let B be a simple subalgebra of A. If B is a central k-algebra, then A = B ⊗k C where C is the (central simple) centralizer of B in A. Proof. We have dimk (A) = dimk (B ⊗k C) by Theorem 8.7.1. By Lemma 8.4.7 the tensor product is simple. Hence the natural map B ⊗k C → A is injective hence an isomorphism.  Lemma 8.7.3. Let A be a finite central simple algebra over k. If K ⊂ A is a subfield, then the following are equivalent (1) [A : k] = [K : k]2 , (2) K is its own centralizer, and (3) K is a maximal commutative subring. Proof. Theorem 8.7.1 shows that (1) and (2) are equivalent. It is clear that (3) and (2) are equivalent.  Lemma 8.7.4. Let A be a finite central skew field over k. Then every maximal subfield K ⊂ A satisfies [A : k] = [K : k]2 . Proof. Special case of Lemma 8.7.3.



8.8. Splitting fields Definition 8.8.1. Let A be a finite central simple k-algebra. We say a field extension k ⊂ k 0 splits A, or k 0 is a splitting field for A if A ⊗k k 0 is a matrix algebra over k 0 .

8.8. SPLITTING FIELDS

609

Another way to say this is that the class of A maps to zero under the map Br(k) → Br(k 0 ). Theorem 8.8.2. Let A be a finite central simple k-algebra. Let k ⊂ k 0 be a finite field extension. The following are equivalent (1) k 0 splits A, and (2) there exists a finite central simple algebra B similar to A such that k 0 ⊂ B and [B : k] = [k 0 : k]2 . Proof. Assume (2). It suffices to show that B ⊗k k 0 is a matrix algebra. We know that B ⊗k B op ∼ = Endk (B). Since k 0 is the centralizer of k 0 in B op by Lemma 8.7.3 we see that B ⊗k k 0 is the centralizer of k ⊗ k 0 in B ⊗k B op = Endk (B). Of course this centralizer is just Endk0 (B) where we view B as a k 0 vector space via the embedding k 0 → B. Thus the result. ∼ Endk0 (V ) for Assume (1). This means that we have an isomorphism A ⊗k k 0 = some k 0 -vector space V . Let B be the commutant of A in Endk (V ). Note that k 0 sits in B. By Lemma 8.7.2 the classes of A and B add up to zero in Br(k). From the dimension formula in Theorem 8.7.1 we see that [B : k][A : k] = dimk (V )2 = [k 0 : k]2 dimk0 (V )2 = [k 0 : k]2 [A : k]. Hence [B : k] = [k 0 : k]2 . Thus we have proved the result for the opposite to the Brauer class of A. However, k 0 splits the Brauer class of A if and only if it splits the Brauer class of the opposite algebra, so we win anyway.  Lemma 8.8.3. A maximal subfield of a finite central skew field K over k is a splitting field for K. Proof. Combine Lemma 8.7.4 with Theorem 8.8.2.



Lemma 8.8.4. Consider a finite central skew field K over k. Let d2 = [K : k]. For any finite splitting field k 0 for K the degree [k 0 : k] is divisible by d. Proof. By Theorem 8.8.2 there exists a finite central simple algebra B in the Brauer class of K such that [B : k] = [k 0 : k]2 . By Lemma 8.5.1 we see that B = Mat(n × n, K) for some n. Then [k 0 : k]2 = n2 d2 whence the result.  Proposition 8.8.5. Consider a finite central skew field K over k. There exists a maximal subfield k ⊂ k 0 ⊂ K which is separable over k. In particular, every Brauer class has a finite separable spitting field. Proof. Since every Brauer class is represented by a finite central skew field over k, we see that the second statement follows from the first by Lemma 8.8.3. To prove the first statement, suppose that we are given a separable subfield k 0 ⊂ K. Then the centralizer K 0 of k 0 in K has center k 0 , and the problem reduces to finding a maximal subfield of K 0 separable over k 0 . Thus it suffices to prove, if k 6= K, that we can find an element x ∈ K, x 6∈ K which is separable over k. This statement is clear in characteristic zero. Hence we may assume that k has characteristic p > 0. If the ground field k is finite then, the result is clear as well (because extensions of finite fields are always separable). Thus we may assume that k is an infinite field of positive characteristic. To get a contradiction assume no element of K is separable over k. By the discussion in Algebra, Section 7.39 this means the minimal polynomial of any x ∈ K is of the

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form T q − a where q is a power of p and a ∈ k. Since it is clear that every element of K has a minimal polynomial of degree ≤ dimk (K) we conclude that there exists a fixed p-power q such that xq ∈ k for all x ∈ K. Consider the map (−)q : K −→ K and write it out in terms of a k-basis {a1 , . . . , an } of K with a1 = 1. So X X ( xi ai )q = fi (x1 , . . . , xn )ai . Since multiplication on A is k-bilinear we see that each fi is a polynomial in x1 , . . . , xn (details omitted). The choice of q above and the fact that k is infinite shows that fi is identically zero for i ≥ 2. Hence we see that it remains zero on extending k to its algebraic closure k. But the algebra A ⊗k k is a matrix algebra, which implies there are some elements whose qth power is not central (e.g., e11 ). This is the desired contradiction.  The results above allow us to characterize finite central simple algebras as follows. Lemma 8.8.6. Let k be a field. For a k-algebra A the following are equivalent (1) A is finite central simple k-algebra, (2) A is a finite dimensional k-vector space, k is the center of A, and A has no nontrivial two-sided ideal, ¯ (3) there exists d ≥ 1 such that A ⊗k k¯ ∼ = Mat(d × d, k), sep ∼ (4) there exists d ≥ 1 such that A ⊗k k = Mat(d × d, k sep ), (5) there exist d ≥ 1 and a finite Galois extension k ⊂ k 0 such that A ⊗k0 k 0 ∼ = Mat(d × d, k 0 ), (6) there exist n ≥ 1 and a finite central skew field K over k such that A ∼ = Mat(n × n, K). The integer d is called the degree of A. Proof. The equivalence of (1) and (2) is a consequence of the definitions, see Section 8.2. Assume (1). By Proposition 8.8.5 there exists a separable splitting field k ⊂ k 0 for A. Of course, then a Galois closure of k 0 /k is a splitting field also. Thus we see that (1) implies (5). It is clear that (5) ⇒ (4) ⇒ (3). Assume (3). Then A ⊗k k is a finite central simple k-algebra for example by Lemma 8.4.5. This trivially implies that A is a finite central simple k-algebra. Finally, the equivalence of (1) and (6) is Wedderburn’s theorem, see Theorem 8.3.3.  8.9. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra

(11) (12) (13) (14) (15) (16) (17) (18) (19) (20)

Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings

8.9. OTHER CHAPTERS

(21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44)

Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces

(48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

611

More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 9

Sites and Sheaves 9.1. Introduction The notion of a site was introduced by Grothendieck to be able to study sheaves in the ´etale topology of schemes. The basic reference for this notion is perhaps [AGV71]. Our notion of a site differs from that in [AGV71]; what we call a site is called a category endowed with a pretopology in [AGV71, Expos´e II, D´efinition 1.3]. The reason we do this is that in algebraic geometry it is often convenient to work with a given class of coverings, for example when defining when a property of schemes is local in a given topology, see Descent, Section 31.11. Our exposition will closely follow [Art62]. We will not use universes. 9.2. Presheaves Let C be a category. A presheaf of sets is a contravariant functor F from C to Sets (see Categories, Remark 4.2.11). So for every object U of C we have a set F(U ). The elements of this set are called the sections of F over U . For every morphism f : V → U the map F(f ) : F(U ) → F(V ) is called the restricton map and is often denoted f ∗ : F(U ) → F(V ). Another way of expressing this is to say that f ∗ (s) is the pullback of s via f . Functoriality means that g ∗ f ∗ (s) = (f ◦ g)∗ (s). Sometimes we use the notation s|V := f ∗ (s). This notation is consistent with the notion of restriction of functions from topology because if W → V → U are morphisms in C and s is a section of F over U then s|W = (s|V )|W by the functorial nature of F. Of course we have to be careful since it may very well happen that there is more than one morphism V → U and it is certainly not going to be the case that the corresponding pullback maps are equal. Definition 9.2.1. A presheaf of sets on C is a contravariant functor from C to Sets. Morphisms of presheaves are transformations of functors. The category of presheaves of sets is denoted PSh(C). Note that for any object U of C the functor of points hU , see Categories, Example 4.3.4 is a presheaf. These are called the representable presheaves. These presheaves have the pleasing property that for any presheaf F we have MorPSh(C) (hU , F) = F(U ). This is similar to the Yoneda lemma (Categories, Lemma 4.3.5) and left as a good exercise to the reader. Similarly, we can define the notion of a presheaf of abelian groups, rings, etc. More generally we may define a presheaf with values in a category. 613

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Definition 9.2.2. Let C, A be categories. A presheaf F on C with values in A is a contravariant functor from C to A, i.e., F : C ◦ → A. A morphism of presheaves F → G on C with values in A is a transformation of functors from F to G. These form the objects and morphisms of the category of presheaves on C with values in A. Remark 9.2.3. As already pointed out we may consider the category presheaves with values in any of the “big” categories listed in Categories, Remark 4.2.2. These will be “big” categories as well and they will be listed in the above mentioned remark as we go along. 9.3. Injective and surjective maps of presheaves Definition 9.3.1. Let C be a category, and let ϕ : F → G be a map of presheaves of sets. (1) We say that ϕ is injective if for every object U of C we have α : F(U ) → G(U ) is injective. (2) We say that ϕ is surjective if for every object U of C we have α : F(U ) → G(U ) is surjective. Lemma 9.3.2. The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of PSh(C). A map is an isomorphism if and only if it is both injective and surjective. Proof. Omitted.



Definition 9.3.3. We say F is a subpresheaf of G if for every object U ∈ Ob(C) the set F(U ) is a subset of G(U ), compatibly with the restriction mappings. In other words, the inclusion maps F(U ) → G(U ) glue together to give an (injective) morphism of presheaves F → G. Lemma 9.3.4. Let C be a category. Suppose that ϕ : F → G is a morphism of presheaves of setson C. There exists a unique subpresheaf G 0 ⊂ G such that ϕ factors as F → G 0 → G and such that the first map is surjective. Proof. Omitted.



Definition 9.3.5. Notation as in Lemma 9.3.4. We say that G 0 is the image of ϕ. 9.4. Limits and colimits of presheaves Let C be a category. Limits and colimits exist in the category PSh(C). In addition, for any U ∈ ob(C) the functor PSh(C) −→ Sets,

F 7−→ F(U )

commutes with limits and colimits. Perhaps the easiest way to prove these statement is the following. Given a diagram F : I → PSh(C) define presheaves Flim : U 7−→ limi∈I Fi (U ) and Fcolim : U 7−→ colimi∈I Fi (U ) There are clearly projection maps Flim → Fi and canonical maps Fi → Fcolim . These maps satisfy the requirements of the maps of a limit (reps. colimit) of Categories, Definition 4.13.1 (resp. Categories, Definition 4.13.2). Finally, if (G, qi : G → Fi ) is another system (as in the definition of a limit), then we get for every

9.5. FUNCTORIALITY OF CATEGORIES OF PRESHEAVES

615

U a system of maps G(U ) → Fi (U ) with suitable functoriality requirements. And thus a unique map G(U ) → Flim (U ). It is easy to verify these are compatible as we vary U and arise from the desired map G → Flim . A similar argument works in the case of the colimit. 9.5. Functoriality of categories of presheaves Let u : C → D be a functor between categories. In this case we denote up : PSh(D) −→ PSh(C) the functor that associates to G on D the presheaf up G = G ◦ u. Note that by the previous section this functor commutes with all limits. For V ∈ ob(D) let IVu denote the category with (9.5.0.1)

Mor

u IV

Ob(IVu ) ((U, φ), (U 0 , φ0 ))

= {(U, φ) | U ∈ Ob(C), φ : V → u(U )} = {f : U → U 0 in C | u(f ) ◦ φ = φ0 }

We sometimes drop the subscript u from the notation and we simply write IV . We will use these categories to define a left adjoint to the functor up . Before we do so we prove a few technical lemmas. Lemma 9.5.1. Let u : C → D be a functor between categories. Suppose that C has fibre products and equalizers, and that u commutes with them. Then the categories (IV )opp satisfy the hypotheses of Categories, Lemma 4.17.3. Proof. There are two conditions to check. First, suppose we are given three objects φ : V → u(U ), φ0 : V → u(U 0 ), and φ00 : V → u(U 00 ) and morphisms a : U 0 → U , b : U 00 → U such that u(a) ◦ φ = φ0 and u(b)◦φ = φ00 . We have to show there exists another object φ000 : V → u(U 000 ) and morphisms c : U 000 → U 0 and d : U 000 → U 00 such that u(c) ◦ φ = φ000 , u(d) ◦ φ = φ000 and a ◦ c = b ◦ d. We take U 000 = U 0 ×U U 00 with c and d the projection morphisms. This works as u commutes with fibre products; we omit the verification. Second, suppose we are given two objects φ : V → u(U ) and φ0 : V → u(U 0 ) and morphisms a, b : (U, φ) → (U 0 , φ0 ). We have to find a morphism c : (U 00 , φ00 ) → (U, φ) which equalizes a and b. Let c : U 00 → U be the equalizer of a and b in the category C. As u commutes with equalizers and since u(a) ◦ φ = u(b) ◦ φ = φ0 we obtain a morphism φ00 : V → u(U 00 ).  Lemma 9.5.2. Let u : C → D be a functor between categories. Assume (1) the category C has a final object X and u(X) is a final object of D , and (2) the category C has fibre products and u commutes with them. Then the index categories (IVu )opp of are filtered (see Categories, Definition 4.17.1). Proof. The assumptions imply that the assumptions of Lemma 9.5.1 are satisfied (see the discussion in Categories, Section 4.16). By Categories, Lemma 4.17.3 we see that IV is a (possibly empty) disjoint union of directed categories. Hence it suffices to show that IV is nonempty and connected. First, we show that IV is nonempty. Namely, let X be the final object of C, which exists by assumption. Let V → u(X) be the morphism coming from the fact that u(X) is final in D by assumption. This gives an object of IV .

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Second, we show that IV is connected. Let φ1 : V → u(U1 ) and φ2 : V → u(U2 ) be in Ob(IV ). By assumption U1 × U2 exists and u(U1 × U2 ) = u(U1 ) × u(U2 ). Consider the morphism φ : V → u(U1 × U2 ) corresponding to (φ1 , φ2 ) by the universal property of products. Clearly the object φ : V → u(U1 × U2 ) maps to both φ1 : V → u(U1 ) and φ2 : V → u(U2 ).  Given g : V 0 → V in D we get a functor g : IV → IV 0 by setting g(U, φ) = (U, φ ◦ g) on objects. Given a presheaf F on C we obtain a functor FV : IVopp −→ Sets,

(U, φ) 7−→ F(U ).

In other words, FV is a presheaf of sets on IV . Note that we have FV 0 ◦ g = FV . We define up F(V ) := colimIVopp FV c(φ)

As a colimit we obtain for each (U, φ) ∈ Ob(IV ) a canonical map F(U ) −−→ up F(V ). For g : V 0 → V as above there is a canonical restriction map g ∗ : up F(V ) → up F(V 0 ) compatible with FV 0 ◦ g = FV by Categories, Lemma 4.13.7. It is the unique map so that for all (U, φ) ∈ Ob(IV ) the diagram F(U ) id

 F(U )

c(φ)

/ up F(V ) g∗

 c(φ◦g) / up F(V 0 )

commutes. The uniquess of these maps implies that we obtain a presheaf. This presheaf will be denoted up F. Lemma 9.5.3. There is a canonical map F(U ) → up F(u(U )), which is compatible with restriction maps (on F and on up F). Proof. This is just the map c(idu(U ) ) introduced above.



Note that any map of presheaves F → F 0 gives rise to compatible systems of maps between functors FY → FY0 , and hence to a map of presheaves up F → up F 0 . In other words, we have defined a functor up : PSh(C) −→ PSh(D) Lemma 9.5.4. The functor up is a left adjoint to the functor up . In other words the formula MorPSh(C) (F, up G) = MorPSh(D) (up F, G) holds bifunctorially in F and G. Proof. Let G be a presheaf on D and let F be a presheaf on C. We will show that the displayed formula holds by constructing maps either way. We will leave it to the reader to verify they are each others inverse. Given a map α : up F → G we get up α : up up F → up G. Lemma 9.5.3 says that there is a map F → up up F. The composition of the two gives the desired map. (The good thing about this construction is that it is clearly functorial in everything in sight.) Conversely, given a map β : F → up G we get a map up β : up F → up up G. We claim that the functor up GY on IY has a canonical map to the constant functor with

9.6. SITES

617

value G(Y ). Namely, for every object (X, φ) of IY , the value of up GY on this object is G(u(X)) which maps to G(Y ) by G(φ) = φ∗ . This is a transformation of functors because G is a functor itself. This leads to a map up up G(Y ) → G(Y ). Another trivial verification shows that this is functorial in Y leading to a map of presheaves up up G → G. The composition up F → up up G → G is the desired map.  Remark 9.5.5. Suppose that A is a category such that any diagram IY → A has a colimit in A. In this case it is clear that there are functors up and up , defined in exactly the same way as above, on the categories of presheaves with values in A. Moreover, the adjointness of the pair up and up continues to hold in this setting. Lemma 9.5.6. Let u : C → D be a functor between categories. For any object U of C we have up hU = hu(U ) . Proof. By adjointness of up and up we have MorPSh(D) (up hU , G) = MorPSh(C) (hU , up G) = up G(U ) = G(u(U )) and hence by Yoneda’s lemma we see that up hU = hu(U ) as presheaves.



9.6. Sites Our notion of a site uses the following type of structures. Definition 9.6.1. Let C be a category, see Conventions, Section 2.3. A family of morphisms with fixed target in C is given by an object U ∈ Ob(C), a set I and for each i ∈ I a morphism Ui → U of C with target U . We use the notation {Ui → U }i∈I to indicate this. It can happen that the set I is empty! This notation is meant to suggest an open covering as in topology. Definition 9.6.2. A site1 is given by a category C and a set Cov(C) of families of morphisms with fixed target {Ui → U }i∈I , called coverings of C, satisfying the following axioms (1) If V → U is an isomorphism then {V → U } ∈ Cov(C). (2) If {Ui → U }i∈I ∈ Cov(C) and for each i we have {Vij → Ui }j∈Ji ∈ Cov(C), then {Vij → U }i∈I,j∈Ji ∈ Cov(C). (3) If {Ui → U }i∈I ∈ Cov(C) and V → U is a morphism of C then Ui ×U V exists for all i and {Ui ×U V → V }i∈I ∈ Cov(C). Remark 9.6.3. (On set theoretic issues – skip on a first reading.) The main reason for introducing sites is to study the category of sheaves on a site, because it is the generalization of the category of sheaves on a topological space that has been so important in algebraic geometry. In order to avoid thinking about things like “classes of classes” and so on, we will not allow sites to be “big” categories, in contrast to what we do for categories and 2-categories. Suppose that C is a category and that Cov(C) is a proper class of coverings satisfying (1), (2) and (3) above. We will not allow this as a site either, mainly because we are going to take limits over coverings. However, there are several natural ways to replace Cov(C) by a set of coverings or a slightly different structure that give rise to the same category of sheaves. For example: 1This notation differs from that of [AGV71], as explained in the introduction.

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(1) In Sets, Section 3.11 we show how to pick a suitable set of coverings that gives the same category of sheaves. (2) Another thing we can do is to take the associated topology (see Definition 9.41.2). The resulting topology on C has the same category of sheaves. Two topologies have the same categories of sheaves if and only if they are equal, see Theorem 9.43.2. A topology on a category is given by a choice of sieves on objects. The collection of all possible sieves and even all possible topologies on C is a set. (3) We could also slightly modify the notion of a site, see Remark 9.41.4 below, and end up with a canonical set of coverings which is contained in the powerset of the set of arrows of C. Each of these solutions has some minor drawback. For the first, one has to check that constructions later on do not depend on the choice of the set of coverings. For the second, one has to learn about topologies and redo many of the arguments for sites. For the third, see the last sentence of Remark 9.41.4. Our approach will be to work with sites as in Definition 9.6.2 above. Given a category C with a proper class of coverings as above, we will replace this by a set of coverings producing a site using Sets, Lemma 3.11.1. It is shown in Lemma 9.8.6 below that the resulting category of sheaves (the topos) is independent of this choice. We leave it to the reader to use one of the other two strategies to deal with these issues if he/she so desires. Example 9.6.4. Let X be a topological space. Let TX be the category whose objects consist of all the open sets U in X and whose morphisms are just the inclusion maps. That is, there is at most one morphism between any two objects S in TX . Now define {Ui → U }i∈I ∈ Cov(TX ) if and only if Ui = U . Conditions (1) and (2) above are clear, and (3) is also clear once we realize that in TX we have U × V = U ∩ V . Note that in particular the empty set has to be an element of TX since otherwise this would not work in general. Furthermore, it is equally important, as we will see later, to allow the empty covering of the empty set as a covering! We turn TX into a site by choosing a suitable set of coverings Cov(TX )κ,α as in Sets, Lemma 3.11.1. Presheaves and sheaves (as defined below) on the site TX will agree exactly with the usual notion of a presheaves and sheaves on a topological space, as defined in Sheaves, Section 6.1. Example 9.6.5. Let G be a group. Consider the category G-Sets whose objects are sets X with a left G-action, with G-equivariant maps as the morphisms. An important example is G G which is the G-set whose underlying set is G and action given by left multiplication. This category has fiber products, S see Categories, Section 4.7. We declare {ϕi : Ui → U }i∈I to be a covering if i∈I ϕi (Ui ) = U . This gives a class of coverings on G-Sets which is easily see to satisfy conditions (1), (2), and (3) of Definition 9.6.2. The result is not a site since both the collection of objects of the underlying category and the collection of coverings form a proper class. We first replace by G-Sets by a full subcategory G-Setsα as in Sets, Lemma 3.10.1. After this the site (G-Setsα , Covκ,α0 (G-Setsα )) gotten by suitably restricting the collection of coverings as in Sets, Lemma 3.11.1 will be denoted TG . Example 9.6.6. Let C be a category. There is a canonical way to turn this into a site where {idU : U → U } are the coverings. Sheaves on this site are the presheaves on C. This corresponding topology is called the chaotic or indiscrete topology.

9.7. SHEAVES

619

9.7. Sheaves Let C be a site. Before we introduce the notion of a sheaf with values in a category we explain what it means for a presheaf of sets to be a sheaf. Let F be a presheaf of sets on C and let {Ui → U }i∈I be an element of Cov(C). By assumption all the fibre products Ui ×U Uj exist in C. There are two natural maps pr∗ 0

Q

i∈I

F(Ui ) pr∗ 1

/Q / (i0 ,i1 )∈I×I F(Ui0 ×U Ui1 )

which we will denote pr∗i , i = 0, 1 as indicated in the displayed equation. Namely, an element of the left hand side corresponds to a family (si )i∈I , where each si is a section of F over Ui . For each pair (i0 , i1 ) ∈ I ×I we have the projection morphisms (i ,i1 )

pri00

(i ,i1 )

: Ui0 ×U Ui1 −→ Ui0 and pri10

: Ui0 ×U Ui1 −→ Ui1 .

Thus we may pull back either the section si0 via the first of these maps or the section si1 via the second. Explicitly the maps we refered to above are   (i ,i ),∗ pr∗0 : (si )i∈I 7−→ pri00 1 (si0 ) (i0 ,i1 )∈I×I

and   (i ,i ),∗ pr∗1 : (si )i∈I 7−→ pri10 1 (si1 ) Finally consider the natural map Y F(U ) −→

i∈I

F(Ui ),

. (i0 ,i1 )∈I×I

s 7−→ (s|Ui )i∈I

where we have used the notation s|Ui to indicate the pullback of s via the map Ui → U . It is clear from the functorial natural of F and the commutativity of the fibre product diagrams that pr∗0 ((s|Ui )i∈I ) = pr∗1 ((s|Ui )i∈I ). Definition 9.7.1. Let C be a site, and let F be a presheaf of sets on C. We say F is a sheaf if for every covering {Ui → U }i∈I ∈ Cov(C) the diagram (9.7.1.1)

F(U )

/

pr∗ 0

Q

i∈I

F(Ui ) pr∗ 1

/Q / (i0 ,i1 )∈I×I F(Ui0 ×U Ui1 )

represents the first arrow as the equalizer of pr∗0 and pr∗1 . Loosely speaking this means that given sections si ∈ F(Ui ) such that si |Ui ×U Uj = sj |Ui ×U Uj in F(Ui ×U Uj ) for all pairs (i, j) ∈ I × I then there exists a unique s ∈ F(U ) such that si = s|Ui . Remark 9.7.2. If the covering {Ui → U }i∈I is the empty family (this means that I = ∅), then the sheaf condition signifies that F(U ) = {∗} is a singleton set. This is true because in (9.7.1.1) the second and third sets are empty products in the category of sets, which are final objects in the category of sets, hence singletons. Example 9.7.3. Let X be a topological space. Let TX be the site constructed in Example 9.6.4. The notion of a sheaf on TX coincides with the notion of a sheaf on X introduced in Sheaves, Definition 6.7.1.

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Example 9.7.4. Let X be a topological space. Let us consider the site TX0 which is the same as the site TX of Example 9.6.4 except that we disallow the empty covering of the empty set. In other words, we do allow the covering {∅ → ∅} but we do not allow the covering whose index set is empty. It is easy to show that this still defines a site. However, we claim that the sheaves on TX0 are different from the sheaves on TX . For example, as an extreme case consider the situation where X = {p} is a singleton. Then the objects of TX0 are ∅, X and the coverings are {{∅ → ∅}, {X → X}}. Clearly, a sheaf on this is given by any choice of a set F(∅) and any choice of a set F(X), together with any restricion map F(X) → F(∅). Thus sheaves on TX0 are the same as usual sheaves on the two point space {η, p} with open sets {∅, {η}, {p, η}}. In general sheaves on TX0 are the same as sheaves on the space X q {η}, with opens given by the empty set and any set of the form U ∪ {η} for U ⊂ X open. Definition 9.7.5. The category Sh(C) of sheaves of sets is the full subcategory of the category PSh(C) whose objects are the sheaves of sets. Let A be a category. If products indexed by I, and I × I exist in A for any I that occurs as an index set for covering families then Definition 9.7.1 above makes sense, and defines a notion of a sheaf on C with values in A. Note that the diagram in A F(U )

pr∗ 0

/Q

i∈I

F(Ui ) pr∗ 1

/Q /

(i0 ,i1 )∈I×I

F(Ui0 ×U Ui1 )

is an equalizer diagram if and only if for every object X of A the diagram of sets MorA (X, F(U ))

/

pr∗ 0

Q

MorA (X, F(Ui )) pr∗ 1

/

/Q

MorA (X, F(Ui0 ×U Ui1 ))

is an equalizer diagram. Suppose A is arbitrary. Let F be a presheaf with values in A. Choose any object X ∈ Ob(A). Then we get a presheaf of sets FX defined by the rule FX (U ) = MorA (X, F(U )). From the above it follows that a good definition is obtained by requiring all the presheaves FX to be sheaves of sets. Definition 9.7.6. Let C be a site, let A be a category and let F be a presheaf on C with values in A. We say that F is a sheaf if for all objects X of A the presheaf of sets FX (defined above) is a sheaf. 9.8. Families of morphisms with fixed target This section is meant to introduce some notions regarding families of morphisms with the same target. Definition 9.8.1. Let C be a category. Let U = {Ui → U }i∈I be a family of morphisms of C with fixed target. Let V = {Vj → V }j∈J be another. (1) A morphism of families of maps with fixed target of C from U to V, or simply a morphism from U to V is given by a morphism U → V , a map

9.8. FAMILIES OF MORPHISMS WITH FIXED TARGET

621

of sets α : I → J and for each i ∈ I a morphism Ui → Vα(i) such that the diagram / Vα(i) Ui  U

 /V

is commutative. (2) In the special case that U = V and U → V is the identity we call U a refinement of the family V. A trivial but important remark is that if U = {Ui → U }i∈I is the empty family of maps, i.e., if I = ∅, then no family V = {Vj → V }j∈J with J 6= ∅ can refine U! Definition 9.8.2. Let C be a category. Let U = {ϕi : Ui → U }i∈I , and V = {ψj : Vj → U }j∈J be two families of morphisms with fixed target. (1) We say U and V are combinatorially equivalent if there exist maps α : I → J and β : J → I such that ϕi = ψα(i) and ψj = ϕβ(j) . (2) We say U and V are tautologically equivalent if there exist maps α : I → J and β : J → I and for all i ∈ I and j ∈ J commutative diagrams / Vα(i)

Ui 

U

/ Uβ(j)

Vj

}



U

}

with isomorphisms as horizontal arrows. Lemma 9.8.3. Let C be a category. Let U = {ϕi : Ui → U }i∈I , and V = {ψj : Vj → U }j∈J be two families of morphisms with the same fixed target. (1) If U and V are combinatorially equivalent then they are tautologically equivalent. (2) If U and V are tautologically equivalent then U is a refinement of V and V is a refinement of U. (3) The relation “being combinatorially equivalent” is an equivalence relation on all families of morphisms with fixed target. (4) The relation “being tautologically equivalent” is an equivalence relation on all families of morphisms with fixed target. (5) The relation “U refines V and V refines U” is an equivalence relation on all families of morphisms with fixed target. Proof. Omitted.



In the following lemma, given a category C, a presheaf F on C, a family U = {Ui → U }i∈I such that all fibre products Ui ×U Ui0 exist, we say that the sheaf condition for F with respect to U holds if the diagram (9.7.1.1) is an equalizer diagram. Lemma 9.8.4. Let C be a category. Let U = {ϕi : Ui → U }i∈I , and V = {ψj : Vj → U }j∈J be two families of morphisms with the same fixed target. Assume that the fibre products Ui ×U Ui0 and Vj ×U Vj 0 exist. If U and V are tautologically equivalent, then for any presheaf F on C the sheaf condition for F with respect to U is equivalent to the sheaf condition for F with respect to V.

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Proof. First, note that if ϕ : A → B is an isomorphism in the category C, then ϕ∗ : F(B) → F(A) is an isomorphism. Let β : J → I be a map and let ψj : Vj → Uβ(j) be isomorphisms over U which are assumed to exist by hypothesis. Let us show that the sheaf condition for V implies the sheaf condition for U. Suppose given sections si ∈ F(Ui ) such that si |Ui ×U Ui0 = si0 |Ui ×U Ui0 in F(Ui ×U Ui0 ) for all pairs (i, i0 ) ∈ I × I. Then we can define sj = ψj∗ sβ(j) . For any pair (j, j 0 ) ∈ J × J 0 the morphism ψj ×idU ψj 0 : Vj ×U Vj 0 → Uβ(j) ×U Uβ(j 0 ) is an isomorphism as well. Hence by transport of structure we see that sj |Vj ×U Vj0 = sj 0 |Vj ×U Vj0 as well. The sheaf condition w.r.t. V implies there exists a unique s such that s|Vj = sj for all j ∈ J. By the first remark of the proof this implies that s|Ui = si for all i ∈ Im(β) as well. Suppose that i ∈ I, i 6∈ Im(β). For such an i we have isomorphisms Ui → Vα(i) → Uβ(α(i)) over U . This gives a morphism Ui → Ui ×U Uβ(α(i)) which is a section of the projection. Because si and sβ(α(i)) restrict to the same element on the fibre product we conclude that sβ(α(i)) pulls back to si via Ui → Uβ(α(i)) . Thus we see that also si = s|Ui as desired.  Lemma 9.8.5. Let C be a category. Let Covi , i = 1, 2 be two sets of families of morphisms with fixed target which each define the structure of a site on C. (1) If every U ∈ Cov1 is tautologically equivalent to some V ∈ Cov2 , then Sh(C, Cov2 ) ⊂ Sh(C, Cov1 ). If also, every U ∈ Cov2 is tautologically quivalent to some V ∈ Cov1 then the category of sheaves are equal. (2) Suppose that for each U ∈ Cov1 there exists a V ∈ Cov2 such that V refines U. In this case Sh(C, Cov2 ) ⊂ Sh(C, Cov1 ). If also for every U ∈ Cov2 there exists a V ∈ Cov1 such that V refines U, then the categories of sheaves are equal. Proof. Part (1) follows directly from Lemma 9.8.4 and the definitions. We advise the reader to skip the proof of (2) on a first reading. Let F be a sheaf of sets for the site (C, Cov2 ). Let U ∈ Cov1 , say U = {Ui → U }i∈I . Choose a refinement V ∈ Cov2 of U, say V = {Vj → U }j∈J and refinement given by α : J → I and fj : Vj → Uα(j) . First let s, s0 ∈ F(U ). If for all i ∈ I we have s|Ui = s0 |Ui , then we also have s|Vj = s0 |Vj for all j ∈ J. This implies that s = s0 by the sheaf condition for F with respect to Cov2 . Hence we see that the unicity in the sheaf condition for F and the site (C, Cov1 ) holds. Next, suppose given si ∈ F(Ui ) such that si |Ui ×U Ui0 = si0 |Ui ×U Ui0 for all i, i0 ∈ I. Set sj = fj∗ (sα(j) ) ∈ F(Vj ). Since the morphisms fj are morphisms over U we obtain induced morphisms fjj 0 : Vj ×U Vj 0 → Uα(i) ×U Uα(i0 ) compatible with the fj , fj 0 via the projection maps. It follows that ∗ ∗ ) = fjj ) = sj 0 |Vj ×U Vj0 sj |Vj ×U Vj0 = fjj 0 (sα(j) |Uα(j) ×U U 0 (sα(j 0 ) |Uα(j) ×U U α(j 0 ) α(j 0 )

for all j, j 0 ∈ J. Hence, by the sheaf condition for F with respect to Cov2 , we get a section s ∈ F(U ) which restricts to sj on each Vj . We are done if we show s restricts to si0 on Ui0 for any i0 ∈ I. For each i0 ∈ I the family U 0 = {Ui ×U Ui0 → Ui0 }i∈I is an element of Cov1 by the axioms of a site. Also, the family V 0 = {Vj ×U Ui0 →

9.9. THE EXAMPLE OF G-SETS

623

Ui0 }j∈J is an element of Cov2 . Then V 0 refines U 0 via α : J → I and the maps fj0 = fj × idUi0 . The element si0 restricts to si |Ui ×U Ui0 on the members of the covering U 0 and hence via (fj0 )∗ to the elements sj |Vj ×U Ui0 on the members of the covering V 0 . By construction of s this is the same as the family of restrictions of s|Ui0 to the members of the covering V 0 . Hence by the sheaf condition for F with respect to Cov2 we see that s|Ui0 = si0 as desired.  Lemma 9.8.6. Let C be a category. Let Cov(C) be a proper class of coverings satisfying conditions (1), (2) and (3) of Definition 9.6.2. Let Cov1 , Cov2 ⊂ Cov(C) be two subsets of Cov(C) which endow C with the structure of a site. If every covering U ∈ Cov(C) is combinatorially equivalent to a covering in Cov1 and combinatorially equivalent to a covering in Cov2 , then Sh(C, Cov1 ) = Sh(C, Cov2 ). Proof. This is clear from Lemmas 9.8.5 and 9.8.3 above as the hypothesis implies that every covering U ∈ Cov1 ⊂ Cov(C) is combinatorially equivalent to an element of Cov2 , and similarly with the roles of Cov1 and Cov2 reversed.  9.9. The example of G-sets As an example, consider the site TG of Example 9.6.5. We will describe the category of sheaves on TG . The answer will turn out to be independent of the choices made in defining TG . In fact, during the proof we will need only the following properties of the site TG : (a) TG is a full subcategory of G-Sets, (b) TG contains the G-set G G, (c) TG has fibre products and they are the same as in G-Sets, (d) given U ∈ Ob(TG ) and a G-invariant subset O ⊂ U , there exists an object of TG isomorphic to O, and (e) any surjective family of maps {Ui → U }i∈I , with U, Ui ∈ Ob(TG ) is combinatorially equivalent to a covering of TG . These properties hold by Sets, Lemmas 3.10.2 and 3.11.1. Remark that the map HomG (G G, G G) −→ Gopp , ϕ 7−→ ϕ(1) is an isomorphism of groups. The inverse map sends g ∈ G to the map Rg : s 7→ sg (i.e. right multiplication). Note that Rg1 g2 = Rg2 ◦ Rg1 so the opposite is necessary. This implies that for every presheaf F on TG the value F(G G) inherets the structure of a G-set as follows: g · s for g ∈ G and s ∈ F(G G) defined by F(Rg )(s). This is a left action because (g1 g2 ) · s = F(Rg1 g2 )(s) = F(Rg2 ◦ Rg1 )(s) = F(Rg1 )(F(Rg2 )(s)) = g1 · (g2 · s). Here we’ve used that F is contravariant. Note that if F → G is a morphism of presheaves of sets on TG then we get a map F(G G) → G(G G) which is compatible with the G-actions we have just defined. All in all we have constructed a functor PSh(TG ) −→ G-Sets,

F 7−→ F(G G).

We leave it to the reader to verify that this construction has the pleasing property that the representable presheaf hU is mapped to something canonically isomorphic to U . In a formula hU (G G) = HomG (G G, U ) ∼ = U.

624

9. SITES AND SHEAVES

Suppose that S is a G-set. We define a presheaf FS by the formula2 FS (U ) = MorG-Sets (U, S). This is clearly a presheaf. On the ` other hand, suppose that {Ui → U }i∈I is a covering in TG . This implies that i Ui → U is surjective. Thus it is clear that the map Y Y FS (U ) = MorG-Sets (U, S) −→ FS (Ui ) = MorG-Sets (Ui , S) is injective. And, given a family of G-equivariant maps si : Ui → S, such that all the diagrams / Uj Ui ×U Uj sj

  si /S Ui commute, there is a unique G-equivariant map s : U → S such that si is the composition Ui → U → S. Namely, we just define s(u) = si (ui ) where i ∈ I is any index such that there exists some ui ∈ Ui mapping to u under the map Ui → U . The commutativity of the diagrams above implies exactly that this construction is well defined. All in all we have constructed a functor G-Sets −→ Sh(TG ),

S 7−→ FS .

We now have the following diagram of categories and functors PSh(TG ) e

F 7→F (G G)

/ G-Sets

S7→FS

z Sh(TG ) It is immediate from the definitions that FS (G G) = MorG (G G, S) = S, the last equality by evaluation at 1. This almost proves the following. Proposition 9.9.1. The functors F 7→ F(G G) and S 7→ FS define quasi-inverse equivalences between Sh(TG ) and G-Sets. Proof. We have already seen that composing the functors one way around is isomorphic to the identity functor. In the other direction, for any sheaf H there is a natural map of sheaves can : H −→ FH(G G) . Namely, for any object U of TG we let canU be the map H(U ) −→ FH(G G) (U ) = MorG (U, H(G G)) s 7−→ (u 7→ αu∗ s). Here αu : G G → U is the map αu (g) = gu and αu∗ : H(U ) → H(G G) is the pullback map. A trivial but confusing verification shows that this is indeed a map of presheaves. We have to show that can is an isomorphism. We do this by showing canU is an isomorphism for all U ∈ ob(TG ). We leave the (important but easy) case that U = ` G G to the reader. A general object U of TG is a disjoint union of Gorbits: U = i∈I Oi . The family of maps {Oi → U }i∈I is tautologically equivalent 2It may appear this is the representable presheaf defined by S. This may not be the case because S may not be an object of TG which was chosen to be a sufficiently large set of G-sets.

9.10. SHEAFIFICATION

625

to a covering in TG (by the properties of TG listed at the beginning of this section). Hence by Lemma 9.8.4 the sheaf H satisfies the sheaf property with Q respect to {Oi → U }i∈I . The sheaf property for this covering implies H(U ) = i H(Oi ). Hence it suffices to show that canU is an isomorphism when U consists of a single G-orbit. Let u ∈ U and let H ⊂ G be its stabilizer. Clearly, MorG (U, H(G G)) = H(G G)H equals the subset of H-invariant elements. On the other hand consider the covering {G G → U } given by g 7→ gu (again it is just combinatorially equivalent to some covering of TG , and again this doesn’t matter). Q Note that the fibre product (G G) ×U (G G) is equal to {(g, gh), g ∈ G, h ∈ H} ∼ = h∈H G G. Hence the sheaf property for this covering reads as / H(G G)

H(U )

pr∗ 0 pr∗ 1

/

/Q

h∈H

H(G G).

The two maps pr∗i into the factor H(G G) differ by multiplication by h. Now the result follows from this and the fact that can is an isomorphism for U = G G.  9.10. Sheafification In order to define the sheafification we study the zeroth Cech cohomology group of a covering and its functoriality properties. Let F be a presheaf of sets on C, and let U = {Ui → U }i∈I be a covering of C. Let us use the notation F(U) to indicate the equalizer Y H 0 (U, F) = {(si )i∈I ∈ F(Ui ) | si |Ui ×U Uj = sj |Ui ×U Uj ∀i, j ∈ I}. i

As we will see later, this is the zeroth Cech cohomology of F over U with respect to the covering U. A small remark is that we can define H 0 (U, F) as soon as all the morphisms Ui → U are representable, i.e., U need not be a covering of the site. There is a canonical map F(U ) → H 0 (U, F). It is clear that a morphism of coverings U → V induces commutative diagrams ; Ui 8

/ Vα(i) .

/ Vα(i) ×V Vα(j)

Ui ×U Uj #

Uj

& / Vα(j)

This in turn produces a map H 0 (V, F) → H 0 (U, F), compatible with the map F(V ) → F(U ). By construction, a presheaf F is a sheaf if and only if for every covering U of C the natural map F(U ) → H 0 (U, F) is bijective. We will use this notion to prove the following simple lemma about limits of sheaves. Lemma 9.10.1. Let F : I → Sh(C) be a diagram. Then limI F exists and is equal to the limit in the category of presheaves.

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9. SITES AND SHEAVES

Proof. Let limi Fi be the limit as a presheaf. We will show that this is a sheaf and then it will trivially follow that it is a limit in the category of sheaves. To prove the sheaf property, let V = {Vj → V }j∈J be a covering. Let (sj )j∈J be an element of H 0 (V, limi Fi ). Using the projection maps we get elements (sj,i )j∈J in H 0 (V, Fi ). By the sheaf poperty for Fi we see that there is a unique si ∈ Fi (V ) such that sj,i = si |Vj . Let φ : i → i0 be a morphism of the index category. We would like to show that F(φ) : Fi → Fi0 maps si to si0 . We know this is true for the sections si,j and si0 ,j for all j and hence by the sheaf property for Fi0 this is true. At this point we have an element s = (si )i∈Ob(I) of (limi Fi )(V ). We leave it to the reader to see this element has the required property that sj = s|Vj .  Example 9.10.2. A particular example is the limit over the empty diagram. This gives the final object in the category of (pre)sheaves. It is the sheaf that associates to each object U of C a singleton set, with unique restriction mappings. We often denote this sheaf by ∗. Let JU be the category of all coverings of U . In other words, the objects of JU are the coverings of U in C, and the morphisms are the refinements. By our conventions on sites this is indeed a category, i.e., the collection of objects and morphisms forms a set. Note that Ob(JU ) is not empty since {idU } is an object of it. According to the remarks above the construction U 7→ H 0 (U, F) is a contravariant functor on JU . We define F + (U ) = colimJUopp H 0 (U, F) See Categories, Section 4.13 for a discussion of limits and colimits. We point out that later we will see that F + (U ) is the zeroth Cech cohomology of F over U . Before we say more about the structure of the colimit, we turn the collection of sets F + (U ), U ∈ Ob(C) into a presheaf. Namely, let V → U be a morphism of C. By the axioms of a site there is a functor3 JU −→ JV ,

{Ui → U } 7−→ {Ui ×U V → V }.

Note that the projection maps furnish a functorial morphism of coverings {Ui ×U V → V } → {Ui → U } and hence, by the construction above, a functorial map of sets H 0 ({Ui → U }, F) → H 0 ({Ui ×U V → V }, F). In other words, there is a transformation of functors from H 0 (−, F) : JU → Sets to the compostion H 0 (−,F )

JU → JV −−−−−−→ Sets. Hence by generalities of colimits we obtain a canonical map F + (U ) → F + (V ). In terms of the description of the set F + (U ) above, it just takes the element associated with s = (si ) ∈ H 0 ({Ui → U }, F) to the element associated with (si |V ×U Ui ) ∈ H 0 ({Ui ×U V → V }, F). Lemma 9.10.3. The constructions above define a presheaf F + together with a canonical map of presheaves F → F + . Proof. All we have to do is to show that given morphisms W → V → U the composition F + (U ) → F + (V ) → F + (W ) equals the map F + (U ) → F + (W ). This can be shown directly by verifying that, given a covering {Ui → U } and s = (si ) ∈ H 0 ({Ui → U }, F), we have canonically W ×U Ui ∼ = W ×V (V ×U Ui ), and si |W ×U Ui corresponds to (si |V ×U Ui )|W ×V (V ×U Ui ) via this isomorphism.  3This construction actually involves a choice of the fibre products U × V and hence the i U axiom of choice. The resulting map does not depend on the choices made, see below.

9.10. SHEAFIFICATION

627

More indirectly, the result of Lemma 9.10.6 shows that we may pullback an element s as above via any morphism from any covering of W to {Ui → U } and we will always end up with the same element in F + (W ). Lemma 9.10.4. The association F 7→ (F → F + ) is a functor. Proof. Instead of proving this we state exactly what needs to be proven. Let F → G be a map of presehaves. Prove the commutativity of: F

/ F+

 G

 / G+ 

The next two lemmas imply that the colimits above are colimits over a directed partially ordered set. Lemma 9.10.5. Given a pair of coverings {Ui → U } and {Vj → U } of a given object U of the site C, there exists a covering which is a common refinement. Proof. Since C is a site we have that for every i the family {Vj ×U Ui → Ui }j is a covering. And, then another axiom implies that {Vj ×U Ui → U }i,j is a covering of U . Clearly this covering refines both given coverings.  Lemma 9.10.6. Any two morphisms f, g : U → V of coverings inducing the same morphism U → V induce the same map H 0 (V, F) → H 0 (U, F). Proof. Let U = {Ui → U }i∈I and V = {Vj → V }j∈J . The morphism f consists of a map U → V , a map α : I → J and maps fi : Ui → Vα(i) . Likewise, g determines a map β : I → J and maps gi : Ui → Vβ(i) . As f and g induce the same map U → V , the diagram V = α(i) fi

!

Ui gi

=V

! Vβ(i)

is commutative for every i ∈ I. Hence f and g factor through the fibre product fi

Ui

Vα(i) O 9

pr1

/ Vα(i) ×V Vβ(i)

ϕ

gi

%



pr2

Vβ(i) .

Now let s = (sj )j ∈ H 0 (V, F). Then for all i ∈ I: (f ∗ s)i = fi∗ (sα(i) ) = ϕ∗ pr∗1 (sα(i) ) = ϕ∗ pr∗2 (sβ(i) ) = gi∗ (sβ(i) ) = (g ∗ s)i ,

628

9. SITES AND SHEAVES

where the middle equality is given by the definition of H 0 (V, F). This shows that the maps H 0 (V, F) → H 0 (U, F) induced by f and g are equal.  Remark 9.10.7. In particular this lemma shows that if U is a refinement of V, and if V is a refinement of U, then there is a canonical identification H 0 (U, F) = H 0 (V, F). From these two lemmas, and the fact that JU is nonempty, it follows that the diagram H 0 (−, F) : JUopp → Sets is filtered, see Categories, Definition 4.17.1. Hence, by Categories, Section 4.17 the colimit F + (U ) may be described in the following straightforward manner. Namely, every element in the set F + (U ) arises from an element s ∈ H 0 (U, F) for some covering U of U . Given a second element s0 ∈ H 0 (U 0 , F) then s and s0 determine the same element of the colimit if and only if there exists a covering V of U and refinements f : V → U and f 0 : V → U 0 such that f ∗ s = (f 0 )∗ s0 in H 0 (V, F). Since the trivial covering {idU } is an object of JU we get a canonical map F(U ) → F + (U ). Lemma 9.10.8. The map θ : F → F + has the following property: For every object U of C and every section s ∈ F + (U ) there exists a covering {Ui → U } such that s|Ui is in the image of θ : F(Ui ) → F + (Ui ). Proof. Namely, let {Ui → U } be a covering such that s arises from the element (si ) ∈ H 0 ({Ui → U }, F). According to Lemma 9.10.6 we may consider the covering {Ui → Ui } and the (obvious) morphism of coverings {Ui → Ui } → {Ui → U } to compute the pullback of s to an element of F + (Ui ). And indeed, using this covering we get exactly θ(si ) for the restriction of s to Ui .  Definition 9.10.9. We say that a presheaf ofQsets F on a site C is separated if, for all coverings of {Ui → U }, the map F(U ) → F(Ui ) is injective. Theorem 9.10.10. With F as above (1) The presheaf F + is separated. (2) If F is separated, then F + is a sheaf and the map of presheaves F → F + is injective. (3) If F is a sheaf, then F → F + is an isomorphism. (4) The presheaf F ++ is always a sheaf. Proof. Proof of (1). Suppose that s, s0 ∈ F + (U ) and suppose that there exists some covering {Ui → U } such that s|Ui = s0 |Ui for all i. We now have three coverings of U : the covering {Ui → U } above, a covering U for s as in Lemma 9.10.8, and a similar covering U 0 for s0 . By Lemma 9.10.5, we can find a common refinement, say {Wj → U }. This means we have sj , s0j ∈ F(Wj ) such that s|Wj = θ(sj ), similarly for s0 |Wj , and such that θ(sj ) = θ(s0j ). This last equality means that there exists some covering {Wjk → Wj } such that sj |Wjk = s0j |Wjk . Then since {Wjk → U } is a covering we see that s, s0 map to the same element of H 0 ({Wjk → U }, F) as desired. Proof of (2). It is clear that F → F + is injective because all the maps F(U ) → H 0 (U, F) are injective. It is also clear that, if U → U 0 is a refinement, then H 0 (U 0 , F) → H 0 (U, F) is injective. Now, suppose that {Ui → U } is a covering, and let (si ) be a family of elements of F + (Ui ) satisfying the sheaf condition si |Ui ×U Uj = sj |Ui ×U Uj for all i, j ∈ I. Choose coverings (as in Lemma 9.10.8) {Uij → Ui } such

9.10. SHEAFIFICATION

629

that si |Uij is the image of the (unique) element sij ∈ F(Uij ). The sheaf condition implies that sij and si0 j 0 agree over Uij ×U Ui0 j 0 because it maps to Ui ×U Ui0 and we have the equality there. Hence (sij ) ∈ H 0 ({Uij → U }, F) gives rise to an element s ∈ F + (U ). We leave it to the reader to verify that s|Ui = si . Proof of (3). This is immediate from the definitions because the sheaf property says exactly that every map F → H 0 (U, F) is bijective (for every covering U of U ). Statement (4) is now obvious.



Definition 9.10.11. Let C be a site and let F be a presheaf of sets on C. The sheaf F # := F ++ together with the canonical map F → F # is called the sheaf associated to F. Proposition 9.10.12. The canonical map F → F # has the following universal property: For any map F → G, where G is a sheaf of sets, there is a unique map F # → G such that F → F # → G equals the given map. Proof. By Lemma 9.10.4 we get a commutative diagram F

/ F+

/ F ++

 G

 / G+

 / G ++

and by Theorem 9.10.10 the lower horizontal maps are isomorphisms. The uniqueness follows from Lemma 9.10.8 which says that every section of F # locally comes from sections of F.  It is clear from this result that the functor F 7→ (F → F # ) is unique up to unique isomorphism of functors. Actually, let us temporarily denote i : Sh(C) → PSh(C) the functor of inclusion. The result above actually says that MorPSh(C) (F, i(G)) = MorSh(C) (F # , G). In other words, the functor of sheafification is the left adjoint to the inclusion functor i. We finish this section with a couple of lemmas. Lemma 9.10.13. Let F : I → Sh(C) be a diagram. Then colimI F exists and is the sheafification of the colimit in the category of presheaves. Proof. Since the sheafification functor is a left adjoint it commutes with all colimits, see Categories, Lemma 4.22.3. Hence, since PSh(C) has colimits, we deduce that Sh(C) has colimits (which are the sheafifications of the colimits in presheaves).  Lemma 9.10.14. The functor PSh(C) → Sh(C), F 7→ F # is exact. Proof. Since it is a left adjoint it is right exact, see Categories, Lemma 4.22.4. On the other hand, by Lemmas 9.10.5 and Lemma 9.10.6 the colimits in the construction of F + are really over the directed partially ordered set Ob(JU ) where U ≥ U 0 if and only if U is a refinement of U 0 . Hence by Categories, Lemma 4.17.2 we see that F → F + commutes with finite limits (as a functor from presheaves to presheaves). Then we conclude using Lemma 9.10.1. 

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Lemma 9.10.15. Let C be a site. Let F be a presheaf of sets on C. Denote θ2 : F → F # the canonical map of F into its sheafification. Let U be an object of C. Let s ∈ F # (U ). There exists a covering {Ui → U } and sections si ∈ F(Ui ) such that (1) s|Ui = θ2 (si ), and (2) for every i, j there exists a covering {Uijk → Ui ×U Uj } of C such that the pullback of si and sj to each Uijk agree. Conversely, given any covering {Ui → U }, elements si ∈ F(Ui ) such that (2) holds, then there exists a unique section s ∈ F # (U ) such that (1) holds. Proof. Omitted.



9.11. Injective and surjective maps of sheaves Definition 9.11.1. Let C be a site, and let ϕ : F → G be a map of sheaves of sets. (1) We say that ϕ is injective if for every object U of C the map ϕ : F(U ) → G(U ) is injective. (2) We say that ϕ is surjective if for every object U of C and every section s ∈ F(U ) there exists a covering {Ui → U } such that for all i the restriction s|Ui is in the image of ϕ : F(Ui ) → G(Ui ). Lemma 9.11.2. The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of the category Sh(C). A map of sheaves is an isomorphism if and only if it is both injective and surjective. Proof. Omitted.

 9.12. Representable sheaves

Let C be a category. The canonical topology is the finest topology such that all representable presheaves are sheaves (it is formally defined in Definition 9.40.12 but we will not need this). This topology is not always the topology associated to the structure of a site on C. We will give a collection of coverings that generates this topology in case C has fibered products. First we give the following general definition. Definition 9.12.1. Let C be a category. We say that a family {Ui → U } is an effective epimorphism if all the morphisms Ui → U are representable (see Categories, Definition 4.6.3), and for any X ∈ Ob(C) the sequence / / MorC (Ui , X) MorC (U, X) / MorC (Ui ×U Uj , X) is an equalizer diagram. We say that a family {Ui → U } is a universal effective epimorphism if for any morphism V → U the base change {Ui ×U V → V } is an effective epimorphism. The class of families which are universal effective epimorphisms satisfies the axioms of Definition 9.6.2. If C has fibre products, then the associated topology is the canonical topology. (In this case, to get a site argue as in Sets, Lemma 3.11.1.) Conversely, suppose that C is a site such that all representable presheaves are sheaves. Then clearly, all coverings are universal effective epimorphisms. Thus the following definition is the “correct” one in the setting of sites.

9.12. REPRESENTABLE SHEAVES

631

Definition 9.12.2. We say that the topology on a site C is weaker than the canonical topology, or that the topology is subcanonical if all the coverings of C are universal effective epimorphisms. A representable sheaf is a representable presheaf which is also a sheaf. Since it is perhap better to avoid this terminology when the topology is not subcanonical, we only define it formally in that case. Definition 9.12.3. Suppose that the topology on the site C is weaker than the canonical topology. The Yoneda embedding h (see Categories, Section 4.3) actually presents C as a full subcategory of the category of sheaves of C. In this case we sometimes write U = hU or simply U for the representable sheaf associated to the object U of C. Note that we have in the situation of the definition MorSh(C) (U , F) = F(U ) for every sheaf F, since after all the same thing was true for presheaves. In general (but only rarely) the presheaves hU are not sheaves and to get a sheaf you have to sheafifiy them. In this case it will still be true that MorSh(C) (h# U , F) = F(U ) for every sheaf F by the adjointness property of #. The next lemma says that, if the topology is weaker than the canonical topology, every sheaf is made up out of representable sheaves in a way. Lemma 9.12.4. Let C be a site. Let F be a sheaf of sets. There exists a diagram of sheaves of sets / /F F1 / F0 which represents F as a coequalizer, such that Fi , i = 0, 1 are coproducts of sheaves of the form h# U. Proof. First we show there is an epimorphism F0 → F of the desired type. Namely, just take a F0 = (hU )# −→ F U ∈Ob(C),s∈F (U )

Here the arrow restricted to the component corresponding to (U, s) maps the element idU ∈ h# U (U ) to the section s ∈ F(U ). This is an epimorphism according to Lemma 9.11.2 above. To construct F1 first set G = F0 ×F F0 and then construct an epimorphism F1 → G as above.  Lemma 9.12.5. Let C be a site. If {Ui → U }i∈I is a covering of the site C, then the morphism of presheaves of sets a hUi → hU i∈I

becomes surjective after sheafification. ` # Proof. By Lemma 9.11.2 above we have to show that i∈I h# Ui → hU is an epimorphism. Let F be a sheaf of sets. A morphism h# → F corresponds to a section U Q # s ∈ F(U ). Hence the injectivity of Mor(h# , F) → U i Mor(hUi , F) follows directly from the sheaf property of F. 

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9. SITES AND SHEAVES

9.13. Continuous functors Definition 9.13.1. Let C and D be sites. A functor u : C → D is called continuous if for every {Vi → V }i∈I ∈ Cov(C) we have the following (1) {u(Vi ) → u(V )}i∈I is in Cov(D), and (2) for any morphism T → V in C the morphism u(T ×V Vi ) → u(T ) ×u(V ) u(Vi ) is an isomorphism. Recall that given a functor u as above, and a presheaf of sets F on D we have defined up F to be simply the presheaf F ◦ u, in other words up F(V ) = F(u(V )) for every object V of C. Lemma 9.13.2. Let C and D be sites. Let u : C → D be a continuous functor. If F is a sheaf on D then up F is a sheaf as well. Proof. Let {Vi → V } be a covering. By assumption {u(Vi ) → u(V )} is a covering in D and u(Vi ×V Vj ) = u(Vi ) ×u(V ) u(Vj ). Hence the sheaf condition for up F and the covering {Vi → V } is precisely the same as the sheaf condition for F and the covering {u(Vi ) → u(V )}.  In order to avoid confusion we sometimes denote us : Sh(D) −→ Sh(C) the functor up restricted to the subcategory of sheaves of sets. Lemma 9.13.3. In the situation of Lemma 9.13.2. The functor us : G 7→ (up G)# is a left adjoint to us . Proof. Follows directly from Lemma 9.5.4 and Proposition 9.10.12.



Here is a technical lemma. Lemma 9.13.4. In the situation of Lemma 9.13.2. For any presheaf G on C we have (up G)# = (up (G # ))# . Proof. For any sheaf F on D we have MorSh(D) (us (G # ), F)

=

MorSh(C) (G # , us F)

=

MorPSh(C) (G # , up F)

=

MorPSh(C) (G, up F)

=

MorPSh(D) (up G, F)

=

MorSh(D) ((up G)# , F)

and the result follows from the Yoneda lemma.



Lemma 9.13.5. Let u : C → D be a continuous functor between sites. For any # object U of C we have us h# U = hu(U ) . Proof. Follows from Lemmas 9.5.6 and 9.13.4.



Remark 9.13.6. (Skip on first reading.) Let C and D be sites. Let us use the definition of tautologically equivalent families of maps, see Definition 9.8.2 to (slightly) weaken the conditions defining continuity. Let u : C → D be a functor. Let us call u quasi-continuous if for every V = {Vi → V }i∈I ∈ Cov(C) we have the following

9.14. MORPHISMS OF SITES

633

(1’) the family of maps {u(Vi ) → u(V )}i∈I is tautologically equivalent to an element of Cov(D), and (2) for any morphism T → V in C the morphism u(T ×V Vi ) → u(T ) ×u(V ) u(Vi ) is an isomorphism. We are going to see that Lemmas 9.13.2 and 9.13.3 hold in case u is quasi-continuous as well. We first remark that the morphisms u(Vi ) → u(V ) are representable, since they are isomorphic to representable morphisms (by the first condition). In particular, the family u(V) = {u(Vi ) → u(V )}i∈I gives rise to a zeroth Cech cohomology group H 0 (u(V), F) for any presheaf F on D. Let U = {Uj → u(V )}j∈J be an element of Cov(D) tautologically equivalent to {u(Vi ) → u(V )}i∈I . Note that u(V) is a refinement of U and vice versa. Hence by Remark 9.10.7 we see that H 0 (u(V), F) = H 0 (U, F). In particular, if F is a sheaf, then F(u(V )) = H 0 (u(V), F) because of the sheaf property expressed in terms of zeroth Cech cohomology groups. We conclude that up F is a sheaf if F is a sheaf, since H 0 (V, up F) = H 0 (u(V), F) which we just observed is equal to F(u(V )) = up F(V ). Thus Lemma 9.13.2 holds. Lemma 9.13.3 follows immediately. 9.14. Morphisms of sites Definition 9.14.1. Let C and D be sites. A morphism of sites f : D → C is given by a continuous functor u : C → D such that the functor us is exact. Notice how the functor u goes in the direction opposite the morphism f . If f ↔ u is a morphism of sites then we use the notation f −1 = us and f∗ = us . The functor f −1 is called the pullback functor and the functor f∗ is called the pushforward functor. As in topology we have the following adjointness property MorSh(D) (f −1 G, F) = MorSh(C) (G, f∗ F) The motivation for this definition comes from the following example. Example 9.14.2. Let f : X → Y be a continuous map of topological spaces. Recall that we have sites TX and TY , see Example 9.6.4. Consider the functor u : TY → TX , V 7→ f −1 (V ). This functor is clearly continuous because inverse images of open coverings are open coverings. (Actually, this depends on how you chose sets of coverings for TX and TY . But in any case the functor is quasi-continuous, see Remark 9.13.6.) It is easy to verify that the functor us equals the usual pushforward functor f∗ from topology. Hence, since us is an adjoint and since the usual topological pullback functor f −1 is an adjoint as well, we get a canonical isomorphism f −1 ∼ = us . Since f −1 is exact we deduce that us is exact. Hence u defines a morphism of sites f : TX → TY , which we may denote f as well since we’ve already seen the functors us , us agree with their usual notions anyway. Lemma 9.14.3. Let Ci , i = 1, 2, 3 be sites. Let u : C2 → C1 and v : C3 → C2 be continuous functors which induce morphisms of sites. Then the functor u ◦ v : C3 → C1 is continuous and defines a morphism of sites C1 → C3 . Proof. It is immediate from the definitions that u ◦ v is a continuous functor. In addition, we clearly have (u ◦ v)p = v p ◦ up , and hence (u ◦ v)s = v s ◦ us . Hence functors (u◦v)s and us ◦vs are both left adjoints of (u◦v)s . Therefore (u◦v)s ∼ = us ◦vs and we conclude that (u ◦ v)s is exact as a composition of exact functors. 

634

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Definition 9.14.4. Let Ci , i = 1, 2, 3 be sites. Let f : C1 → C2 and g : C2 → C3 be morphisms of sites given by continuous functors u : C2 → C1 and v : C3 → C2 . The composition g ◦ f is the morphism of sites corresponding to the functor u ◦ v. In this situation we have (g ◦ f )∗ = g∗ ◦ f∗ and (g ◦ f )−1 = f −1 ◦ g −1 (see proof of Lemma 9.14.3). Lemma 9.14.5. Let C and D be sites. Let u : C → D be continuous. Assume all the categories (IVu )opp of Section 9.5 are filtered. Then u defines a morphism of sites D → C, in other words us is exact. Proof. Since us is the left adjoint of us we see that us is right exact, see Categories, Lemma 4.22.4. Hence it suffices to show that us is left exact. In other words we have to show that us commutes with finite limits. Because the categories IYopp are filtered we see that up commutes with finite limits, see Categories, Lemma 4.17.2 (this also uses the description of limits in PSh, see Section 9.4). And since sheafification commutes with finite limits as well (Lemma 9.10.14) we conclude because us = # ◦ up .  Proposition 9.14.6. Let C and D be sites. Let u : C → D be continuous. Assume furthermore the following: (1) the category C has a final object X and u(X) is a final object of D , and (2) the category C has fibre products and u commutes with them. Then u defines a morphism of sites D → C, in other words us is exact. Proof. This follows from Lemmas 9.5.2 and 9.14.5.



Remark 9.14.7. The conditions of Proposition 9.14.6 above are equivalent to saying that u is left exact, i.e., commutes with finite limits. See Categories, Lemmas 4.16.4 and 4.21.2. It seems more natural to phrase it in terms of final objects and fibre products since this seems to have more geometric meaning in the examples. Remark 9.14.8. (Skip on first reading.) Let C and D be sites. Analogously to Definition 9.14.1 we say that a quasi-morphism of sites f : D → C is given by a quasi-continuous functor u : C → D (see Remark 9.13.6) such that us is exact. The analogue of Proposition 9.14.6 in this setting is obtained by replacing the word “continuous” by the word “quasi-continuous”, and replacing the word “morphism” by “quasi-morphism”. The proof is literally the same. 9.15. Topoi Here is a definition of a topos which is suitable for our purposes. Namely, a topos is the category of sheaves on a site. In order to specify a topos you just specify the site. The real difference between a topos and a site lies in the definition of morphisms. Namely, it turns out that there are lots of morphisms of topoi which do not come from morphisms of the underlying sites. Definition 9.15.1. Topoi. (1) A topos is the category Sh(C) of sheaves of sets on a site C. (2) Let C, D be sites. A morphism of topoi f from Sh(D) to Sh(C) is given by a pair of functors f∗ : Sh(D) → Sh(C) and f −1 : Sh(C) → Sh(D) such that

9.15. TOPOI

635

(a) we have MorSh(D) (f −1 G, F) = MorSh(C) (G, f∗ F) bifunctorially, and (b) the functor f −1 commutes with finite limits, i.e., is left exact. (3) Let C, D, E be sites. Given morphisms of topoi f : Sh(D) → Sh(C) and g : Sh(E) → Sh(D) the composition f ◦ g is the morphism of topoi defined by the functors (f ◦ g)∗ = f∗ ◦ g∗ and (f ◦ g)−1 = g −1 ◦ f −1 . Note that, being an adjoint pair, the functor f∗ commutes with all limits and that f −1 commutes with all colimits, see Categories, Lemma 4.22.3. In particular, f −1 is exact. Suppose that α : S1 → S2 is an equivalence of (possibly “big”) categories. If S1 , S2 are topoi, then setting f∗ = α and f −1 equal to the quasi-inverse of α gives a morphism f : S1 → S2 of topoi. Moreover this morphism is an equivalence in the 2-category of topoi (see Section 9.32). Thus it makes sense to say “S is a topos” if S is equivalent to the category of sheaves on a site (and not necessarily equal to the the category of sheaves on a site). We will occasionally use this abuse of notation. Remark 9.15.2. (Set theoretical issues related to morphisms of topoi. Skip on a first reading.) A morphism of topoi as defined above is not a set but a class. In other words it is given by a mathematical formula rather than a mathematical object. Allthough we may contemplate the collection of all morphisms between two given topoi, it is not a good idea to introduce it as a mathematical object. On the other hand, suppose C and D are given sites. Consider a functor Φ : C → Sh(D). Such a thing is a set, in other words, it is a mathematical object. We may, in succession, ask the following questions on Φ. (1) Is it true, given a sheaf F on D, that the rule U 7→ MorSh(D) (Φ(U ), F) defines a sheaf on C? If so, this defines a functor Φ∗ : Sh(D) → Sh(C). (2) Is it true that Φ∗ has a left adjoint? If so, write Φ−1 for this left adjoint. (3) Is it true that Φ−1 is exact? If the last question still has the answer “yes”, then we obtain a morphism of topoi (Φ∗ , Φ−1 ). Moreover, given any morphism of topoi (f∗ , f −1 ) we may set Φ(U ) = −1 ∼ −1 ∼ f −1 (h# = Φ (compatible U ) and obtain a functor Φ as above with f∗ = Φ∗ and f with adjoint property). The upshot is that by working with the collection of Φ instead of morphisms of topoi, we (a) replaced the notion of a morphism of topoi by a mathematical object, and (b) the collection of Φ forms a class (and not a collection of classes). Of course, more can be said, for example one can work out more precisely the significance of condition (2) above for example; we do this in the case of points of topoi in Section 9.28. Most geometrically interesting morphisms of topoi come about via Lemma 9.19.1 and the following lemma. Lemma 9.15.3. Given a morphism of sites f : D → C corresponding to the functor u : C → D the pair of functors (f −1 = us , f∗ = us ) is a morphism of topoi. Proof. This is obvious from Definition 9.14.1.



The simplest example of a site is perhaps the site whose category has exactly one object pt and one morphism idpt and whose only covering is the covering {idpt }.

636

9. SITES AND SHEAVES

We will simply write pt for this site. It is clear that the category of sheaves = the category of presheaves = the category of sets. In a formula Sh(pt) = Sets. Remark 9.15.4. There are many sites that give rise to the topos Sh(pt). A useful example is the following. Suppose that S is a set (of sets) which contains at least one nonempty element. Let S be the category whose objects are elements of S and whose morphisms are arbitrary set maps. Assume that S has fibre products. For example this will be the case if S = P(infinite set) is the power set of any infinite set (exercise in set theory). Make S into a site by declaring surjective families of maps to be coverings (and choose a suitable sufficiently large set of covering families as in Sets, Section 3.11). We claim that Sh(S) is equivalent to the category of sets. We first prove this in case S contains e ∈ S which is a singleton. In this case, there is an equivalence of topoi i : Sh(pt) → Sh(S) given by the functors (9.15.4.1)

i−1 F = F(e),

i∗ E = (U 7→ MorSets (U, E))

Namely, suppose that F is a sheaf on S. For any U ∈ Ob(S) = S we can find a covering {ϕu : e → U }u∈U , where ϕu maps the unique Q element of e to u ∈ U . The sheaf condition implies in this case that F(U ) = u∈U F(e). In other words F(U ) = MorSets (U, F(e)). Moreover, this rule is compatible with restriction mappings. Hence the functor i∗ : Sets = Sh(pt) −→ Sh(S),

E 7−→ (U 7→ MorSets (U, E))

is an equivalence of categories, and its inverse is the functor i−1 given above. If S does not contain a singleton, then the functor i∗ as defined above still makes sense. To show that it is still an equivalence in this case, choose any nonempty e˜ ∈ S and a map ϕ : e˜ → e˜ whose image is a singleton. For any sheaf F set F(e) := Im(F(ϕ) : F(˜ e) −→ F(˜ e)) and show that this is a quasi-inverse to i∗ . Details omitted. Remark 9.15.5. (Skip on first reading.) Let C and D be sites. A quasi-morphism of sites f : D → C (see Remark 9.14.8) gives rise to a morphis of topoi f from Sh(D) to Sh(C) exactly as in Lemma 9.15.3. 9.16. G-sets and morphisms Let ϕ : G → H be a homomorphism of groups. Choose (suitable) sites TG and TH as in Example 9.6.5 and Section 9.9. Let u : TH → TG be the functor which assigns to a H-set U the G-set Uϕ which has the same underlying set but G action defined by g · u = ϕ(g)u. It is clear that u commutes with finite limits and is continuous4. Applying Proposition 9.14.6 and Lemma 9.15.3 we obtain a morphism of topoi f : Sh(TG ) −→ Sh(TH ) associated with ϕ. Using Proposition 9.9.1 we see that we get a pair of adjoint functors f∗ : G-Sets −→ H-Sets, f −1 : H-Sets −→ G-Sets. Let’s work out what are these functors in this case. 4Set theoretical remark: First choose T . Then choose T to contain u(T ) and such that H G H every covering in TH corresponds to a covering in TG . This is possible by Sets, Lemmas 3.10.1, 3.10.2 and 3.11.1.

9.17. MORE FUNCTORIALITY OF PRESHEAVES

637

We first work out a formula for f∗ . Recall that given a G-set S the corresponding sheaf FS on TG is given by the rule FS (U ) = MorG (U, S). And on the other hand, given a sheaf G on TH the corresponding H-set is given by the rule G(H H). Hence we see that f∗ S = MorG-Sets ((H H)ϕ , S). If we work this out a little bit more then we get f∗ S = {a : H → S | a(gh) = ga(h)} with left H-action given by (h · a)(h0 ) = a(h0 h) for any element a ∈ f∗ S. Next, we explicitly compute f −1 . Note that since the topology on TG and TH is subcanonical, all representable presheaves are sheaves. Moreover, given an object V of TH we see that f −1 hV is equal to hu(V ) (see Lemma 9.13.5). Hence we see that f −1 S = Sϕ for representable sheaves. Since every sheaf on TH is a coproduct of representable sheaves we conclude that this is true in general. Hence we see that for any H-set T we have f −1 T = Tϕ . The adjunction between f −1 and f∗ is evidenced by the formula MorG-Sets (Tϕ , S) = MorH-Sets (T, f∗ S) with f∗ S as above. This can be proved directly. Moreover, it is then clear that (f −1 , f∗ ) form an adjoint pair and that f −1 is exact. So alternatively to the above the morphism of topoi f : G-Sets → H-Sets can be defined directly in this manner. 9.17. More functoriality of presheaves In this section we revisit the material of Section 9.5. Let u : C → D be a functor between categories. Recall that up : PSh(D) −→ PSh(C) is the functor that associates to G on D the presheaf up G = G ◦ u. It turns out that this functor not only has a left adjoint (namely up ) but also a right adjoint. Namely, for any V ∈ Ob(D) we define a category V I = uV I. Its objects are pairs (U, ψ : u(U ) → V ). Note that the arrow is in the opposite direction from the arrow we used in defining the category IVu in Section 9.5. A morphism (U, ψ) → (U 0 , ψ 0 ) is given by a morphism α : U → U 0 such that ψ = ψ 0 ◦ u(α). In addition, given any presheaf of sets F on C we introduce the functor V F : V I opp → Sets, which is defined by the rule V F(U, ψ) = F(U ). We define p u(F)(V

) := limV I opp V F

As a limit there are projection maps c(ψ) : p u(F)(V ) → F(U ) for every object (U, ψ) of V I. In fact,    collections s(U,ψ) ∈ F(U )  ∀β : (U1 , ψ1 ) → (U2 , ψ2 ) in V I p u(F)(V ) =   we have β ∗ s(U2 ,ψ2 ) = s(U1 ,ψ1 ) where the correspondence is given by s 7→ s(U,ψ) = c(ψ)(s). We leave it to the reader to define the restriction mappings p u(F)(V ) → p u(F)(V 0 ) associated to any morphism V 0 → V of D. The resulting presheaf will be denoted p uF.

638

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Lemma 9.17.1. There is a canonical map p uF(u(U )) → F(U ), which is compatible with restriction maps. Proof. This is just the projection map c(idu(U ) ) above.



0

Note that any map of presheaves F → F gives rise to compatible systems of maps between functors V F → V F 0 , and hence to a map of presheaves p uF → p uF 0 . In other words, we have defined a functor pu

: PSh(C) −→ PSh(D)

Lemma 9.17.2. The functor p u is a right adjoint to the functor up . In other words the formula MorPSh(C) (up G, F) = MorPSh(D) (G, p uF) holds bifunctorially in F and G. Proof. This is proved in exactly the same way as the proof of Lemma 9.5.4. We note that the map up p uF → F from Lemma 9.17.1 is the map that is used to go from the right to the left. Alternately, think of a presheaf of sets F on C as a presheaf F 0 on C opp with values in Setsopp , and similarly on D. Check that (p uF)0 = up (F 0 ), and that (up G)0 = up (G 0 ). By Remark 9.5.5 we have the adjointness of up and up for presheaves with values in Setsopp . The result then follows formally from this.  9.18. Cocontinuous functors There is another way to construct morphisms of topoi. This involves using cocontinuous functors defined as follows. Definition 9.18.1. Let C and D be sites. Let u : C → D be a functor. The functor u is called cocontinuous if for every U ∈ Ob(C) and every covering {Vj → u(U )}j∈J of D there exists a covering {Ui → U }i∈I of C such that the family of maps {u(Ui ) → u(U )}i∈I refines the covering {Vj → u(U )}j∈J . Note that {u(Ui ) → u(U )}i∈I is in general not a covering of the site D. Lemma 9.18.2. Let C and D be sites. Let u : C → D be cocontinuous. Let F be a sheaf on C. Then p uF is a sheaf on D, which we will denote s uF. Proof. Let {Vj → V }j∈J be a covering of the site D. We have to show that Q /Q / p uF(Vj ) p uF(V ) / p uF(Vj ×V Vj 0 ) is an equalizer diagram. Since p u is right adjoint to up we have p uF(V

) = MorPSh(D) (hV , p uF) = MorPSh(C) (up hV , F) = MorSh(C) ((up hV )# , F)

Hence it suffices to show that ` p (9.18.2.1) u hVj ×V Vj0

/` p / u hVj

/ up hV

becomes a coequalizer diagram after sheafification. (Recall that a coproduct in the category of sheaves is the sheafification of the coproduct in the category of presheaves, see Lemma 9.10.13.) We first show that the second arrow of (9.18.2.1) becomes surjective after sheafification. To do this we use Lemma 9.11.2. Thus it suffices to show a section s

9.18. COCONTINUOUS FUNCTORS

639

` p of up hV over U lifts to a section of u hVj on the members of a covering of U . Note that s is a morphism s : u(U ) → V . Then {Vj ×V,s u(U ) → u(U )} is a covering of D. Hence, as u is cocontinuous, there is a covering {Ui → U } such that {u(Ui ) → u(U )} refines {Vj ×V,s u(U ) → u(U )}. This means that each restriction s|Ui : u(Ui ) → V factors through a morphism si : u(Ui ) → Vj for some j, i.e., s|Ui is in the image of up hVj (Ui ) → up hV (Ui ) as desired. ` Let s, s0 ∈ ( up hVj )# (U ) map to the same element of (up hV )# (U ). To finish the proof of the lemma we show that after replacing`U by the members of a covering that s, s0 are the image of the same section of up hVj ×V Vj0 by the two maps of (9.18.2.1). We may first replace U by the members of a covering and assume that s ∈ up hVj (U ) and s0 ∈ up hVj0 (U ). A second such replacement guarantees that s and s0 have the same image in up hV (U ) instead of in the sheafification. Hence s : u(U ) → Vj and s0 : u(U ) → Vj 0 are morphisms of D such that u(U )

s0

/ Vj 0

s

 Vj

 /V

is commutative. Thus we obtain t = (s, s0 ) : u(U ) → Vj ×V Vj 0 , i.e., a section t ∈ up hVj ×V Vj0 (U ) which maps to s, s0 as desired.  Lemma 9.18.3. Let C and D be sites. Let u : C → D be cocontinuous. The functor Sh(D) → Sh(C), G 7→ (up G)# is a left adjoint to the functor s u introduced in Lemma 9.18.2 above. Moreover, it is exact. Proof. Let us prove the adjointness property as follows MorSh(C) ((up G)# , F)

=

MorPSh(C) (up G, F)

=

MorPSh(D) (G, p uF)

=

MorSh(D) (G, s uF).

Thus it is a left adjoint and hence right exact, see Categories, Lemma 4.22.4. We have seen that sheafification is left exact, see Lemma 9.10.14. Moreover, the inclusion i : Sh(D) → PSh(D) is left exact by Lemma 9.10.1. Finally, the functor up is left exact because it is a right adjoint (namely to up ). Thus the functor is the composition # ◦ up ◦ i of left exact functors, hence left exact.  We finish this section with a technical lemma. Lemma 9.18.4. In the situation of Lemma 9.18.3. For any presheaf G on D we have (up G)# = (up (G # ))# . Proof. For any sheaf F on C we have MorSh(C) ((up (G # ))# , F)

=

MorSh(D) (G # , s uF)

=

MorPSh(D) (G # , p uF)

=

MorPSh(D) (G, p uF)

=

MorPSh(C) (up G, F)

=

MorSh(C) ((up G)# , F)

and the result follows from the Yoneda lemma.



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9. SITES AND SHEAVES

9.19. Cocontinuous functors and morphisms of topoi It is clear from the above that a cocontinuous functor u gives a morphism of topoi in the same direction as u. Thus this is in the opposite direction from the morphism of topoi associated (under certain conditions) to a continuous u as in Definition 9.14.1, Proposition 9.14.6, and Lemma 9.15.3. Lemma 9.19.1. Let C and D be sites. Let u : C → D be cocontinuous. The functors g∗ = s u and g −1 = (up )# define a morphism of topoi g from Sh(C) to Sh(D). Proof. This is exactly the content of Lemma 9.18.3.



Lemma 9.19.2. Let u : C → D, and v : D → E be cocontinuous functors. Then v ◦ u is cocontinuous and we have h = g ◦ f where f : Sh(C) → Sh(D), resp. g : Sh(D) → Sh(E), resp. h : Sh(C) → Sh(E) is the morphism of topoi associated to u, resp. v, resp. v ◦ u. Proof. Let U ∈ Ob(C). Let {Ei → v(u(U ))} be a covering of U in E. By assumption there exists a covering {Dj → u(U )} in D such that {v(Dj ) → v(u(U ))} refines {Ei → v(u(U ))}. Also by assumption there exists a covering {Cl → U } in C such that {u(Cl ) → u(U )} refines {Dj → u(U )}. Then it is true that {v(u(Cl )) → v(u(U ))} refines the covering {Ei → v(u(U ))}. This proves that v ◦ u is cocontinuous. To prove the last assertion it suffices to show that s v ◦ s u = s (v ◦u). It suffices to prove that p v ◦ p u = p (v ◦ u), see Lemma 9.18.2. Since p u, resp. p v, resp. p (v ◦ u) is right adjoint to up , resp. v p , resp. (v ◦ u)p it suffices to prove that up ◦ v p = (v ◦ u)p . And this is direct from the definitions.  Example 9.19.3. Let X be a topological space. Let j : U → X be the inclusion of an open subspace. Recall that we have sites TX and TU , see Example 9.6.4. Recall that we have the functor u : TX → TU associated to j which is continuous and gives rise to a morphism of sites TU → TX , see Example 9.14.2. This also gives a morphism of topoi (j∗ , j −1 ). Next, consider the functor v : TU → TX , V 7→ v(V ) = V (just the same open but now thought of as an object of TX ). This S functor is cocontinuous. Namely, if v(V ) = j∈J Wj is an open covering in X, then S each Wj must be a subset of U and hence is of the form v(Vj ), and trivially V = j∈J Vj is an open covering in U . We conclude by Lemma 9.19.1 above that there is a morphism of topoi associated to v Sh(U ) −→ Sh(X) given by s v and (v p )# . We claim that actually (v p )# = j −1 and that s v = j∗ , in other words, that this is the same morphism of topoi as the one given above. Perhaps the easiest way to see this is to realize that for any sheaf G on X we have v p G(V ) = G(V ) which according to Sheaves, Lemma 6.31.1 is a description of j −1 G (and hence sheafification is superfluous in this case). The equality of s v and j∗ follows by uniqueness of adjoint functors (but may also be computed directly). Example 9.19.4. This example is a slight generalization of Example 9.19.3. Let f : X → Y be a continuous map of topological spaces. Assume that f is open. Recall that we have sites TX and TY , see Example 9.6.4. Recall that we have the functor u : TY → TX associated to f which is continuous and gives rise to a morphism of sites TX → TY , see Example 9.14.2. This also gives a morphism

9.19. COCONTINUOUS FUNCTORS AND MORPHISMS OF TOPOI

641

of topoi (f∗ , f −1 ). Next, consider the functor v : S TX → TY , U 7→ v(U ) = f (U ). This functor is cocontinuous. Namely, if f (U ) = j∈J Vj is an open covering in S Y , then S setting Uj = f −1 (Vj ) ∩ U we get an S open covering U = Uj such that f (U ) = f (Uj ) is a refinement of f (U ) = Vj . We conclude by Lemma 9.19.1 above that there is a morphism of topoi associated to v Sh(X) −→ Sh(Y ) given by s v and (v p )# . We claim that actually (v p )# = f −1 and that s v = f∗ , in other words, that this is the same morphism of topoi as the one given above. For any sheaf G on Y we have v p G(U ) = G(f (U )). On the other hand, we may compute up G(U ) = colimf (U )⊂V G(V ) = G(f (U )) because clearly (f (U ), U ⊂ f −1 (f (U ))) is an initial object of the category IUu of Section 9.5. Hence up = v p and we conclude f −1 = us = (v p )# . The equality of s v and f∗ follows by uniqueness of adjoint functors (but may also be computed directly). In the first Example 9.19.3 the functor v is also continuous. But in the second Example 9.19.4 it is generally not continuous because condition (2) of Definition 9.13.1 may fail. Hence the following lemma applies to the first example, but not to the second. Lemma 9.19.5. Let C and D be sites. Let u : C → D be a functor. Assume that (a) u is cocontinuous, and (b) u is continuous. Let g : Sh(C) → Sh(D) be the associated morphism of topoi. Then (1) sheafification in the formula g −1 = (up )# is unnecessary, in other words g −1 (G)(U ) = G(u(U )), (2) g −1 has a left adjoint g! = (up )# , and (3) g −1 commutes with arbitrary limits and colimits. Proof. By Lemma 9.13.2 for any sheaf G on D the presheaf up G is a sheaf on C. And then we see the adjointness by the following string of equalities MorSh(C) (F, g −1 G)

=

MorPSh(C) (F, up G)

=

MorPSh(D) (up F, G)

=

MorSh(D) (g! F, G)

The statement on limits and colimits follows from the discussion in Categories, Section 4.22.  In the situation of Lemma 9.19.5 above we see that we have a sequence of adjoint functors g! , g −1 , g∗ . The functor g! is not exact in general, because it does not transform a final object of Sh(C) into a final object of Sh(D) in general. See Sheaves, Remark 6.31.13. On the other hand, in the topological setting of Example 9.19.3 the functor j! is exact on abelian sheaves, see Modules, Lemma 15.3.5. The following lemma gives the generalization to the case of sites. Lemma 9.19.6. Let C and D be sites. Let u : C → D be a functor. Assume that (a) u is cocontinuous, (b) u is continuous, and

642

9. SITES AND SHEAVES

(c) fibre products and equalizers exist in C and u commutes with them. In this case the functor g! above commutes with fibre products and equalizers (and more generally with any finite, nonempty connected limits). Proof. Assume (a), (b), and (c). We have g! = (up )# . Recall (Lemma 9.10.1) that limits of sheaves are equal to the corresponding limits as presheaves. And sheafification commutes with finite limits (Lemma 9.10.14). Thus it suffices to show that up commutes with fibre products and equalizers. To do this it suffices that colimits over the categories (IVu )opp of Section 9.5 commute with fibre products and equalizers. This follows from Lemma 9.5.1 and Categories, Lemma 4.17.4.  The following lemma deals with a case that is even more like the morphism associated to an open immersion of topological spaces. Lemma 9.19.7. Let C and D be sites. Let u : C → D be a functor. Assume that (a) u is cocontinuous, (b) u is continuous, and (c) u is fully faithful. For g! , g −1 , g∗ as above the canonical maps F → g −1 g! F and g −1 g∗ F → F are isomorphisms for all sheaves F on C. Proof. Let X be an object of C. In Lemmas 9.18.2 and 9.19.5 we have seen that sheafification is not necessary for the functors g −1 = (up )# and g∗ = (p u )# . We may compute (g −1 g∗ F)(X) = g∗ F(u(X)) = lim F(Y ). Here the limit is over the category of pairs (Y, u(Y ) → u(X)) where the morphisms u(Y ) → u(X) are not required to be of the form u(α) with α a morphism of C. By assumption (c) we see that they automatically come from morphisms of C and we deduce that the limit is the value on (X, u(idX )), i.e., F(X). This proves that g −1 g∗ F = F. On the other hand, (g −1 g! F)(X) = g! F(u(X)) = (up F)# (u(X)), and up F(u(X)) = colim F(Y ). Here the colimit is over the category of pairs (Y, u(X) → u(Y )) where the morphisms u(X) → u(Y ) are not required to be of the form u(α) with α a morphism of C. By assumption (c) we see that they automatically come from morphisms of C and we deduce that the colimit is the value on (X, u(idX )), i.e., F(X). Thus for every X ∈ Ob(C) we have up F(u(X)) = F(X). Since u is cocontinuous and continuous any covering of u(X) in D can be refined by a covering (!) {u(Xi ) → u(X)} of D where {Xi → X} is a covering in C. This implies that (up F)+ (u(X)) = F(X) also, since in the colimit defining the value of (up F)+ on u(X) we may restrict to the cofinal system of coverings {u(Xi ) → u(X)} as above. Hence we see that (up F)+ (u(X)) = F(X) for all objects X of C as well. Repeating this argument one more time gives the equality (up F)# (u(X)) = F(X) for all objects X of C. This produces the desired equality g −1 g! F = F.  Finally, here is a case that does not have any corresponding topological example. Namely, this lemma explains what happens when we enlarge a “partial universe” of schemes keeping the same topology. Lemma (a) (b) (c)

9.19.8. Let C and D be sites. Let u : C → D be a functor. Assume that u is cocontinuous, u is continuous, u is fully faithful,

9.20. COCONTINUOUS FUNCTORS WHICH HAVE A RIGHT ADJOINT

643

(d) fibre products exist in C and u commutes with them, and (e) there exist final objects eC ∈ Ob(C), eD ∈ Ob(D) such that u(eC ) = eD . Let g! , g −1 , g∗ be as above. Then, u defines a morphism of sites f : D → C with f∗ = g −1 , f −1 = g! . The composition Sh(C)

g

/ Sh(D)

f

/ Sh(C)

is isomorphic to the identity morphism of the topos Sh(C). Moreover, the functor f −1 is fully faithful. Proof. By assumption the functor u satisfies the hypotheses of Proposition 9.14.6. Hence u defines a morphism of sites and hence a morphism of topoi f as in Lemma 9.15.3. The formulas f∗ = g −1 and f −1 = g! are clear from the lemma cited and Lemma 9.19.5. We have f∗ ◦ g∗ = g −1 ◦ g∗ ∼ = id, and g −1 ◦ f −1 = g −1 ◦ g! ∼ = id by Lemma 9.19.7. We still have to show that f −1 is fully faithful. Let F, G ∈ Ob(Sh(C)). We have to show that the map MorSh(C) (F, G) −→ MorSh(D) (f −1 F, f −1 G) is bijective. But the right hand side is equal to MorSh(D) (f −1 F, f −1 G) = MorSh(C) (f∗ f −1 F, G) = MorSh(C) (g −1 f −1 F, G) = MorSh(C) (F, G) (the first equality by adjunction) which proves what we want.



Example 9.19.9. Let X be a topological space. Let i : Z → X be the inclusion of a subset (with induced topology). Consider the functor u : TX → TZ , U 7→ u(U ) = Z ∩ U . At first glance it may appear that this functor is cocontinuous as well. After all, since Z has the induced topology, shouldn’t any covering of U ∩ Z it come from a covering of U in X? Not so! Namely, what if U ∩ Z = ∅? In that case, the empty covering is a covering of U ∩ Z, and the empty covering can only be refined by the empty covering. Thus we conclude that u cocontinuous ⇒ every nonempty open U of X has nonempty intersection with Z. But this is not sufficient. For example, if X = R the real number line with the usual topology, S and Z = R \ {0}, then there is an open covering of Z, namely Z = {x < 0} ∪ n {1/n < x} which cannot be refined by the restriction of any open covering of X. 9.20. Cocontinuous functors which have a right adjoint It may happen that a cocontinuous functor u has a right adjoint v. In this case it is often the case that v is continuous, and if so, then it defines a morphism of topoi (which is the same as the one defined by u). Lemma 9.20.1. Let C and D be sites. Let u : C → D, and v : D → C be functors. Assume that u is cocontinuous, and that v is a right adjoint to u. Let g : Sh(C) → Sh(D) be the morphism of topoi associated to u, see Lemma 9.19.1. Then g∗ F is equal to the presheaf v p F, in other words, (g∗ F)(V ) = F(v(V )).

644

9. SITES AND SHEAVES

Proof. Let V be an object of D. We have up hV = hv(V ) because up hV (U ) = MorD (u(U ), V ) = MorC (U, v(V )) by assumption. By Lemma 9.18.4 this implies # p # # p # that g −1 (h# V ) = (u hV ) = (u hV ) = hv(V ) . Hence for any sheaf F on C we have (g∗ F)(V )

=

MorSh(D) (h# V , g∗ F)

=

MorSh(C) (g −1 (h# V ), F)

=

MorSh(C) (h# v(V ) , F)

= F(v(V )) which proves the lemma.



In the situation of the lemma we see that v p transforms sheaves into sheaves. Hence we can define v s = v p restricted to sheaves. Just as in Lemma 9.13.3 we see that vs : G 7→ (vp G)# is a left adjoint to v s . On the other hand, we have v s = g∗ and g −1 is a left adjoint of g∗ as well. We conclude that g −1 = vs is exact. Lemma 9.20.2. In the situation of Lemma 9.20.1. We have g∗ = v s = v p and g −1 = vs = (vp )# . If v is continuous then v defines a morphism of sites f from C to D whose associated morphism of topoi is equal to the morphism g associated to the cocontinuous functor u. Proof. Clear from the discussion above the lemma and Definitions 9.14.1 and Lemma 9.15.3.  9.21. Localization Let C be a site. Let U ∈ Ob(C). See Categories, Example 4.2.13 for the definition of the category C/U of objects over U . We turn C/U into a site by declaring a family of morphisms {Vj → V } of objects over U to be a covering of C/U if and only if it is a covering in C. Consider the forgetful functor jU : C/U −→ C. This is clearly cocontinuous and continuous. Hence by the results of the previous sections we obtain a morphism of topoi jU : Sh(C/U ) −→ Sh(C) given by jU−1 and jU ∗ , as well as a functor jU ! . Definition 9.21.1. Let C be a site. Let U ∈ Ob(C). (1) The site C/U is called the localization of the site C at the object U . (2) The morphism of topoi jU : Sh(C/U ) → Sh(C) is called the localization morphism. (3) The functor jU ∗ is called the direct image functor. (4) For a sheaf F on C the sheaf jU−1 F is called the restriction of F to C/U . (5) For a sheaf G on C/U the sheaf jU ! G is called the extension of G by the empty set. The restriction jU−1 F is the sheaf defined by the rule jU−1 F(X/U ) = F(X) as expected. The extension by the empty set also has a very easy description in this case; here it is.

9.21. LOCALIZATION

645

Lemma 9.21.2. Let C be a site. Let U ∈ Ob(C). Let G be a presheaf on C/U . Then jU ! (G # ) is the sheaf associated to the presheaf a ϕ V 7−→ G(V − → U) ϕ∈MorC (V,U )

with obvious restriction mappings. Proof. By Lemma 9.19.5 we have jU ! (G # ) = ((jU )p G # )# . By Lemma 9.13.4 this is equal to ((jU )p G)# . Hence it suffices to prove that (jU )p is given by the formula above for any presheaf G on C/U . OK, and by the definition in Section 9.5 we have (jU )p G(V ) = colim(W/U,V →W ) G(W ) Now it is clear that the category of pairs (W/U, V → W ) has an object Oϕ = (ϕ : V → U, id : V → V ) for every ϕ : V → U , and moreover for any object there is a unique morphism from one of the Oϕ into it. The result follows.  Lemma 9.21.3. Let C be a site. Let U ∈ Ob(C). Let X/U be an object of C/U . # Then we have jU ! (h# X/U ) = hX . Proof. Denote p : X → U the structure morphism of X. By Lemma 9.21.2 we see jU ! (h# X/U ) is the sheaf associated to the presheaf a V 7−→ {ψ : V → X | p ◦ ψ = ϕ} ϕ∈MorC (V,U )

This is clearly the same thing as MorC (V, X). Hence the lemma follows.



We have jU ! (∗) = h# U by either of the two lemmas above. Hence for every sheaf G over C/U there is a canonical map of sheaves jU ! G → h# U . This characterizes sheaves in the essential image of jU ! . Lemma 9.21.4. Let C be a site. Let U ∈ Ob(C). The functor jU ! gives an equivalence of categories Sh(C/U ) −→ Sh(C)/h# U Proof. We explain how to get a functor from Sh(C)/h# U to Sh(C/U ). Suppose # that ϕ : F → hU is given. For any object a : X → U of C/U we consider the set Fϕ (X → U ) of elements s ∈ F(X) which under ϕ map to the image of a ∈ MorC (X, U ) = hU (X) in h# U (X). It is easy to see that (X → U ) 7→ Fϕ (X → U ) is a sheaf on C/U . The verification that (F, ϕ) 7→ Fϕ is an inverse to the functor jU ! is omitted.  The lemma says the functor jU ! is the composition Sh(C/U ) → Sh(C)/h# U → Sh(C) where the first arrow is an equivalence. Lemma 9.21.5. Let C be a site. Let U ∈ Ob(C). The functor jU ! commutes with with fibre products and equalizers (and more generally finite, nonempty, connected limits). In particular, if F ⊂ F 0 in Sh(C/U ), then jU ! F ⊂ jU ! F 0 . Proof. This follows from the fact that an isomorphism of categories commutes with all limits and the functor Sh(C)/h# U → Sh(C) commutes with fibre products and equalizers. Alternatively, one can prove this directly using the description of jU ! in Lemma 9.21.2 using that sheafification is exact. (Also, in case C has fibre products and equalizers, the result follows from Lemma 9.19.6.) 

646

9. SITES AND SHEAVES

Lemma 9.21.6. Let C be a site. Let U ∈ Ob(C). For any sheaf F on C we have jU ! jU−1 F = F × h# U. Proof. This is clear from the description of jU ! in Lemma 9.21.2.



Lemma 9.21.7. Let C be a site. Let f : V → U be a morphism of C. Then there exists a commutative diagram C/V

/ C/U

j jV

!

C

~

jU

of cocontinuous functors. Here j : C/V → C/U , (a : W → V ) 7→ (f ◦ a : W → U ) is identified with the functor jV /U : (C/U )/(V /U ) → C/U via the identification (C/U )/(V /U ) = C/V . Moreover we have jV ! = jU ! ◦ j! , jV−1 = j −1 ◦ jU−1 , and jV ∗ = jU ∗ ◦ j∗ . Proof. The commutativity of the diagram is immediate. The agreement of j with jV /U follows from the definitions. By Lemma 9.19.2 we see that the following diagram of morphisms of topoi Sh(C/V ) (9.21.7.1)

/ Sh(C/U )

j

$ z Sh(C)

jV

jU

is commutative. This proves that jV−1 = j −1 ◦ jU−1 and jV ∗ = jU ∗ ◦ j∗ . The equality jV ! = jU ! ◦ j! follows formally from adjointness properties.  Lemma 9.21.8. Notation C, f : V → U , jU , jV , and j as in Lemma 9.21.7. # Via the identifications Sh(C/V ) = Sh(C)/h# V and Sh(C/U ) = Sh(C)/hU of Lemma −1 9.21.4 the functor j has the following description ϕ

# # j −1 (H − → h# U ) = (H ×ϕ,h# ,f hV → hV ). U

h# U

Proof. Suppose that ϕ : H → is an object of Sh(C)/h# U . By the proof of Lemma 9.21.4 this corresponds to the sheaf Hϕ on C/U defined by the rule (a : W → U ) 7−→ {s ∈ H(W ) | ϕ(s) = a} on C/U . The pullback j −1 Hϕ to C/V is given by the rule (a : W → V ) 7−→ {s ∈ H(W ) | ϕ(s) = f ◦ a} −1 by the description of j −1 = jU/V as the restriction of Hϕ to C/V . On the other hand, applying the rule to the object

H0 = H ×ϕ,h# ,f h# V

ϕ0

U

/ h# V

0 of Sh(C)/h# V we get Hϕ0 given by

(a : W → V ) 7−→{s0 ∈ H0 (W ) | ϕ0 (s0 ) = a} 0 0 ={(s, a0 ) ∈ H(W ) × h# V (W ) | a = a and ϕ(s) = f ◦ a }

which is exactly the same rule as the one describing j −1 Hϕ above.



9.22. GLUEING SHEAVES

647

Remark 9.21.9. Localization and presheaves. Let C be a category. Let U be an object of C. Strictly speaking the functors jU−1 , jU ∗ and jU ! have not been defined for presheaves. But of course, we can think of a presheaf as a sheaf for the chaotic topology on C (see Example 9.6.6). Hence we also obtain a functor jU−1 : PSh(C) −→ PSh(C/U ) and functors jU ∗ , jU ! : PSh(C/U ) −→ PSh(C) which are right, left adjoint to jU−1 . By Lemma 9.21.2 we see that jU ! G is the presheaf a ϕ V 7−→ G(V − → U) ϕ∈MorC (V,U )

In addition the functor jU ! commutes with fibre products and equalizers. 9.22. Glueing sheaves This section is the analogue of Sheaves, Section 6.33. Lemma 9.22.1. Let C be a site. Let {Ui → U } be a covering of C. Let F, G be sheaves on C. Given a collection ϕi : F|C/Ui −→ G|C/Ui of maps of sheaves such that for all i, j ∈ I the maps ϕi , ϕj restrict to the same map F|C/Ui ×U Uj → G|C/Ui ×U Uj then there exists a unique map of sheaves ϕ : F|C/U −→ G|C/U whose restriction to each C/Ui agrees with ϕi . Proof. Omitted. Note that the restrictions are always those of Lemma 9.21.7.



The previous lemma implies that given two sheaves F, G on a site C the rule U 7−→ MorSh(C/U ) (F|C/U , G|C/U ) defines a sheaf. This is a kind of internal hom sheaf. It is seldom used in the setting of sheaves of sets, and more usually in the setting of sheaves of modules, see Modules on Sites, Section 16.25. Let C be a site. Let {Ui → U }i∈I be a covering of C. For each i ∈ I let Fi be a sheaf of sets on C/Ui . For each pair i, j ∈ I, let ϕij : Fi |C/Ui ×U Uj −→ Fj |C/Ui ×U Uj be an isomorphism of sheaves of sets. Assume in addition that for every triple of indices i, j, k ∈ I the following diagram is commutative Fi |C/Ui ×U Uj ×U Uk

/ Fk |C/U × U × U 5 i U j U k

ϕik ϕij

)

ϕjk

Fj |C/Ui ×U Uj ×U Uk We will call such a collection of data (Fi , ϕij ) a glueing data for sheaves of sets with respect to the covering {Ui → U }i∈I .

648

9. SITES AND SHEAVES

Lemma 9.22.2. Let C be a site. Let {Ui → U }i∈I be a covering of C. Given any glueing data (Fi , ϕij ) for sheaves of sets with respect to the covering {Ui → U }i∈I there exists a sheaf of sets F on C/U together with isomorphisms ϕi : F|C/Ui → Fi such that the diagrams F|C/Ui ×U Uj id

ϕi

/ Fi |C/U × U i U j ϕij



F|C/Ui ×U Uj

ϕj



/ Fj |C/U × U i U j

are commutative. Proof. Let us describe how to construct the sheaf F on C/U . Let a : V → U be an object of C/U . Then Y F(V /U ) = {(si )i∈I ∈ Fi (Ui ×U V /Ui ) | ϕij (si |Ui ×U Uj ×U V ) = sj |Ui ×U Uj ×U V } i∈I

We omit the construction of the restriction mappings. We omit the verification that this is a sheaf. We omit the construction of the isomorphisms ϕi , and we omit proving the commutativity of the diagrams of the lemma.  Let C be a site. Let {Ui → U }i∈I be a covering of C. Let F be a sheaf on C/U . Associated to F we have its canonical glueing data given by the restrictions F|C/Ui and the canonical isomorphisms

F|C/Ui |C/Ui ×U Uj = F|C/Uj |C/Ui ×U Uj coming from the fact that the composition of the functors C/Ui ×U Uj → C/Ui → C/U and C/Ui ×U Uj → C/Uj → C/U are equal. Lemma 9.22.3. Let C be a site. Let {Ui → U }i∈I be a covering of C. The category Sh(C/U ) is equivalent to the category of glueing data via the functor that associates to F on C/U the canonical glueing data. Proof. In Lemma 9.22.1 we saw that the functor is fully faithful, and in Lemma 9.22.2 we proved that it is essentially surjective (by explicitly constructing a quasiinverse functor).  9.23. More localization In this section we prove a few lemmas on localization where we impose some additional hypotheses on the site on or the object we are localizing at. Lemma 9.23.1. Let C be a site. Let U ∈ Ob(C). If the topology on C is subcanonical, see Definition 9.12.2, and if G is a sheaf on C/U , then a ϕ jU ! (G)(V ) = G(V − → U ), ϕ∈MorC (V,U )

in other words sheafification is not necessary in Lemma 9.21.2.

9.23. MORE LOCALIZATION

649

Proof. Let V = {Vi → V }i∈I be a covering of V in the site C. We are going to check the sheaf condition for the presheaf H of Lemma 9.21.2 directly. Let (si , ϕi )i∈I ∈ Q ϕi i H(Vi ), This means ϕi : Vi → U is a morphism in C, and si ∈ G(Vi −→ U ). The restriction of the pair (si , ϕi ) to Vi ×V Vj is the pair (si |Vi ×V Vj /U , pr1 ◦ ϕi ), and likewise the restriction of the pair (sj , ϕj ) to Vi ×V Vj is the pair (sj |Vi ×V Vj /U , pr2 ◦ ˇ 0 (V, H), then we see that pr1 ◦ ϕi = ϕj ). Hence, if the family (si , ϕi ) lies in H pr2 ◦ϕj . The condition that the topology on C is weaker than the canonical topology then implies that there exists a unique morphism ϕ : V → U such that ϕi is the composition of Vi → V with ϕ. At this point the sheaf condition for G garantees ϕ that the sections si glue to a unique section s ∈ G(V − → U ). Hence (s, ϕ) ∈ H(V ) as desired.  Lemma 9.23.2. Let C be a site. Let U ∈ Ob(C). Assume C has products of pairs of objects. Then (1) the functor jU has a continuous right adjoint, namely the functor v(X) = X × U/U , (2) the functor v defines a morphism of sites C/U → C whose associated morphism of topoi equals jU : Sh(C/U ) → Sh(C), and (3) we have jU ∗ F(X) = F(X × U/U ). Proof. The functor v being right adjoint to jU means that given Y /U and X we have MorC (Y, X) = MorC/U (Y /U, X × U/U ) which is clear. To check that v is continous let {Xi → X} be a convering of C. By the third axiom of a site (Definition 9.6.2) we see that {Xi ×X (X × U ) → X ×X (X × U )} = {Xi × U → X × U } is a covering of C also. Hence v is continuous. The other statements of the lemma follow from Lemmas 9.20.1 and 9.20.2.  A fundamental property of an open immersion is that the restriction of the pushforward and the restriction of the extension by the empty set produces back the original sheaf. This is not always true for the functors associated to jU above. It is true when U is a “subobject of the final object”. Lemma 9.23.3. Let C be a site. Let U ∈ Ob(C). Assume that every X in C has at most one morphism to U . Let F be a sheaf on C/U . The canonical maps F → jU−1 jU ! F and jU−1 jU ∗ F → F are isomorphisms. Proof. If C has fibre products, then this is a special case of Lemma 9.19.7. In general we have the following direct proof. Let X/U be an object over U . In Lemmas 9.18.2 and 9.19.5 we have seen that sheafification is not necessary for the functors jU−1 = (up )# and jU ∗ = (p u)# . We may compute (jU−1 jU ∗ F)(X/U ) = jU ∗ F(X) = lim F(Y /U ). Here the limit is over the category of pairs (Y /U, Y → X) where the morphisms Y → X are not required to be over U . By our assumption however we see that they are automatically morphisms over U and we deduce that the limit is the value on idX , i.e., F(X/U ). This proves that jU−1 jU ∗ F = F. On the other hand, (jU−1 jU ! F)(X/U ) = jU ! F(X) = (up F)# (X), and up F(X) = colim F(Y /U ). Here the colimit is over the category of pairs (Y /U, X → Y ) where

650

9. SITES AND SHEAVES

the morphisms X → Y are not required to be over U . By our assumption however we see that they are automatically morphisms over U and we deduce that the colimit is the value on idX , i.e., F(X/U ). This shows that the sheafification is not necessary (since any object over X is automatically in a unique way an object over U ) and the result follows.  9.24. Localization and morphisms The following lemma is important in order to understand relation between localization and morphisms of sites and topoi. Lemma 9.24.1. Let f : C → D be a morphism of sites corresponding to the continuous functor u : D → C. Let V ∈ Ob(D) and set U = u(V ). Then the functor u0 : D/V → C/U , V 0 /V 7→ u(V 0 )/U determines a morphism of sites f 0 : C/U → D/V . The morphism f 0 fits into a commutative diagram of topoi Sh(C/U )

jU

f0

 Sh(D/V )

/ Sh(C) f

jV

 / Sh(D).

# Using the identifications Sh(C/U ) = Sh(C)/h# U and Sh(D/V ) = Sh(D)/hV of Lemma 0 −1 9.21.4 the functor (f ) is described by the rule f −1 ϕ

ϕ

−1 (f 0 )−1 (H − → h# H −−−→ h# V ) = (f U ).

Finally, we have f∗0 jU−1 = jV−1 f∗ . Proof. It is clear that u0 is continuous, and hence we get functors f∗0 = (u0 )s = (u0 )p (see Sections 9.5 and 9.13) and an adjoint (f 0 )−1 = (u0 )s = ((u0 )p )# . The assertion f∗0 jU−1 = jV−1 f∗ follows as (jV−1 f∗ F)(V 0 /V ) = f∗ F(V 0 ) = F(u(V 0 )) = (jU−1 F)(u(V 0 )/U ) = (f∗0 jU−1 F)(V 0 /V ) which holds even for presheaves. What isn’t clear a priori is that (f 0 )−1 is exact, that the diagram commutes, and that the description of (f 0 )−1 holds. Let H be a sheaf on D/V . Let us compute jU ! (f 0 )−1 H. We have jU ! (f 0 )−1 H

= ((jU )p (u0p H)# )# = ((jU )p u0p H)# = (up (jV )p H)# = f −1 jV ! H

The first equality by unwinding the definitions. The second equality by Lemma 9.13.4. The third equality because u ◦ jV = jU ◦ u0 . The fourth equality by Lemma 9.13.4 again. All of the equalities above are isomorphisms of functors, and hence we may interpret this as saying that the following diagram of categories and functors is commutative / Sh(C)/h# / Sh(C) Sh(C/U ) jU ! O O O U (f 0 )−1

Sh(D/V )

f −1 jV !

f −1

/ Sh(D)/h# V

/ Sh(D)

9.24. LOCALIZATION AND MORPHISMS

651

# # The middle arrow makes sense as f −1 h# V = (hu(V ) ) = hU , see Lemma 9.13.5. In 0 −1 particular this proves the description of (f ) given in the statement of the lemma. Since by Lemma 9.21.4 the left horizontal arrows are equivalences and since f −1 is exact by assumption we conclude that (f 0 )−1 = u0s is exact. Namely, because it is a left adjoint it is already right exact (Categories, Lemma 4.22.3). Hence we only need to show that it transforms a final object into a final object and commutes with fibre products (Categories, Lemma 4.21.2). Both are clear for the induced functor # 0 f −1 : Sh(D)/h# V → Sh(C)/hU . This proves that f is a morphism of sites.

We still have to verify that (f 0 )−1 jV−1 = jU−1 f −1 . To see this use the formula above and the description in Lemma 9.21.6. Namely, combined these give, for any sheaf G on D, that −1 −1 −1 jU ! (f 0 )−1 jV−1 G = f −1 jV ! jV−1 G = f −1 (G × h# G × h# G. V)=f U = jU ! jU f

Since the functor jU ! induces an equivalence Sh(C/U ) → Sh(C)/h# U we conclude.  The following lemma is a special case of the more general Lemma 9.24.1 above. Lemma 9.24.2. Let C, D be sites. Let u : D → C be a functor. Let V ∈ Ob(D). Set U = u(V ). Assume that (1) C and D have all finite limits, (2) u is continuous, and (3) u commutes with finite limits. There exists a commutative diagram of morphisms of sites C/U

jU

f0

 D/V

/C f

jV

 /D

where the right vertical arrow corresponds to u, the left vertical arrow corresponds to the functor u0 : D/V → C/U , V 0 /V 7→ u(V 0 )/u(V ) and the horizontal arrows correspond to the functors C → C/U , X 7→ X × U and D → D/V , Y 7→ Y × V as in Lemma 9.23.2. Moreover, the associated diagram of morphisms of topoi is equal to the diagram of Lemma 9.24.1. In particular we have f∗0 jU−1 = jV−1 f∗ . Proof. Note that u satisfies the assumptions of Proposition 9.14.6 and hence induces a morphism of sites f : C → D by that proposition. It is clear that u induces a functor u0 as indicated. It is clear that this functor also satisfies the assumptions of Proposition 9.14.6. Hence we get a morphism of sites f 0 : C/U → D/V . The diagram commutes by our definition of composition of morphisms of sites (see Definition 9.14.4) and because u(Y × V ) = u(Y ) × u(V ) = u(Y ) × U which shows that the diagram of categories and functors opposite to the diagram of the lemma commutes.  At this point we can localize a site, we know how to relocalize, and we can localize a morphism of sites at an object of the site downstairs. If we combine these then we get the following kind of diagram.

652

9. SITES AND SHEAVES

Lemma 9.24.3. Let f : C → D be a morphism of sites corresponding to the continuous functor u : C → D. Let V ∈ Ob(D), U ∈ Ob(C) and c : U → u(V ) a morphism of C. There exists a commutative diagram of topoi Sh(C/U )

jU

f

fc

 Sh(D/V )

/ Sh(C)

jV

 / Sh(D).

We have fc = f 0 ◦ jU/u(V ) where f 0 : Sh(C/u(V )) → Sh(D/V ) is as in Lemma 9.24.1 and jU/u(V ) : Sh(C/U ) → Sh(C/u(V )) is as in Lemma 9.21.7. Using the # identifications Sh(C/U ) = Sh(C)/h# U and Sh(D/V ) = Sh(D)/hV of Lemma 9.21.4 −1 the functor (fc ) is described by the rule ϕ

−1 (fc )−1 (H − → h# H ×f −1 ϕ,h# V ) = (f

u(V )

,c

# h# U → hU ).

Finally, given any morphisms b : V 0 → V , a : U 0 → U and c0 : U 0 → u(V 0 ) such that / u(V 0 ) U0 c0

a

 U

u(b)

c

 / u(V )

commutes, then the diagram Sh(C/U 0 )

jU 0 /U

fc0

 Sh(D/V 0 )

/ Sh(C/U ) fc

jV 0 /V

 / Sh(D/V ).

commutes. Proof. This lemma proves itself, and is more a collection of things we know at this stage of the development of theory. For example the commutativity of the first square follows from the commutativity of Diagram (9.21.7.1) and the commutativity of the diagram in Lemma 9.24.1. The description of fc−1 follows on combining Lemma 9.21.8 with Lemma 9.24.1. The commutativity of the last square then follows from the equality f −1 H ×h#

u(V )

,c

# −1 h# (H ×h# h# U ×h# hU 0 = f V 0 ) ×h# U

V

u(V 0 ),c0

h# U0

# −1 # which is formal using that f −1 h# hV 0 = h# V = hu(V ) and f u(V 0 ) , see Lemma 9.13.5. 

In the following lemma we find another kind of functoriality of localization, in case the morphism of topoi comes from a cocontinuous functor. This is a kind of diagram which is different from the diagram in Lemma 9.24.1, and in particular, in general the equality f∗0 jU−1 = jV−1 f∗ seen in Lemma 9.24.1 does not hold in the situation of the following lemma.

9.25. MORPHISMS OF TOPOI

653

Lemma 9.24.4. Let C, D be sites. Let u : C → D be a cocontinuous functor. Let U be an object of C, and set V = u(U ). We have a commutative diagram C/U

jU

u0

 D/V

/C u

jV

 /D

where the left vertical arrow is u0 : C/U → D/V , U 0 /U 7→ V 0 /V . Then u0 is cocontinuous also and we get a commutative diagram of topoi Sh(C/U )

jU

f0

 Sh(D/V )

/ Sh(C) f

jV

 / Sh(D)

where f (resp. f 0 ) corresponds to u (resp. u0 ). Proof. The commutativity of the first diagram is clear. It implies the commutativity of the second diagram provided we show that u0 is cocontinuous. Let U 0 /U be an object of C/U . Let {Vj /V → u(U 0 )/V }j∈J be a covering of u(U 0 )/V in D/V . Since u is cocontinuous there exists a covering {Ui0 → U 0 }i∈I such that the family {u(Ui0 ) → u(U 0 )} refines the covering {Vj → u(U 0 )} in D. In other words, there exists a map of index sets α : I → J and morphisms φi : u(Ui0 ) → Vα(i) over U 0 . Think of Ui0 as an object over U via the composition Ui0 → U 0 → U . Then {Ui0 /U → U 0 /U } is a covering of C/U such that {u(Ui0 )/V → u(U 0 )/V } refines {Vj /V → u(U 0 )/V } (use the same α and the same maps φi ). Hence u0 : C/U → D/V is cocontinuous.  9.25. Morphisms of topoi In this section we show that any morphism of topoi is equivalent to a morphism of topoi which comes from a morphism of sites. 9.25.1. Let C, D be sites. Let u : C → D be a functor. Assume that u is cocontinuous, u is continuous, given a, b : U 0 → U in C such that u(a) = u(b), then there exists a covering {fi : Ui0 → U 0 } in C such that a ◦ fi = b ◦ fi , (4) given U 0 , U ∈ Ob(C) and a morphism c : u(U 0 ) → u(U ) in D there exists a covering {fi : Ui0 → U 0 } in C and morphisms ci : Ui0 → U such that u(ci ) = c ◦ u(fi ), and (5) given V ∈ Ob(D) there exists a covering of V in D of the form {u(Ui ) → V }i∈I . Then the morphism of topoi

Lemma (1) (2) (3)

g : Sh(C) −→ Sh(D) associated to the cocontinuous functor u by Lemma 9.19.1 is an equivalence. Proof. Assume u satisfies properties (1) – (5). We will show that the adjunction mappings G −→ g∗ g −1 G and g −1 g∗ F −→ F

654

9. SITES AND SHEAVES

are isomorphisms. Note that Lemma 9.19.5 applies and we have g −1 G(U ) = G(u(U )) for any sheaf G on D. Next, let F be a sheaf on C, and let V be an object of D. By definition we have g∗ F(V ) = limu(U )→V F(U ). Hence g −1 g∗ F(U ) = limU 0 ,u(U 0 )→u(U ) F(U 0 ) where the morphisms ψ : u(U 0 ) → u(U ) need not be of the form u(α). The category of such pairs (U 0 , ψ) has a final object, namely (U, id), which gives rise to the map from the limit into F(U ). Let (s(U 0 ,ψ) ) be an element of the limit. We want to show that s(U 0 ,ψ) is uniquely determined by the value s(U,id) ∈ F(U ). By property (4) given any (U 0 , ψ) there exists a covering {Ui0 → U 0 } such that the compositions u(Ui0 ) → u(U 0 ) → u(U ) are of the form u(ci ) for some ci : Ui0 → U in C. Hence s(U 0 ,ψ) |Ui0 = c∗i (sU,id ). Since F is a sheaf it follows that indeed s(U 0 ,ψ) is determined by s(U,id) . This proves uniqueness. For existence, assume given any s ∈ F(U ), ψ : u(U 0 ) → u(U ), {fi : Ui0 → U 0 } and ci : Ui0 → U such that ψ ◦ u(fi ) = u(ci ) as above. We claim there exists a (unique) element s(U 0 ,ψ) ∈ F(U 0 ) such that s(U 0 ,ψ) |Ui0 = c∗i (s). Namely, a priori it is not clear the elements c∗i (s)|Ui0 ×U 0 Uj0 and c∗j (s)|Ui0 ×U 0 Uj0 agree, since the diagram / U0 U 0 ×U 0 U 0 i

j

pr2

j

cj

pr1

 Ui0

ci

 /U

need not commute. But condition (3) of the lemma garantees that there exist 0 coverings {fijk : Uijk → Ui0 ×U 0 Uj0 }k∈Kij such that ci ◦ pr1 ◦ fijk = cj ◦ pr2 ◦ fijk . Hence     ∗ ∗ fijk c∗i s|Ui0 ×U 0 Uj0 = fijk c∗j s|Ui0 ×U 0 Uj0 Hence c∗i (s)|Ui0 ×U 0 Uj0 = c∗j (s)|Ui0 ×U 0 Uj0 by the sheaf condition for F and hence the existence of sU 0 ,ψ also by the sheaf condition for F. The uniqueness garantees that the collection (sU 0 ,ψ ) so obtained is an element of the limit with s(U,ψ) = s. This proves that g −1 g∗ F → F is an isomorphism. Let G be a sheaf on D. Let V be an object of D. Then we see that g∗ g −1 G(V ) = limU,ψ:u(U )→V G(u(U )) By the preceding paragraph we see that the value of the sheaf g∗ g −1 G on an object V of the form V = u(U ) is equal to G(u(U )). (Formally, this holds because we have g −1 g∗ g −1 ∼ = g −1 , and the description of g −1 given at the beginning of the proof; informally just by comparing limits here and above.) Hence the adjunction mapping G → g∗ g −1 G has the property that it is a bijection on sections over any object of the form u(U ). Since by axiom (5) there exists a covering of V by objects of the form u(U ) we see easily that the adjunction map is an isomorphism.  It will be convenient to give cocontinuous functors as in Lemma 9.25.1 a name.

9.25. MORPHISMS OF TOPOI

655

Definition 9.25.2. Let C, D be sites. A special cocontinuous functor u from C to D is a cocontinuous functor u : C → D satisfying the assumptions and conclusions of Lemma 9.25.1. Lemma 9.25.3. Let C, D be sites. Let u : C → D be a special cocontinuous functor. For every object U of C we have a commutative diagram C/U  D/u(U )

jU

/C u

ju(U )

 /D

as in Lemma 9.24.4. The left vertical arrow is a special cocontinuous functor. Hence in the commutative diagram of topoi Sh(C/U )  Sh(D/u(U ))

jU

/ Sh(C) u

ju(U )

 / Sh(D)

the vertical arrows are equivalences. Proof. We have seen the existence and commutativity of the diagrams in Lemma 9.24.4. We have to check hypotheses (1) – (5) of Lemma 9.25.1 for the induced functor u : C/U → D/u(U ). This is completely mechanical. Property (1). This is Lemma 9.24.4. Property (2). Let {Ui0 /U 0 → U 0 /U }i∈I be a covering of U 0 /U in C/U . Because u is continuous we see that {u(Ui0 )/u(U 0 ) → u(U 0 )/u(U )}i∈I is a covering of u(U 0 )/u(U ) in D/u(U ). Hence (2) holds for u : C/U → D/u(U ). Property (3). Let a, b : U 00 /U → U 0 /U in C/U be morphisms such that u(a) = u(b) in D/u(U ). Because u satisfies (3) we see there exists a covering {fi : Ui00 → U 00 } in C such that a ◦ fi = b ◦ fi . This gives a covering {fi : Ui00 /U → U 00 /U } in C/U such that a ◦ fi = b ◦ fi . Hence (3) holds for u : C/U → D/u(U ). Property (4). Let U 00 /U, U 0 /U ∈ Ob(C/U ) and a morphism c : u(U 00 )/u(U ) → u(U 0 )/u(U ) in D/u(U ) be given. Because u satisfies property (4) there exists a covering {fi : Ui00 → U 00 } in C and morphisms ci : Ui00 → U 0 such that u(ci ) = c ◦ u(fi ). We think of Ui00 as an object over U via the composition Ui00 → U 00 → U . It may not be true that ci is a morphism over U ! But since u(ci ) is a morphism over 00 u(U ) we may apply property (3) for u and find coverings {fik : Uik → Ui00 } such 00 0 00 that cik = ci ◦ fik : Uik → U are morphisms over U . Hence {fi ◦ fik : Uik /U → U 00 /U } is a covering in C/U such that u(cik ) = c ◦ u(fik ). Hence (4) holds for u : C/U → D/u(U ). Property (5). Let h : V → u(U ) be an object of D/u(U ). Because u saitisfies property (5) there exists a covering {ci : u(Ui ) → V } in D. By property (3) we can find coverings {fij : Uij → Ui } and morphisms cij : Uij → U such that u(cij ) = h ◦ ci ◦ u(fij ). Hence {u(Uij )/u(U ) → V /u(U )} is a covering in D/u(U ) of the desired shape and we conclude that (5) holds for u : C/U → D/u(U ).  Lemma 9.25.4. Let C be a site. Let C 0 ⊂ Sh(C) be a full subcategory (with a set of objects) such that

656

9. SITES AND SHEAVES 0 (1) h# U ∈ Ob(C ) for all U ∈ Ob(C), and 0 (2) C is preserved under fibre products in Sh(C).

Declare a covering of C 0 to be any family {Fi → F}i∈I of maps such that F is a surjective map of sheaves. Then

`

i∈I

Fi →

(1) C 0 is a site (after choosing a set of coverings, see Sets, Lemma 3.11.1), (2) representable presheaves on C 0 are sheaves (i.e., the topology on C 0 is subcanonical, see Definition 9.12.2), (3) the functor v : C → C 0 , U 7→ h# U is a special cocontinuous functor, hence induces an equivalence g : Sh(C) → Sh(C 0 ), (4) for any F ∈ Ob(C 0 ) we have g −1 hF = F, and (5) for any U ∈ Ob(C) we have g∗ h# U = hv(U ) = hh# . U

Proof. Warning: Some of the statements above may look be a bit confusing at first; this is because objects of C 0 can also be viewed as sheaves on C! We omit the proof that the coverings of C 0 as described in the lemma satisfy the conditions of Definition 9.6.2. Suppose that {Fi → F} is a surjective family of morphisms of sheaves. Let G be another sheaf. Part (2) of the lemma says that the equalizer of ` ` / MorSh(C) ( i∈I Fi , G) / MorSh(C) ( (i0 ,i1 )∈I×I Fi0 ×F Fi1 , G) is MorSh(C) (F, G). This is clear (for example use Lemma 9.11.2). To prove (3) we have to check conditions (1) – (5) of Lemma 9.25.1. The fact that v is cocontinuous is equivalent to the description of surjective maps of sheaves in Lemma 9.11.2. The functor v is continuous because U 7→ h# U commutes with fibre products, and transforms coverings into coverings (see Lemma 9.10.14, and Lemma 9.12.5). Properties (3), (4) of Lemma 9.25.1 are statements about morphisms f : # # h# U → hU 0 . Such a morphism is the same thing as an element of hU 0 (U ). Hence (3) and (4) are immediate from the construction of the sheafification. Property (5) of Lemma 9.25.1 is Lemma 9.12.4. Denote g : Sh(C) → Sh(C 0 ) the equivalence of topoi associated with v by Lemma 9.25.1. Let F be as in part (4) of the lemma. For any U ∈ Ob(C) we have g −1 hF (U ) = hF (v(U )) = MorSh(C) (h# U , F) = F(U ) The first equality by Lemma 9.19.5. Thus part (4) holds. Let F ∈ Ob(C 0 ). Let U ∈ Ob(C). Then # g∗ h# U (F) = MorSh(C 0 ) (hF , g∗ hU )

= MorSh(C) (g −1 hF , h# U) = MorSh(C) (F, h# U) = MorC 0 (F, h# U) as desired (where the third equality was shown above). Using this we can massage any topos to live over a site having all finite limits.



9.25. MORPHISMS OF TOPOI

657

Lemma 9.25.5. Let Sh(C) be a topos. There exists an equivalence of topoi g : Sh(C) → Sh(C 0 ) induced by a special cocontinuous functor u : C → C 0 such that C 0 is a site with a subcanonical topology having fibre products and a final object (in other words, C 0 has all finite limits). Moreover, given a set of sheaves {Fi }i∈I we may choose C 0 such that each g∗ Fi is a representable sheaf. Proof. Consider the full subcategory C1 ⊂ Sh(C) consisting of all h# U for all U ∈ Ob(C), the given sheaves Fi and the final sheaf ∗ (see Example 9.10.2). LetSCn+1 be a full subcategory consisting of all fibre products of objects of Cn . Set C 0 = n≥1 Cn . ` A covering in C 0 is any family {Fi → F}i∈I such that i∈I Fi → F is surjective as a map of sheaves on C. The functor v : C → C 0 is given by U 7→ h# U . Apply Lemma 9.25.4.  Here is the goal of the current section. Lemma 9.25.6. Let C, D be sites. Let f : Sh(C) → Sh(D) be a morphism of topoi. Then there exists a site C 0 and a diagram of functors C

v

/ C0 o

u

D

such that (1) the functor v is a special cocontinuous functor, (2) the functor u commutes with fibre products, is continuous and defines a morphism of sites C 0 → D0 , and (3) the morphism of topoi f agrees with the composition of morphisms of topoi Sh(C) −→ Sh(C 0 ) −→ Sh(D) where the first arrow comes from v via Lemma 9.25.1 and the second arrow from u via Lemma 9.15.3. −1 # Proof. Consider the full subcategory C1 ⊂ Sh(C) consisting of all h# hV U and all f for all U ∈ Ob(C) and all V ∈ Ob(D). Let Cn+1Sbe a full subcategory consisting of all fibre products of objects of Cn . Set C 0 = n≥1 Cn . A covering in C 0 is any ` family {Fi → F}i∈I such that i∈I Fi → F is surjective as a map of sheaves on 0 C. The functor v : C → C 0 is given by U 7→ h# U . The functor u : D → C is given by −1 # V 7→ f hV .

Part (1) follows from Lemma 9.25.4. Proof of (2) and (3) of the lemma. The functor u commutes with fibre products −1 as both V 7→ h# do. Moreover, since f −1 is exact and commutes with V and f arbitrary colimits we see that it transforms a covering into a surjective family of morphisms of sheaves. Hence u is continuous. To see that it defines a morphism of sites we still have to see that us is exact. In order to do this we will show that g −1 ◦ us = f −1 . Namely, then since g −1 is an equivalence and f −1 is exact we will conclude. Because g −1 is adjoint to g∗ , and us is adjoint to us , and f −1 is adjoint to f∗ it also suffices to prove that us ◦ g∗ = f∗ . Let U be an object of C and let V

658

9. SITES AND SHEAVES

be an object of D. Then # −1 # (us g∗ h# hV ) U )(V ) = g∗ hU (f # = MorSh(C) (f −1 h# V , hU ) # = MorSh(D) (h# V , f∗ hU )

= f∗ h# U (V ) The first equality because us = up . The second equality by Lemma 9.25.4 (5). The third equality by adjointness of f∗ and f −1 and the final equality by properties of sheafification and the Yoneda lemma. We omit the verification that these identities are functorial in U and V . Hence we see that we have us ◦ g∗ = f∗ for sheaves of s the form h# U . This implies that u ◦ g∗ = f∗ and we win (some details omitted).  Remark 9.25.7. Notation and assumptions as in Lemma 9.25.6. If the site D has a final object and fibre products then the functor u : D → C 0 satisfies all the assumptions of Proposition 9.14.6. Namely, in addition to the properties mentioned in the lemma u also transforms the final object of D into the final object of C 0 . This is clear from the construction of u. Hence, if we first apply Lemmas 9.25.5 to D and then Lemma 9.25.6 to the resulting morphism of topoi Sh(C) → Sh(D0 ) we obtain the following statement: Any morphism of topoi f : Sh(C) → Sh(D) fits into a commutative diagram / Sh(D) Sh(C) f

g

 Sh(C 0 )

e

f0

 / Sh(D0 )

where the following properties hold: (1) the morphisms e and g are equivalences given by special cocontinuous functors C → C 0 and D → D0 , (2) the sites C 0 and D0 have fibre products, final objects and have subcanonical topologies, (3) the morphism f 0 : C 0 → D0 comes from a morphism of sites corresponding to a functor u : D0 → C 0 to which Proposition 9.14.6 applies, and (4) given any set of sheaves Fi (resp. Gj ) on C (resp. D) we may assume each of these is a representable sheaf on C 0 (resp. D0 ). It is often useful to replace C and D by C 0 and D0 . Remark 9.25.8. Notation and assumptions as in Lemma 9.25.6. Suppose that in addition the original morphism of topoi Sh(C) → Sh(D) is an equivalence. Then the construction in the proof of Lemma 9.25.6 gives two functors C → C0 ← D which are both special continuous functors. Hence in this case we can actually factor the morphism of topoi as a composition Sh(C) → Sh(C 0 ) = Sh(D0 ) ← Sh(D) as in Remark 9.25.7, but with the middle morphism an identity.

9.26. LOCALIZATION OF TOPOI

659

9.26. Localization of topoi We repeat some of the material on localization to the apparantly more general case of topoi. In reality this is not more general since we may always enlarge the underlying sites to assume that we are localizing at objects of the site. Lemma 9.26.1. Let C be a site. Let F be a sheaf on C. Then the category Sh(C)/F is a topos. There is a canonical morphism of topoi jF : Sh(C)/F −→ Sh(C) which is a localization as in Section 9.21 such that −1 (1) the functor jF is the functor H 7→ H × F/F, and (2) the functor jF ! is the forgetful functor G/F 7→ G. Proof. Apply Lemma 9.25.5. This means we may assume C is a site with subcanonical topology, and F = hU = h# U for some U ∈ Ob(C). Hence the material of Section 9.21 applies. In particular, there is an equivalence Sh(C/U ) = Sh(C)/h# U such that the composition Sh(C/U ) → Sh(C)/h# U → Sh(C) is equal to jU ! , see Lemma 9.21.4. Denote a : Sh(C)/h# U → Sh(C/U ) the inverse −1 functor, so jF ! = jU ! ◦ a, jF = jU−1 ◦ a and jF ,∗ = jU,∗ ◦ a. The description of jF ! −1 follows from the above. The description of jF follows from Lemma 9.21.6.  Remark 9.26.2. In the situation of Lemma 9.26.1 we can also describe the functor jF ,∗ . It is the functor which associates to ϕ : G → F the sheaf U 7−→ {α : F|U → G|U such that α is a right inverse to ϕ|U } In order to prove that this works the introduction of Hom-sheaves is desirable, hence we postpone this to a later time. Lemma 9.26.3. Let C be a site. Let F be a sheaf on C. Let C/F be the category of pairs (U, s) where U ∈ Ob(C) and s ∈ F(U ). Let a covering in C/F be a family {(Ui , si ) → (U, s)} such that {Ui → U } is a covering of C. Then j : C/F → C is a continuous and cocontinuous functor of sites which induces a morphism of topi j : Sh(C/F) → Sh(C). In fact, there is an equivalence Sh(C/F) = Sh(C)/F which turns j into jF . Proof. We omit the verification that C/F is a site and that j is continuous and cocontinuous. By Lemma 9.19.5 there exists a morphism of topoi j as indicated, with j −1 G(U, s) = G(U ), and there is a left adjoint j! to j −1 . A morphism ϕ : ∗ → g −1 G on C/F is the same thing as a rule which assigns to every pair (U, s) a section ϕ(s) ∈ G(U ) compatible with restriction maps. Hence this is the same thing as a morphism ϕ : F → G over C. We conclude that j! ∗ = F. In particular, for every H ∈ Sh(C/F) there is a canonical map j! H → j! ∗ = F j!0

i.e., we obtain a functor : Sh(C/F) → Sh(C)/F. An inverse to this functor is the rule which assigns to an object ϕ : G → F of Sh(C)/F the sheaf a(G/F) : (U, s) 7−→ {t ∈ G(U ) | ϕ(t) = s} We omit the verification that a(G/F) is a sheaf and that a is inverse to j!0 .



660

9. SITES AND SHEAVES

Definition 9.26.4. Let C be a site. Let F be a sheaf on C. (1) The topos Sh(C)/F is called the localization of the topos Sh(C) at F. (2) The morphism of topoi jF : Sh(C)/F → Sh(C) of Lemma 9.26.1 is called the localization morphism. We are going to show that whenever the sheaf F is equal to h# U for some object U of the site, then the localization of the topos is equal to the category of sheaves on the localization of the site at U . Moreover, we are going to check that any functorialities are compatible with this identification. Lemma 9.26.5. Let C be a site. Let F = h# U for some object U of C. Then jF : Sh(C)/F → Sh(C) constructed in Lemma 9.26.1 agrees with the morphism of topoi jU : Sh(C/U ) → Sh(C) constructed in Section 9.21 via the identification Sh(C/U ) = Sh(C)/h# U of Lemma 9.21.4. Proof. We have seen in Lemma 9.21.4 that the composition Sh(C/U ) → Sh(C)/h# U → Sh(C) is jU ! . The functor Sh(C)/h# → Sh(C) is j by Lemma 9.26.1. Hence F! U −1 jF ! = jU ! via the identification. So jF = jU−1 (by adjointness) and so jF ,∗ = jU,∗ (by adjointness again).  Lemma 9.26.6. Let C be a site. If s : G → F is a morphism of sheaves on C then there exists a natural commutative diagram of morphisms of topoi Sh(C)/G

/ Sh(C)/F

j

$ z Sh(C)

jG

jF

where j = jG/F is the localization of the topos Sh(C)/F at the object G/F. In particular we have j −1 (H → F) = (H ×F G → G). and e

s◦e

j! (E − → F) = (E −−→ G). Proof. The description of j −1 and j! comes from the description of those functors in Lemma 9.26.1. The equality of functors jG! = jF ! ◦j! is clear from the description of these functors (as forgetful functors). By adjointness we also obtain the equalities −1 jG−1 = j −1 ◦ jF , and jG,∗ = jF ,∗ ◦ j∗ .  Lemma 9.26.7. Assume C and s : G → F are as in Lemma 9.26.6. If G = h# V and F = h# and s : G → F comes from a morphism V → U of C then the U diagram in Lemma 9.26.6 is identified with diagram (9.21.7.1) via the identifications # Sh(C/V ) = Sh(C)/h# V and Sh(C/U ) = Sh(C)/hU of Lemma 9.21.4. Proof. This is true because the descriptions of j −1 agree. See Lemma 9.21.8 and Lemma 9.26.6.  9.27. Localization and morphisms of topoi This section is the analogue of Section 9.24 for morphisms of topoi.

9.27. LOCALIZATION AND MORPHISMS OF TOPOI

661

Lemma 9.27.1. Let f : Sh(C) → Sh(D) be a morphism of topoi. Let G be a sheaf on D. Set F = f −1 G. Then there exists a commutative diagram of topoi Sh(C)/F

jF

f0

 Sh(D)/G

/ Sh(C) f

jG

 / Sh(D).

The morphism f 0 is characterized by the property that f −1 ϕ

ϕ

(f 0 )−1 (H − → G) = (f −1 H −−−→ F) −1 and we have f∗0 jF = jG−1 f∗ .

Proof. Since the statement is about topoi and does not refer to the underlying sites we may change sites at will. Hence by the discussion in Remark 9.25.7 we may assume that f is given by a continuous functor u : D → C satisfying the assumptions of Proposition 9.14.6 between sites having all finite limits and subcanonical topologies, and such that G = hV for some object V of D. Then F = f −1 hV = hu(V ) by Lemma 9.13.5. By Lemma 9.24.1 we obtain a commutative diagram of morphisms of topoi / Sh(C) Sh(C/U ) jU

f

0

 Sh(D/V )

f

jV

 / Sh(D),

and we have f∗0 jU−1 = jV−1 f∗ . By Lemma 9.26.5 we may identify jF and jU and jG and jV . The description of (f 0 )−1 is given in Lemma 9.24.1.  Lemma 9.27.2. Let f : C → D be a morphism of sites given by the continuous functor u : D → C. Let V be an object of D. Set U = u(V ). Set G = h# V , and −1 # F = h# = f h (see Lemma 9.13.5). Then the diagram of morphisms of topoi U V of Lemma 9.27.1 agrees with the diagram of morphisms of topoi of Lemma 9.24.1 via the identifications jF = jU and jG = jV of Lemma 9.26.5. Proof. This is not a complete triviality as the choice of morphism of sites giving rise to f made in the proof of Lemma 9.27.1 may be different from the morphisms of sites given to us in the lemma. But in both cases the functor (f 0 )−1 is described by the same rule. Hence they agree and the associated morphism of topoi is the same. Some details omitted.  Lemma 9.27.3. Let f : Sh(C) → Sh(D) be a morphism of topoi. Let G ∈ Sh(D), F ∈ Sh(C) and s : F → f −1 G a morphism of sheaves. There exists a commutative diagram of topoi / Sh(C) Sh(C)/F jF

fs

 Sh(D)/G

f

jG

 / Sh(D).

We have fs = f 0 ◦ jF /f −1 G where f 0 : Sh(C)/f −1 G → Sh(D)/F is as in Lemma 9.27.1 and jF /f −1 G : Sh(C)/F → Sh(C)/f −1 G is as in Lemma 9.26.6. The functor

662

9. SITES AND SHEAVES

(fs )−1 is described by the rule ϕ

(fs )−1 (H − → G) = (f −1 H ×f −1 ϕ,f −1 G,s F → F). Finally, given any morphisms b : G 0 → G, a : F 0 → F and s0 : F 0 → f −1 G 0 such that / f −1 G 0 F0 0 s

a

 F

s



f −1 b

/ f −1 G

commutes, then the diagram Sh(C)/F 0

jF 0 /F

fs0

 Sh(D)/G 0

/ Sh(C)/F fs

jG 0 /G

 / Sh(D)/G.

commutes. Proof. The commutativity of the first square follows from the commutativity of the diagram in Lemma 9.26.6 and the commutativity of the diagram in Lemma 9.27.1. The description of fs−1 follows on combining the descriptions of (f 0 )−1 in Lemma 9.27.1 with the description of (jF /f −1 G )−1 in Lemma 9.26.6. The commutativity of the last square then follows from the equality f −1 H ×f −1 G,s F ×F F 0 = f −1 (H ×G G 0 ) ×G 0 ,s0 F 0 which is formal.



Lemma 9.27.4. Let f : C → D be a morphism of sites given by the continuous functor u : D → C. Let V be an object of D. Let c : U → u(V ) be a morphism. # −1 # Set G = h# hV . Let s : F → f −1 G be the map induced by c. V and F = hU = f Then the diagram of morphisms of topoi of Lemma 9.24.3 agrees with the diagram of morphisms of topoi of Lemma 9.27.3 via the identifications jF = jU and jG = jV of Lemma 9.26.5. Proof. This follows on combining Lemmas 9.26.7 and 9.27.2.



9.28. Points Definition 9.28.1. Let C be a site. A point of the topos Sh(C) is a morphism of topoi p from Sh(pt) to Sh(C). We will define a point of a site in terms of a functor u : C → Sets. It will turn out later that u will define a morphism of sites which gives rise to a point of the topos associated to C, see Lemma 9.28.8. Let C be a site. Let p = u be a functor u : C → Sets. This curious language is introduced because it seems funny to talk about neighbourhoods of functors; so we think of a “point” p as a geometric thing which is given by a categorical datum, namely the functor u. The fact that p is actually equal to u does not matter. A neighbourhood of p is a pair (U, x) with U ∈ Ob(C) and x ∈ u(U ). A morphism of neighbourhoods (V, y) → (U, x) is given by a morphism α : V → U of C such that u(α)(y) = x. Note that the category of neighbourhoods isn’t a “big” category.

9.28. POINTS

663

We define the stalk of a presheaf F at p as (9.28.1.1)

Fp = colim{(U,x)}opp F(U ).

The colimit is over the opposite of the category of neighbourhoods of p. In other words, an element of Fp is given by a triple (U, x, s), where (U, x) is a neighbourhood of p and s ∈ F(U ). Equality of triples is the equivalence relation generated by (U, x, s) ∼ (V, y, α∗ s) when α is as above. Note that if ϕ : F → G is a morphism of presheaves of sets, then we get a canonical map of stalks ϕp : Fp → Gp . Thus we obtain a stalk functor PSh(C) −→ Sets,

F 7−→ Fp .

We have defined the stalk functor using any functor p = u : C → Sets. No conditions are necessary for the definition to work5. On the other hand, it is probably better not to use this notion unless p actually is a point (see definition below), since in general the stalk functor does not have good properties. Definition 9.28.2. Let C be a site. A point p of the site C is given by a functor u : C → Sets such that ` (1) For every covering {Ui → U } of C the map u(Ui ) → u(U ) is surjective. (2) For every covering {Ui → U } of C and every morphism V → U the maps u(Ui ×U V ) → u(Ui ) ×u(U ) u(V ) are bijective. (3) The stalk functor Sh(C) → Sets, F → Fp is left exact. The conditions should be familiar since they are modeled after those of Definitions 9.13.1 and 9.14.1. Note that (3) implies that ∗p = {∗}, see Example 9.10.2. Hence u(U ) 6= ∅ for at least some U (because the empty colimit produces the empty set). We will show below (Lemma 9.28.7) that this does give rise to a point of the topos Sh(C). Before we do so, we prove some lemmas for general functors u. Lemma 9.28.3. Let C be a site. Let p = u : C → Sets be a functor. There are functorial isomorphisms (hU )p = u(U ) for U ∈ Ob(C). Proof. An element of (hU )p is given by a triple (V, y, f ), where V ∈ Ob(C), y ∈ u(V ) and f ∈ hU (V ) = MorC (V, U ). Two such (V, y, f ), (V 0 , y 0 , f 0 ) determine the same object if there exists a morphism φ : V → V 0 such that u(φ)(x) = x0 and f 0 ◦ φ = f , and in general you have to take chains of identities like this to get the correct equivalence relation. In any case, every (V, y, f ) is equivalent to the element (U, u(f )(y), idU ). If φ exists as above, then the triples (V, y, f ), (V 0 , y 0 , f 0 ) determine the same triple (U, u(f )(y), idU ) = (U, u(f 0 )(y 0 ), idU ). This proves that the map u(U ) → (hU )p , x 7→ class of (U, x, idU ) is bijective.  Let C be a site. Let p = u : C → Sets be a functor. In analogy with the constructions in Section 9.5 given a set E we define a presheaf up E by the rule (9.28.3.1)

U 7−→ up E(U ) = MorSets (u(U ), E) = Map(u(U ), E).

This defines a functor up : Sets → PSh(C), E 7→ up E. Lemma 9.28.4. For any functor u : C → Sets. The functor up is a right adjoint to the stalk functor on presheaves. 5One should try to avoid the case where u(U ) = ∅ for all U .

664

9. SITES AND SHEAVES

Proof. Let F be a presheaf on C. Let E be a set. A morphism F → up E is given by a compatible system of maps F(U ) → Map(u(U ), E), i.e., a compatible system of maps F(U ) × u(U ) → E. And by definition of Fp a map Fp → E is given by a rule associating with each triple (U, x, σ) an element in E such that equivalent triples map to the same element, see discussion surrounding Equation (9.28.1.1). This also means a compatible system of maps F(U ) × u(U ) → E.  In analogy with Section 9.13 we have the following lemma. Lemma 9.28.5. Let C be a site. Let p = u : C → Sets be a functor. Suppose that for every covering {Ui → U } of C ` (1) the map u(Ui ) → u(U ) is surjective and (2) the maps u(Ui ×U Uj ) → u(Ui ) ×u(U ) u(Uj ) are surjective. Then we have (1) the presheaf up E is a sheaf for all sets E, denote it us E, (2) the stalk functor Sh(C) → Sets and the functor us : Sets → Sh(C) are adjoint, and (3) we have Fp = Fp# for every presheaf of sets F. Proof. The first assertion is immediate from the definition of a sheaf, assumptions (1) and (2), and the definition of up E. The second is a restatement of the adjointness of up and the stalk functor (but now restricted to sheaves). The third assertion follows as, for any set E, we have Map(Fp , E) = MorPSh(C) (F, up E) = MorSh(C) (F # , us E) = Map(Fp# , E) by the adjointness property of sheafification.



In particular Lemma 9.28.5 holds when p = u is a point. In this case we think of the sheaf us E as the “skyscraper” sheaf with value E at p. Definition 9.28.6. Let p be a point of the site C given by the functor u. For a set E we define p∗ E = us E the sheaf described in Lemma 9.28.5 above. We sometimes call this a skyscraper sheaf. In particular we have the following adjointness property of skyscraper sheaves and stalks: MorSh(C) (F, p∗ E) = Map(Fp , E) This motivates the notation p−1 F = Fp which we will sometimes use. Lemma 9.28.7. Let C be a site. (1) Let p be a point of the site C. Then the pair of functors (p∗ , p−1 ) introduced above define a morphism of topoi Sh(pt) → Sh(C). (2) Let p = (p∗ , p−1 ) be a point of the topos Sh(C). Then the functor u : U 7→ 0 p−1 (h# U ) gives rise to a point p of the site C whose associated morphism 0 0 −1 of topoi (p∗ , (p ) ) is equal to p. Proof. Proof of (1). By the above the functors p∗ and p−1 are adjoint. The functor p−1 is required to be exact by Definition 9.28.2. Hence the conditions imposed in Definition 9.15.1 are all satisfied and we see that (1) holds. ` Proof of (2). Let {Ui → U } be a covering of C. Then (hUi )# → h# U is surjective, see Lemma 9.12.5. Since p−1 is exact (by definition of a morphism of topoi) we

9.28. POINTS

665

` conclude that u(Ui ) → u(U ) is surjective. This proves part (1) of Definition 9.28.2. Sheafification is exact, see Lemma 9.10.14. Hence if U ×V W exists in C, then # # h# U ×V W = hU ×h# hW V

and we see that u(U ×V W ) = u(U ) ×u(V ) u(W ) since p−1 is exact. This proves part (2) of Definition 9.28.2. Let p0 = u, and let Fp0 be the stalk functor defined by Equation (9.28.1.1) using u. There is a canonical comparison map c : Fp0 → Fp = p−1 F. Namely, given a triple (U, x, σ) representing an element ξ of Fp0 we think of σ −1 as a map σ : h# (σ)(x) since x ∈ u(U ) = p−1 (h# U → F and we can set c(ξ) = p U ). By Lemma 9.28.3 we see that (hU )p0 = u(U ). Since conditions (1) and (2) of Definition 9.28.2 hold for p0 we also have (h# U )p0 = (hU )p0 by Lemma 9.28.5. Hence we have −1 # (h# (hU ) U )p0 = (hU )p0 = u(U ) = p −1 # We claim this bijection equals the comparison map c : (h# (hU ) (verificaU )p0 → p tion omitted). Any sheaf on C is a coequalizer of maps of coproducts of sheaves of −1 the form h# U , see Lemma 9.12.4. The stalk functor F 7→ Fp0 and the functor p commute with arbitrary colimits (as they are both left adjoints). We conclude c is an isomorphism for every sheaf F. Thus the stalk functor F 7→ Fp0 is isomorphic to p−1 and we in particular see that it is exact. This proves condition (3) of Definition 9.28.2 holds and p0 is a point. The final assertion has already been shown above, since we saw that p−1 = (p0 )−1 . 

Actually a point always corresponds to a morphism of sites as we show in the following lemma. Lemma 9.28.8. Let C be a site. Let p be a point of C given by u : C → Sets. Let S0 be an infinite set such that u(U ) ⊂ S0 for all U ∈ Ob(C). Let S be the site constructed out of the powerset S = P(S0 ) in Remark 9.15.4. Then (1) there is an equivalence i : Sh(pt) → Sh(S), (2) the functor u : C → S induces a morphism of sites f : S → C, and (3) the composition Sh(pt) → Sh(S) → Sh(C) is the morphism of topoi (p∗ , p−1 ) of Lemma 9.28.7. Proof. Part (1) we saw in Remark 9.15.4. Moreover, recall that the equivalence associates to the set E the sheaf i∗ E on S defined by the rule V 7→ MorSets (V, E). Part (2) is clear from the definition of a point of C (Definition 9.28.2) and the definition of a morphism of sites (Definition 9.14.1). Finally, consider f∗ i∗ E. By construction we have f∗ i∗ E(U ) = i∗ E(u(U )) = MorSets (u(U ), E) which is equal to p∗ E(U ), see Equation (9.28.3.1). This proves (3).



Contrary to what happens in the topological case it is not always true that the stalk of the skyscraper sheaf with value E is E. Here is what is true in general. Lemma 9.28.9. Let C be a site. Let p : Sh(pt) → Sh(C) be a point of the topos associated to C. For any set E there are canonical maps E −→ (p∗ E)p −→ E

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whose composition is idE . Proof. There is always an adjunction map (p∗ E)p = p−1 p∗ E → E. This map is an isomorphism when E = {∗} because p∗ and p−1 are both left exact, hence transform the final object into the final object. Hence given e ∈ E we can consider the map ie : {∗} → E which gives p−1 p∗ {∗}

p−1 p∗ ie

∼ =

 {∗}

ie

/ p−1 p∗ E  /E

whence the map E → (p∗ E)p = p−1 p∗ E as desired.



Lemma 9.28.10. Let C be a site. Let p : Sh(pt) → Sh(C) be a point of the topos associated to C. The functor p∗ : Sets → Sh(C) has the following properties: It commutes with arbitrary limits, it is left exact, it is faithful, it transforms surjections into surjections, it commutes with coequalizers, it reflects injections, it reflects surjections, and it reflects isomorpisms. Proof. Because p∗ is a right adjoint it commutes with arbitrary limits and it is left exact. The fact that p−1 p∗ E → E is a canonically split surjection implies that p∗ is faithful, reflects injections, reflects surjections, and reflects isomorphisms. By Lemma 9.28.7 we may assume that p comes from a point u : C → Sets of the underlying site C. In this case the sheaf p∗ E is given by p∗ E(U ) = MorSets (u(U ), E) see Equation (9.28.3.1) and Definition 9.28.6. It follows immediately from this formula that p∗ transforms surjections into surjections and coequalizers into coequalizers.  9.29. Constructing points In this section we give criteria for when a functor from a site to the category of sets defines a point of that site. Lemma 9.29.1. Let C be a site. Assume that C has a final object X and fibred products. Let p = u : C → Sets be a functor such that (1) u(X) is a singleton set, and (2) for every pair of morphisms U → W and V → W with the same target the map u(U ×W V ) → u(U ) ×u(W ) u(V ) is bijective. Then the opposite of the category of neighbourhoods of p is filtered. Moreover, the stalk functor Sh(C) → Sets, F → Fp commutes with finite limits. Proof. This is analogous to the proof of Lemma 9.5.2 above. The assumptions on C imply that C has finite limits. See Categories, Lemma 4.16.4. Assumption (1) implies that the category of neighbourhoods is nonempty. Suppose (U, x) and (V, y) are neighbourhoods. Then u(U × V ) = u(U ×X V ) = u(U ) ×u(X) u(V ) = u(U ) × u(V ) by (2). Hence there exists a neighbourhood (U ×X V, z) mapping to both (U, x) and (V, y). Let a, b : (V, y) → (U, x) be two morphisms in the category

9.29. CONSTRUCTING POINTS

667

of neighbourhoods. Let W be the equalizer of a, b : V → U . As in the proof of Categories, Lemma 4.16.4 we may write W in terms of fibre products: W = (V ×a,U,b V ) ×(pr1 ,pr2 ),V ×V,∆ V The bijectivity in (2) garantees there exists an element z ∈ u(W ) which maps to ((y, y), y). Then (W, z) → (V, y) equalizes a, b as desired. Let I → Sh(C), i 7→ Fi be a finite diagram of sheaves. We have to show that the stalk of the limit of this system agrees with the limit of the stalks. Let F be the limit of the system as a presheaf. According to Lemma 9.10.1 this is a sheaf and it is the limit in the category of sheaves. Hence we have to show that Fp = limI Fi,p . Recall also that F has a simple description, see Section 9.4. Thus we have to show that limi colim{(U,x)}opp Fi (U ) = colim{(U,x)}opp limi Fi (U ). This holds, by Categories, Lemma 4.17.2, because we just showed the opposite of the category of neighbourhoods is filtered.  Proposition 9.29.2. Let C be a site. Assume that finite limits exist in C. (I.e., C has fibre products, and a final object.) A point p of such a site C is given by a functor u : C → Sets such that (1) u commutes with finite limits, and ` (2) if {Ui → U } is a covering, then i u(Ui ) → u(U ) is surjective. Proof. Suppose first that p is a point (Definition 9.28.2) given by a functor u. Condition (2) is satisfied directly from the definition of a point. By Lemma 9.28.3 we have (hU )p = u(U ). By Lemma 9.28.5 we have (h# U )p = (hU )p . Thus we see that u is equal to the composition of functors h

#

()p

C− → PSh(C) − → Sh(C) −−→ Sets Each of these functors is left exact, and hence we see u satisfies (1). Conversely, suppose that u satisfies (1) and (2). In this case we immediately see that u satisfies the first two conditions of Definition 9.28.2. And its stalk functor is exact, because it is a left adjoint by Lemma 9.28.5 and it commutes with finite limits by Lemma 9.29.1.  Remark 9.29.3. In fact, let C be a site. Assume C has a final object X and fibre products. Let p = u : C → Sets be a functor such that (1) u(X) = {∗} a singleton, and (2) for every pair of morphisms U → W and V → W with the same target the map u(U ×W V ) → u(U ) ×u(W ) u(V ` ) is surjective. (3) for every covering {Ui → U } the map u(Ui ) → u(U ) is surjective. Then, in general, p is not a point of C. An example is the category C with two objects {U, X} and exactly one non-identity arrow, namely U → X. We endow C with the trivial topology, i.e., the only coverings are {U → U } and {X → X}. A sheaf F is the same thing as a presheaf and consists of a triple (A, B, A → B): namely A = F(X), B = F(U ) and A → B is the restriction mapping corresponding to U → X. Note that U ×X U = U so fibre products exist. Consider the functor

668

9. SITES AND SHEAVES

u = p with u(X) = {∗} and u(U ) = {∗1 , ∗2 }. This satisfies (1), (2), and (3), but the corresponding stalk functor (9.28.1.1) is the functor a (A, B, A → B) 7−→ B B A

which isn’t exact. Namely, consider (∅, {1}, ∅ → {1}) → ({1}, {1}, {1} → {1}) which is an injective map of sheaves, but is transformed into the noninjective map of sets a a {1} {1} −→ {1} {1} {1}

by the stalk functor. Example 9.29.4. Let X be a topological space. Let TX be the site of Example 9.6.4. Let x ∈ X be a point. Consider the functor  ∅ if x 6∈ U u : TX −→ Sets, U 7→ {∗} if x ∈ U This functor commutes with product and fibred products, and turns coverings into surjective families of maps. Hence we obtain a point p of the site TX . It is immediately verified that the stalk functor agrees with the stalk at x defined in Sheaves, Section 6.11. Example 9.29.5. Let X be a topological space. What are the points of the topos Sh(X)? To see this, let TX be the site of Example 9.6.4. By Lemma 9.28.7 a point of Sh(X) corresponds to a point of this site. Let p be a point of the site TX given by the functor u : TX → Sets. We are going to use the characterization of such a u in Proposition 9.29.2. This implies immediately that u(∅) = ∅ and u(U ∩ V ) = u(U ) × u(V ). In particular we have u(U ) = u(U ) × u(U ) via the diagonal S map which implies that u(U ) is either a singleton or empty. Moreover, if U = Ui is an open covering then u(U ) = ∅ ⇒ ∀i, u(Ui ) = ∅

and u(U ) 6= ∅ ⇒ ∃i, u(Ui ) 6= ∅.

We conclude that there is a unique largest open W ⊂ X with u(W ) = ∅, namely the union of all the opens U with u(U ) = ∅. Let Z = X \ W . If Z = Z1 ∪ Z2 with Zi ⊂ Z closed, then W = (X \ Z1 ) ∩ (X \ Z2 ) so ∅ = u(W ) = u(X \ Z1 ) × u(X \ Z2 ) and we conclude that u(X \ Z1 ) = ∅ or that u(X \ Z2 ) = ∅. This means that X \ Z1 = W or that X \ Z2 = W . In other words, Z is irreducible. Now we see that u is described by the rule  ∅ if Z ∩ U = ∅ u : TX −→ Sets, U 7→ {∗} if Z ∩ U 6= ∅ Note that for any irreducible closed Z ⊂ X this functor satisfies assumptions (1), (2) of Proposition 9.29.2 and hence defines a point. In other words we see that points of the site TX are in one-to-one correspondence with irreducible closed subsets of X. In particular, if X is a sober topological space, then points of TX and points of X are in one to one correspondence, see Example 9.29.4. Example 9.29.6. Consider the site TG described in Example 9.6.5 and Section 9.9. The forgetful functor u : TG → Sets commutes with products and fibred products and turns coverings into surjective families. Hence it defines a point of TG . We identify Sh(TG ) and G-Sets. The stalk functor p−1 : Sh(TG ) = G-Sets −→ Sets

9.30. POINTS AND AND MORPHISMS OF TOPOI

669

is the forgetful functor. The pushforward p∗ is the functor Sets −→ Sh(TG ) = G-Sets which maps a set S to the G-set Map(G, S) with action g · ψ = ψ ◦ Rg where Rg is right multiplication. In particular we have p−1 p∗ S = Map(G, S) as a set and the maps S → Map(G, S) → S of Lemma 9.28.9 are the obvious ones. 9.30. Points and and morphisms of topoi In this section we make a few remarks about points and morphisms of topoi. Lemma 9.30.1. Let f : D → C be a morphism of sites given by a continuous functor u : C → D. Let p be a point of D given by the functor v : D → Sets, see Definition 9.28.2. Then the functor v ◦ u : C → Sets defines a point q of C and moreover there is a canonical identification (f −1 F)p = Fq for any sheaf F on C. First proof Lemma 9.30.1. Note that since u is continuous and since v defines a point, it is immediate that v ◦ u satisfies conditions (1) and (2) of Definition 9.28.2. Let us prove the displayed equality. Let F be a sheaf on C. Then Fq = colim(U,x) F(U ) where the colimit is over objects U in C and elements x ∈ v(u(U )). Similarly, we have (f −1 F)p = (up F)p = colim(V,x) colimU,φ:V →u(U ) F(U ) = colim(V,x,U,φ:V →u(U )) F(U ) = colim(U,x) F(U ) = Fq Explanation: The first equality holds because f −1 F = (up F)# and because Gp = Gp# for any presheaf G, see Lemma 9.28.5. The second equality holds by the definition of up . In the third equality we simply combine colimits. To see the fourth equality we apply Categories, Lemma 4.15.5 to the functor F of diagram categories defined by the rule F ((V, x, U, φ : V → u(U ))) = (U, v(φ)(x)). The lemma applies, because F has a right inverse, namely (U, x) 7→ (u(U ), x, U, id : u(U ) → u(U )) and because there is always a morphism (V, x, U, φ : V → u(U )) −→ (u(U ), v(φ)(x), U, id : u(U ) → u(U )) in the fibre category over (U, x) which shows the fibre categories are nonempty and connected. The fifth equality is clear. Hence now we see that q also satisfies condition (3) of Definition 9.28.2 because it is a composition of exact functors. This finishes the proof.  Second proof Lemma 9.30.1. By Lemma 9.28.8 we may factor (p∗ , p−1 ) as i

h

Sh(pt) → − Sh(S) − → Sh(D) where the second morphism of topoi comes from a morphism of sites h : S → D induced by the functor v : D → S (which makes sense as S ⊂ Sets is a full

670

9. SITES AND SHEAVES

subcategory containing every object in the image of v). By Lemma 9.14.3 the composition v ◦ u : C → S defines a morphism of sites g : S → C. In particular, the functor v ◦ u : C → S is continuous which by the definition of the coverings in S, see Remark 9.15.4, means that v ◦ u satisfies conditions (1) and (2) of Definition 9.28.2. On the other hand, we see that g∗ i∗ E(U ) = i∗ E(v(u(U )) = MorSets (v(u(U )), E) by the construction of i in Remark 9.15.4. Note that this is the same as the formula for which is equal to (v◦u)p E, see Equation (9.28.3.1). By Lemma 9.28.5 the functor g∗ i∗ = (v ◦ u)p = (v ◦ u)s is right adjoint to the the stalk functor F 7→ Fq . Hence we see that the stalk functor q −1 is canonically isomorphic to i−1 ◦ g −1 . Hence it is exact and we conclude that q is a point. Finally, as we have g = f ◦ h by construction we see that q −1 = i−1 ◦ h−1 ◦ f −1 = p−1 ◦ f −1 , i.e., we have the displayed formula of the lemma.  Lemma 9.30.2. Let f : Sh(D) → Sh(C) be a morphism of topoi. Let p : Sh(pt) → Sh(D) be a point. Then q = f ◦ p is a point of the topos Sh(C) and we have a canonical identification (f −1 F)p = Fq for any sheaf F on C. Proof. This is immediate from the definitions and the fact that we can compose morphisms of topoi.  9.31. Localization and points In this section we show that points of a localization C/U are constructed in a simple manner from the points of C. Lemma 9.31.1. Let C be a site. Let p be a point of C given by u : C → Sets. Let U be an object of C and let x ∈ u(U ). The functor v : C/U −→ Sets,

(ϕ : V → U ) 7−→ {y ∈ u(V ) | u(ϕ)(y) = x}

defines a point q of the site C/U such that the diagram Sh(pt) q

y Sh(C/U )

p

jU

 / Sh(C)

commutes. In other words Fp = (jU−1 F)q for any sheaf on C. Proof. Choose S and S as in Lemma 9.28.8. We may identify Sh(pt) = Sh(S) as in that lemma, and we may write p = f : Sh(S) → Sh(C) for the morphism of topoi induced by u. By Lemma 9.24.1 we get a commutative diagram of topoi Sh(S/u(U ))

ju(U )

p0

 Sh(C/U )

/ Sh(S) p

jU

 / Sh(C),

where p0 is given by the functor u0 : C/U → S/u(U ), V /U 7→ u(V )/u(U ). Consider the functor jx : S ∼ = S/x obtained by assigning to a set E the set E endowed with

9.31. LOCALIZATION AND POINTS

671

the constant map E → u(U ) with value x. Then jx is a fully faithful cocontinuous functor which has a continuous right adjoint vx : (ψ : E → u(U )) 7→ ψ −1 ({x}). Note that jU ◦ jx = idS , and vx ◦ u0 = v. These observations imply that we have the following commutative diagram of topoi Sh(S) a

q

& Sh(S/u(U ))

ju(U )

p0

  Sh(C/U )

/ Sh(S)

p

p jU

 / Sh(C) o

Namely: (1) The morphism a : Sh(S) → Sh(S/u(U )) is the morphism of topoi assoicated to the cocontinuous functor jx , which equals the morphism associated to the continuous functor vx , see Lemma 9.19.1 and Section 9.20. (2) The composition p ◦ ju(U ) ◦ a = p since ju(U ) ◦ jx = idS . (3) The compostion p0 ◦ a gives a morphism of topoi. Moreover, it is the morphism of topoi associated to the continuous functor vx ◦ u0 = v. Hence v does indeed define a point q of C/U which fits into the diagram above by construction. This ends the proof of the lemma.



Lemma 9.31.2. Let C, p, u, U be as in Lemma 9.31.1. The construction of Lemma 9.31.1 gives a one to one correspondence between points q of C/U lying over p and elements x of u(U ). Proof. Let q be a point of C/U given by the functor v : C/U → Sets such that jU ◦ q = p as morphisms of topoi. Recall that u(V ) = p−1 (h# V ) for any object V of −1 # C, see Lemma 9.28.7. Similarly v(V /U ) = q (hV /U ) for any object V /U of C/U . Consider the following two diagrams MorC/U (W/U, V /U )

/ MorC (W, V )

h# V /U

/ j −1 (h# ) U V

 MorC/U (W/U, U/U )

 / MorC (W, U )

 h# U/U

 / j −1 (h# ) U U

The right hand diagram is the sheafification of the diagram of presheaves on C/U which maps W/U to the left hand diagram of sets. (There is a small technical point to make here, namely, that we have (jU−1 hV )# = jU−1 (h# V ) and similarly for hU , see Lemma 9.18.4.) Note that the left hand diagram of sets is cartesian. Since sheafification is exact (Lemma 9.10.14) we conclude that the right hand diagram is cartesian.

672

9. SITES AND SHEAVES

Apply the exact functor q −1 to the right hand diagram to get a cartesian diagram v(V /U )

/ u(V )

 v(U/U )

 / u(U )

of sets. Here we have used that q −1 ◦ j −1 = p−1 . Since U/U is a final object of C/U we see that v(U/U ) is a singleton. Hence the image of v(U/U ) in u(U ) is an element x, and the top horizontal map gives a bijection v(V /U ) → {y ∈ u(V ) | y 7→ x in u(U )} as desired.  Lemma 9.31.3. Let C be a site. Let p be a point of C given by u : C → Sets. Let U be an object of C. For any sheaf G on C/U we have a (jU ! G)p = Gq q

where the coproduct is over the points q of C/U associated to elements x ∈ u(U ) as in Lemma 9.31.1. Proof. ` We use the description of jU ! G as the sheaf associated to the presheaf V 7→ ϕ∈MorC (V,U ) G(V /ϕ U ) of Lemma 9.21.2. Also, the stalk of jU ! G at p is equal to the stalk of this presheaf, see Lemma 9.28.5. Hence we see that a G(V /ϕ U ) (jU ! G)p = colim(V,y) ϕ:V →U

To each element (V, y, ϕ, s) of this colimit, we can assign x = u(ϕ)(y) ∈ u(U ). Hence we obtain a (jU ! G)p = colim(ϕ:V →U,y), u(ϕ)(y)=x G(V /ϕ U ). x∈u(U )

This is equal to the expression of the lemma by our construction of the points q.  Remark 9.31.4. Warning: The result of Lemma 9.31.3 has no analogue for jU,∗ . 9.32. 2-morphisms of topoi This is a brief section concerning the notion of a 2-morphism of topoi. Definition 9.32.1. Let f, g : Sh(C) → Sh(D) be two morphisms of topoi. A 2-morphism from f to g is given by a transformation of functors t : f∗ → g∗ . Pictorially we sometimes represent t as follows: f

Sh(C)



t

+

3 Sh(D)

g

Note that since f −1 is adjoint to f∗ and g −1 is adjoint to g∗ we see that t induces also a transformation of functors t : g −1 → f −1 (usually denoted by the same symbol) uniquely characterized by the condition that the diagram MorSh(C) (G, f∗ F)

MorSh(C) (f −1 G, F) −◦t

t◦−



MorSh(C) (G, g∗ F)

 MorSh(C) (g −1 G, F)

9.33. MORPHISMS BETWEEN POINTS

673

commutes. Because of set theoretic difficulties (see Remark 9.15.2) we do not obtain a 2-category of topoi. But we can still define horizontal and vertical composition and show that the axioms of a strict 2-category listed in Categories, Section 4.26 hold. Namely, vertical composition of 2-morphisms is clear (just compose transformations of functors), composition of 1-morphisms has been defined in Definition 9.15.1, and horizontal composition of f



Sh(C)

+

t

f0



3 Sh(D)

+

3 Sh(E)

s

g0

g

is defined by the transformation of functors s?t introduced in Categories, Definition 4.25.1. Explicitly, s ? t is given by f∗0 f∗ F

f∗0 t

/ f∗0 g∗ F

s

/ g∗0 g∗ F

s

f∗0 f∗ F

or

/ g∗0 f∗ F

g∗0 t

/ g∗0 g∗ F

(these maps are equal). Since these definitions agree with the ones in Categories, Section 4.25 it follows from Categories, Lemma 4.25.2 that the axioms of a strict 2-category hold with these definitions. 9.33. Morphisms between points Lemma 9.33.1. Let C be a site. Let u, u0 : C → Sets be two functors, and let t : u0 → u be a transformation of functors. Then we obtain a canonical transformation of stalk functors tstalk : Fp0 → Fp which agrees with t via the identifications of Lemma 9.28.3. Proof. Omitted.



Definition 9.33.2. Let C be a site. Let p, p0 be points of C given by functors u, u0 : C → Sets. A morphism f : p → p0 is given by a transformation of functors fu : u0 → u. Note how the transformation of functors goes the other way. This makes sense, as we will see later, by thinking of the morphism f as a kind of 2-arrow pictorially as follows: p

Sets = Sh(pt)

 p

f

+

3 Sh(C)

0

Namely, we will see later that fu induces a canonical transformation of functors p∗ → p0∗ between the skyscraper sheaf constructions. This is a fairly important notion, and deserves a more complete treatment here. List of desiderata (1) Describe the automorphisms of the point of TG described in Example 9.29.6. (2) Describe Mor(p, p0 ) in terms of Mor(p∗ , p0∗ ). (3) Specialization of points in topological spaces. Show that if x0 ∈ {x} in the topological space X, then there is a morphism p → p0 , where p (resp. p0 ) is the point of TX associated to x (resp. x0 ).

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9.34. Sites with enough points Definition 9.34.1. Let C be a site. (1) A family of points {pi }i∈I is called conservative if for every map of sheaves φ : F → G which is an isomorphism on all the fibres Fpi → Gpi is an isomorphism. (2) We say that C has enough points if there exists a conservative family of points. It turns out that you can then check “exactness” at the stalks. Lemma 9.34.2. Let C be a site and let {pi }i∈I be a conservative family of points. Then (1) Given any map of sheaves ϕ : F → G we have ∀i, ϕpi injective implies ϕ injective. (2) Given any map of sheaves ϕ : F → G we have ∀i, ϕpi surjective implies ϕ surjective. (3) Given any pair of maps of sheaves ϕ1 , ϕ2 : F → G we have ∀i, ϕ1,pi = ϕ2,pi implies ϕ1 = ϕ2 . (4) Given a finite diagram G : J → Sh(C), a sheaf F and morphisms qj : F → Gj then (F, qj ) is a limit of the diagram if and only if for each i the stalk (Fpi , (qj )pi ) is one. (5) Given a finite diagram F : J → Sh(C), a sheaf G and morphisms ej : Fj → G then (G, ej ) is a colimit of the diagram if and only if for each i the stalk (Gpi , (ej )pi ) is one. Proof. We will use over and over again that all the stalk functors commute with any finite limits and colimits and hence with products, fibred products, etc. We will also use that injective maps are the monomorphisms and the surjective maps are the epimorphisms. A map of sheaves ϕ : F → G is injective if and only if F → F ×G F is an isomorphism. Hence (1). Similarly, ϕ : F → G is surjective if and only if G qF G → G is an isomorphism. Hence (2). The maps a, b : F → G are equal if and only if F ×a,G,b F → F × F is an isomorphism. Hence (3). The assertions (4) and (5) follow immediately from the definitions and the remarks at the start of this proof.  Lemma 9.34.3. Let C be a site and let {(pi , ui )}i∈I be a family of points. The family is conservative if and only if for every sheaf F and every U ∈ Ob(C) and every pair of distinct sections s, s0 ∈ F(U ), s 6= s0 there exists an i and x ∈ ui (U ) such that the triples (U, x, s) and (U, x, s0 ) define distinct elements of Fpi . Proof. Suppose that the family is conservative and that F, U , and s, s0 are as in the lemma. The sections s, s0 define maps a, a0 : (hU )# → F which are distinct. Hence, by Lemma 9.34.2 there is an i such that api 6= a0pi . Recall that (hU )# pi = ui (U ), by Lemmas 9.28.3 and 9.28.5. Hence there exists an x ∈ ui (U ) such that api (x) 6= a0pi (x) in Fpi . Unwinding the definitions you see that (U, x, s) and (U, x, s0 ) are as in the statement of the lemma. To prove the converse, assume the condition on the existence of points of the lemma. Let φ : F → G be a map of sheaves which is an isomorphism at all the stalks. We have to show that φ is both injective and surjective, see Lemma 9.11.2. Injectivity is an immediate consequence of the assumption. Let ∗ denote the final object of

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675

` the category of sheaves, see Example 9.10.2. Consider the sheaf H = G F ∗. The map F → G is surjective if and only if the map ∗ → H is an isomorphism. By construction all the maps on stalks ∗pi = {∗} → Hpi are bijective. If φ is not surjective, then there exists a U and a section s ∈ H(U ) which is not equal to the section ∗. By assumption we see there exists an index i and x ∈ ui (U ) such that (U, x, s) and (U, x, ∗) define distinct points of Hpi . This is a contradiction.  In the following lemma the points qi,x are exactly all the points of C/U lying over the point pi according to Lemma 9.31.2. Lemma 9.34.4. Let C be a site. Let U be an object of C. let {(pi , ui )}i∈I be a family of points of C. For x ∈ ui (U ) let qi,x be the point of C/U constructed in Lemma 9.31.1. If {pi } is a conservative family of points, then {qi,x }i∈I,x∈ui (U ) is a conservative family of points of C/U . In particular, if C has enough points, then so does every localization C/U . Proof. We know that jU ! induces an equivalence jU ! ` : Sh(C/U ) → Sh(C)/h# U , see Lemma 9.21.4. Moreover, we know that (jU ! G)pi = x Gqi,x , see Lemma 9.31.3. Hence the result follows formally.  The following lemma tells us we can check the existence of points locally on the site. Lemma 9.34.5. Let C be a site. Let {Ui }i∈I be a family of objects of C. Assume ` # (1) hUi → ∗ is a surjective map of sheaves, and (2) each localization C/Ui has enough points. Then C has enough points. Proof. For each i ∈ I let {pj }j∈Ji be a conservative family of points of C/Ui . For j ∈ Ji denote qj : Sh(pt) → Sh(C) the composition of pj with the localization morphism Sh(C/Ui ) → Sh(C). Then qj is a point, see Lemma 9.30.2. We claim that the family of points {qj }j∈` Ji is conservative. Namely, let F →`G be a map of sheaves on C such that Fqj → Gqj is an isomorphism for all j ∈ Ji . Let W be an object of C. By assumption (1) there exists a covering {Wa → W } and morphisms Wa → Ui(a) . Since (F|C/Ui(a) )pj = Fqj and (G|C/Ui(a) )pj = Gqj by Lemma 9.30.2 we see that F|Ui(a) → G|Ui(a) is an isomorphism since the family of points {pj }j∈Ji(a) is conservative. Hence F(Wa ) → G(Wa ) is bijective for each a. Similarly F(Wa ×W Wb ) → G(Wa ×W Wb ) is bijective for each a, b. By the sheaf condition this shows that F(W ) → G(W ) is bijective, i.e., F → G is an isomorphism.  9.35. Criterion for existence of points This section corresponds to Deligne’s appendix to [AGV71, Expos´e VI]. In fact it is almost literally the same. Let C be a site. Suppose that (I, ≥) is a directed partially ordered set, and that (Ui , fii0 ) is an inverse system over I, see Categories, Definition 4.19.1. Given the data (I, ≥, Ui , fii0 ) we define u : C −→ Sets,

u(V ) = colimi MorC (Ui , V )

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Let F 7→ Fp be the stalk functor associated to u as in Section 9.28. It is direct from the definition that actually Fp = colimi F(Ui ) in this special case. Note that u commutes with all finite limits (I mean those that are representable in C) because each of the functors V 7→ MorC (Ui , V ) do, see Categories, Lemma 4.17.2. We say that a system (I, ≥, Ui , fii0 ) is a refinement of (J, ≥, Vj , gjj 0 ) if J ⊂ I, the ordening on J induced from that of I and Vj = Uj , gjj 0 = fjj 0 (in words, the inverse system over J is induced by that over I). Let u be the functor associated to (I, ≥, Ui , fii0 ) and let u0 be the functor associated to (J, ≥, Vj , gjj 0 ). This induces a transformation of functors u0 −→ u simply because the colimits for u0 are over a subsystem of the systems in the colimits for u. In particular we get an associated transformation of stalk functors Fp0 → Fp , see Lemma 9.33.1. Lemma 9.35.1. Let C be a site. Let (J, ≥, Vj , gjj 0 ) be a system as above with associated pair of functors (u0 , p0 ). Let F be a sheaf on C. Let s, s0 ∈ Fp0 be distinct elements. Let {Wk → W } be a finite covering of C. Let f ∈ u(W ). There exists a refinement (I, ≥, Ui , fii0 ) of (J, ≥, Vj , gjj 0 ) such that s, s0 map to distinct elements of Fp and that the image of f in u0 (W ) is in the image of one of the u0 (Wk ). Proof. There exists a j0 ∈ J such that f is defined by f 0 : Vj0 → W . For j ≥ j0 we set Vj,k = VjQ ×f 0 ◦fjj0 ,W Wk . Then {Vj,k → Vj } is a finite covering in the site C. Hence F(Vj ) ⊂ k F(Vj,k ). By Categories, Lemma 4.17.2 once again we see that Y Fp0 = colimj F(Vj ) −→ colimj F(Vj,k ) k

is injective. Hence there exists a k such that s and s0 have distinct image in colimj F(Vj,k ). Let J0 = {j ∈ J, j ≥ j0 } and I = J q J0 . We order I so that no element of the second summand is smaller than any element of the first, but otherwise using the ordering on J. If j ∈ I is in the first summand then we use Vj and if j ∈ I is in the second summand then we use Vj,k . We omit the definition of the transition maps of the inverse system. By the above it follows that s, s0 have distinct image in Fp . Moreover, the restriction of f 0 to Vj,k factors through Wk by construction.  Lemma 9.35.2. Let C be a site. Let (J, ≥, Vj , gjj 0 ) be a system as above with associated pair of functors (u0 , p0 ). Let F be a sheaf on C. Let s, s0 ∈ Fp0 be distinct elements. There exists a refinement (I, ≥, Ui , fii0 ) of (J, ≥, Vj , gjj 0 ) such that s, s0 map to distinct elements of Fp and such that for every finite covering {Wk → W } of the site C, and any f ∈ u0 (W ) the image of f in u(W ) is in the image of one of the u(Wk ). Proof. Let E be the set of pairs ({Wk → W }, f ∈ u0 (W )). Consider pairs (E 0 ⊂ E, (I, ≥, Ui , fii0 )) such that (1) (I, ≥, Ui , gii0 ) is a refinement of (J, ≥, Vj , gjj 0 ), (2) s, s0 map to distinct elements of Fp , and (3) for every pair ({Wk → W }, f ∈ u0 (W )) ∈ E 0 we have that the image of f in u(W ) is in the image of one of the u(Wk ).

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We order such pairs by inclusion in the first factor and by refinement in the second. Denote S the class of all pairs (E 0 ⊂ E, (I, ≥, Ui , fii0 )) as above. We claim that 0 the hypothesis of Zorn’s lemma holds for S. Namely, suppose that a, ≥ S (Ea , (I 0 , Ui , fii0 ))a∈A is S a totally ordered subset of S. Then we can define E = a∈A Ea0 and we can set I = a∈A Ia . We claim that the corresponding pair (E 0 , (I, ≥, Ui , fii0 )) is an element of S. Conditions (1) and (3) are clear. For condition (2) you note that u = colima∈A ua and correspondingly Fp = colima∈A Fpa The distinctness of the images of s, s0 in this stalk follows from the description of a directed colimit of sets, see Categories, Section 4.17. We will simply write S (E 0 , (I, . . .)) = a∈A (Ea0 , (Ia , . . .)) in this situation. OK, so Zorn’s Lemma would apply if S was a set, and this would, combined with Lemma 9.35.1 above easily prove the lemma. It doesn’t since S is a class. In order to circumvent this we choose a well ordering on E. For e ∈ E set Ee0 = {e0 ∈ E | e0 ≤ e}. By transfinite induction we construct pairs (Ee0 , (Ie , . . .)) ∈ S such that e1 ≤ e2 ⇒ (Ee0 1 , (Ie1 , . . .)) ≤ (Ee0 2 , (Ie2 , . . .)). Let e ∈ E, say e = ({Wk → W }, f ∈ u0 (W )). If e has a predecessor e−1, then we let (Ie , . . .) be a refinement of (Ie−1 , . . .) as in Lemma 9.35.1 with respect to the system e = ({Wk → W }, f ∈Su0 (W )). If e does not have a predecessor, then we let (Ie , . . .) be a refinement of e0 <e (Ie0 , . . .) with respect to the system e = ({Wk → W }, f ∈ u0 (W )). Finally, the union S  e∈E Ie will be a solution to the problem posed in the lemma. Proposition 9.35.3. Let C be a site. Assume that (1) finite limits exist in C, and (2) every covering {Ui → U }i∈I has a refinement by a finite covering of C. Then C has enough points. Proof. We have to show that given any sheaf F on C, any U ∈ Ob(C), and any distinct sections s, s0 ∈ F(U ), there exists a point p such that s, s0 have distinct image in Fp . See Lemma 9.34.3. Consider the system (J, ≥, Vj , gjj 0 ) with J = {1}, V1 = U , g11 = idU . Apply Lemma 9.35.2. By the result of that lemma we get a system (I, ≥, Ui , fii0 ) refining our system such that sp 6= s0p and ` such that moreover for every finite covering {Wk → W } of the site C the map k u(Wk ) → u(W ) is surjective. Since every covering of C can be refined by a finite covering we conclude ` that k u(Wk ) → u(W ) is surjective for any covering {Wk → W } of the site C. This implies that u = p is a point, see Proposition 9.29.2 (and the discussion at the beginning of this section which garantees that u commutes with finite limits).  9.36. Exactness properties of pushforward Let f be a morphism of topoi. The functor f∗ in general is only left exact. There are many additional conditions one can impose on this functor to single out particular classes of morphisms of topoi. We collect them here and note some of the logical dependencies. Some parts of the following lemma are purely category theoretical (i.e., they do not depend on having a morphism of topoi, just having a pair of adjoint functors is enough). Lemma 9.36.1. Let f : Sh(C) → Sh(D) be a morphism of topoi. Consider the following properties (on sheaves of sets): (1) f∗ is faithful,

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(2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

f∗ is fully faithful, f −1 f∗ F → F is surjective for all F in Sh(C), f∗ transforms surjections into surjections, f∗ commutes with coequalizers, f∗ commutes with pushouts, f −1 f∗ F → F is an isomorphism for all F in Sh(C), f∗ reflects injections, f∗ reflects surjections, f∗ reflects bijections, and for any surjection F → f −1 G there exists a surjection G 0 → G such that f −1 G 0 → f −1 G factors through F → f −1 G.

Then we have the following implications (a) (b) (c) (d) (e) (f) (g)

(2) (3) (7) (3) (6) (4) (8)

⇒ (1), ⇒ (1), ⇒ (1), (2), (3), (8), (9), (10). ⇔ (9), ⇒ (4), ⇔ (11), and + (9) ⇒ (10).

Proof. Proof of (a): This is immediate from the definitions. Proof of (b). Suppose that a, b : F → F 0 are maps of sheaves on C. If f∗ a = f∗ b, then f −1 f∗ a = f −1 f∗ b. Consider the commutative diagram FO f −1 f∗ F

/

/ FO

0

/ −1 0 / f f∗ F

If the bottom two arrows are equal and the vertical arrows are surjective then the top two arrows are equal. Hence (b) follows. Proof of (c). Suppose that a : F → F 0 is a map of sheaves on C. Consider the commutative diagram FO

/ F0 O

f −1 f∗ F

/ f −1 f∗ F 0

If (7) holds, then the vertical arrows are isomorphisms. Hence if f∗ a is injective (resp. surjective, resp. bijective) then the bottom arrow is injective (resp. surjective, resp. bijective) and hence the top arrow is injective (resp. surjective, resp. bijective). Thus we see that (7) implies (8), (9), (10). It is clear that (7) implies (3) and hence (1). Finally, if β : f∗ F → f∗ F 0 is a map of sheaves, then α = f −1 β : F = f −1 f∗ F → f −1 f∗ F 0 = F 0 is a map of sheaves on C. Chasing diagrams we see that

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the following diagram f∗O F f∗ f −1O f∗ F f∗ F

/ f∗ F 0 O

f∗ α

f∗ f −1 β

/ f∗ f −1 f∗ F 0 O

β

/ f∗ F 0

is commutative, in other words f∗ α = β. Hence we see that (2) holds. Proof of (d). Assume (3). Suppose that a : F → F 0 is a map of sheaves on C such that f∗ a is surjective. As f −1 is exact this implies that f −1 f∗ a : f −1 f∗ F → f −1 f∗ F 0 is surjective. Combined with (3) this implies that a is surjective. This means that (9) holds. Assume (9). Let F be a sheaf on C. We have to show that the map f −1 f∗ F → F is surjective. It suffices to show that f∗ f −1 f∗ F → f∗ F is surjective. And this is true because there is a canonical map f∗ F → f∗ f −1 f∗ F which is a one-sided inverse. Proof of (e). If F → F 0 is surjective then the map F 0 qF F 0 → F 0 is injective. Hence (6) implies that f∗ F 0 qf∗ F f∗ F 0 → f∗ F 0 is injective also. And this in turn implies that f∗ F → f∗ F 0 is surjective. Hence we see that (6) implies (4). Proof of (f). Assume (4). Let F → f −1 G be a surjective map of sheaves on C. By (4) we see that f∗ F → f∗ f −1 G is surjective. Let G 0 be the fibre product f∗O F

/ f∗ f −1 G O

G0

/G

so that G 0 → G is surjective also. Consider the commutative diagram FO

/ f −1 G O

f −1 fO ∗ F

/ f −1 f∗ f −1 G O

f −1 G 0

/ f −1 G

and we see the required result. Conversely, assume (11). Let a : F → F 0 be surjective map of sheaves on C. Consider the fibre product diagram FO

/ F0 O

F 00

/ f −1 f∗ F 0

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Because the lower horizontal arrow is surjective and by (11) we can find a surjection γ : G 0 → f∗ F 0 such that f −1 γ factors through F 00 → f −1 f∗ F 0 :

f −1 G 0

FO

/ F0 O

/ F 00

/ f −1 f∗ F 0

Pushing this down using f∗ we get a commutative diagram

0 f∗ f −1 O G

f∗O F

/ f∗ F 0 O

/ f∗ F 00

/ f∗ f −1 f∗ F 0 O / f∗ F 0

G0 which proves that (4) holds.

Proof of (g). This is immediate from the definitions.



Here is a condition on a morphism of sites which garantees that the functor f∗ transforms surjective maps into surjective maps. Lemma 9.36.2. Let f : D → C be a morphism of sites associated to the continuous functor u : C → D. Assume that for any object U of C and any covering {Vj → u(U )} in D there exists a covering {Ui → U } in C such that the map of sheaves a

# h# u(Ui ) → hu(U )

factors through the map of sheaves a

# h# Vj → hu(U ) .

Then f∗ transforms surjective maps of sheaves into surjective maps of sheaves. Proof. Let a : F → G be a surjective map of sheaves on D. Let U be an object of C and let s ∈ f∗ G(U ) = G(u(U )). By assumption there exists a covering {Vj → u(U )} and sections sj ∈ F(Vj ) with a(sj ) = s|Vj . Now we may think of the sections s, sj and a as giving a commutative diagram of maps of sheaves `

h# Vj 

h# u(U )

`

sj

/F a

s

 /G

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681

By assumption there exists a covering {Ui → U } such that we can enlarge the commutative diagram above as follows ` # hVj ` / F sj ;

`

h# u(Ui )



/ h# u(U )

a s

 /G

Because F is a sheaf the map from the left lower corner to the right upper corner corresponds to a family of sections si ∈ F(u(Ui )), i.e., sections si ∈ f∗ F(Ui ). The commutativity of the diagram implies that a(si ) is equal to the restriction of s to Ui . In other words we have shown that f∗ a is a surjective map of sheaves.  Example 9.36.3. Assume f : D → C satisfies the assumptions of Lemma 9.36.2. Then it is in general not the case that f∗ commutes with coequalizers or pushouts. Namely, suppose that f is the morphism of sites associated to the morphism of topological spaces X = {1, 2} → Y = {∗} (see Example 9.14.2), where Y is a singleton space, and X = {1, 2} is a discrete space with two points. A sheaf F on X is given by a pair (A1 , A2 ) of sets. Then f∗ F corresponds to the set A1 × A2 . Hence if a = (a1 , a2 ), b = (b1 , b2 ) : (A1 , A2 ) → (B1 , B2 ) are maps of sheaves on X, then the coequalizer of a, b is (C1 , C2 ) where Ci is the coequalizer of ai , bi , and the coequalizer of f∗ a, f∗ b is the coequalizer of a1 × a2 , b1 × b2 : A1 × A2 −→ B1 × B2 which is in general different from C1 ×C2 . Namely, if A2 = ∅ then A1 ×A2 = ∅, and hence the coequalizer of the displayed arrows is B1 × B2 , but in general C1 6= B1 . A similar example works for pushouts. The following lemma gives a criterion for when a morphism of sites has a functor f∗ which reflects injections and surjections. Note that this also implies that f∗ is faithful and that the map f −1 f∗ F → F is always surjective. Lemma 9.36.4. Let f : D → C be a morphism of sites given by the functor u : C → D. Assume that for every object V of D there exist objects Ui of C and morphisms u(Ui ) → V such that {u(Ui ) → V } is a covering of D. In this case the functor f∗ : Sh(D) → Sh(C) reflects injections and surjections. Proof. Let α : F → G be Q maps of sheaves Q on D. By assumption for every object V of D we get F(V ) ⊂ F(u(Ui )) = (us F)(u(Ui )) by the sheaf condition for some Ui objects of C and similarly for G. Hence it is clear that if f∗ α is injective, then α is injective. In other words f∗ reflects injections. Suppose that f∗ α is surjective. Then for V, Ui , u(Ui ) → V as above and a section s ∈ G(V ), there exist coverings {Uij → Ui } such that s|u(Uij ) is in the image of F(u(Uij )). Since {u(Uij ) → V } is a covering (as u is continuous and by the axioms of a site) we conclude that s is locally in the image. Thus α is surjective. In other words f∗ reflects surjections.  9.37. Almost cocontinuous functors Let C be a site. The category PSh(C) has an initial object, namely the presheaf which assigns the empty set to each object of C. Let us denote this presheaf by ∅.

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It follows from the properties of sheafification that the sheafification ∅# of ∅ is an initial object of the category Sh(C) of sheaves on C. Definition 9.37.1. Let C be a site. We say an object U of C is sheaf theoretically empty if ∅# → h# U is an isomorphism of sheaves. The following lemma makes this notion more explicit. Lemma 9.37.2. Let C be a site. Let U be an object of C. The following are equivalent: (1) U is sheaf theoretically empty, (2) F(U ) is a singleton for each sheaf F, (3) ∅# (U ) is a singleton, (4) ∅# (U ) is nonempty, and (5) the empty family is a covering of U in C. Moreover, if U is sheaf theoretically empty, then for any morphism U 0 → U of C the object U 0 is sheaf theoretically empty. Proof. For any sheaf F we have F(U ) = MorSh(C) (h# U , F). Hence, we see that (1) and (2) are equivalent. It is clear that (2) implies (3) implies (4). If every covering of U is given by a nonempty family, then ∅+ (U ) is empty by definition of the plus construction. Note that ∅+ = ∅# as ∅ is a separated presheaf, see Theorem 9.10.10. Thus we see that (4) implies (5). If (5) holds, then F(U ) is a singleton for every sheaf F by the sheaf condition for F, see Remark 9.7.2. Thus (5) implies (2) and (1) – (5) are equivalent. The final assertion of the lemma follows from Axiom (3) of Definition 9.6.2 applied the the empty covering of U .  Definition 9.37.3. Let C, D be sites. Let u : C → D be a functor. We say u is almost cocontinuous if for every object U of C and every covering {Vj → u(U )}j∈J there exists a covering {Ui → U }i∈I in C such that for each i in I we have at least one of the following two conditions (1) u(Ui ) is sheaf theoretically empty, or (2) the morphism u(Ui ) → u(U ) factors through Vj for some j ∈ J. The motivation for this definition comes from a closed immersion i : Z → X of topological spaces. As discussed in Example 9.19.9 the continuous functor TX → TZ , U 7→ Z ∩ U is not cocontinuous. But it is almost cocontinuous in the sense defined above. We know that i∗ while not exact on sheaves of sets, is exact on sheaves of abelian groups, see Sheaves, Remark 6.32.5. And this holds in general for continuous and almost cocontinuous functors. Lemma 9.37.4. Let C, D be sites. Let u : C → D be a functor. Assume that u is continuous and almost cocontinuous. Let G be a presheaf on D such that G(V ) is a singleton whenever V is sheaf theoretically empty. Then (up G)# = up (G # ). Proof. Let U ∈ Ob(C). We have to show that (up G)# (U ) = up (G # )(U ). It suffices to show that (up G)+ (U ) = up (G + )(U ) since G + is another presheaf for which the assumption of the lemma holds. We have ˇ 0 (V, G) up (G + )(U ) = G + (u(U )) = colimV H where the colimit is over the coverings V of u(U ) in D. On the other hand, we see that ˇ 0 (u(U), G) up (G)+ (U ) = colimU H

9.38. SHEAVES OF ALGEBRAIC STRUCTURES

683

where the colimit is over the category of coverings U = {Ui → U }i∈I of U in C and u(U) = {u(Ui ) → u(U )}i∈I . The condition that u is continuous means that each u(U) is a covering. Write I = I1 q I2 , where I2 = {i ∈ I | u(Ui ) is sheaf theoretically empty} 0

Then u(U) = {u(Ui ) → u(U )}i∈I1 is still a covering of because each of the other pieces can be covered by the empty family and hence can be dropped by Axiom ˇ 0 (u(U), G) = H ˇ 0 (u(U)0 , G) by our assumption (2) of Definition 9.6.2. Moreover, H on G. Finally, the condition that u is almost cocontinuous implies that for every covering V of u(U ) there exists a covering U of U such that u(U)0 refines V. It follows that the two colimits displayed above have the same value as desired.  Lemma 9.37.5. Let C, D be sites. Let u : C → D be a functor. Assume that u is continuous and almost cocontinuous. Then us = up : Sh(D) → Sh(C) commutes with pushouts and coequalizers (and more generally finite, nonempty, connected colimits). Proof. Let I be a finite, nonempty, connected index category. Let I → Sh(D), i 7→ Gi by a diagram. We know that the colimit of this diagram is the sheafification of the colimit in the category of presheaves, see Lemma 9.10.13. Denote colimP sh the colimit in the category of presheaves. Since I is finite, nonempty and connected sh we see that colimP Gi is a presheaf satisfying the assumptions of Lemma 9.37.4 i (because a finite nonempty connected colimit of singleton sets is a singleton). Hence that lemma gives sh us (colimi Gi ) = us ((colimP Gi )# ) i sh Gi ))# = (up (colimP i Sh p u (Gi ))# = (colimP i

= colimi us (Gi ) as desired.



Lemma 9.37.6. Let f : D → C be a morphism of sites associated to the continuous functor u : C → D. If u is almost cocontinuous then f∗ commutes with pushouts and coequalizers (and more generally finite, nonempty, connected colimits). Proof. This is a special case of Lemma 9.37.5.



9.38. Sheaves of algebraic structures In Sheaves, Section 6.15 we introduced a type of algebraic struture to be a pair (A, s), where A is a category, and s : A → Sets is a functor such that (1) s is faithful, (2) A has limits and s commutes with limits, (3) A has filtered colimits and s commutes with them, and (4) s reflects isomorphisms. For such a type of algebraic structure we saw that a presheaf F with values in A on a space X is a sheaf if and only if the associated presheaf of sets is a sheaf. Moreover, we worked out the notion of stalk, and given a continuous map f : X → Y we defined adjoint functors pushforward and pullback on sheaves of algebraic structures which agrees with pushforward and pullback on the underlying sheaves of sets. In

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addition extending a sheaf of algebraic structures from a basis to all opens of a space, works as expected. Part of this material still works in the setting of sites and sheaves. Let (A, s) be a type of algebraic structure. Let C be a site. Let us denote PSh(C, A), resp. Sh(C, A) the category of presheaves, resp. sheaves with values in A on C. (α) A presheaf with values in A is a sheaf if and only if its underlying presheaf of sets is a sheaf. See the proof of Sheaves, Lemma 6.9.2. (β) Given a presheaf F with values in A the presheaf F # = (F + )+ is a sheaf. This is true since the colimits in the sheafification process are filtered, and even colimits over directed partially ordered sets (see Section 9.10, especially the proof of Lemma 9.10.14) and since s commutes with filtered colimits. (γ) We get the following commutative diagram / PSh(C, A) Sh(C, A) o #

s

 Sh(C) o

s

/

 PSh(C)

(δ) We have F = F # if and only if F is a sheaf of algebraic structures. () The functor # is adjoint to the inclusion functor: MorPSh(C,A) (G, F) = MorSh(C,A) (G # , F) The proof is the same as the proof of Proposition 9.10.12. (ζ) The functor F 7→ F # is left exact. The proof is the same as the proof of Lemma 9.10.14. Definition 9.38.1. Let f : D → C be a morphism of sites given by a functor u : C → D. We define the pushforward functor for presheaves of algebraic structures by the rule up F(U ) = F(uU ), and for sheaves of algebraic structures by the same rule, namely f∗ F(U ) = F(uU ). The problem comes with trying the define the pullback. The reason is that the colimits defining the functor up in Section 9.5 may not be filtered. Thus the axioms above are not enough in general to define the pullback of a (pre)sheaf of algebraic structures. Nonetheless, in almost all cases the following lemma is sufficient to define pushfoward, and pullback of (pre)sheaves of algebraic structures. Lemma 9.38.2. Suppose the functor u : C → D satisfies the hypotheses of Proposition 9.14.6, and hence gives rise to a morphism of sites f : D → C. In this case the pullback functor f −1 (resp. up ) and the pushforward functor f∗ (resp. up ) extend to an adjoint pair of functors on the categories of sheaves (resp. presheaves) of algebraic structures. Moreover, these functors commute with taking the underlying sheaf (resp. presheaf ) of sets. Proof. We have defined f∗ = up above. In the course of the proof of Proposition 9.14.6 we saw that all the colimits used to define up are filtered under the assumptions of the proposition. Hence we conclude from the definition of a type of algebraic structure that we may define up by exactly the same colimits as a functor on presheaves of algebraic structures. Adjointness of up and up is proved

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in exactly the same way as the proof of Lemma 9.5.4. The discussion of sheafification of presheaves of algebraic structures above then implies that we may define f −1 (F) = (up F)# .  We briefly discuss a method for dealing with pullback and pushforward for a general morphism of sites, and more generally for any morphism of topoi. Let C be a site. In the case A = Ab, we may think of an abelian (pre)sheaf on C as a quadruple (F, +, 0, i). Here the data are (D1) F is a sheaf of sets, (D2) + : F × F → F is a morphism of sheaves of sets, (D3) 0 : ∗ → F is a morphism from the singleton sheaf (see Example 9.10.2) to F, and (D4) i : F → F is a morphism of sheaves of sets. These data have to satisfy the following axioms (A1) + is associative and commutative, (A2) 0 is a unit for +, and (A3) + ◦ (1, i) = 0 ◦ (F → ∗). Compare Sheaves, Lemma 6.4.3. Let f : D → C be a morphism of sites. Note that since f −1 is exact we have f −1 ∗ = ∗ and f −1 (F ×F) = f −1 F ×f −1 F. Thus we can define f −1 F simply as the quadruple (f −1 F, f −1 +, f −1 0, f −1 i). The axioms are going to be preserved because f −1 is a functor which commutes with finite limits. Finally it is not hard to check that f∗ and f −1 are adjoint as usual. In [AGV71] this method is used. They introduce something called an “esp`ece the structure alg´ebrique d´efinie par limites projectives finie”. For such an esp`ece you can use the method described above to define a pair of adjoint functors f −1 and f∗ as above. This clearly works for most algebraic structures that one encounters in practice. Instead of formalizing this construction we simply list those algebraic structures for which this method works (to be verified case by case). In fact, this method works for any morphism of topoi. Proposition 9.38.3. Let C, D be sites. Let f = (f −1 , f∗ ) be a morphism of topoi from Sh(D) → Sh(C). The method introduced above gives rise to an adjoint pair of functors (f −1 , f∗ ) on sheaves of algebraic structures compatible with taking the underlying sheaves of sets for the following types of algebraic structures: (1) pointed sets, (2) abelian groups, (3) groups, (4) monoids, (5) rings, (6) modules over a fixed ring, and (7) lie algebras over a fixed field. Moreover, in each of these cases the results above labeled (α), (β), (γ), (δ), (), and (ζ) hold. Proof. The final statement of the proposition holds simply since each of the listed categories, endowed with the obvious forgetful functor, is indeed a type of algebraic structure in the sense explained at the beginning of this section. See Sheaves, Lemma 6.15.2.

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Proof of (2). We think of a sheaf of abelian groups as a quadruple (F, +, 0, i) as explained in the discussion preceding the proposition. If (F, +, 0, i) lives on C, then its pullback is defined as (f −1 F, f −1 +, f −1 0, f −1 i). If (G, +, 0, i) lives on D, then its pushforward is defined as (f∗ G, f∗ +, f∗ 0, f∗ i). This works because f∗ (G × G) = f∗ G × f∗ G. Adjointness follows from adjointness of the set based functors, since   ϕ ∈ MorSh(C) (F1 , F2 ) MorAb(C) ((F1 , +, 0, i), (F2 , +, 0, i)) = ϕ is compatible with +, 0, i Details left to the reader. This method also works for sheaves of rings by thinking of a sheaf of rings (with unit) as a sixtuple (O, +, 0, i, ·, 1) satisfying a list of axioms that you can find in any elementary algebra book. A sheaf of pointed sets is a pair (F, p), where F is a sheaf of sets, and p : ∗ → F is a map of sheaves of sets. A sheaf of groups is given by a quadruple (F, ·, 1, i) with suitable axioms. A sheaf of monoids is given by a pair (F, ·) with suitable axiom. Let R be a ring. An sheaf of R-modules is given by a quintuple (F, +, 0, i, {λr }r∈R ), where the quadruple (F, +, 0, i) is a sheaf of abelian groups as above, and λr : F → F is a family of morphisms of sheaves of sets such that λr ◦0 = 0, λr ◦+ = +◦(λr , λr ), λr+r0 = + ◦ λr × λr0 ◦ (id, id), λrr0 = λr ◦ λr0 , λ1 = id, λ0 = 0 ◦ (F → ∗).  We will discuss the category of sheaves of modules over a sheaf of rings in Modules on Sites, Section 16.10. Remark 9.38.4. Let C, D be sites. Let u : D → C be a continuous functor which gives rise to a morphism of sites C → D. Note that even in the case of abelian groups we have not defined a pullback functor for presheaves of abelian groups. Since all colimits are representable in the category of abelian groups, we certainly may define a functor uab p on abelian presheaves by the same colimits as we have used to define up on presheaves of sets. It will also be the case that uab p is adjoint to up on the categories of abelian presheaves. However, it will not always be the case that uab p agrees with up on the underlying presheaves of sets. 9.39. Pullback maps It sometimes happens that a site C does not have a final object. In this case we define the global section functor as follows. Definition 9.39.1. The global sections of a presheaf of sets F over a site C is the set Γ(C, F) = MorPSh(C) (∗, F) where ∗ is the final object in the category of presheaves on C, i.e., the presheaf which associates to every object a singleton. Of course the same definition applies to sheaves as well. Here is one way to compute global sections.

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Lemma 9.39.2. Let C be a site. Let a, b : V → U be objects of C such that / # /∗ h# / hU V is a coequalizer in Sh(C). Then Γ(C, F) is the equalizer of a∗ , b∗ : F(U ) → F(V ). Proof. Since MorSh(C) (h# U , F) = F(U ) this is clear from the definitions.



Now, let f : Sh(D) → Sh(C) be a morphism of topoi. Then for any sheaf G on D there is a pullback map f −1 : Γ(D, F) −→ Γ(C, f −1 F) Namely, as f −1 is exact it transforms ∗ into ∗. We can generalize this a bit by considering a pair of sheaves F, G on C, D together with a map f −1 F → G. Then we compose to get a map Γ(D, F) −→ Γ(C, G) A slightly more general construction which occurs frequently in nature is the following. Suppose that we have a commutative diagram of morphisms of topoi Sh(D)

/ Sh(C)

f h

{ $ Sh(B)

g

Next, suppose that we have a sheaf F on D. Then there is a pullback map f −1 : g∗ F −→ h∗ f −1 F Namely, it is just the map coming from the identification h∗ f −1 F = g∗ f∗ f −1 F together with the canonical map F → f∗ f −1 F pushed down to B. Again, if we have a pair of sheaves F, G on C, D together with a map f −1 F → G, then we compose to get a map g∗ F −→ h∗ G Restricting to sections over an object of B one recovers the pullback map on global sections in many cases, see (insert future reference here). A seemingly more general situation is where we have a commutative diagram of topoi Sh(D)

f

g

h

 Sh(B)

/ Sh(C)

e

 / Sh(A)

and a sheaf G on C. Then there is a map e−1 g∗ G → h∗ f −1 G. Namely, this map is adjoint to a map g∗ G → e∗ h∗ f −1 G = (e ◦ h)∗ f −1 G which is the pullback map just described. 9.40. Topologies In this section we define what a topology on a category is as defined in [AGV71]. One can develop all of the machinery of sheaves and topoi in this language. A modern exposition of this material can be found in [KS06]. However, the case of most interest for algebraic geometry is the topology defined by a site on its underlying category. Thus we strongly suggest the first time reader skip this section and all other sections of this chapter!

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Definition 9.40.1. Let C be a category. Let U ∈ Ob(C). A sieve S on U is a subpresheaf S ⊂ hU . In other words, a sieve on U picks out for each object T ∈ Ob(C) a subset S(T ) of the set of all morphisms T → U . In fact, the only condition on the collection of subsets S(T ) ⊂ hU (T ) = MorC (T, U ) is the following rule  (α : T → U ) ∈ S(T ) (9.40.1.1) ⇒ (α ◦ g : T 0 → U ) ∈ S(T 0 ) g : T0 → T A good mental picture to keep in mind is to think of the map S → hU as a “morphism from S to U ”. 9.40.2. Let C be a category. Let U ∈ Ob(C). The collection of sieves on U is a set. Inclusion defines a partial ordering on this set. Unions and intersections of sieves are sieves. Given a family of morphisms {Ui → U }i∈I of C with target U there exists a unique smallest sieve S on U such that each Ui → U belongs to S(Ui ). (5) The sieve S = hU is the maximal sieve. (6) The empty subpresheaf is the minimal sieve.

Lemma (1) (2) (3) (4)

Proof. By our definition ofQsubpresheaf, the collection of all subpresheaves of a presheaf F is a subset of U ∈Ob(C) P(F(U )). And this is a set. (Here P(A) denotes the powerset of A.) Hence the collection of sieves on U is a set. The partial ordering is defined by: S ≤ S 0 if and only if S(T ) ⊂ S 0 (T ) for all T → U . Notation: S ⊂ S 0 . S Given aScollection of S sieves Si , i ∈ I on U we can define Si as the sieve T with values ( Si )(T ) = Si (T ) for all T ∈ Ob(C). We define the intersection Si in the same way. Given {Ui → U }i∈I as in the statement, consider the morphisms of presheaves hUi → hU . We simply define S as the union of the images (Definition 9.3.5) of these maps of presheaves. The last two statements of the lemma are obvious.



Definition 9.40.3. Let C be a category. Given a family of morphisms {fi : Ui → U }i∈I of C with target U we say the sieve S on U described in Lemma 9.40.2 part (4) is the sieve on U generated by the morphisms fi . Definition 9.40.4. Let C be a category. Let f : V → U be a morphism of C. Let S ⊂ hU be a sieve. We define the pullback of S by f to be the sieve S ×U V of V defined by the rule (α : T → V ) ∈ (S ×U V )(T ) ⇔ (f ◦ α : T → U ) ∈ S(T ) We leave it to the reader to see that this is indeed a sieve (hint: use Equation 9.40.1.1). We also sometimes call S ×U V the base change of S by f : V → U . Lemma 9.40.5. Let C be a category. Let U ∈ Ob(C). Let S be a sieve on U . If f : V → U is in S, then S ×U V = hV is maximal. Proof. Trivial from the definitions.



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Definition 9.40.6. Let C be a category. A topology on C is given by the following datum: For every U ∈ Ob(C) a subset J(U ) of the set of all sieves on U . These sets J(U ) have to satisfy the following conditions (1) For every morphism f : V → U in C, and every element S ∈ J(U ) the pullback S ×U V is an element of J(V ). (2) If S and S 0 are sieves on U ∈ Ob(C), if S ∈ J(U ), and if for all f ∈ S(V ) the pullback S 0 ×U V belongs to J(V ), then S 0 belongs to J(U ). (3) For every U ∈ Ob(C) the maximal sieve S = hU belongs to J(U ). In this case, the sieves belonging to J(U ) are called the covering sieves. Lemma 9.40.7. Let C be a category. Let J be a topology on C. Let U ∈ Ob(C). (1) Finite intersections of elements of J(U ) are in J(U ). (2) If S ∈ J(U ) and S 0 ⊃ S, then S 0 ∈ J(U ). Proof. Let S, S 0 ∈ J(U ). Consider S 00 = S ∩ S 0 . For every V → U in S(U ) we have S 0 ×U V = S 00 ×U V simply because V → U already is in S. Hence by the second axiom of the definition we see that S 00 ∈ J(U ). Let S ∈ J(U ) and S 0 ⊃ S. For every V → U in S(U ) we have S 0 ×U V = hV by Lemma 9.40.5. Thus S 0 ×U V ∈ J(V ) by the third axiom. Hence S 0 ∈ J(U ) by the second axiom.  Definition 9.40.8. Let C be a category. Let J, J 0 be two topologies on C. We say that J is finer than J 0 if and only if for every object U of C we have J 0 (U ) ⊂ J(U ). In other words, any covering sieve of J 0 is a covering sieve of J. There exists a finest topology on C, namely that topology where any sieve is a covering sieve. This is called the discrete topology of C. There also exists a coarsest topology. Namely, the topology where J(U ) = {hU } for all objects U . This is called the chaotic or indiscrete topology. Lemma 9.40.9. Let C be a category. Let {Ji }i∈I be a set of topologies. T (1) The rule J(U ) = Ji (U ) defines a topology on C. (2) There is a coarsest topology finer than all of the topologies Ji . Proof. The first part is direct from the definitions. The second follows by taking the intersection of all topologies finer than all of the Ji .  At this point we can define without any motivation what a sheaf is. Definition 9.40.10. Let C be a category endowed with a topology J. Let F be a presheaf of sets on C. We say that F is a sheaf on C if for every U ∈ Ob(C) and for every covering sieve S of U the canonical map MorPSh(C) (hU , F) −→ MorPSh(C) (S, F) is bijective.

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Recall that the left hand side of the displayed formula equals F(U ). In other words, F is a sheaf if and only if a section of F over U is the same thing as a compatible collection of sections sT,α ∈ F(T ) parametrized by (α : T → U ) ∈ S(T ), and this for every covering sieve S on U . Lemma 9.40.11. Let C be a category. Let {Fi }i∈I be a collection of presheaves of sets on C. For each U ∈ Ob(C) denote J(U ) the set of sieves S with the following property: For every morphism V → U , the maps MorPSh(C) (hV , Fi ) −→ MorPSh(C) (S ×U V, Fi ) are bijective for all i ∈ I. Then J defines a topology on C. This topology is the finest topology in which all of the Fi are sheaves. Proof. If we show that J is a topology, then the last statement of the lemma immediately follows. The first and second axioms of a topology are immediately verified. Thus, assume that we have an object U , and sieves S, S 0 of U such that S ∈ J(U ), and for all V → U in S(V ) we have S 0 ×U V ∈ J(V ). We have to show that S 0 ∈ J(U ). In other words, we have to show that for any f : W → U , the maps Fi (W ) = MorPSh(C) (hW , Fi ) −→ MorPSh(C) (S 0 ×U W, Fi ) are bijective for all i ∈ I. Pick an element i ∈ I and pick an element ϕ ∈ MorPSh(C) (S 0 ×U W, Fi ). We will construct a section s ∈ Fi (W ) mapping to ϕ. Suppose α : V → W is an element of S ×U W . According to the definition of pullbacks we see that the composition f ◦ α : V → W → U is in S. Hence S 0 ×U V is in J(W ) by assumption on the pair of sieves S, S 0 . Now we have a commutative diagram of presheaves / hV S 0 ×U V  S 0 ×U W

 / hW

The restriction of ϕ to S 0 ×U V corresponds to an element sV,α ∈ Fi (V ). This we see from the definition of J, and because S 0 ×U V is in J(W ). We leave it to the reader to check that the rule (V, α) 7→ sV,α defines an element ψ ∈ MorPSh(C) (S ×U W, Fi ). Since S ∈ J(U ) we see immediately from the definition of J that ψ corresponds to an element s of Fi (W ). We leave it to the reader to verify that the construction ϕ 7→ s is inverse to the natural map displayed above.  Definition 9.40.12. Let C be a category. The finest topology on C such that all representable presheaves are sheaves, see Lemma 9.40.11, is called the canonical topology of C. 9.41. The topology defined by a site Suppose that C is a category, and suppose that Cov1 (C) and Cov2 (C) are sets of coverings that define the structure of a site on C. In this situation it can happen that the categories of sheaves (of sets) for Cov1 (C) and Cov2 (C) are the same, see for example Lemma 9.8.5.

9.41. THE TOPOLOGY DEFINED BY A SITE

691

It turns out that the category of sheaves on C with respect to some topology J determines and is determined by the topology J. This is a nontrivial statement which we will address later, see Theorem 9.43.2. Accepting this for the moment it makes sense to study the topology determined by a site. Lemma 9.41.1. Let C be a site with coverings Cov(C). For every object U of C, let J(U ) denote the set of sieves S on U with the following property: there exists a covering {fi : Ui → U }i∈I ∈ Cov(C) so that the sieve S 0 generated by the fi (see Definition 9.40.3) is contained in S. (1) This J is a topology on C. (2) A presheaf F is a sheaf for this topology (see Definition 9.40.10) if and only if it is a sheaf on the site (see Definition 9.7.1). Proof. To prove the first assertion we just note that axioms (1), (2) and (3) of the definition of a site (Definition 9.6.2) directly imply the axioms (3), (2) and (1) of the definition of a topology (Definition 9.40.6). As an example we prove J has property (2). Namely, let U be an object of C, let S, S 0 be sieves on U such that S ∈ J(U ), and such that for every V → U in S(V ) we have S 0 ×U V ∈ J(V ). By definition of J(U ) we can find a covering {fi : Ui → U } of the site such that S the image of hUi → hU is contained in S. Since each S 0 ×U Ui is in J(Ui ) we see that there are coverings {Uij → Ui } of the site such that hUij → hUi is contained in S 0 ×U Ui . By definition of the base change this means that hUij → hU is contained in the subpresheaf S 0 ⊂ hU . By axiom (2) for sites we see that {Uij → U } is a covering of U and we conclude that S 0 ∈ J(U ) by definition of J. Let F be a presheaf. Suppose that F is a sheaf in the topology J. We will show that F is a sheaf on the site as well. Let {fi : Ui → U }i∈I be a covering of the site. Let si ∈ F(Ui ) be a family of sections such that si |Ui ×U Uj = sj |Ui ×U Uj for all i, j. We have to show that there exists a unique section s ∈ F(U ) restricting back to the si on the Ui . Let S ⊂ hU be the sieve generated by the fi . Note that S ∈ J(U ) by definition. In stead of constructing s, by the sheaf condition in the topology, it suffices to construct an element ϕ ∈ MorPSh(C) (S, F). Take α ∈ S(T ) for some object T ∈ U. This means exactly that α : T → U is a morphism which factors through fi for some i ∈ I (and maybe more than 1). Pick such an index i and a factorization α = fi ◦ αi . Define ϕ(α) = αi∗ si . If i0 , α = fi ◦αi0 0 is a second choice, then αi∗ si = (αi0 0 )∗ si0 exactly because of our condition si |Ui ×U Uj = sj |Ui ×U Uj for all i, j. Thus ϕ(α) is well defined. We leave it to the reader to verify that ϕ, which in turn determines s is correct in the sense that s restricts back to si . Let F be a presheaf. Suppose that F is a sheaf on the site (C, Cov(C)). We will show that F is a sheaf for the topology J as well. Let U be an object of C. Let S be a covering sieve on U with respect to the topology J. Let ϕ ∈ MorPSh(C) (S, F). We have to show there is a unique element in F(U ) = MorPSh(C) (hU , F) which restricts back to ϕ. By definition there exists a covering {fi : Ui → U }i∈I ∈ Cov(C) such that fi : Ui ∈ U belongs to S(Ui ). Hence we can set si = ϕ(fi ) ∈ F(Ui ).

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Then it is a pleasant exercise to see that si |Ui ×U Uj = sj |Ui ×U Uj for all i, j. Thus we obtain the desired section s by the sheaf condition for F on the site (C, Cov(C)). Details left to the reader.  Definition 9.41.2. Let C be a site with coverings Cov(C). The topology associated to C is the topology J contructed in Lemma 9.41.1 above. Let C be a category. Let Cov1 (C) and Cov2 (C) be two coverings defining the structure of a site on C. It may very well happen that the topologies defined by these are the same. If this happens then we say Cov1 (C) and Cov2 (C) define the same topology on C. And if this happens then the categories of sheaves are the same, by Lemma 9.41.1. It is usually the case that we only care about the topology defined by a collection of coverings, and we view the possibility of choosing different sets of coverings as a tool to study the topology. Remark 9.41.3. Enlarging the class of coverings. Clearly, if Cov(C) defines the structure of a site on C then we may add to C any set of families of morphisms with fixed target tautologically equivalent (see Definition 9.8.2) to elements of Cov(C) without changing the topology. Remark 9.41.4. Shrinking the class of coverings. Let C be a site. Consider the power set S = P (Arrow(C)) (power set) of the set of morphisms, i.e., the set of all sets of morphisms. Let Sτ ⊂ S be the subset consisting of those T ∈ S such that (a) all ϕ ∈ T have the same target, (b) the collection {ϕ}ϕ∈T is tautologically equivalent (see Definition 9.8.2) to some covering in Cov(C). Clearly, considering the elements of Sτ as the coverings, we do not get exactly the notion of a site as defined in Definition 9.6.2. The structure (C, Sτ ) we get satisfies slightly modified conditions. The modified conditions are: (0’) Cov(C) ⊂ P (Arrow(C)), (1’) If V → U is an isomorphism then {V → U } ∈ Cov(C). (2’) If {Ui → U }i∈I ∈ Cov(C) and for each i we have {Vij → Ui }j∈Ji ∈ Cov(C), then {Vij → U }i∈I,j∈Ji is tautologically equivalent to an element of Cov(C). (3’) If {Ui → U }i∈I ∈ Cov(C) and V → U is a morphism of C then Ui ×U V exists for all i and {Ui ×U V → V }i∈I is tautologically equivalent to an element of Cov(C). And it is easy to verify that, given a structure satisfying (0’) – (3’) above, then after suitably enlarging Cov(C) (compare Sets, Section 3.11) we get a site. Obviously there is little difference between this notion and the actual notion of a site, at least from the point of view of the topology. There are two benefits: because of condition (0’) above the coverings automatically form a set, and because of (0’) the totality of all structures of this type forms a set as well. The price you pay for this is that you have to keep writing “tautologically equivalent” everywhere. 9.42. Sheafification in a topology In this section we explain the analogue of the sheafification construction in a topology.

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693

Let C be a category. Let J be a topology on C. Let F be a presheaf of sets. For every U ∈ Ob(C) we define LF(U ) = colimS∈J(U )opp MorPSh(C) (S, F) as a colimit. Here we think of J(U ) as a partially ordered set, ordered by inclusion, see Lemma 9.40.2. The transition maps in the system are defined as follows. If S ⊂ S 0 are in J(U ), then S → S 0 is a morphism of presheaves. Hence there is a natural restriction mapping MorPSh(C) (S, F) −→ MorPSh(C) (S 0 , F). Thus we see that S 7→ MorPSh(C) (S, F) is a directed system as in Categories, Definition 4.19.1 provided we reverse the ordering on J(U ) (which is what the superscript opp is supposed to indicate). In particular, since hU ∈ J(U ) there is a canonical map ` : F(U ) −→ LF(U ) coming from the identification F(U ) = MorPSh(C) (hU , F). In addition, the colimit defining LF(U ) is directed since for any pair of covering sieves S, S 0 on U the sieve S ∩ S 0 is a covering sieve too, see Lemma 9.40.2. Let f : V → U be a morphism in C. Let S ∈ J(U ). There is a commutative diagram / hV S ×U V   / hU S We can use the left vertical map to get canonical restriction maps MorPSh(C) (S, F) → MorPSh(C) (S ×U V, F). Base change S 7→ S ×U V induces an order preserving map J(U ) → J(V ). And the restriction maps define a transformation of functors as in Categories, Lemma categories-lemma-functorial-colimit. Hence we get a natural retriction map LF(U ) −→ LF(V ). Lemma 9.42.1. In the situation above. (1) The assignment U 7→ LF(U ) combined with the restriction mappings defined above is a presheaf. (2) The maps ` glue to give a morphism of presheaves ` : F → LF. `

(3) The rule F 7→ (F → − LF) is a functor. (4) If F is a subpresheaf of G, then LF is a subpresheaf of LG. (5) The map ` : F → LF has the following property: For every section s ∈ LF(U ) there exists a covering sieve S on U and an element ϕ ∈ MorPSh(C) (S, F) such that `(ϕ) equals the restriction of s to S. Proof. Omitted.



Definition 9.42.2. Let C be a category. Let J be a topology on C. We say that a presheaf of sets F is separated if for every object U and every covering sieve S on U the canonical map F(U ) → MorPSh(C) (S, F) is injective. Theorem 9.42.3. Let C be a category. Let J be a topology on C. Let F be a presheaf of sets.

694

9. SITES AND SHEAVES

(1) The presheaf LF is separated. (2) If F is separated, then LF is a sheaf and the map of presheaves F → LF is injective. (3) If F is a sheaf, then F → LF is an isomorphism. (4) The presheaf LLF is always a sheaf. Proof. Part (3) is trivial from the definition of L and the definition of a sheaf (Definition 9.40.10). Part (4) follows formally from the others. We sketch the proof of (1). Suppose S is a covering sieve of the object U . Suppose that ϕi ∈ LF(U ), i = 1, 2 map to the same element in MorPSh(C) (S, LF). We may find a single covering sieve S 0 on U such that both ϕi are represented by elements ϕi ∈ MorPSh(C) (S 0 , F). We may assume that S 0 = S by replacing both S and S 0 by S 0 ∩ S which is also a covering sieve, see Lemma 9.40.2. Suppose V ∈ Ob(C), and α : V → U in S(V ). Then we have S ×U V = hV , see Lemma 9.40.5. Thus the restrictions of ϕi via V → U correspond to sections si,V,α of F over V . The assumption is that there exist a covering sieve SV,α of V such that si,V,α restrict to the same element of MorPSh(C) (SV,α , F). Consider the sieve S 00 on U defined by the rule (f : T → U ) ∈ S 00 (T ) ⇔ (9.42.3.1)

∃ V, α : V → U, α ∈ S(V ), ∃ g : T → V, g ∈ SV,α (T ), f =α◦g

By axiom (2) of a topology we see that S 00 is a covering sieve on U . By construction we see that ϕ1 and ϕ2 restrict to the same element of MorPSh(C) (S 00 , LF) as desired. We sketch the proof of (2). Assume that F is a separated presheaf of sets on C with respect to the topology J. Let S be a covering sieve of the object U of C. Suppose that ϕ ∈ MorC (S, LF). We have to find an element s ∈ LF(U ) restricting to ϕ. Suppose V ∈ Ob(C), and α : V → U in S(V ). The value ϕ(α) ∈ LF(V ) is given by a covering sieve SV,α of V and a morphism of presheaves ϕV,α : SV,α → F. As in the proof above, define a covering sieve S 00 on U by Equation (9.42.3.1). We define ϕ00 : S 00 −→ F by the following simple rule: For every f : T → U , f ∈ S 00 (T ) choose V, α, g as in Equation (9.42.3.1). Then set ϕ00 (f ) = ϕV,α (g). We claim this is independent of the choice of V, α, g. Consider a second such choiceV 0 , α0 , g 0 . The restrictions of ϕV,α and ϕV 0 ,α0 to the intersection of the following covering sieves on T (SV,α ×V,g T ) ∩ (SV 0 ,α0 ×V 0 ,g0 T ) agree. Namely, these restrictions both correspond to the restriction of ϕ to T (via f ) and the desired equality follows because F is separated. Denote the common restriction ψ. The independence of choice follows because ϕV,α (g) = ψ(idT ) = ϕV 0 ,α0 (g 0 ). OK, so now ϕ00 gives an element s ∈ LF(U ). We leave it to the reader to check that s restricts to ϕ. 

9.43. TOPOLOGIES AND SHEAVES

695

Definition 9.42.4. Let C be a category endowed with a topology J. Let F be a presheaf of sets on C. The sheaf F # := LLF together with the canonical map F → F # is called the sheaf associated to F. Proposition 9.42.5. Let C be a category endowed with a topology. Let F be a presheaf of sets on C. The canonical map F → F # has the following universal property: For any map F → G, where G is a sheaf of sets, there is a unique map F # → G such that F → F # → G equals the given map. Proof. Same as the proof of Proposition 9.10.12.



9.43. Topologies and sheaves Lemma 9.43.1. Let C be a category endowed with a topology J. Let U be an object of C. Let S be a sieve on U . The following are equivalent (1) The sieve S is a covering sieve. (2) The sheafifcation S # → h# U of the map S → hU is an isomorphism. Proof. First we make a couple of general remarks. We will use that S # = LLS, # # and h# U = LLhU . In particular, by Lemma 9.42.1, we see that S → hU is injective. Note that idU ∈ hU (U ). Hence it gives rise to sections of LhU and h# U = LLhU over U which we will also denote idU . Suppose S is a covering sieve. It clearly suffices to find a morphism hU → S # such that the composition hU → h# U is the canonical map. To find such a map it suffices to find a section s ∈ S # (U ) wich restricts to idU . But since S is a covering sieve, the element idS ∈ MorPSh(C) (S, S) gives rise to a section of LS over U which restricts to idU in LhU . Hence we win. # Suppose that S # → h# U is an isomorphism. Let 1 ∈ S (U ) be the element corre# # sponding to idU in hU (U ). Because S = LLS there exists a covering sieve S 0 on U such that 1 comes from a

ϕ ∈ MorPSh(C) (S 0 , LS). This in turn means that for every α : V → U , α ∈ S 0 (V ) there exists a covering sieve SV,α on V such that ϕ(idV ) corresponds to a morphism of presheaves SV,α → S. In other words SV,α is contained in S ×U V . By the second axiom of a topology we see that S is a covering sieve.  Theorem 9.43.2. Let C be a category. Let J, J 0 be topologies on C. The following are equivalent (1) J = J 0 , (2) sheaves for the topology J are the same as sheaves for the topology J 0 . Proof. It is a tautology that if J = J 0 then the notions of sheaves are the same. Conversely, Lemma 9.43.1 characterizes covering sieves in terms of the sheafification functor. But the sheafification functor PSh(C) → Sh(C, J) is the right adjoint of the inclusion functor Sh(C, J) → PSh(C). Hence if the subcategories Sh(C, J) and Sh(C, J 0 ) are the same, then the sheafification functors are the same and hence the collections of covering sieves are the same.  Lemma 9.43.3. Assumption and notation as in Theorem 9.43.2. Then J ⊂ J 0 if and only if every sheaf for the topology J 0 is a sheaf for the topology J.

696

9. SITES AND SHEAVES

Proof. One direction is clear. For the other direction suppose that Sh(C, J 0 ) ⊂ Sh(C, J). By formal nonsense this implies that if F is a presheaf of sets, and F → F # , resp. F → F #,0 is the sheafification wrt J, resp. J 0 then there is a canonical map F # → F #,0 such that F → F # → F #,0 equals the canonical map F → F #,0 . Of course, F # → F #,0 identifies the second sheaf as the sheafification of the first with respect to the topology J 0 . Apply this to the map S → hU of Lemma 9.43.1. We get a commutative diagram S

/ S#

/ S #,0

 hU

 / h#

 / h#,0 U

U

And clearly, if S is a covering sieve for the topology J then the middle vertical map is an isomorphism (by the lemma) and we conclude that the right vertical map is an isomorphism as it is the sheafification of the one in the middle wrt J 0 . By the lemma again we conclude that S is a covering sieve for J 0 as well.  9.44. Topologies and continuous functors Explain how a continous functor gives an adjoint pair of functors on sheaves. 9.45. Points and topologies Recall from Section 9.28 that given a functor p = u : C → Sets we can define a stalk functor PSh(C) −→ Sets, F 7−→ Fp . Definition 9.45.1. Let C be a category. Let J be a topology on C. A point p of the topology is given by a functor u : C → Sets such that (1) For every covering sieve S on U the map Sp → (hU )p is surjective. (2) The stalk functor Sh(C) → Sets, F → Fp is exact. 9.46. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites

(17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules

9.46. OTHER CHAPTERS

(33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53)

More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces

(54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

697

More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 10

Homological Algebra 10.1. Introduction Basic homological algebra will be explained in this document. We add as needed in the other parts, since there is clearly an infinite amount of this stuff around. A reference is [Mac63]. 10.2. Basic notions The following notions are considered basic and will not be defined, and or proved. This does not mean they are all necessarily easy or well known. (1) Nothing yet. 10.3. Abelian categories An abelian category will be a category satisfying just enough axioms so the snake lemma holds. Definition 10.3.1. A category A is called preadditive if each morphism set MorA (x, y) is endowed with the structure of an abelian group such that the compositions Mor(x, y) × Mor(y, z) −→ Mor(x, z) are bilinear. A functor F : A → B of preadditive categories is called additive if and only if F : Mor(x, y) → Mor(F (x), F (y)) is a homomorphism of abelian groups for all x, y ∈ Ob(A). In particular for every x, y there exists at least one morphism x → y, namely the zero map. Lemma 10.3.2. Let A be a preadditive category. Let x be an object of A. The following are equivalent (1) x is an initial object, (2) x is a final object, and (3) idx = 0 in MorA (x, x). Furthermore, if such an object 0 exists, then a morphism α : x → y factors through 0 if and only if α = 0. Proof. Omitted.



Definition 10.3.3. In a preadditive category A we call zero object, and we denote it 0 any final and initial object as in Lemma 10.3.2 above. Lemma 10.3.4. Let A be a preadditive category. Let x, y ∈ Ob(A). ` ` If the product x × y exists, then so does the coproduct x y. If the coproduct x y exists, then ` so does the product x × y. In this case also x y ∼ = x × y. 699

700

10. HOMOLOGICAL ALGEBRA

Proof. Suppose that z = x × y with projections p : z → x and q : z → y. Denote i : x → z the morphism corresponding to (1, 0). Denote j : y → z the morphism corresponding to (0, 1). Thus we have the commutative diagram p

i

j

y

/x ?

1

x

?z 1

q

/ y

where the diagonal compositions are zero. It follows that i ◦ p + j ◦ q : z → z is the identity since it is a morphism which upon composing with p gives p and upon composing with q gives q. Suppose given morphisms a : x → w and b : y → w. Then we can form the map a ◦ p + b ◦ q : z → w. In this ` way we get a bijection Mor(z, w) = Mor(x, w) × Mor(y, w) which show that z = x y. ` We leave it to the reader to construct the morphisms p, q given a coproduct x y instead of a product.  Definition 10.3.5. Given a pair of objects x, y in a preadditive category A we call direct sum, and we denote it x ⊕ y the product x × y endowed with the morphisms i, j, p, q as in Lemma 10.3.4 above. Remark 10.3.6. Note that the proof of Lemma 10.3.4 shows that given p and q the morphisms i, j are uniquely determined by the rules p ◦ i = idx , q ◦ j = idy , p ◦ j = 0, q ◦ i = 0. Moreover, we automatically have i ◦ p + j ◦ q = idx⊕y . Similarly, given i, j the morphisms p and q are uniquely determined. Finally, given objects x, y, z and morphisms i : x → z, j : y → z, p : z → x and q : z → y such that p ◦ i = idx , q ◦ j = idy , p ◦ j = 0, q ◦ i = 0 and i ◦ p + j ◦ q = idz , then z is the direct sum of x and y with the four morphisms equal to i, j, p, q. Lemma 10.3.7. Let A, B be preadditive categories. Let F : A → B be an additive functor. Then F transforms direct sums to direct sums and zero to zero. Proof. Suppose F is additive. A direct sum z of x and y is characterized by having morphisms i : x → z, j : y → z, p : z → x and q : z → y such that p ◦ i = idx , q ◦ j = idy , p ◦ j = 0, q ◦ i = 0 and i ◦ p + j ◦ q = idz , according to Remark 10.3.6. Clearly F (x), F (y), F (z) and the morphisms F (i), F (j), F (p), F (q) satisfy exactly the same relations (by additivity) and we see that F (z) is a direct sum of F (x) and F (y).  Definition 10.3.8. A category A is called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums. Namely the empty product is a finite product and if it exists, then it is a final object. Definition 10.3.9. Let A be a preadditive category. Let f : x → y be a morphism. (1) A kernel of f is a morphism i : z → x such that (a) f ◦ i = 0 and (b) for any i0 : z 0 → x such that f ◦ i0 = 0 there exists a unique morphism g : z 0 → z such that i0 = i ◦ g. (2) If the kernel of f exists, then we denote this Ker(f ) → x.

10.3. ABELIAN CATEGORIES

701

(3) A cokernel of f is a morphism p : y → z such that (a) p ◦ f = 0 and (b) for any p0 : y → z 0 such that p0 ◦ f = 0 there exists a unique morphism g : z → z 0 such that p0 = g ◦ p. (4) If a cokernel of f exists we denote this y → Coker(f ). (5) If a kernel of f exists, then a coimage of f is a cokernel for the morphism Ker(f ) → x. (6) If a kernel and coimage exist then we denote this x → Coim(f ). (7) If a cokernel of f exists, then the image of f is a kernel of the morphism y → Coker(f ). (8) If a cokernel and image of f exist then we denote this Im(f ) → y. Lemma 10.3.10. Let f : x → y be a morphism in a preadditive category such that the kernel, cokernel, image and coimage all exist. Then f can be factored uniquely as x → Coim(f ) → Im(f ) → y. Proof. There is a canonical morphism Coim(f ) → y because Ker(f ) → x → y is zero. The composition Coim(f ) → y → Coker(f ) is zero, because it is the unique morphism which gives rise to the morphism x → y → Coker(f ) which is zero. Hence Coim(f ) → y factors uniquely through Im(f ) → y, which gives us the desired map.  Example 10.3.11. Let k be a field. Consider the category of filtered vector spaces over k. (See Definition 10.13.1.) Consider the filtered vector spaces (V, F ) and (W, F ) with V = W = k and   V if i < 0 W if i ≤ 0 i i F V = and F W = 0 if i ≥ 0 0 if i > 0 The map f : V → W corresponding to idk on the underlying vector spaces has trivial kernel and cokernel but is not an isomorphism. Note also that Coim(f ) = V and Im(f ) = W . This means that the category of filtered vector spaces over k is not abelian. Definition 10.3.12. A category A is abelian if it is additive, if all kernels and cokernels exist, and if the natural map Coim(f ) → Im(f ) is an isomorphism for all morphisms f of A. Lemma 10.3.13. Let A be a preadditive category. The additions on sets of morphisms make Aopp into a preadditive category. Furthermore, A is additive if and only if Aopp is additive, and A is abelian if and only if Aopp is abelian. Proof. Omitted.



Definition 10.3.14. Let f : x → y be a morphism in an abelian category. (1) We say f is injective if Ker(f ) = 0. (2) We say f is surjective if Coker(f ) = 0. If x → y is injective, then we say that x is a subobject of y and we use the notation x ⊂ y. If x → y is surjective, then we say that y is a quotient of x. Lemma 10.3.15. Let f : x → y be a morphism in an abelian category. Then (1) f is injective if and only if f is a monomorphism, and (2) f is surjective if and only if f is an epimorphism. Proof. Omitted.



702

10. HOMOLOGICAL ALGEBRA

In an abelian category, if x ⊂ y is a subobject, then we denote x/y = Coker(x → y). Lemma 10.3.16. Let A be an abelian category. All finite limits and finite colimits exist in A. Proof. To show that finite limits exist it suffices to show that finite products and equalizers exist, see Categories, Lemma 4.16.4. Finite products exist by definition and the equalizer of a, b : x → y is the kernel of a − b. The argument for finite colimits is similar but dual to this.  Example 10.3.17. Let A be an abelian category. Pushouts and fibre products in A have the following simple descriptions: (1) If a : x → y, b : z → y are morphisms in A, then we have the fibre product: x ×y z = Ker((a, −b) : x ⊕ z → y). (2) If a : y → x, b : y → z are morphisms in A, then we have the pushout: x qy z = Coker((a, −b) : y → x ⊕ z). Definition 10.3.18. Let A be an additive category. We say a sequence of morphisms ... → x → y → z → ... in A is a complex if the composition of any two (drawn) arrows is zero. If A is abelian then we say a sequence as above is exact at y if Im(x → y) = Ker(y → z). We say it is exact if it is exact at every object. A short exact sequence is an exact complex of the form 0 → A → B → C → 0. In the following lemma we asssume the reader knows what it means for a sequence of abelian groups to be exact. Lemma 10.3.19. Let A be an abelian category. Let 0 → M1 → M2 → M3 → 0 be a complex of A. (1) M1 → M2 → M3 → 0 is exact if and only if 0 → HomA (M3 , N ) → HomA (M2 , N ) → HomA (M1 , N ) is an exact sequence of abelian groups for all objects N of A, and (2) 0 → M1 → M2 → M3 is exact if and only if 0 → HomA (N, M1 ) → HomA (N, M2 ) → HomA (N, M1 ) is an exact sequence of abelian groups for all objects N of A. Proof. Omitted. Hint: See Algebra, Lemma 7.10.1.



Definition 10.3.20. Let A be an abelian category. Let i : A → B and q : B → C be morphisms of A such that 0 → A → B → C → 0 is a short exact sequence. We say the short exact sequence is split if there exist morphisms j : C → B and p : B → A such that (B, i, j, p, q) is the direct sum of A and C. Lemma 10.3.21. Let A be an abelian category. Let 0 → A → B → C → 0 be a short exact sequence. (1) Given a morphism s : C → B left inverse to B → C, there exists a unique π : B → A such that (s, π) splits the short exact sequence as in Definition 10.3.20.

10.3. ABELIAN CATEGORIES

703

(2) Given a morphism π : B → A right inverse to A → B, there exists a unique s : C → B such that (s, π) splits the short exact sequence as in Definition 10.3.20. Proof. Omitted.



Lemma 10.3.22. Let A be an abelian category. (1) If x → y is surjective, then for every z → y the projection z ×y z → z is surjective. (2) If x → y is injective, then for every x → z the morpism z → z qx y is injective. Proof. We prove (1). Assume a : x → y surjective and b : z → y arbitrary. Let c : z → t be a morphism of A such that z ×y z → z → t is zero. Note that 0 → x ×y z → x ⊕ z → y → 0 is a short exact sequence, use Example 10.3.17 and the fact that a is surjective. Consider the map c˜ = (0, c) : x ⊕ z → t. By assumption the composition x ×y z → x ⊕ z → t is zero hence we see that c˜ can be factored as x ⊕ z → y → t for some morphism c0 : y → t, see Lemma 10.3.19. This means that c = c0 ◦ b and that 0 = c0 ◦ a. As a is surjective we conclude that c0 = 0, hence c = 0 as desired. The proof of (2) is dual to the proof of (1) and is omitted.



Lemma 10.3.23. Let A be an abelian category. Suppose given a commutative diagram /z /0 /y x α

β

γ

/ u / v / w 0 with exact rows, then there is a canonical exact sequence Ker(α) → Ker(β) → Ker(γ) → Coker(α) → Coker(β) → Coker(γ) Moreover, if x → y is injective, then the first map is injective, and if v → w is surjective, then the last map is surjective. Proof. Omitted. Let us sketch the construction of the map δ : Ker(γ) → Coker(α) is. Let T ∈ Ob(A). Consider a morphism a : T → z with γ ◦ a = 0. In other words a maps T into Ker(γ). We have to construct δ ◦ a : T → Coker(α). Because y → z is surjective, the fibre product T 0 = T ×z y surjects onto T , see Lemma 10.3.22. Denote a0 : T 0 → y the second projection. Consider the morphism β ◦ a0 : T 0 → v. Composing this morphism with v → w gives the same morphism as the composition T 0 → T → z → w in other words, it gives the zero morphism. Because u → v is the kernel of v → w we conclude that a0 factors through a morphism a00 : T 0 → u. Note that the kernel T 00 of T 0 → T maps to zero under the composition T 0 → y → z, and hence maps into Im(x → y). Thus a00 |T 00 : T 00 → u maps into the image of α. We conclude that there exists a factorization /u T0 a00

pr1

 T

δ◦a

 / Coker(α)

704

10. HOMOLOGICAL ALGEBRA

which gives the desired map δ ◦ a : T → Coker(α).



Lemma 10.3.24. Let A be an abelian category. Let /x /y /z w α

γ

β

 w0

 / x0

δ

 / z0

 / y0

be a commutative diagram with exact rows. (1) If α, γ are surjective and δ is injective, then β is surjective. (2) If β, δ are injective and α is surjective, then γ is injective. Proof. Assume α, γ are surjective and δ is injective. We may replace w0 by Im(w0 → x0 ), i.e., we may assume that w0 → x0 is injective. We may replace z by Im(y → z), i.e., we may assume that y → z is surjective. Then we may apply Lemma 10.3.23 to /z /0 /y Ker(y → z)  / Ker(y 0 → z 0 )

0

 / y0

 / z0

to conclude that Ker(y → z) → Ker(y 0 → z 0 ) is surjective. Finally, we apply Lemma 10.3.23 to /x / Ker(y → z) /0 w  / w0

0

 / x0

 / Ker(y 0 → z 0 )

to conclude that x → x0 is surjective. This proves (1). The proof of (2) is dual to this.  Lemma 10.3.25. Let A be an abelian category. Let /w /x /y v α

 v0

β

 / w0

γ

 / x0

δ

 / y0

/z 

 / z0

be a commutative diagram with exact rows. If β, δ are isomorphisms,  is injective, and α is surjective then γ is an isomorphism. Proof. Immediate consequence of Lemma 10.3.24.



10.4. Extensions Definition 10.4.1. Let A be an abelian category. Let A, C ∈ Ob(A). An extension E of B by A is a short exact sequence 0 → A → E → B → 0. By abuse of language we often omit mention of the morphisms A → E and E → B, allthough they are definitively part of the structure of an extension.

10.4. EXTENSIONS

705

Definition 10.4.2. Let A be an abelian category. Let A, C ∈ Ob(A). The set of isomorphism classes of extensions of B by A is denoted ExtA (B, A). This is called the Ext-group. This definition works, because by our conventions A is a set, and hence ExtA (B, A) is a set. In any of the cases of “big” abelian categories listed in Categories, Remark 4.2.2. one can check by hand that ExtA (B, A) is a set as well. Also, we will see later that this is always the case when A has either enough projectives or enough injectives. Insert future reference here. Actually we can turn ExtA (−, −) into a functor Aopp × A −→ Sets,

(A, B) 7−→ ExtA (A, B)

as follows: (1) Given a morphism B 0 → B and an extension E of B by A we define E 0 = E ×B B 0 so that we have the following commutative diagram of short exact sequences 0

/A

/ E0

/ B0

/0

0

 /A

 /E

 /B

/0

The extension E 0 is called the pullback of E via B 0 → B. (2) Given a morphism A → A0 and an extension E of B by A we define ` 0 0 E = A A E so that we have the following commutative diagram of short exact sequences 0

/A

/E

/B

/0

0

 / A0

 / E0

 /B

/0

The extension E 0 is called the pushout of E via A → A0 . To see that this defines a functor as indicated above there are several things to verify. First of all functoriality in the variable B requires that (E ×B B 0 )×B 0 B 00 = E ×B B 00 which is a general property of fibre products. Dually one deals with functoriality in the variable A. Finally, given A → A0 and B 0 → B we have to show that a a A0 (E ×B B 0 ) ∼ E) ×B B 0 = (A0 A

0

0

A

0

as extensions of B by A . Recall that A A E is a quotient of A0 ⊕ E. Thus the right hand side is a quotient of A0 ⊕ E ×B B 0 , and it is straightforward to see that the kernel is exactly what you need in order to get the left hand side. `

Note that if E1 and E2 are extensions of B by A, then E1 ⊕ E2 is an extension of B ⊕ B by A ⊕ A. We pull back by the diagonal map B → B ⊕ B and we push out

706

10. HOMOLOGICAL ALGEBRA

by the sum map A ⊕ A → A to get an extension E1 + E2 of B by A. / A⊕A / E1 ⊕ E2 / B⊕B /0 0 P

0

 / E0 O

 /A O

 / B⊕B O

/0

∆

0

/A

/ E1 + E2

/B

/0

The extension E1 + E2 is called the Baer sum of the given extensions. Lemma 10.4.3. The construction (E1 , E2 ) 7→ E1 +E2 above defines a commutative group law on ExtA (B, A) which is functorial in both variables. Proof. Omitted.



Lemma 10.4.4. Let A be an abelian category. Let 0 → M1 → M2 → M3 → 0 be a short exact sequence in A. (1) There is a canonical six term exact sequence of abelian groups 0

/ HomA (M3 , N )

ExtA (M3 , N )

r

/ HomA (M2 , N )

/ HomA (M1 , N )

/ ExtA (M2 , N )

/ ExtA (M1 , N )

for all objects N of A, and (2) there is a canonical six term exact sequence of abelian groups 0

/ HomA (N, M1 )

ExtA (N, M1 )

r

/ HomA (N, M2 )

/ HomA (N, M1 )

/ ExtA (N, M2 )

/ ExtA (N, M1 )

for all objects N of A. Proof. Omitted. Hint: The boundary maps are defined using either the pushout or pullback of the given short exact sequence.  10.5. Additive functors Recall that we defined, in Categories, Definition 4.21.1 the notion of a “right exact”, “left exact” and “exact” functor in the setting of a functor between categories that have finite (co)limits. Thus this applies in particular to functors between abelian categories. Lemma 10.5.1. Let A and B be abelian categories. Let F : A → B be a functor. (1) If F is either left or right exact, then it is additive. (2) If F is additive then it is left exact if and only if for every short exact sequence 0 → A → B → C → 0 the sequence 0 → F (A) → F (B) → F (C) is exact. (3) If F is additive then it is right exact if and only if for every short exact sequence 0 → A → B → C → 0 the sequence F (A) → F (B) → F (C) → 0 is exact.

10.5. ADDITIVE FUNCTORS

707

(4) If F is additive then it is exact if and only if for every short exact sequence 0 → A → B → C → 0 the sequence 0 → F (A) → F (B) → F (C) → 0 is exact. Proof. Let us first note that if F commutes with the empty limit or the empty colimit, then F (0) = 0. In particular F applied to the zero morphism is zero. We will use this below without mention. Suppose that F is left exact, i.e., commutes with finite limits. Then F (A × A) = F (A)×F (A) with projections F (p) and F (q). Hence F (A⊕A) = F (A)⊕F (A) with all four morphisms F (i), F (j), F (p), F (q) equal to their counterparts in B as they satisfy the same relations, see Remark 10.3.6. Then f = F (p + q) is a morphism f : F (A) ⊕ F (A) → F (A) such that f ◦ F (i) = F (p ◦ i + q ◦ i) = F (idA ) = idF (A) . And similarly f ◦ F (j) = idA . We conclude that F (p + q) = F (p) + F (q). For any pair of morphisms a, b : B → A the map g = F (i ◦ a + j ◦ b) : F (B) → F (A) ⊕ F (A) is a morphism such that F (p) ◦ g = F (p ◦ (i ◦ a + j ◦ b)) = F (a) and similarly F (q) ◦ g = F (b). Hence g = F (i) ◦ F (a) + F (j) ◦ F (b). The sum of a and b is the composition B

i◦a+j◦b

/ A⊕A

p+q

/ A.

Applying F we get F (B)

F (i)◦F (a)+F (j)◦F (b)

/ F (A) ⊕ F (A)

F (p)+F (q)

/ A.

where we used the expressions for f and g obtained above. Hence F is additive.1 Denote f : B → C a map from B to C. Exactness of 0 → A → B → C just means that A = Ker(f ). Clearly the kernel of f is the equalizer of the two maps f and 0 from B to C. Hence if F commutes with limits, then F (Ker(f )) = Ker(F (f )) which exactly means that 0 → F (A) → F (B) → F (C) is exact. Conversely, suppose that F is additive and transforms any short exact sequence 0 → A → B → C into an exact sequence 0 → F (A) → F (B) → F (C). Because it is additive it commutes with direct sums and hence finite products in A. To show it commutes with finite limits it therefore suffices to show that it commutes with equalizers. But equalizers in an abelian category are the same as the kernel of the difference map, hence it suffices to show that F commutes with taking kernels. Let f : A → B be a morphism. Factor f as A → I → B with f 0 : A → I surjective and i : I → B injective. (This is possible by the definition of an abelian category.) Then it is clear that Ker(f ) = Ker(f 0 ). Also 0 → Ker(f 0 ) → A → I → 0 and 0 → I → B → B/I → 0 are short exact. By the condition imposed on F we see that 0 → F (Ker(f 0 )) → F (A) → F (I) and 0 → F (I) → F (B) → F (B/I) are exact. Hence it is also the case that F (Ker(f 0 )) is the kernel of the map F (A) → F (B), and we win. The proof of (3) is similar to the proof of (2). Statement (4) is a combination of (2) and (3).  Lemma 10.5.2. Let A and B be abelian categories. Let F : A → B be an exact functor. For every pair of objects A, B of A the functor F induces an abelian group homomorphism ExtA (B, A) −→ ExtB (F (B), F (A)) 1I’m sure there is an infinitely slicker proof of this.

708

10. HOMOLOGICAL ALGEBRA

which maps the extension E to F (E). Proof. Omitted.



The following lemma is used in the proof that the category of abelian sheaves on a site is abelian, where the functor b is sheafification. Lemma 10.5.3. Let a : A → B and b : B → A be functors. Assume that (1) A, B are additive categories, a, b are additive functors, and a is right adjoint to b, (2) B is abelian and b is left exact, and (3) ba ∼ = idA . Then A is abelian. Proof. As B is abelian we see that all finite limits and colimits exist in B by Lemma 10.3.16. Since b is a left adjoint we see that b is also right exact and hence exact, see Categories, Lemma 4.22.4. Let ϕ : B1 → B2 be a morphism of B. In particular, if K = Ker(B1 → B2 ), then K is the equalizer of 0 and ϕ and hence bK is the equalizer of 0 and bϕ, hence bK is the kernel of bϕ. Similarly, if Q = Coker(B1 → B2 ), then Q is the coequalizer of 0 and ϕ and hence bQ is the coequalizer of 0 and bϕ, hence bQ is the cokernel of bϕ. Thus we see that every morphism of the form bϕ in A has a kernel and a cokernel. However, since ba ∼ = id we see that every morphism of A is of this form, and we conclude that kernels and cokernels exist in A. In fact, the argument shows that if ψ : A1 → A2 is a morphism then Ker(ψ) = bKer(aψ), and Coker(ψ) = bCoker(aψ). Now we still have to show that Coim(ψ) = Im(ψ). We do this as follows. First note that since A has kernels and cokernels it has all finite limits and colimits (see proof of Lemma 10.3.16). Hence we see by Categories, Lemma 4.22.4 that a is left exact and hence transforms kernels (=equalizers) into kernels. Coim(ψ) = Coker(Ker(ψ) → A1 )

by definition

= bCoker(a(Ker(ψ) → A1 ))

by formula above

= bCoker(Ker(aψ) → aA1 ))

a preserves kernels

= bCoim(aψ)

by definition B is abelian

= bIm(aψ) = bKer(aA2 → Coker(aψ)) = Ker(baA2 → bCoker(aψ)) = Ker(A2 → bCoker(aψ)) = Ker(A2 → Coker(ψ)) = Im(ψ)

by definition b preserves kernels ba = idA by formula above by definition

Thus the lemma holds.

 10.6. Localization

In this section we note how Gabriel-Zisman localization interacts with the additive structure on a category.

10.6. LOCALIZATION

709

Lemma 10.6.1. Let C be a preadditive category. Let S be a left or right multiplicative system. There exists a canonical preadditive structure on S −1 C such that the localization functor Q : C → S −1 C is additive. Proof. We will prove this in the case S is a left multiplicative system. The case where S is a right multiplicative system is dual. Suppose that X, Y are objects of C and that α, β : X → Y are morphisms in S −1 C. According to Categories, Lemma 4.24.3 we may represent these by pairs s−1 f, s−1 g with common denominator s. In this case we define α + β to be the equivalence class of s−1 (f + g). In the rest of the proof we show that this is well defined and that composition is bilinear. Once this is done it is clear that Q is an additive functor. Let us show construction above is well defined. An abstract way of saying this is that filtered colimits of abelian groups agree with filtered colimits of sets and to use Categories, Equation (4.24.5.1). We can work this out in a bit more detail as follows. Say s : Y → Y1 and f, g : X → Y1 . Suppose we have a second representation of α, β as (s0 )−1 f 0 , (s0 )−1 g 0 with s0 : Y → Y2 and f 0 , g 0 : X → Y2 . By Categories, Remark 4.24.5 we can find a morphism s3 : Y → Y3 and morphisms a1 : Y1 → Y3 , a2 : Y2 → Y3 such that a1 ◦ s = s3 = a2 ◦ s0 and also a1 ◦ f = a2 ◦ f 0 and a1 ◦ g = a2 ◦ g 0 . Hence we see that s−1 (f + g) is equivalent to −1 s−1 3 (a1 ◦ (f + g)) = s3 (a1 ◦ f + a2 ◦ g) 0 0 = s−1 3 (a2 ◦ f + a2 ◦ g ) 0 0 = s−1 3 (a2 ◦ (f + g ))

which is equivalent to (s0 )−1 (f 0 + g 0 ). Fix s : Y → Y 0 and f, g : X → Y 0 with α = s−1 f and β = s−1 g as morphisms X → Y in S −1 C. To show that composition is bilinear first consider the case of a morphism γ : Y → Z in S −1 C. Say γ = t−1 h for some h : Y → Z 0 and t : Z → Z 0 in S. Using LMS2 we choose morphisms a : Y 0 → Z 00 and t0 : Z 0 → Z 00 in S such that a ◦ s = t0 ◦ h. Picture Z t

Y

h

 / Z0

a

 / Z 00

s

X

f,g

 / Y0

t

Then γ ◦ α = (t0 ◦ t)−1 (a ◦ f ) and γ ◦ β = (t0 ◦ t)−1 (a ◦ g). Hence we see that γ ◦ (α + β) is represented by (t0 ◦ t)−1 (a ◦ (f + g)) = (t0 ◦ t)−1 (a ◦ f + a ◦ g) which represents γ ◦ α + γ ◦ β. Finally, assume that δ : W → X is another morphism of S −1 C. Say δ = r−1 i for some i : W → X 0 and r : X → X 0 in S. We claim that we can find a morphism s : Y 0 → Y 00 in S and morphisms a00 , b00 : X 0 → Y 00 such that the following diagram

710

10. HOMOLOGICAL ALGEBRA

commutes Y s

X s

f,g,f +g

 / Y0

s0

 a00 ,b00 ,a00 +b00  i / / Y 00 W X0 Namely, using LMS2 we can first choose s1 : Y 0 → Y1 , s2 : Y 0 → Y2 in S and a : X 0 → Y1 , b : X 0 → Y2 such that a ◦ s = s1 ◦ f and b ◦ s = s2 ◦ f . Then using that the category Y 0 /S is filtered (see Categories, Remark 4.24.5), we can find a s0 : Y 0 → Y 00 and morphisms a0 : Y1 → Y 00 , b0 : Y2 → Y 00 such that s0 = a0 ◦ s1 and s0 = b0 ◦ s2 . Setting a00 = a0 ◦ a and b00 = b0 ◦ b works. At this point we see that the compositions α ◦ δ and β ◦ δ are represented by (s0 ◦ s)−1 a00 and (s0 ◦ s)−1 b00 . Hence α ◦ δ + β ◦ δ is represented by (s0 ◦ s)−1 (a00 + b00 ) which by the diagram again is a representative of (α + β) ◦ δ.  Lemma 10.6.2. Let C be an additive category. Let S be a left or right multiplicative system. Then S −1 C is an additive category and the localization functor Q : C → S −1 C is additive. Proof. By Lemma 10.6.1 we see that S −1 C is preadditive and that Q is additive. Recall that the functor Q commutes with finite colimits (resp. finite limits), see Categories, Lemmas 4.24.7 and 4.24.14. We conclude that S −1 C has a zero object and direct sums, see Lemmas 10.3.2 and 10.3.4.  The following lemma describes the kernel (see Definition 10.7.5) of the localization functor in case we invert a multiplicative system. Lemma 10.6.3. Let C be an additive category. Let S be a multiplicative system compatible with the triangulated structure. Let X be an object of D. The following are equivalent (1) Q(X) = 0 in S −1 C, (2) there exists Y ∈ Ob(C) such that 0 : X → Y is an element of S, and (3) there exists Z ∈ Ob(C) such that 0 : Z → X is an element of S. Proof. If (2) holds we see that 0 = Q(0) : Q(X) → Q(Y ) is an isomorphism. In the additive category S −1 C this implies that Q(X) = 0. Hence (2) ⇒ (1). Similarly, (3) ⇒ (1). Suppose that Q(X) = 0. This implies that the morphism f : 0 → X is transformed into an isomorphism in S −1 C. Hence by Categories, Lemma 4.24.18 there exists a morphism g : Z → 0 such that f g ∈ S. This proves (1) ⇒ (3). Similarly, (1) ⇒ (2).  Lemma 10.6.4. Let A be an abelian category. (1) If S is a left multiplicative system, then the category S −1 A has cokernels and the functor Q : A → S −1 A commutes with them. (2) If S is a right multiplicative system, then the category S −1 A has kernels and the functor Q : A → S −1 A commutes with them. (3) If S is a multiplicative system, then the category S −1 A is abelian and the functor Q : A → S −1 A is exact.

10.7. SERRE SUBCATEGORIES

711

Proof. Assume S is a left multiplicative system. Let a : X → Y be a morphism of S −1 A. Then a = s−1 f for some s : Y → Y 0 in S and f : X → Y 0 . Since Q(s) is an isomorphism we see that the existence of Coker(a : X → Y ) is equivalent to the existence of Coker(Q(f ) : X → Y 0 ). Since Coker(Q(f )) is the coequalizer of 0 and Q(f ) we see that Coker(Q(f )) is represented by Q(Coker(f )) by Categories, Lemma 4.24.7. This proves (1). Part (2) is dual to part (1). If S is a multiplicative system, then S is both a left and a right multiplicative system. Thus we see that S −1 A has kernels and cokernels and Q commutes with kernels and cokernels. To finish the proof of (3) we have to show that Coim = Im in S −1 A. Again using that any arrow in S −1 A is isomorphic to an arrow Q(f ) we see that the result follows from the result for A.  10.7. Serre subcategories In [Ser53, Chapter I, Section 1] a notion of a “class” of abelian groups is defined. This notion has been extended to abelian categories by many authors (in slightly different ways). We will use the following variant which is virtually identical to Serre’s original definition. Definition 10.7.1. Let A be an abelian category. (1) A Serre subcategory of A is a nonempty full subcategory C of A such that given an exact sequence A→B→C with A, C ∈ Ob(C), then also B ∈ Ob(C). (2) A weak Serre subcategory of A is a nonempty full subcategory C of A such that given an exact sequence A0 → A1 → A2 → A3 → A4 with A0 , A1 , A3 , A4 in C, then also A2 in C. In some references the second notion is called a “thick” subcategory and in other references the first notion is called a “thick” subcategory. However, it seems that the notion of a Serre subcagegory is universally accepted to be the one defined above. Note that in both cases the category C is abelian and that the inclusion functor C → A is a fully faithful exact functor. Let’s characterize these types of subcategories in more detail. Lemma 10.7.2. Let A be an abelian category. Let C be a subcategory of A. Then C is a Serre subcategory if and only if the following conditions are satisfied: (1) (2) (3) (4)

0 ∈ Ob(C), C is a strictly full subcategory of A, any subobject or quotient of an object of C is an object of C, if A ∈ Ob(A) is an extension of objects of C then also A ∈ Ob(C).

Moreover, a Serre subcategory is an abelian category and the inclusion functor is exact. Proof. Omitted.



712

10. HOMOLOGICAL ALGEBRA

Lemma 10.7.3. Let A be an abelian category. Let C be a subcategory of A. Then C is a weak Serre subcategory if and only if the following conditions are satisfied: (1) 0 ∈ Ob(C), (2) C is a strictly full subcategory of A, (3) kernels and cokernels in A of morphisms between objects of C are in C, (4) if A ∈ Ob(A) is an extension of objects of C then also A ∈ Ob(C). Moreover, a weak Serre subcategory is an abelian category and the inclusion functor is exact. Proof. Omitted.



Lemma 10.7.4. Let A, B be abelian categories. Let F : A → B be an exact functor. Then the full subcategory of objects C of A such that F (C) = 0 forms a Serre subcategory of A. Proof. Omitted.



Definition 10.7.5. Let A, B be abelian categories. Let F : A → B be an exact functor. Then the full subcategory of objects C of A such that F (C) = 0 is called the kernel of the functor F , and is sometimes denoted Ker(F ). Lemma 10.7.6. Let A be an abelian category. Let C ⊂ A be a Serre subcategory. There exists an abelian category A/C and an exact functor F : A −→ A/C which is essentially surjective and whose kernel is C. The category A/C and the functor F are characterized by the following universal property: For any exact functor G : A → B such that C ⊂ Ker(G) there exists a factorization G = H ◦ F for a unique exact functor H : A/C → B. Proof. Consider the set of arrows of A defined by the following formula S = {f ∈ Arrows(A) | Ker(f ), Coker(f ) ∈ Ob(C)}. We claim that S is a multiplicative system. To prove this we have to check MS1, MS2, MS3, see Categories, Definition 4.24.1. It is clear that identities are elements of S. Suppose that f : A → B and g : B → C are elements of S. There are exact sequences 0 → Ker(f ) → Ker(gf ) → Ker(g) → 0 0 → Coker(f ) → Coker(gf ) → Coker(g) → 0 Hence it follows that gf ∈ S. This proves MS1. Consider a solid diagram A

g

s

t

 C

/B

f

 / C qA B

with t ∈ S. Set W = C qA B = Coker((1, −1) : A → C ⊕B). Then Ker(t) → Ker(s) is surjective and Coker(t) → Coker(s) is an isomorphism. Hence s is an element of S. This proves LMS2 and the proof of RMS2 is dual. Finally, consider morphisms f, g : B → C and a morphism s : A → B in S such that f ◦ s = g ◦ s. This means that (f − g) ◦ s = 0. In turn this means

10.8. K-GROUPS

713

that I = Im(f − g) ⊂ C is a quotient of Coker(s) hence an object of C. Thus t : C → C 0 = C/I is an element of S such that t ◦ (f − g) = 0, i.e., such that t ◦ f = t ◦ g. This proves LMS3 and the proof of RMS3 is dual. Having proved that S is a multiplicative system we set A/C = S −1 A, and we set F equal to the localization functor Q. By Lemma 10.6.4 the category A/C is abelian and F is exact. If X is in the kernel of F = Q, then by Lemma 10.6.3 we see that 0 : X → Z is an element of S and hence X is an object of C, i.e., the kernel of F is C. Finally, if G is as in the statement of the lemma, then G turns every element of S into an isomorphism. Hence we obtain the functor H : A/C → B from the universal property of localization, see Categories, Lemma 4.24.6.  Lemma 10.7.7. Let A, B be abelian categories. Let F : A → B be an exact functor. Let C = Ker(F ). Then the induced functor F : A/C → B is faithful. Proof. This is true because the kernel of F is zero by construction. Namely, if f : X → Y is a morphism in A/C such that F (f ) = 0, then Ker(f ) → X and Y → Coker(f ) are transformed into isomorphisms by F , hence are isomorphisms by the remark on the kernel of F . Thus f = 0.  10.8. K-groups Definition 10.8.1. Let A be an abelian category. We denote K0 (A) the zeroth K-group of A. It is the abelian group constructed as follows. Take the free abelian group on the objects on A and for every short exact sequence 0 → A → B → C → 0 impose the relation [B] − [A] − [C] = 0. Another way to say this is that there is a presentation M M Z[A → B → C] −→ Z[A] −→ K0 (A) −→ 0 A→B→C ses

A∈Ob(A)

with [A → B → C] 7→ [B] − [A] − [C] of K0 (A). The short exact sequence 0 → 0 → 0 → 0 → 0 leads to the relation [0] = 0 in K0 (A). There are no settheoretical issues as all of our categories are “small” if not mentioned otherwise. Some examples of K-groups for categories of modules over rings where computed in Algebra, Section 7.52. Lemma 10.8.2. Let F : A → B be an exact functor between abelian categories. Then F induces a homomorphism of K-groups K0 (F ) : K0 (A) → K0 (B) by simply setting K0 (F )([A]) = [F (A)]. Proof. Proves itself.



Suppose we are given an object M of an abelian category A and a complex of the form (10.8.2.1)

...

/M

ϕ

/M

ψ

/M

ϕ

/M

/ ...

In this situation we define H 0 (M, ϕ, ψ) = Ker(ψ)/Im(ϕ),

and H 1 (M, ϕ, ψ) = Ker(ϕ)/Im(ψ).

Lemma 10.8.3. Let A be an abelian category. Let C ⊂ A be a Serre subcategory and set B = A/C.

714

10. HOMOLOGICAL ALGEBRA

(1) The exact functors C → A and A → B induce an exact sequence K0 (C) → K0 (A) → K0 (B) → 0 of K-groups, and (2) the kernel of K0 (C) → K0 (A) is equal to the collection of elements of the form [H 0 (M, ϕ, ψ)] − [H 1 (M, ϕ, ψ)] where (M, ϕ, ψ) is a complex as in (10.8.2.1) with the property that it becomes exact in B; in other words that H 0 (M, ϕ, ψ) and H 1 (M, ϕ, ψ) are objects of C. Proof. We omit the proof of (1). The proof of (2) is in a sense completely combinatorial. First we remark that any class of the type [H 0 (M, ϕ, ψ)] − [H 1 (M, ϕ, ψ)] is zero in K0 (A) by the following calculation 0 = [M ] − [M ] = [Ker(ϕ)] + [Im(ϕ)] − [Ker(ψ)] − [Im(ψ)] = [Ker(ϕ)/Im(ψ)] − [Ker(ψ)/Im(ϕ)] = [H 1 (M, ϕ, ψ)] − [H 0 (M, ϕ, ψ)] as desired. Hence it suffices to show that any element in the kernel of K0 (C) → K0 (A) is of this form. Any element x in K0 (C) can be represented as the difference x = [P ] − [Q] of two objects of C (fun exercise). Suppose that this element maps to zero in K0 (A). This means that there exist ` (1) a finite set I = I + I − , (2) for each i ∈ I a short exact sequence 0 → Ai → Bi → Ci → 0 in the abelian category A such that X X [P ] − [Q] = ([Bi ] − [Ai ] − [Ci ]) − +

i∈I −

i∈I

([Bi ] − [Ai ] − [Ci ])

in the free abelian group on the objects of A. We can rewrite this as X X X X [P ] + ([Ai ] + [Ci ]) + [Bi ] = [Q] + ([Ai ] + [Ci ]) + + − − i∈I

i∈I

i∈I

i∈I +

[Bi ].

Since the right and left hand side should contain the same objects of A counted with multiplicity, this means there should be a bijection τ between the terms which occur above. Set a a T + = {p} {a, c} × I + {b} × I − and Set T = T +

`

T−

a a T − = {q} {a, c} × I − {b} × I + . ` = {p, q} {a, b, c} × I. For t ∈ T define  P if t=p     t=q  Q if O(t) = Ai if t = (a, i)   Bi if t = (b, i)    Ci if t = (c, i)

10.9. COHOMOLOGICAL DELTA-FUNCTORS

715

Hence we can view τ : T + → T − as a bijection such that O(t) = O(τ (t)) for all + + t ∈ T + . Let t− be the unique element such that τ (t+ 0 = τ (p) and let t0 ∈ T 0 ) = q. Consider the object M O(t) M+ = + t∈T

By using τ we see that it is equal to the object M M− = O(t) − t∈T

Consider the map ϕ : M + −→ M − which on the summand O(t) = Ai corresponding to t = (a, i), i ∈ I + uses the map Ai → Bi into the summand O((b, i)) = Bi of M − and on the summand O(t) = Bi corresponding to (b, i), i ∈ I − uses the map Bi → Ci into the summand O((c, i)) = Ci of M − . The map is zero on the summands corresponding to p and (c, i), i ∈ I + . Similarly, consider the map ψ : M − −→ M + which on the summand O(t) = Ai corresponding to t = (a, i), i ∈ I − uses the map Ai → Bi into the summand O((b, i)) = Bi of M + and on the summand O(t) = Bi corresponding to (b, i), i ∈ I + uses the map Bi → Ci into the summand O((c, i)) = Ci of M + . The map is zero on the summands corresponding to q and (c, i), i ∈ I − . Note that the kernel of ϕ is equal to the direct sum of the summand P and the summands O((c, i)) = Ci , i ∈ I + and the subobjects Ai inside the summands O((b, i)) = Bi , i ∈ I − . The image of ψ is equal to the direct sum of the summands O((c, i)) = Ci , i ∈ I + and the subobjects Ai inside the summands O((b, i)) = Bi , i ∈ I − . In other words we see that P ∼ = Ker(ϕ)/Im(ψ). In exactly the same way we see that Q∼ = Ker(ψ)/Im(ϕ). Since as we remarked above the existence of the bijection τ shows that M + = M − we see that the lemma follows.  10.9. Cohomological delta-functors Definition 10.9.1. Let A, B be abelian categories. A cohomological δ-functor or simply a δ-functor from A to B is given by the following data: (1) a collection F n : A → B, n ≥ 0 of additive functors, and (2) for every short exact sequence 0 → A → B → C → 0 of A a collection δA→B→C : F n (C) → F n+1 (A), n ≥ 0 of morphisms of B. These data are assumed to satisfy the following axioms

716

10. HOMOLOGICAL ALGEBRA

(1) for every short exact sequence as above the sequence / F 0 (A)

0

/ F 0 (B) u

2

u

δA→B→C

1

/ F 1 (B)

F (A)

F (A)

/ F 0 (C)

/ F 1 (C)

δA→B→C

/ F 2 (B)

/ ...

is exact, and (2) for every morphism (A → B → C) → (A0 → B 0 → C 0 ) of short exact sequences of A the diagrams F n (C)  F n (C 0 )

δA→B→C

δA0 →B 0 →C 0

/ F n+1 (A)  / F n+1 (A0 )

are commutative. Note that this in particular implies that F 0 is left exact. Definition 10.9.2. Let A, B be abelian categories. Let (F n , δF ) and (Gn , δG ) be δ-functors from A to B. A morphism of δ-functors from F to G is a collection of transformation of functors tn : F n → Gn , n ≥ 0 such that for every short exact sequence 0 → A → B → C → 0 of A the diagrams F n (C)

δF,A→B→C

tn

 Gn (C)

δG,A→B→C

/ F n+1 (A) 

tn+1

/ Gn+1 (A)

are commutative. Definition 10.9.3. Let A, B be abelian categories. Let F = (F n , δF ) be a δfunctor from A to B. We say F is a universal δ-functor if an only if for every δ-functor G = (Gn , δG ) and any morphism of functors t : F 0 → G0 there exists a unique morphism of δ-functors {tn }n≥0 : F → G such that t = t0 . Lemma 10.9.4. Let A, B be abelian categories. Let F = (F n , δF ) be a δ-functor from A to B. Suppose that for every n > 0 and any A ∈ Ob(A) there exists an injective morphism u : A → B (depending on A and n) such that F n (u) : F n (A) → F n (B) is zero. Then F is a universal δ-functor. Proof. Let G = (Gn , δG ) be a δ-functor from A to B and let t : F 0 → G0 be a morphism of functors. We have to show there exists a unique morphism of δfunctors {tn }n≥0 : F → G such that t = t0 . We construct tn by induction on n. For n = 0 we set t0 = t. Suppose we have already constructed a unique sequence of transformation of functors ti for i ≤ n compatible with the maps δ in degrees ≤ n. Let A ∈ Ob(A). By assumption we may choose a embedding u : A → B such that F n+1 (u) = 0. Let C = B/u(A). The long exact cohomology sequence for

10.10. COMPLEXES

717

the short exact sequence 0 → A → B → C → 0 and the δ-functor F gives that F n+1 (A) = Coker(F n (B) → F n (C)) by our choice of u. Since we have already defined tn we can set tn+1 : F n+1 (A) → Gn+1 (A) A equal to the unique map such that Coker(F n (B) → F n (C)) δF,A→B→C



/ Coker(Gn (B) → Gn (C))

tn



tn+1 A

F n+1 (A)

δG,A→B→C

/ Gn+1 (A)

commutes. This is clearly uniquely determined by the requirements imposed. We omit the verification that this defines a transformation of functors.  Lemma 10.9.5. Let A, B be abelian categories. Let F : A → B be a functor. If there exists a universal δ-functor (F n , δF ) from A to B with F 0 = F , then it is determined up to unique isomorphism of δ-functors. Proof. Immediate from the definitions.



10.10. Complexes Of course the notions of a chain complex and a cochain complex are dual and you only have to read one of the two parts of this section. So pick the one you like. (Actually, this doesn’t quite work right since the conventions on numbering things are not adapted to an easy transition between chain and cochain complexes.) A chain complex A• in an additive category A is a complex dn+1

d

n An−1 → . . . . . . → An+1 −−−→ An −→

of A. In other words, we are given an object Ai of A for all i ∈ Z and for all i ∈ Z a morphism di : Ai → Ai−1 such that di−1 ◦ di = 0 for all i. A morphism of chain complexes f : A• → B• is given by a family of morphisms fi : Ai → Bi such that all the diagrams / Ai−1 Ai di

fi

 Bi

di



fi−1

/ Bi−1

commute. The category of chain complexes of A is denoted Ch(A). The full subcategory consisting of objects of the form . . . → A2 → A1 → A0 → 0 → 0 → . . . is denoted Ch≥0 (A). In other words, a chain complex A• belongs to Ch≥0 (A) if and only if Ai = 0 for all i < 0. A homotopy h between a pair of morphisms of chain complexes f, g : A• → B• is is a collection of morphisms hi : Ai → Bi+1 such that we have fi − gi = di+1 ◦ hi + hi−1 ◦ di for all i. Clearly, the notions of chain complex, morphism of chain complexes, and homotopies between morphisms of chain complexes makes sense even in a preadditive category.

718

10. HOMOLOGICAL ALGEBRA

Lemma 10.10.1. Let A be an additive category. Let f, g : B• → C• be morphisms of chain complexes. Suppose given morphisms of chain complexes a : A• → B• , and c : C• → D• . If {hi : Bi → Ci+1 } defines a homotopy between f and g, then {ci+1 ◦ hi ◦ ai } defines a homotopy between c ◦ f ◦ a and c ◦ g ◦ a. Proof. Omitted.



In particular this means that it makes sense to define the category of chain complexes with maps up to homotopy. We’ll return to this later. Definition 10.10.2. Let A be an additive category. We say a morphism a : A• → B• is a homotopy equivalence if there exists a morphism b : B• → A• such that there exists a homotopy between a ◦ b and idA and there exists a homotopy between b ◦ a and idB . If there exists such a morphism between A• and B• , then we say that A• and B• are homotopy equivalent. In other words, two complexes are homotopy equivalent if they become isomorphic in the category of complexes up to homotopy. Lemma 10.10.3. Let A be an abelian category. (1) The category of chain complexes in A is abelian. (2) A morphism of complexes f : A• → B• is injective if and only if each fn : An → Bn is injective. (3) A morphism of complexes f : A• → B• is surjective if and only if each fn : An → Bn is surjective. (4) A sequence of chain complexes f

g

A• − → B• − → C• is exact at B• if and only if each sequence fi

gi

Ai −→ Bi −→ Ci is exact at Bi . Proof. Omitted.



For any i ∈ Z the ith homology group of a chain complex A• in an abelian category is defined by the following formula Hi (A• ) = Ker(di )/Im(di+1 ). If f : A• → B• is a morphism of chain complexes of A then we get an induced morphism Hi (f ) : Hi (A• ) → Hi (B• ) because clearly fi (Ker(di : Ai → Ai−1 )) ⊂ Ker(di : Bi → Bi−1 ), and similarly for Im(di+1 ). Thus we obtain a functor Hi : Ch(A) −→ A. Definition 10.10.4. Let A be an abelian category. (1) A morphism of chain complexes f : A• → B• is called a quasi-isomorphism if the induced maps Hi (f ) : Hi (A• ) → Hi (B• ) is an isomorphism for all i ∈ Z. (2) A chain complex A• is called acyclic if all of its homology objects Hi (A• ) are zero. Lemma 10.10.5. Let A be an abelian category.

10.10. COMPLEXES

719

(1) If the maps f, g : A• → B• are homotopic, then the induced maps Hi (f ) and Hi (g) are equal. (2) If the map f : A• → B• is a homotopy equivalence, then f is a quisiisomorphism. Proof. Omitted.



Lemma 10.10.6. Let A be an abelian category. Suppose that 0 → A• → B • → C • → 0 is a short exact sequence of chain complexes of A. Then there is a canonical long exact homology sequence ...

Hi (A• )

s

Hi−1 (A• )

s

...

...

/ Hi (B• )

/ Hi (C• )

/ Hi−1 (B• )

/ Hi−1 (C• )

...

...

... s

Proof. Omitted. The maps come from the Snake Lemma 10.3.23 applied to the diagrams / Bi /Im(dB,i+1 )

Ai /Im(dA,i+1 ) 

0

dA,i



/ Ker(dA,i−1 )

/ Ci /Im(dC,i+1 )

dB,i

/ Ker(dB,i−1 )



/0

dC,i

/ Ker(dC,i−1 ) 

A cochain complex A• in an additive category A is a complex dn−1

dn

. . . → An−1 −−−→ An −→ An+1 → . . . of A. In other words, we are given an object Ai of A for all i ∈ Z and for all i ∈ Z a morphism di : Ai → Ai+1 such that di+1 ◦ di = 0 for all i. A morphism of cochain complexes f : A• → B • is given by a family of morphisms f i : Ai → B i such that all the diagrams / Ai+1 Ai di

f

i

f i+1

  di / Bi B i+1 commute. The category of cochain complexes of A is denoted CoCh(A). The full subcategory consisting of objects of the form . . . → 0 → 0 → A0 → A1 → A2 → . . . is denoted CoCh≥0 (A). In other words, a cochain complex A• belongs to the subcategory CoCh≥0 (A) if and only if Ai = 0 for all i < 0. A homotopy h between

720

10. HOMOLOGICAL ALGEBRA

a pair of morphisms of cochain complexes f, g : A• → B • is is a collection of morphisms hi : Ai → B i−1 such that we have f i − g i = di−1 ◦ hi + hi+1 ◦ di for all i. Clearly, the notions of cochain complex, morphism of cochain complexes, and homotopies between morphisms of cochain complexes makes sense even in a preadditive category. Lemma 10.10.7. Let A be an additive category. Let f, g : B • → C • be morphisms of cochain complexes. Suppose given morphisms of cochain complexes a : A• → B • , and c : C • → D• . If {hi : B i → C i−1 } defines a homotopy between f and g, then {ci−1 ◦ hi ◦ ai } defines a homotopy between c ◦ f ◦ a and c ◦ g ◦ a. Proof. Omitted.



In particular this means that it makes sense to define the category of cochain complexes with maps up to homotopy. We’ll return to this later. Definition 10.10.8. Let A be an additive category. We say a morphism a : A• → B • is a homotopy equivalence if there exists a morphism b : B • → A• such that there exists a homotopy between a ◦ b and idA and there exists a homotopy between b ◦ a and idB . If there exists such a morphism between A• and B • , then we say that A• and B • are homotopy equivalent. In other words, two complexes are homotopy equivalent if they become isomorphic in the category of complexes up to homotopy. Lemma 10.10.9. Let A be an abelian category. (1) The category of cochain complexes in A is abelian. (2) A morphism of cochain complexes f : A• → B • is injective if and only if each f n : An → B n is injective. (3) A morphism of cochain complexes f : A• → B • is surjective if and only if each f n : An → B n is surjective. (4) A sequence of cochain complexes f

g

A• − → B• − → C• is exact at B • if and only if each sequence fi

gi

Ai −→ B i −→ C i is exact at B i . Proof. Omitted.

 •

For any i ∈ Z the ith cohomology group of a cochain complex A is defined by the following formula H i (A• ) = Ker(di )/Im(di−1 ). If f : A• → B • is a morphism of cochain complexes of A then we get an induced morphism H i (f ) : H i (A• ) → H i (B • ) because clearly f i (Ker(di : Ai → Ai+1 )) ⊂ Ker(di : B i → B i+1 ), and similarly for Im(di−1 ). Thus we obtain a functor H i : CoCh(A) −→ A. Definition 10.10.10. Let A be an abelian category.

10.11. TRUNCATION OF COMPLEXES

721

(1) A morphism of cochain complexes f : A• → B • of A is called a quasiisomorphism if the induced maps H i (f ) : H i (A• ) → H i (B • ) is an isomorphism for all i ∈ Z. (2) A cochain complex A• is called acyclic if all of its cohomology objects H i (A• ) are zero. Lemma 10.10.11. Let A be an abelian category. (1) If the maps f, g : A• → B • are homotopic, then the induced maps H i (f ) and H i (g) are equal. (2) If f : A• → B • is a homotopy equivalence, then f is a quasi-isomorphism. Proof. Omitted.



Lemma 10.10.12. Let A be an abelian category. Suppose that 0 → A• → B • → C • → 0 is a short exact sequence of chain complexes of A. Then there is a canonical long exact homology sequence ...

H i (A• )

s

H i+1 (A• )

s

...

...

/ H i (B • )

/ H i (C • )

/ H i+1 (B • )

/ H i+1 (C • )

...

...

... s

Proof. Omitted. The maps come from the Snake Lemma 10.3.23 applied to the diagrams Ai /Im(di−1 A ) diA

 / Ker(di+1 ) A

0

/ B i /Im(di−1 ) B diB

 / Ker(di+1 ) B

/ C i /Im(di−1 ) C

/0

diC

 / Ker(di+1 ) C 

10.11. Truncation of complexes Let A be an abelian category. Let A• be a chain complex. There are several ways to truncate the complex A• . (1) The “stupid” truncation σ≤n is the the subcomplex σ≤n A• defined by the rule (σ≤n A• )i = 0 if i > n and (σ≤n A• )i = Ai if i ≤ n. In a picture σ≤n A•

...

/0

/ An

/ An−1

/ ...

 A•

...

 / An+1

 / An

 / An−1

/ ...

Note the property σ≤n A• /σ≤n−1 A• = An [−n].

722

10. HOMOLOGICAL ALGEBRA

(2) The “stupid” truncation σ≥n is the the quotient complex σ≥n A• defined by the rule (σ≥n A• )i = Ai if i ≥ n and (σ≥n A• )i = 0 if i < n. In a picture A•

...

/ An+1

/ An

/ An−1

/ ...

 σ≥n A•

...

 / An+1

 / An

 /0

/ ...

The map of complexes σ≥n A• → σ≥n+1 A• is surjective with kernel An [−n]. (3) The canonical truncation τ≥n A• is defined by the picture τ≥n A•

...

/ An+1

/ Ker(dn )

/0

/ ...

 A•

...

 / An+1

 / An

 / An−1

/ ...

Note that these complexes have the property that  Hi (A• ) if i ≥ n Hi (τ≥n A• ) = 0 if i < n (4) The canonical truncation τ≤n A• is defined by the picture A•

...

/ An+1

/ An

/ An−1

/ ...

 τ≤n A•

...

 /0

 / Coker(dn+1 )

 / An−1

/ ...

Note that these complexes have the property that  Hi (A• ) if i ≤ n Hi (τ≤n A• ) = 0 if i > n Let A be an abelian category. Let A• be a cochain complex. There are four ways to truncate the complex A• . (1) The “stupid” truncation σ≥n is the subcomplex σ≥n A• defined by the rule (σ≥n A• )i = 0 if i < n and (σ≥n A• )i = Ai if i ≥ n. In a picture σ≥n A•

...

/0

/ An

/ An+1

/ ...

 A•

...

 / An−1

 / An

 / An+1

/ ...

Note the property σ≥n A• /σ≥n+1 A• = An [−n]. (2) The “stupid” truncation σ≤n is the quotient complex σ≤n A• defined by the rule (σ≥n A• )i = 0 if i > n and (σ≥n A• )i = Ai if i ≤ n. In a picture A•

...

/ An−1

/ An

/ An+1

/ ...

 σ≤n A•

...

 / An−1

 / An

 /0

/ ...

The map of complexes σ≤n A• → σ≤n−1 A• is surjective with kernel An [−n].

10.12. HOMOTOPY AND THE SHIFT FUNCTOR

723

(3) The canonical truncation τ≤n A• is defined by the picture τ≤n A•

...

/ An−1

/ Ker(dn )

/0

/ ...

 A•

...

 / An−1

 / An

 / An+1

/ ...

Note that these complexes have the property that  i • H (A ) if i ≤ n i • H (τ≤n A ) = 0 if i > n (4) The canonical truncation τ≥n A• is defined by the picture A•

...

/ An−1

/ An

/ An+1

/ ...

 τ≥n A•

...

 /0

 / Coker(dn−1 )

 / An+1

/ ...

Note that these complexes have the property that  0 if i < n i • H (τ≤n A ) = H i (A• ) if i ≥ n 10.12. Homotopy and the shift functor It is an annoying feature that signs and indices have to be part of any discussion of homological algebra2. Definition 10.12.1. Let A be an additive category. Let A• be a chain complex with boundary maps dA,n : An → An−1 . For any k ∈ Z we define the k-shifted chain complex A[k]• as follows: (1) we set A[k]n = An+k , and (2) we set dA[k],n : A[k]n → A[k]n−1 equal to dA[k],n = (−1)k dA,n+k . If f : A• → B• is a morphism of chain complexes, then we let f [k] : A[k]• → B[k]• be the morphism of chain complexes with f [k]n = fk+n . Of course this means we have functors [k] : Ch(A) → Ch(A) which mutually commute (on the nose, without any intervening isomorphisms of functors), such that A[k][l]• = A[k + l]• and with [0] = idCh(A) . Definition 10.12.2. Let A be an abelian category. Let A• be a chain complex with boundary maps dA,n : An → An−1 . For any k ∈ Z we identify Hi+k (A• ) → Hi (A[k]• ) via the identification Ai+k = A[k]i . This identification is functorial in A• . Note that since no signs are involved in this definition we actually get a compatible system of identifications of all the homology objects Hi−k (A[k]• ), which are further compatible with the identifications A[k][l]• = A[k + l]• and with [0] = idCh(A) . Let A be an additive category. Suppose that A• and B• are chain complexes, a, b : A• → B• are morphsms of chain complexes, and {hi : Ai → Bi+1 } is a homotopy between a and b. Recall that this means that ai −bi = di+1 ◦hi +hi−1 ◦di . 2I am sure you think that my conventions are wrong. If so and if you feel strongly about it then drop me an email with an explanation.

724

10. HOMOLOGICAL ALGEBRA

What if a = b? Then we obtain the formula 0 = di+1 ◦ hi + hi−1 ◦ di , in other words, −di+1 ◦ hi = hi−1 ◦ di . By definition above this means the collection {hi } above defines a morphism of chain complexes A• −→ B[1]• . Such a thing is the same as a morphism A[−1]• → B• by our remarks above. This proves the following lemma. Lemma 10.12.3. Let A be an additive category. Suppose that A• and B• are chain complexes. Given any morphism of chain complexes a : A• → B• there is a bijection between the set of homotopies from a to a and MorCh(A) (A• , B[1]• ). More generally, the set of homotopies between a and b is either empty or a principal homogenous space under the group MorCh(A) (A• , B[1]• ). Proof. See above.



Lemma 10.12.4. Let A be an abelian category. Let 0 → A• → B • → C • → 0 be a sort exact sequence of complexes. Suppose that {sn : Cn → Bn } is a family of morphisms which split the short exact sequences 0 → An → Bn → Cn → 0. Let πn : Bn → An be the associated projections, see Lemma 10.3.21. Then the family of morphisms πn−1 ◦ dB,n ◦ sn : Cn → An−1 define a morphism of complexes δ(s) : C• → A[−1]• . Proof. Denote i : A• → B• and q : B• → C• the maps of complexes in the short exact sequence. Then in−1 ◦ πn−1 ◦ dB,n ◦ sn = dB,n ◦ sn − sn−1 ◦ dC,n . Hence in−2 ◦ dA,n−1 ◦ πn−1 ◦ dB,n ◦ sn = dB,n−1 ◦ (dB,n ◦ sn − sn−1 ◦ dC,n ) = −dB,n−1 ◦ sn−1 ◦ dC,n as desired.  Lemma 10.12.5. Notation and assumptions as in Lemma 10.12.4 above. The morphism of complexes δ(s) : C• → A[−1]• induces the maps Hi (δ(s)) : Hi (C• ) −→ Hi (A[−1]• ) = Hi−1 (A• ) which occur in the long exact homology sequence associated to the short exact sequence of chain complexes by Lemma 10.10.6. Proof. Omitted.



Lemma 10.12.6. Notation and assumptions as in Lemma 10.12.4 above. Suppose {s0n : Cn → Bn } is a second choice of splittings. Write s0n = sn + πn ◦ hn for some unique morphisms hn : Cn → An . The family of maps {hn : Cn → A[−1]n+1 } is a homotopy between the associated morphisms δ(s), δ(s0 ) : C• → A[−1]• . Proof. Omitted.



Definition 10.12.7. Let A be an additive category. Let A• be a cochain complex with boundary maps dnA : An → An−1 . For any k ∈ Z we define the k-shifted cochain complex A[k]• as follows: (1) we set A[k]n = An+k , and (2) we set dnA[k] : A[k]n → A[k]n−1 equal to dnA[k] = (−1)k dn+k A .

10.12. HOMOTOPY AND THE SHIFT FUNCTOR

725

If f : A• → B • is a morphism of cochain complexes, then we let f [k] : A[k]• → B[k]• be the morphism of cochain complexes with f [k]n = f k+n . Of course this means we have functors [k] : CoCh(A) → CoCh(A) which mutually commute (on the nose, without any intervening isomorphisms of functors) and such that A[k][l]• = A[k + l]• and with [0] = idCoCh(A) . Definition 10.12.8. Let A be an abelian category. Let A• be a cochain complex with boundary maps dnA : An → An+1 . For any k ∈ Z we identify H i+k (A• ) −→ H i (A[k]• ) via the identification Ai+k = A[k]i . This identification is functorial in A• . Note that since no signs are involved in this definition we actually get a compatible system of identifications of all the homology objects H i−k (A[k]• ), which are further compatible with the identifications A[k][l]• = A[k + l]• and with [0] = idCoCh(A) . Let A be an additive category. Suppose that A• and B • are cochain complexes, a, b : A• → B • are morphsms of cochain complexes, and {hi : Ai → B i−1 } is a homotopy between a and b. Recall that this means that ai −bi = di−1 ◦hi +hi+1 ◦di . What if a = b? Then we obtain the formula 0 = di−1 ◦ hi + hi+1 ◦ di , in other words, −di−1 ◦ hi = hi+1 ◦ di . By definition above this means the collection {hi } above defines a morphism of cochain complexes A• −→ B[−1]• . Such a thing is the same as a morphism A[1]• → B • by our remarks above. This proves the following lemma. Lemma 10.12.9. Let A be an additive category. Suppose that A• and B • are cochain complexes. Given any morphism of cochain complexes a : A• → B • there is a bijection between the set of homotopies from a to a and MorCoCh(A) (A• , B[−1]• ). More generally, the set of homotopies between a and b is either empty or a principal homogenous space under the group MorCoCh(A) (A• , B[−1]• ). Proof. See above.



Lemma 10.12.10. Let A be an additive category. Let 0 → A• → B • → C • → 0 be a complex (!) of complexes. Suppose that we are given splittings B n = An ⊕ C n compatible with the maps in the displayed sequence. Let sn : C n → B n and π n : B n → An be the corresponding maps. Then the family of morphisms π n+1 ◦ dnB ◦ sn : C n → An+1 define a morphism of complexes δ : C • → A[1]• . Proof. Denote i : A• → B • and q : B • → C • the maps of complexes in the short exact sequence. Then in+1 ◦ π n+1 ◦ dnB ◦ sn = dnB ◦ sn − sn+1 ◦ dnC . Hence in+2 ◦ dn+1 ◦ π n+1 ◦ dnB ◦ sn = dn+1 ◦ (dnB ◦ sn − sn+1 ◦ dnC ) = −dn+1 ◦ sn+1 ◦ dnC as A B B desired.  Lemma 10.12.11. Notation and assumptions as in Lemma 10.12.10 above. Assume in addition that A is abelian. The morphism of complexes δ : C • → A[1]• induces the maps H i (δ) : H i (C • ) −→ H i (A[1]• ) = H i+1 (A• )

726

10. HOMOLOGICAL ALGEBRA

which occur in the long exact homology sequence associated to the short exact sequence of cochain complexes by Lemma 10.10.12. Proof. Omitted.



Lemma 10.12.12. Notation and assumptions as in Lemma 10.12.10 above. Let α : A• , β : B • → C n be the given morphisms of complexes. Suppose (s0 )n : C n → B n and (π 0 )n : B n → An is a second choice of splittings. Write (s0 )n = sn +αn ◦hn and (π 0 )n = π n + g n ◦ β n for some unique morphisms hn : C n → An and g n : C n → An . Then (1) g n = −hn , and (2) the family of maps {g n : C n → A[1]n−1 } is a homotopy between δ, δ 0 : n C • → A[1]• , more precisely (δ 0 )n = δ n + g n+1 ◦ dnC + dn−1 A[1] ◦ g . Proof. As (s0 )n and (π 0 )n are splittings we have (π 0 )n ◦ (s0 )n = 0. Hence 0 = (π n + g n ◦ β n ) ◦ (sn + αn ◦ hn ) = g n ◦ β n ◦ sn + π n ◦ αn ◦ hn = g n + hn which proves (1). We compute (δ 0 )n as follows (π n+1 + g n+1 ◦ β n+1 ) ◦ dnB ◦ (sn + αn ◦ hn ) = δ n + g n+1 ◦ dnC + dnA ◦ hn n Since hn = −g n and since dn−1 A[1] = −dA we conclude that (2) holds.



10.13. Filtrations A nice reference for this material is [Del71, Section 1]. (Note that our conventions regarding abelian categories are different.) Definition 10.13.1. Let A be an abelian category. (1) A decreasing filtration F on an object A is a family (F n A)n∈Z of subobjects of A such that A ⊃ . . . ⊃ F n A ⊃ F n+1 A ⊃ . . . ⊃ 0 (2) A filtered object of A is pair (A, F ) consisting of an object A of A and a decreasing filtration F on A. (3) A morphism (A, F ) → (B, F ) of filtered objects is given by a morphism ϕ : A → B of A such that ϕ(F i A) ⊂ F i B for all i ∈ Z. (4) The category of filtered objects is denoted Fil(A). (5) Given a filtered object (A, F ) and a subobject X ⊂ A the induced filtration on X is the filtration with F n X = X ∩ F n A. (6) Given a filtered object (A, F ) and a surjection π : A → Y the quotient filtration is the filtration with F n Y = π(F n A). (7) A filtration F on an object A is said to be finite if there exist n, m such that F n A = A and F m A = 0. T i (8) The filtration on a filtered S i object (A, F ) is said to be separated if i F A = 0 and exhaustive if F A = A. By abuse of notation we say that a morphism f : (A, F ) → (B, F ) of filtered objects is injective if f : A → B is injective in the abelian category A. Similarly we say f is surjective if f : A → B is surjective in the category A. Being injective (resp. surjective) is equivalent to being a monomorphism (resp. epimorphism) in Fil(A). By Lemma 10.13.2 this is also equivalent to having zero kernel (resp. cokernel).

10.13. FILTRATIONS

727

Lemma 10.13.2. Let A be an abelian category. The category of filtered objects Fil(A) has the following properties: (1) (2) (3) (4)

It is an additive category. It has a zero object. It has kernels and cokernels, images and coimages. In general it is not an abelian category.

Proof. It is clear that Fil(A) is additive with direct sum given by (A, F )⊕(B, F ) = (A⊕B, F ) where F p (A⊕B) = F p A⊕F p B. The kernel of a morphism f : (A, F ) → (B, F ) of filtered objects is the injection Ker(f ) ⊂ A where Ker(f ) is endowed with the induced filtration. The cokernel of a morphism f : A → B of filtered objects is the surjection B → Coker(f ) where Coker(f ) is endowed with the quotient filtration. Since all kernels and cokernels exist, so do all coimages and images. See Example 10.3.11 for the last statement.  Definition 10.13.3. Let A be an abelian category. A morphism f : A → B of filtered objects of A is said to be strict if f (F i A) = f (A) ∩ F i B for all i ∈ Z. This also equivalent to requiring that f −1 (F i B) = F i A + Ker(f ) for all i ∈ Z. We characterize strict morphisms as follows. Lemma 10.13.4. Let A be an abelian category. Let f : A → B be a morphism of filtered objects of A. The following are equivalent (1) f is strict, (2) the morphism Coim(f ) → Im(f ) of Lemma 10.3.10 is an isomorphism. Proof. Note that Coim(f ) → Im(f ) is an isomorphism of objects of A, and that part (2) signifies that it is an isomorphism of filtered objects. By the description of kernels and cokernels in the proof of Lemma 10.13.2 we see that the filtration on Coim(f ) is the quotient filtration coming from A → Coim(f ). Similarly, the filtration on Im(f ) is the induced filtration coming from the injection Im(f ) → B. The definition of strict is exactly that the quotient filtration is the induced filtration.  Lemma 10.13.5. Let A be an abelian category. A composition of strict morphisms of filtered objects is strict. Proof. Let f : A → B, g : B → C be strict morphisms of filtered objects. Then g(f (F p A)) = g(f (A) ∩ F p B) ⊃ g(f (A)) ∩ g(F p (B)) = (g ◦ f )(A) ∩ (g(B) ∩ F p C) = (g ◦ f )(A) ∩ F p C. The inclusion g(f (F p A)) ⊂ (g ◦ f )(A) ∩ F p C is always true.



Lemma 10.13.6. Let A be an abelian category. Let f : A → B be a strict monomorphism of filtered objects. Let g : A → C be a morphism of filtered objects. Then f ⊕ g : A → B ⊕ C is a strict monomorphism. Proof. Clear from the definitions.



728

10. HOMOLOGICAL ALGEBRA

Lemma 10.13.7. Let A be an abelian category. Let f : B → A be a strict epimorphism of filtered objects. Let g : C → A be a morphism of filtered objects. Then f ⊕ g : B ⊕ C → A is a strict epimorphism. Proof. Clear from the definitions.



Lemma 10.13.8. Let A be an abelian category. Let (A, F ), (B, F ) be filtered objects. Let u : A → B be a morphism of filtered objects. If u is injective then u is strict if and only if the filtration on A is the induced filtration. If u is surjective then u is strict if and only if the filtration on B is the quotient filtration. Proof. This is immediate from the definition.



The following lemma says that subobjects of a filtered object have a well defined filtration independent of a choice of writing the object as a cokernel. Lemma 10.13.9. Let A be an abelian category. Let (A, F ) be a filtered object of A. Let X ⊂ Y ⊂ A be subobjects of A. On the object Y /X = Ker(A/X → A/Y ) the quotient filtration coming from the induced filtration on Y and the induced filtration coming from the quotient filtration on A/X agree. Any of the morphisms X → Y , X → A, Y → A, Y → A/X, Y → Y /X, Y /X → A/X are strict (with induced/quotient filtrations). Proof. The quotient filtration Y /X is given by F p (Y /X) = F p Y /(X ∩ F p Y ) = F p Y /F p X because F p Y = Y ∩ F p A and F p X = X ∩ F p A. The induced filtration from the injection Y /X → A/X is given by F p (Y /X) = Y /X ∩ F p (A/X) = Y /X ∩ (F p A + X)/X = (Y ∩ F p A)/(X ∩ F p A) = F p Y /F p X. Hence the first statement of the lemma. The proof of the other cases is similar.  Lemma 10.13.10. Let A be an abelian category. Let A, B, C ∈ Fil(A). Let f : A → B and g : A → C be morphisms Then there exists a pushout A

f

g0

g

 C

/B

f0

 / C qA B

in Fil(A). If f is strict, so is f 0 . Proof. Set C qA B equal to Coker((1, −1) : A → C ⊕ B) in Fil(A). This cokernel exists, by Lemma 10.13.2. It is a pushout, see Example 10.3.17. Note that F p (C ×A B) is the image of F p C ⊕ F p B. Hence (f 0 )−1 (F p (C ×A B)) = g(f −1 (F p B))) + F p C Whence the last statement.



10.13. FILTRATIONS

729

Lemma 10.13.11. Let A be an abelian category. Let A, B, C ∈ Fil(A). Let f : B → A and g : C → A be morphisms Then there exists a pushout B ×A C

f0

g0

 C

/B g

f

 /A

in Fil(A). If f is strict, so is f 0 . Proof. This lemma is dual to Lemma 10.13.10.



Definition 10.13.12. Let A be an abelian category. A graded object of A is pair (A, k) consisting of an object A of A and a direct sum decomposition M A= ki A i∈Z

by subobjects indexed by Z. A morphism (A, k) → (B, k) of graded objects is given by a morphism ϕ : A → B of A such that ϕ(k i A) ⊂ k i B for all i ∈ Z. The category of graded objects is denoted Gr(A). With our definitions an abelian category does not necessarily have all (countable) direct sums. Of course the definition above still makes sense, but may be a little misleading in case A does not have infinite direct sums. For example, if A = Vectk is the category of finite dimensional vector spaces over a field k, then Gr(Vectk ) is the category of finite dimensional vector spaces with a given gradation, and not the category of graded vector spaces all of whose graded pieces are finite dimensional. Lemma 10.13.13. Let A be an abelian category. The category of graded objects Gr(A) is abelian. Proof. Omitted.



Let A be an abelian category. Let (A, F ) be a filtered object of A. We denote grpF (A) = grp (A) the object F p A/F p+1 A of A. This defines an additive functor grp : Fil(A) −→ A,

(A, F ) 7−→ grp (A).

Assume A has countable direct sums. For (A, F ) in Fil(A) we may set M M gr(A) = grp (A) = F p A/F p+1 A. p∈Z

p∈Z

This defines an additive functor gr : Fil(A) −→ Gr(A),

(A, F ) 7−→ gr(A).

If A does not have all countable direct sums this functor is still defined on the subcategory of Fil(A) consisting of all filtered objects whose filtrations are finite. Lemma 10.13.14. Let A be an abelian category. (1) Let A be a filtered object and X ⊂ A. Then for each p the sequence 0 → grp (X) → grp (A) → grp (A/X) → 0 is exact (with induced filtration on X and quotient filtration on A/X).

730

10. HOMOLOGICAL ALGEBRA

(2) Let f : A → B be a morphism of filtered objects of A. Then for each p the sequences 0 → grp (Ker(f )) → grp (A) → grp (Coim(f )) → 0 and 0 → grp (Im(f )) → grp (B) → grp (Coker(f )) → 0 are exact. Proof. We have F p+1 X = X ∩ F p+1 A, hence map grp (X) → grp (A) is injective. Dually the map grp (A) → grp (A/X) is surjective. The kernel of F p A/F p+1 A → A/X + F p+1 A is clearly F p+1 A + X ∩ F p A/F p+1 A = F p X/F p+1 X hence exactness in the middle. The two short exact sequence of (2) are special cases of the short exact sequence of (1).  Lemma 10.13.15. Let A be an abelian category. Let f : A → B be a morphism of finite filtered objects of A. The following are equivalent (1) f is strict, (2) the morphism Coim(f ) → Im(f ) is an isomorphism, (3) gr(Coim(f )) → gr(Im(f )) is an isomorphism, (4) the sequence gr(Ker(f )) → gr(A) → gr(B) is exact, (5) the sequence gr(A) → gr(B) → gr(Coker(f )) is exact, and (6) the sequence 0 → gr(Ker(f )) → gr(A) → gr(B) → gr(Coker(f )) → 0 is exact. Proof. The equivalence of (1) and (2) is Lemma 10.13.4. By Lemma 10.13.14 we see that (4), (5), (6) imply (3) and that (3) implies (4), (5), (6). Hence it suffices to show that (3) implies (2). Thus we have to show that if f : A → B is an injective and surjective map of finite filtered objects which induces and isomorphism gr(A) → gr(B), then f induces an isomorphism of filtered objects. In other words, we have to show that f (F p A) = F p B for all p. As the filtrations are finite we may prove this by descending induction on p. Suppose that f (F p+1 A) = F p+1 B. Then commutative diagram 0

/ F p+1 A 

0

/ F pA

f

/ F p+1 B

f



/ F pB

and the five lemma imply that f (F p A) = F p B.

/ grp (A)

/0

grp (f )

 / grp (B)

/0 

Lemma 10.13.16. Let A be an abelian category. Let A → B → C be a complex of filtered objects of A. Assume α : A → B and β : B → C are strict morphisms of filtered objects. Then gr(Ker(β)/Im(α)) = Ker(gr(β))/Im(gr(α))). Proof. This follows formally from Lemma 10.13.14 and the fact that Coim(α) ∼ = Im(α) and Coim(β) ∼  = Im(β) by Lemma 10.13.4. Lemma 10.13.17. Let A be an abelian category. Let A → B → C be a complex of filtered objects of A. Assume A, B, C have finite filtrations and that gr(A) → gr(B) → gr(C) is exact. Then

10.14. SPECTRAL SEQUENCES

(1) (2) (3) (4) (5)

731

for each p ∈ Z the sequence grp (A) → grp (B) → grp (C) is exact, for each p ∈ Z the sequence F p (A) → F p (B) → F p (C) is exact, for each p ∈ Z the sequence A/F p (A) → B/F p (B) → C/F p (C) is exact, the maps A → B and B → C are strict, and A → B → C is exact (as a sequence in A).

Proof. Part (1) is immediate from the definitions. We will prove (3) by induction on the length of the filtrations. If each of A, B, C has only one nonzero graded part, then (3) holds as gr(A) = A, etc. Let n be the largest integer such that at least one of F n A, F n B, F n C is nonzero. Set A0 = A/F n A, B 0 = B/F n B, C 0 = C/F n C with induced filtrations. Note that gr(A) = F n A ⊕ gr(A0 ) and similarly for B and C. The induction hypothesis applies to A0 → B 0 → C 0 , which implies that A/F p (A) → B/F p (B) → C/F p (C) is exact for p ≥ n. To conclude the same for p = n + 1, i.e., to prove that A → B → C is exact we use the commutative diagram 0

/ F nA

/A

/ A0

/0

0

 / F nB

 /B

 / B0

/0

0

 / F nC

 /C

 / C0

/0

whose rows are short exact sequences of objects of A. The proof of (2) is dual. Of course (5) follows from (2). To prove (4) denote f : A → B and g : B → C the given morphisms. We know that f (F p (A)) = Ker(F p (B) → F p (C)) by (2) and f (A) = Ker(g) by (5). Hence f (F p (A)) = Ker(F p (B) → F p (C)) = Ker(g) ∩ F p (B) = f (A) ∩ F p (B) which proves that f is strict. The proof that g is strict is dual to this.  10.14. Spectral sequences A nice discussion of spectral sequences may be found in [Eis95]. See also [McC01], [Lan02], etc. Definition 10.14.1. Let A be an abelian category. (1) A spectral sequence in A is given by a system (Er , dr )r≥1 where each Er is an object of A, each dr : Er → Er is a morphism such that dr ◦ dr = 0 and Er+1 = Ker(dr )/Im(dr ) for r ≥ 1. (2) A morphism of spectral sequences f : (Er , dr )r≥1 → (Er0 , d0r )r≥1 is given by a family of morphisms fr : Er → Er0 such that fr ◦ dr = d0r ◦ fr and such that fr+1 is the morphism induced by fr via the identifications 0 Er+1 = Ker(dr )/Im(dr ) and Er+1 = Ker(d0r )/Im(d0r ). We will sometimes loosen this definition somewhat and allow Er+1 to be an object with a given isomorphism Er+1 → Ker(dr )/Im(dr ). In addition we sometimes have a system (Er , dr )r≥r0 for some r0 satsifying the properties of the definition above for indices ≥ r. We will also call this a spectral sequence since by a simple renumbering it falls under the definition anyway. In fact, sometimes it makes sense to allow r0 = 0 or even r0 = −1 due to conventions in the literature.

732

10. HOMOLOGICAL ALGEBRA

Given a spectral sequence (Er , dr )r≥1 we define 0 = B1 ⊂ B2 ⊂ . . . ⊂ Br ⊂ . . . ⊂ Zr ⊂ . . . ⊂ Z2 ⊂ Z1 = E1 by the following simple procedure. Set B2 = Im(d1 ) and Z2 = Ker(d1 ). Then it is clear that d2 : Z2 /B2 → Z2 /B2 . Hence we can define B3 as the unique subobject of E1 containing B2 such that B3 /B2 is the image of d2 . Similarly we can define Z3 as the unique subobject of E1 containing B2 such that Z3 /B2 is the kernel of d2 . And so on and so forth. In particular we have Er = Zr /Br for all r ≥ 1. I case the spectral sequence starts at r = r0 then we can similarly construct Bi , Zi as subobjects in Er0 . Definition 10.14.2. Let A be an abelian category. Let (Er , dr )r≥1 be a spectral sequence. T S (1) If the subobjects Z∞ = Zr and B∞ = Br of E1 exist then we define the limit of the spectral sequence to be the object E∞ = Z∞ /B∞ . (2) We say that the spectral sequence collapses at Er , or degenerates at Er if the differentials dr , dr+1 , . . . are all zero. Note that if the spectral sequence collapses at Er , then we have Er = Er+1 = . . . = E∞ (and the limit exists of course). Also, almost any abelian category we will encounter has countable sums and intersections. 10.15. Spectral sequences: exact couples Definition 10.15.1. Let A be an abelian category. (1) An exact couple is a datum (A, E, α, f, g) where A, E are objects of A and α, f , g are morphisms as in the following diagram /A A_ α f

E



g

with the property that the kernel of each arrow is the image of its predecessor. So Ker(α) = Im(f ), Ker(f ) = Im(g), and Ker(g) = Im(α). (2) A morphism of exact couples t : (A, E, α, f, g) → (A0 , E 0 , α0 , f 0 , g 0 ) is given by morphisms tA : A → A0 and tE : E → E 0 such that α0 ◦ tA = tA ◦ α, f 0 ◦ tE = tA ◦ f , and g 0 ◦ tA = tE ◦ g. Lemma 10.15.2. Let A be an abelian category. Let (A, E, α, f, g) be an exact couple. Set (1) d = g ◦ f : E → E so that d ◦ d = 0, (2) E 0 = Ker(d)/Im(d), (3) A0 = Im(α), (4) α0 : A0 → A0 induced by α, (5) f 0 : E 0 → A0 induced by f , (6) g 0 : A0 → E 0 induced by “g ◦ α−1 ”. Then we have

10.16. SPECTRAL SEQUENCES: DIFFERENTIAL OBJECTS

733

(1) Ker(d) = f −1 (ker(g)) = f −1 (Im(α)), (2) Im(d) = g(Im(f )) = g(Ker(α)), (3) (A0 , E 0 , α0 , f 0 , g 0 ) is an exact couple. Proof. Omitted.



Hence it is clear that given an exact couple (A, E, α, f, g) we get a spectral sequence by setting E1 = E, d1 = d, E2 = E 0 , d2 = d0 = g 0 ◦ f 0 , E3 = E 00 , d3 = d00 = g 00 ◦ f 00 , and so on. Definition 10.15.3. Let A be an abelian category. Let (A, E, α, f, g) be an exact couple. The spectral sequence associated to the exact couple is the spectral sequence (Er , dr )r≥1 with E1 = E, d1 = d, E2 = E 0 , d2 = d0 = g 0 ◦ f 0 , E3 = E 00 , d3 = d00 = g 00 ◦ f 00 , and so on. Lemma 10.15.4. Let A be an abelian category. Let (A, E, α, f, g) be an exact couple. Let (Er , dr )r≥1 be the spectral sequence associated to the exact couple. In this case we have 0 = B1 ⊂ . . . ⊂ Br+1 = g(ker(αr )) ⊂ . . . ⊂ Zr+1 = f −1 (Im(αr )) ⊂ . . . ⊂ Z1 = E and the map dr+1 : Er+1 → Er+1 is described by the following rule: For any (test) object T of A and any elements x : T → Zr+1 and y : T → A such that f ◦x = αr ◦y we have dr ◦ x = g ◦ y where x : T → Er+1 is the induced morphism. Proof. Omitted.



Note that in the situation of the lemma we obviously have [  \  B∞ = g Ker(αr ) ⊂ Z∞ = f −1 Im(αr ) r

r

provided this exist and in this case E∞ = Z∞ /B∞ . 10.16. Spectral sequences: differential objects Definition 10.16.1. Let A be an abelian category. A differential object of A is a pair (A, d) consisting of an object A of A endowed with a selfmap d such that d ◦ d = 0. A morphism of differential objects (A, d) → (B, d) is given by a morphism α : A → B such that d ◦ α = α ◦ d. Lemma 10.16.2. Let A be an abelian category. The category of differential objects of A is abelian. Proof. Omitted.



Definition 10.16.3. For a differential object (A, d) we denote H(A, d) = Ker(d)/Im(d) its homology. Lemma 10.16.4. Let A be an abelian category. Let 0 → (A, d) → (B, d) → (C, d) → 0 be a short exact sequence of differential objects. Then we get an exact homology sequence . . . → H(C, d) → H(A, d) → H(B, d) → H(C, d) → . . .

734

10. HOMOLOGICAL ALGEBRA

Proof. Apply Lemma 10.10.12 to the short exact sequence of complexes 0 0 0

→ A ↓ → A ↓ → A

→ B ↓ → B ↓ → B

→ → →

C ↓ C ↓ C

→

0

→

0

→

0 

We come to an important example of a spectral sequence. Let A be an abelian category. Let (A, d) be a differential object of A. Let α : (A, d) → (A, d) be an endomorphism of this differential object. If we assume α injective, then we get a short exact sequence 0 → (A, d) → (A, d) → (A/αA, d) → 0 of differential objects. By the Lemma 10.16.4 we get an exact couple H(A, d) f

/ H(A, d)

α f

x H(A/αA, d)

g

where g is the canonical map and f is the map defined in the snake lemma. Thus we get an associated spectral sequence! Since in this case we have E1 = H(A/αA, d) we see that it makes sense to define E0 = A/αA and d0 = d. In other words, we start the spectral sequence with r = 0. According to our conventions in Section 10.14 we define a sequence of subobjects 0 = B0 ⊂ . . . ⊂ Br ⊂ . . . ⊂ Zr ⊂ . . . ⊂ Z0 = E0 property that Er = Zr /Br . Namely we have for r ≥ 1 that Br is the image of (αr−1 )−1 (dA) under the natural map A → A/αA, Zr is the image of d−1 (αr A) under the natural map A → A/αA, and dr : Er → Er is given as follows: given an element z ∈ Zr choose an element y ∈ A such that d(z) = αr (y). Then dr (z+Br +αA) = y+Br +αA. Warning: It is not necessarily the case that αA ⊂ (αr−1 )−1 (dA), nor αA ⊂ d−1 (αr A). It is true that (αr−1 )−1 (dA) ⊂ d−1 (αr A). We have

with the (1) (2) (3)

Er =

d−1 (αr A) + αA . (αr−1 )−1 (dA) + αA

It is not hard to verify directly that (1) – (3) give a spectral sequence. Definition 10.16.5. Let A be an abelian category. Let (A, d) be a differential object of A. Let α : A → A be an injective selfmap of A which commutes with d. The spectral sequence associated to (A, d, α) is the spectral sequence (Er , dr )r≥0 described above. 10.17. Spectral sequences: filtered differential objects Definition 10.17.1. Let A be an abelian category. A filtered differential object (K, F, d) is a filtered object (K, F ) of A endowed with an endomorphism d : (K, F ) → (K, F ) whose square is zero: d ◦ d = 0.

10.17. SPECTRAL SEQUENCES: FILTERED DIFFERENTIAL OBJECTS

735

Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. Note that each F n K is a differential by itself. Assume A has countable L object direct sums. In this case set A = F n K and endow it with a differential d by using d on each summand. Consider the map α:A→A n

which maps the summand F A into the summand F n−1 A. This is clearly an injective morphism of differential modules α : (A, d) → (A, d). Hence, by Definition 10.16.5 we get a spectral sequence. We will call this the spectral sequence associated to the filtered differential object (K, F, d). Let us figure out the terms of this spectral sequence. First, note that A/αA = gr(K) endowed with its differential d = gr(d). Hence we see that E0 = gr(K),

d0 = gr(d).

Hence the homology of the graded differential object gr(K) is the next term: E1 = H(gr(K), gr(d)). In addition we see that E0 is a graded object of A and that d0 is compatible with the grading. Hence clearly E1 is a graded object as well. But it turns out that the differential d1 does not preserve this grading; instead it shifts the degree by 1. To work this out precisely, we define Zrp =

F p K ∩ d−1 (F p+r K) + F p+1 K F p+1 K

and F p K ∩ d(F p−r+1 K) + F p+1 K . F p+1 K This notation, allthough quite natural, seems to be different from the notation in most places in the literature. Perhaps it does not matter, since the literature does not seem to have a consistent choice of notation either. With these choices we see L that Br ⊂ E0 , resp. Zr ⊂ E0 (as defined in Section 10.16) is equal to p Brp , resp. L p p Zr . Hence if we define Brp =

Erp = Zrp /Brp L p for r ≥ 0 and p ∈ Z, then we have Er = p Er . We can define a differential dpr : Erp → Erp+r by the rule z + F p+1 K 7−→ dz + F p+r+1 K where z ∈ F p K ∩ d−1 (F p+r K). Lemma 10.17.2. Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. Assume A has countable direct sums. The spectral sequence (Er , dr )r≥0 associated to (K, F, d) has terms M M Er = Erp , dr = dpr . p∈Z

Furthermore, we have

E0p

p∈Z

= gr K, d0 = gr(d), and E1p = H(grp (K), d). p

Proof. Follows from the discussion above.



736

10. HOMOLOGICAL ALGEBRA

Lemma 10.17.3. Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. Assume A has countable direct sums. The spectral sequence (Er , dr )r≥0 associated to (K, F, d) has dp1 : E1p = H(grp (K)) −→ E1p+1 = H(grp+1 (K)) equal to the boundary map in homology associated to the short exact sequence of differential objects 0 → grp+1 (K) → F p K/F p+2 K → grp+1 (K) → 0. Proof. Omitted.



Definition 10.17.4. Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. The induced filtration on H(K, d) is the filtration defined by F p H(K, d) = Im(H(F p K, d) → H(K, d)). Lemma 10.17.5. Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. The associated graded gr(H(K)) L p of the cohomology of K is a graded subquotient of the graded object E∞ = E∞ . S T Proof. RecallLthat E∞ = Z∞ /B∞ by definition, with S B∞ = Br and Z Zr . ∞ = T p p p p p p = Zrp . Thus = Brp and Z∞ with B∞ /B∞ = Z∞ with E∞ Hence E∞ = E∞ T (F p K ∩ d−1 (F p+r K) + F p+1 K) p E∞ = Sr p . p−r+1 K) + F p+1 K) r (F K ∩ d(F On the other hand, we have grp H(K) =

Ker(d) ∩ F p K + F p+1 K Im(d) ∩ F p K + F p+1 K

The result follows since (10.17.5.1)

Ker(d) ∩ F p K + F p+1 K ⊂

[ r

F p K ∩ d−1 (F p+r K) + F p+1 K




and (10.17.5.2)

\ r


F p K ∩ d(F p−r+1 K) + F p+1 K ⊂ Im(d) ∩ F p K + F p+1 K. 

Definition 10.17.6. Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. We say the spectral sequence associated to (K, F, d) converges if gr(H(K)) = E∞ via Lemma 10.17.5. In this case we also say that (Er , dr )r≥0 abuts to or converges to H(K). In the literature one finds more refined notions distinguishing between “weakly converging”, “abutting” and “converging”. Namely, one can require the filtration on H(K) to be either “arbitrary”, or “exhaustive and separated”, or “exhaustive and complete” in addition to the condition that gr(H(K)) = E∞ . We try to avoid introducing this notation by simply adding the relevant information in the statements of the results. Lemma 10.17.7. Let A be an abelian category. Let (K, F, d) be a filtered differential object of A. The associated spectral sequence converges if and only if for every p ∈ Z we have equality in equations (10.17.5.2) and (10.17.5.1). Proof. Immediate from the discussions above.



10.18. SPECTRAL SEQUENCES: FILTERED COMPLEXES

737

10.18. Spectral sequences: filtered complexes Definition 10.18.1. Let A be an abelian category. A filtered complex K • of A is a complex of Fil(A) (see Definition 10.13.1). We will denote the filtration on the objects by F . Thus F p K n denotes the pth step in the filtration of the nth term of the complex. Note that each F p K • is a complex of A. Hence we could also have defined a filtered complex as a filtered object in the (abelian) category of complexes of A. In particular grK • is a graded object of the category of complexes of A. Let denote d the differential of K. Forgetting the grading we can think of L us K n as a filtered differential object of A. Hence according to Section 10.17 we obtain a spectral sequence (Er , dr )r≥0 . In this section we work out the terms of this spectral sequence, and we endow the terms of this spectral seqeunce with additional structure coming from the grading of K. First we point out that E0p = grp K • is a complex and hence is graded. Thus E0 is bigraded in a natural way. It is customary to use the bigrading M E0 = E0p,q , E0p,q = grp K p+q p,q

The idea is that p + q should be thought of as the total degree of the (co)homology classes. Also, p is called the filtration degree, and q is called the complementary degree. The differential d0 is compatible with this bigrading in the following way M p,q p,q p,q+1 d0 = d0 , dp,q . 0 : E0 → E0 Namely, dp0 is just the differential on the complex grp K • (which occurs as grp E0 just shifted a bit). To go further we identify the objects Brp and Zrp introduced in Section 10.17 as graded objects and we work out the corresponding decompositions of the differentials. We do this in a completely straightforward manner, but again we warn the reader that our notation is not the same as notation found elsewhere. We define F p K p+q ∩ d−1 (F p+r K p+q+1 ) + F p+1 K p+q Zrp,q = F p+1 K p+q and F p K p+q ∩ d(F p−r+1 K p+q−1 ) + F p+1 K p+q Brp,q = . F p+1 K p+q p,q p,q p,q and ofL course Er = L Zr /Br . With these L definitions it is completely clear that Zrp = q Zrp,q , Brp = q Brp,q , and Erp = q Erp,q . Moreover, 0 ⊂ . . . ⊂ Brp,q ⊂ . . . ⊂ Zrp,q ⊂ . . . ⊂ E0p,q T S p,q p,q p,q and hence it makes sense to define Z∞ = r Zrp,q and B∞ = r Brp,q and E∞ = p,q p,q p Z∞ /B∞ . Also, the map dr decomposes as the direct sum of the maps p,q dp,q −→ Erp+r,q−r+1 , r : Er

z + F p+1 K p+q 7→ dz + F p+r+1 K p+q+1

where z ∈ F p K p+q ∩ d−1 (F p+r K p+q+1 ). Lemma 10.18.2. Let A be an abelian category. Let (K • , F ) be a filtered complex of A. Assume A has countable direct sums. The spectral sequence (Er , dr )r≥0 associated to (K • , F ) has bigraded terms M M Er = Erp,q , dr = dp,q r .

738

10. HOMOLOGICAL ALGEBRA

= with dr of bidegree (r, −r + 1). Furthermore, we have E0p,q = grp (K p+q ), dp,q 0 grp (dp+q ), and E1p,q = H p+q (grp (K • )). Proof. Follows from the discussion above.



Lemma 10.18.3. Let A be an abelian category. Let (K • , F ) be a filtered complex of A. Assume A has countable direct sums. Let (Er , dr )r≥0 be the spectral sequence associated to (K • , F ). (1) The map p,q p+q (grp (K • )) −→ E1p+1,q = H p+q+1 (grp+1 (K • )) dp,q 1 : E1 = H

is equal to the boundary map in cohomology associated to the short exact sequence of complexes 0 → grp+1 (K • ) → F p K • /F p+2 K • → grp+1 (K • ) → 0. (2) Assume that d(F p K) ⊂ F p+1 K for all p ∈ Z. Then d induces the zero differential on grp (K • ) and hence E1p,q = grp (K • )p+q . Furthermore, in this case p,q p • p+q dp,q −→ E1p,q = grp+1 (K • )p+q+1 1 : E1 = gr (K )

is the morphism induced by d. Proof. Omitted. But compare Lemma 10.17.3.



Lemma 10.18.4. Let A be an abelian category. Let α : (K • , F ) → (L• , F ) be a morphism of filtered complexes of A. Assume A has countable direct sums. Let (Er (K), dr )r≥0 , resp. (Er (L), dr )r≥0 be the spectral sequence associated to (K • , F ), resp. (L• , F ). The morphism α induces a canonical morphism of spectral sequences {αr : Er (K) → Er (L)}r≥0 compatible with the bigradings. Proof. Obvious from the explicit representation of the terms of the spectral sequences.  Definition 10.18.5. Let A be an abelian category. Let (K • , F ) be a filtered complex of A. The induced filtration on H n (K • ) is the filtration defined by F p H n (K • ) = Im(H n (F p K • ) → H n (K • )). Lemma 10.18.6. Let A be an abelian category. Let (K • , F ) be a filtered complex n • • of A. The associated graded gr(H L (K ))p,qof the cohomology of K is a graded subquotient of the graded object p+q=n E∞ . Proof. Let q = n − p. As in the proof of Lemma 10.17.5 we see that T (F p K n ∩ d−1 (F p+r K n+1 ) + F p+1 K n ) p,q E∞ = Sr p n . p−r+1 K n−1 ) + F p+1 K n ) r (F K ∩ d(F On the other hand, we have (10.18.6.1)

grp H n (K) =

The result follows since (10.18.6.2) [ Ker(d) ∩ F p K n + F p+1 K n ⊂

r

Ker(d) ∩ F p K n + F p+1 K n Im(d) ∩ F p K n + F p+1 K n

F p K n ∩ d−1 (F p+r K n+1 ) + F p+1 K n




10.19. SPECTRAL SEQUENCES: DOUBLE COMPLEXES

739

and (10.18.6.3) \
F p K n ∩ d(F p−r+1 K n−1 ) + F p+1 K n ⊂ Im(d) ∩ F p K n + F p+1 K n . r

 Definition 10.18.7. Let A be an abelian category. Let (K • , F ) be a filtered complex of A. the spectral sequence associated to (K • , F ) converges if L We say p,q n • grH (K ) = p+q=n E∞ for every n ∈ Z. This is often symbolized by the notation Erp,q ⇒ H p+q (K • ). Please read the remarks following Definition 10.17.6. Lemma 10.18.8. Let A be an abelian category. Let (K • , F ) be a filtered complex of A. The associated spectral sequence converges if and only if for every p, q ∈ Z we have equality in equations (10.18.6.3) and (10.18.6.2). Proof. Immediate from the discussions above.



Lemma 10.18.9. Let A be an abelian category. Let (K • , F ) be a filtered complex of A. Assume that the filtration on each K n is finite (see Definition 10.13.1). Then (1) the filtration on each H n (K • ) is finite, and (2) the spectral sequence associated to (K • , F ) converges. Proof. Part (1) is clear from Equation (10.18.6.1). We will use Lemma 10.18.8 to prove part (2). Fix p, n ∈ Z. Look at the left hand side of Equation (10.18.6.3). The expression is equal to the right hand side since F m K n−1 = 0 for m  0. Similarly, use F m K n+1 = K n+1 for m  0 to prove equality in Equation (10.18.6.2). 

10.19. Spectral sequences: double complexes Definition 10.19.1. Let A be an additive category. A double complex in A is p,q p,q is an object of A and given by a system ({Ap,q , dp,q 1 , d2 }p,q∈Z ), where each A p,q p,q p,q p,q+1 p,q p+1,q are morphisms of A such that the d1 : A → A and d2 : A → A following rules hold: ◦ dp,q (1) dp+1,q 1 1 =0 ◦ dp,q (2) dp,q+1 2 2 =0 p,q p+1,q ◦ dp,q (3) dp,q+1 ◦ d 1 1 2 = d2 for all p, q ∈ Z. This is just the cochain version of the definition. It says that each Ap,• is a cochain complex and that each dp,• is a morphism of complexes Ap,• → Ap+1,• such that 1 p+1,• p,• d1 ◦ d1 = 0 as morphisms of complexes. In other words a double complex can

740

10. HOMOLOGICAL ALGEBRA

be seen as a complex of complexes. So in the diagram ...

. .O .

...

/ Ap,q+1 O

. .O .

...

/ Ap+1,q+1 O

/ ...

dp,q+1 1

dp,q 2

dp+1,q 2

/ Ap,q O

...

dp,q 1

/ Ap+1,q O

/ ...

... ... ... ... any square commutes. Warning: In the literature one encouters a different definition where a “bicomplex” or a “double complex” has the property that the squares in the diagram anti-commute. p−1,q It is customary to denote HIp (K •,• ) the complex with terms Ker(dp,q ) 1 )/Im(d1 q p (varying q) and differential induced by d2 . Then HII (HI (K •,• )) denotes its coq homology in degree q. It is also customary to denote HII (K •,• ) the complex p,q p,q−1 with terms Ker(d2 )/Im(d2 ) (varying p) and differential induced by d1 . Then q HIp (HII (K •,• )) denotes its cohomology in degree q.

Definition 10.19.2. Let A be an additive category. Let A•,• be a double complex. The associated simple complex sA• , also sometimes called the associated total complex is given by M sAn = Ap,q n=p+q

(if it exists) with differential dnsA =

X n=p+q

p p,q (dp,q 1 + (−1) d2 )

Alternatively, we sometimes write Tot(A•,• ) to denote this complex. If countable direct sums exist in A or if for each n at most finitely many Ap,n−p are nonzero, then sA• exists. Note that the definition is not symmetric in the indices (p, q). There are two natural filtrations on the simple complex sA• associated to the double complex A•,• . Namely, we define M M p FIp (sAn ) = Ai,j and FII (sAn ) = Ai,j . i+j=n, i≥p

i+j=n, j≥p

It is immediately verified that (sA• , FI ) and (sA• , FII ) are filtered complexes. By Section 10.18 we obtain two spectral sequences. It is customary to denote (0 Er , 0 dr )r≥0 the spectral sequence associated to the filtration FI and to denote (00 Er , 00 dr )r≥0 the spectral sequence associated to the filtration FII . Here is a description of these spectral sequences. Lemma 10.19.3. Let A be an abelian category. Let K •,• be a double complex. The spectral sequences associated to K •,• have the following terms: p p,q p,q (1) 0 E0p,q = K p,q with 0 dp,q → K p,q+1 , 0 = (−1) d2 : K q,p 00 p,q q,p 00 p,q q,p (2) E0 = K with d0 = d1 : K → K q+1,p , 0 p,q q q p,• (3) E1 = H (K p,• ) with 0 dp,q 1 = H (d1 ), 00 p,q q •,p 00 p,q (4) E1 = H (K ) with d1 = (−1)q H q (d•,p 2 ),

10.19. SPECTRAL SEQUENCES: DOUBLE COMPLEXES

741

q (K •,• )), (5) 0 E2p,q = HIp (HII p q 00 p,q (6) E2 = HII (HI (K •,• )).

Proof. Omitted.



These spectral sequences define two filtrations on H n (sK • ). We will denote these FI and FII . Definition 10.19.4. Let A be an abelian category. Let K •,• be a double complex. We say the spectral sequence (0 Er , 0 dr )r≥0 converges if Definition 10.18.7 applies. In other words, for all n p,q grFI (H n (sK • )) = ⊕p+q=n 0 E∞

via the canonical comparison of Lemma 10.18.6. Similarly we say the spectral sequence (00 Er , 00 dr )r≥0 converges if Definition 10.18.7 applies. In other words for all n p,q grFII (H n (sK • )) = ⊕p+q=n 00 E∞ via the canonical comparison of Lemma 10.18.6. Same caveats as those following Definition 10.17.6. Lemma 10.19.5 (First quadrant spectral sequence). Let A be an abelian category. Let K •,• be a double complex. Assume that for some i  0 we have K p,q = 0 whenever either p < i or q < i. Then (1) the filtrations FI , FII on each H n (K • ) are finite, (2) the spectral sequence (0 Er , 0 dr )r≥0 converges, and (3) the spectral sequence (00 Er , 00 dr )r≥0 converges. Proof. Follows immediately from Lemma 10.18.9.



Here is our first application of spectral sequences. Lemma 10.19.6. Let A be an abelian category. Let K • be a complex. Let A•,• be a double complex. Let αp : K p → Ap,0 be morphisms. Assume that (1) There exists a i  0 such that K p = Ap,q = 0 for all p < i and all q. (2) We have Ap,q = 0 if q < 0. (3) The morphisms αp give rise to a morphism of complexes α : K • → A•,0 . (4) The complex Ap,• is exact in all degrees q 6= 0 and the morphism K p → Ap,0 induces an isomorphism K p → Ker(dp,0 2 ). Then α induces a quasi-isomorphism K • −→ sA• of complexes. Moreover, there is a variant of this lemma involving the second variable q instead of p. Proof. The map is simply the map given by the morphisms K n → An,0 → sAn , which are easily seen to define a morphism of complexes. Consider the spectral sequence (0 Er , 0 dr )r≥0 associated to the double complex A•,• . By Lemma 10.19.5 this spectral sequence converges and the induced filtration on H n (sA• ) is finite for each n. By Lemma 10.19.3 and assumption (4) we have 0 E1p,q = 0 unless q = 0 and 0 E1p,0 = K p with differential 0 dp,0 identified with dpK . Hence 0 E2p,0 = 1 p,q p • H (K ) and zero otherwise. This clearly implies dp,q 2 = d3 = . . . = 0 for degree n • n • reasons. Hence we conclude that H (sA ) = H (K ). We omit the verification

742

10. HOMOLOGICAL ALGEBRA

that this identification is given by the morphism of complexes K • → sA• introduced above.  Remark 10.19.7. Let A be an abelian category. Let C ⊂ A be a weak Serre subcategory (see Definition 10.7.1). Suppose that K •,• is a double complex to which Lemma 10.19.5 applies such that for some r ≥ 0 all the objects 0 Erp,q belong to C. We claim all the cohomology groups H n (sK • ) belong to C. Namely, the assumptions imply that the kernels and images of 0 dp,q r are in C. Whereupon we see p,q p,q that each 0 Er+1 is in C. By induction we see that each 0 E∞ is in C. Hence each H n (sK • ) has a finite filtration whose subquotients are in C. Using that C is closed under extensions we conclude that H n (sK • ) is in C as claimed. The same result holds for the second spectral sequence associated to K •,• . Similarly, if (K • , F ) is a filtered complex to which Lemma 10.18.9 applies and for some r ≥ 0 all the objects Erp,q belong to C, then each H n (K • ) is an object of C. 10.20. Injectives Definition 10.20.1. Let A be an abelian category. An object J ∈ Ob(A) is called injective if for every injection A ,→ B and every morphism A → J there exists a morphism B → J making the following diagram commute /B A   J Here is the obligatory characterization of injective objects. Lemma 10.20.2. Let A be an abelian category. Let I be an object of A. The following are equivalent: (1) The object I is injective. (2) The functor B 7→ HomA (B, I) is exact. (3) Any short exact sequence 0→I→A→B→0 in A is split. (4) We have ExtA (B, I) = 0 for all B ∈ Ob(A). Proof. Omitted.



Lemma 10.20.3. Let AQbe an abelian category. Suppose Iω , ω ∈ Ω is a set of injective objects of A. If ω∈Ω Iω exists then it is injective. Proof. Omitted.



Definition 10.20.4. Let A be an abelian category. We say A has enough injectives if every object A has an injective morphism A → J into an injective object J. Definition 10.20.5. Let A be an abelian category. We say that A has functorial injective embeddings if there exists a functor J : A −→ Arrows(A) such that (1) s ◦ J = idA ,

10.21. PROJECTIVES

743

(2) for any object A ∈ Ob(A) the morphism J(A) is injective, and (3) for any object A ∈ Ob(A) the object t(J(A)) is an injective object of A. We will denote such a functor by A 7→ (A → J(A)). 10.21. Projectives Definition 10.21.1. Let A be an abelian category. An object P ∈ Ob(A) is called projective if for every surjection A → B and every morphism P → B there exists a morphism P → A making the following diagram commute AO

/B ?

P Here is the obligatory characterization of projective objects. Lemma 10.21.2. Let A be an abelian category. Let P be an object of A. The following are equivalent: (1) The object P is projective. (2) The functor B 7→ HomA (P, B) is exact. (3) Any short exact sequence 0→A→B→P →0 in A is split. (4) We have ExtA (P, A) = 0 for all A ∈ Ob(A). Proof. Omitted.



Lemma 10.21.3. Let A ` be an abelian category. Suppose Pω , ω ∈ Ω is a set of projective objects of A. If ω∈Ω Pω exists then it is projective. Proof. Omitted.



Definition 10.21.4. Let A be an abelian category. We say A has enough projectives if every object A has an surjective morphism P → A from an projective object P onto it. Definition 10.21.5. Let A be an abelian category. We say that A has functorial projective surjections if there exists a functor P : A −→ Arrows(A) such that (1) t ◦ J = idA , (2) for any object A ∈ Ob(A) the morphism P (A) is surjective, and (3) for any object A ∈ Ob(A) the object s(P (A)) is an projective object of A. We will denote such a functor by A 7→ (P (A) → A).

744

10. HOMOLOGICAL ALGEBRA

10.22. Injectives and adjoint functors Here are some lemmas on adjoint functors and their relationship with injectives. See also Lemma 10.5.3. Lemma 10.22.1. Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors. (1) u is right adjoint to v, and (2) v transforms injective maps into injective maps. Then u transforms injectives into injectives. Proof. Let I be an injective object of A. Let ϕ : N → M be an injective map in B and let α : N → uI be a morphism. By adjointness we get a morphism α : vN → I and by assumption vϕ : vN → vM is injective. Hence as I is an injective object we get a morphism β : vM → I extending α. By adjointness again this corresponds to a morphism β : M → uI as desired.  Remark 10.22.2. Let A, B, u : A → B and v : B → A be as in Lemma 10.22.1. In the presence of assumption (1) assumption (2) is equivalent to requiring that v is exact. Moreover, condition (2) is necessary. Here is an example. Let A → B be a ring map. Let u : ModB → ModA be u(N ) = NA and let v : ModA → ModB be v(M ) = M ⊗A B. Then u is right adjoint to v, and u is exact and v is right exact, but v does not transform injective maps into injective maps in general (i.e., v is not left exact). Moreover, it is not the case that u transforms injective B-modules into injective A-modules. For example, if A = Z and B = Z/pZ, then the injective B-module Z/pZ is not an injective Z-module. In fact, the lemma applies to this example if and only if the ring map A → B is flat. Lemma 10.22.3. Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors. Assume (1) u is right adjoint to v, (2) v transforms injective maps into injective maps, (3) A has enough injectives, and (4) vB = 0 implies B = 0 for any B ∈ Ob(B). Then B has enough injectives. Proof. Pick B ∈ Ob(B). Pick an injection vB → I for I an injective object of A. According to Lemma 10.22.1 and the assumptions the corresponding map B → uI is the injection of B into an injective object.  Remark 10.22.4. Let A, B, u : A → B and v : B → A be as In Lemma 10.22.3. In the presence of conditions (1) and (2) condition (4) is equivalent to v being faithful. Moreover, condition (4) is needed. An example is to consider the case where the functors u and v are both the zero functor. Lemma 10.22.5. Let A and B be abelian categories. Let u : A → B and v : B → A be additive functors. Assume (1) u is right adjoint to v, (2) v transforms injective maps into injective maps, (3) A has enough injectives, (4) vB = 0 implies B = 0 for any B ∈ Ob(B), and (5) A has functorial injective hulls.

10.23. INVERSE SYSTEMS

745

Then B has functorial injective hulls. Proof. Let A 7→ (A → J(A)) be a functorial injective hull on A. Then B 7→ (B → uJ(vB)) is a functorial injective hull on B. Compare with the proof of Lemma 10.22.3.  Lemma 10.22.6. Let A and B be abelian categories. Let u : A → B be a functor. If there exists a subset P ⊂ Ob(B) such that (1) every object of B is a quotient of an element of P, and (2) for every P ∈ P there exists an object Q of A such that HomA (Q, A) = HomB (P, u(A)) functorially in A, then there exists a left adjoint v of u. Proof. By the Yoneda lemma the object Q of A corresponding to P is defined up to unique isomorphism by the formula HomA (Q, A) = HomB (P, u(A)). Let us write Q = v(P ). Denote iP : P → u(v(P )) the map corresponding to idv(P ) in HomA (v(P ), v(P )). Functoriality in (2) implies that the bijection is given by HomA (v(P ), A) → HomB (P, u(A)),

ϕ 7→ u(ϕ) ◦ iP

For any pair of elements P1 , P2 ∈ P there is a canonical map HomB (P2 , P1 ) → HomA (v(P2 ), v(P1 )),

ϕ 7→ v(ϕ)

which is characterized, using by u(v(ϕ)) ◦ iP2 = iP1 ◦ ϕ in HomB (P2 , u(v(P1 ))). Note that ϕ 7→ v(ϕ) is additive and compatible with composition; this can be seen directly from the characterization. Hence P 7→ v(P ) is a functor from the full subcategory opf B whose objects are the elements of P. Given an arbitrary object B of B choose an exact sequence P2 → P1 → B → 0 which is possible by assumption (1). Define v(B) to be the object of A fitting into the exact sequence v(P2 ) → v(P1 ) → v(B) → 0 Then HomA (v(B), A) = Ker(HomA (v(P1 ), A) → HomA (v(P2 ), A)) = Ker(HomB (P1 , u(A)) → HomB (P2 , u(A))) = HomB (B, u(A)) Hence we see that we may take P = Ob(B), i.e., we see that v is everywhere defined.  10.23. Inverse systems Let C be a category. In Categories, Section 4.19 we defined the notion of an inverse system over a partially ordered set (with values in the category C). If the partially ordered set is N = {1, 2, 3, . . .} with the usual ordering such an inverse system over N is often simply called an inverse system. It consists quite simply of a pair (Mi , fii0 ) where each Mi , i ∈ N is an object of C, and for each i > i0 , i, i0 ∈ N a morphism fii0 : Mi → Mi0 such that moreover fi0 i00 ◦ fii0 = fii00 whenever this

746

10. HOMOLOGICAL ALGEBRA

makes sense. It is clear that in fact it suffices to give the morphisms M2 → M1 , M3 → M2 , and so on. Hence an inverse system is frequently pictured as follows ϕ2

ϕ3

M1 ←− M2 ←− M3 ← . . . Moreover, we often omit the transition maps ϕi from the notation and we simply say “let (Mi ) be an inverse system”. The collection of all inverse systems with values in C forms a category with the obvious notion of morphism. Lemma 10.23.1. Let C be a category. (1) If C is an additive category, then the category of inverse systems with values in C is an additive cateogry. (2) If C is an abelian category, then the category of inverse systems with values in C is an abelian cateogry. A sequence (Ki ) → (Li ) → (Mi ) of inverse systems is exact if and only if each Ki → Li → Ni is exact. Proof. Omitted.



The limit (see Categories, Section 4.19) of such an inverse system is denoted lim Mi , or limi Mi . If C is the category of abelian groups (or sets), then the limit always exists and in fact can be described as follows Y limi Mi = {(xi ) ∈ Mi | ϕi (xi ) = xi−1 , i = 2, 3, . . .} see Categories, Section 4.14. However, given a short exact sequence 0 → (Ai ) → (Bi ) → (Ci ) → 0 of inverse systems of abelian groups it is not always the case that the associated system of limits is exact. In order to discuss this further we introduce the following notion. Definition 10.23.2. Let C be an abelian category. We say the inverse system (Ai ) satisfies the Mittag-Leffler condition, or for short is ML, if for every i there exists a c = c(i) ≥ i such that Im(Ak → Ai ) = Im(Ac → Ai ) for all k ≥ c. It turns out that the Mittag-Leffler condition is good enough to ensure that the limfunctor is exact, provided one works within the abelian category of abelian groups, or abelian sheaves, etc. It is shown in a paper by A. Neeman (see [Nee02]) that this condition is not strong enough in a general abelian category (where limits of inverse systems exist). Lemma 10.23.3. Let 0 → (Ai ) → (Bi ) → (Ci ) → 0 be a short exact sequence of inverse systems of abelian groups. (1) In any case the sequence 0 → limi Ai → limi Bi → limi Ci is exact. (2) If (Bi ) is ML, then also (Ci ) is ML.

10.23. INVERSE SYSTEMS

747

(3) If (Ai ) is ML, then 0 → limi Ai → limi Bi → limi Ci → 0 is exact. Proof. Nice exercise. See Algebra, Lemma 7.82.1 for part (3).



Lemma 10.23.4. Let (Ai ) → (Bi ) → (Ci ) → (Di ) be an exact sequence of inverse systems of abelian groups. If the system (Ai ) is ML, then the sequence limi Bi → limi Ci → limi Di is exact. Proof. Let Zi = Ker(Ci → Di ) and Ii = Im(Ai → Bi ). Then lim Zi = Ker(lim Ci → lim Di ) and we get a short exact sequence of systems 0 → (Ii ) → (Bi ) → (Zi ) → 0 Moreover, by Lemma 10.23.3 we see that (Ii ) has (ML), thus another application of Lemma 10.23.3 shows that lim Bi → lim Zi is surjective which proves the lemma.  The following characterization of essentially constant inverse systems shows in particular that they have ML. Lemma 10.23.5. Let A be an abelian category. Let (Ai ) be an inverse system in A with limit A = lim Ai . Then (Ai ) is essentially constant (see Categories, Definition 4.20.1) if and only if there exists an i and for all j ≥ i a direct sum decomposition Aj = A ⊕ Zj such that (a) the maps Aj 0 → Aj are compatible with the direct sum decompositions, (b) for all j there exists some j 0 ≥ j such that Zj 0 → Zj is zero. Proof. Assume (Ai ) is essentially constant. Then there exists an i and a morphism Ai → A such that for all j ≥ i there exists a j 0 ≥ j such that Aj 0 → Aj factors as Aj 0 → Ai → A → Aj (the last map comes from A = lim Ai ). Hence setting Zj = Ker(Aj → A) for all j ≥ i works. Proof of the converse is omitted.  Lemma 10.23.6. Let 0 → (Ai ) → (Bi ) → (Ci ) → 0 be an exact sequence of inverse systems of abelian groups. If (Ai ) has ML and (Ci ) is essentially constant, then (Bi ) has ML. Proof. After renumbering we may assume that Ci = C ⊕ Zi compatible with transition maps and that for all i there exists an i0 ≥ i such that Zi0 → Zi is zero, see Lemma 10.23.5. Pick i. Let c ≥ i by an integer such that Im(Ac → A) = Im(Ai0 → Ai ) for all i0 ≥ c. Let c0 ≥ c be an integer such that Zc0 → Zc is zero.

748

10. HOMOLOGICAL ALGEBRA

For i0 ≥ c0 consider the maps / Ai 0 0

/ Bi0

/ C ⊕ Z i0

/0

0

 / Ac0

 / Bc0

 / C ⊕ Zc 0

/0

0

 / Ac

 / Bc

 / C ⊕ Zc

/0

0

 / Ai

 / Bi

 / C ⊕ Zi

/0

Because Zc0 → Zc is zero the image Im(Bc0 → Bc ) is an extension C by a subgroup A0 ⊂ Ac which contains the image of Ac0 → Ac . Hence Im(Bc0 → Bi ) is an extension of C by the image of A0 which is the image of Ac → Ai by our choice of c. In exactly the same way one shows that Im(Bi0 → Bi ) is an extension of C by the image of Ac → Ai . Hence Im(Bc0 → Bi ) = Im(Bi0 → Bi ) and we win.  Lemma 10.23.7. Let (A−2 → A−1 → A0i → A1i ) i i be an inverse system of complexes of abelian groups and denote A−2 → A−1 → A0 → A1 its limit. Denote (Hi−1 ), (Hi0 ) the inverse systems of cohomologies, and −1 denote H −1 , H 0 the cohomologies of A−2 → A−1 → A0 → A1 . If (A−2 i ) and (Ai ) −1 0 0 are ML and (Hi ) is essentially constant, then H = lim Hi . Proof. Let Zij = Ker(Aji → Aj+1 ) and Iij = Im(Aij−1 → Aji ). Note that lim Zi0 = i 0 1 Ker(lim Ai → lim Ai ) as taking kernels commutes with limits. The systems (Ii−1 ) −1 and (Ii0 ) have ML as quotients of the systems (A−2 i ) and (Ai ), see Lemma 10.23.3. Thus an exact sequence 0 → (Ii−1 ) → (Zi−1 ) → (Hi−1 ) → 0 of inverse systems where (Ii−1 ) has ML and where (Hi−1 ) is essentially constant by assumption. Hence (Zi−1 ) has ML by Lemma 10.23.6. The exact sequence 0 0 → (Zi−1 ) → (A−1 i ) → (Ii ) → 0

and an application of Lemma 10.23.3 shows that lim A−1 → lim Ii0 is surjective. i Finally, the exact sequence 0 → (Ii0 ) → (Zi0 ) → (Hi0 ) → 0 and Lemma 10.23.3 show that lim Ii0 → lim Zi0 → lim Hi0 → 0 is exact. Putting everything together we win.  10.24. Exactness of products Lemma 10.24.1. Let I be a set. For i ∈ I let Li → Mi → Ni be a complex of abelian groups. Let Hi = Ker(Mi → Ni )/Im(Li → Mi ) be the cohomology. Then Y Y Y Li → Mi → Ni Q is a complex of abelian groups with homology Hi . Proof. Omitted.



10.26. OTHER CHAPTERS

749

10.25. Differential graded algebras Definition 10.25.1. Let R be a (commutative) ring. A differential graded algebra is either (1) a chain complex A• of R-modules endowed with R-bilinear maps An × Am → An+m , (a, b) 7→ ab such that dn+m (ab) = dn (a)b + (−1)n adm (b) L and such that An becomes an associative and unital R-algebra, or (2) a cochain complex A• of R-modules endowed with R-bilinear maps An × Am → An+m , (a, b) 7→ ab such that dn+m (ab) = dn (a)b + (−1)n adm (b) L n and such that A becomes an associative and unital R-algebra. L L n We often just write A = An or A = A and think of this as an associative unital R-algebra endowed with a Z-grading and an additive operator d whose square is zero and which satisfies the Leibniz rule as explained above. In this case we often say “Let (A, d) be a differential graded algebra”. Definition 10.25.2. A homomorphism of differential graded algebras f : (A, d) → (B, d) is an algebra map f : A → B compatible with the gradings and d. Definition 10.25.3. A differential graded algebra (A, d) is commutative if ab = (−1)nm ba for a in degree n and b in degree m. We say A is strictly commutative if in addition a2 = 0 for deg(a) odd. The following definition makes sense in general but is perhaps “correct” only when tensoring commutative differential graded algebras. Definition 10.25.4. Let R be a ring. Let (A, d), (B, d) be differential graded algebras over R. The tensor product differential graded algebra of A and B is the algebra A ⊗R B with multiplication defined by 0

(a ⊗ b)(a0 ⊗ b0 ) = (−1)deg(a ) deg(b) aa0 ⊗ bb0 endowed with differential d defined by the rule d(a ⊗ b) = d(a) ⊗ b + (−1)m a ⊗ d(b) where m = deg(b). Lemma 10.25.5. Let R be a ring. Let (A, d), (B, d) be differential graded algebras over R. Denote A• , B • the underlying cochain complexes. As cochain complexes of R-modules we have (A ⊗R B)• = Tot(A• ⊗A B • ). p p,q Proof. Recall that the differential of the total complex is given by dp,q 1 + (−1) d2 on Ap ⊗R B q . And this is exactly the same as the rule for the differential on A ⊗R B in Definition 10.25.4. 

10.26. Other chapters (1) (2) (3) (4)

Introduction Conventions Set Theory Categories

(5) (6) (7) (8)

Topology Sheaves on Spaces Commutative Algebra Brauer Groups

750

10. HOMOLOGICAL ALGEBRA

(9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces

(43) Decent Algebraic Spaces (44) Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces (48) More on Morphisms of Spaces (49) Quot and Hilbert Spaces (50) Spaces over Fields (51) Stacks (52) Formal Deformation Theory (53) Groupoids in Algebraic Spaces (54) More on Groupoids in Spaces (55) Bootstrap (56) Examples of Stacks (57) Quotients of Groupoids (58) Algebraic Stacks (59) Sheaves on Algebraic Stacks (60) Criteria for Representability (61) Artin’s Axioms (62) Properties of Algebraic Stacks (63) Morphisms of Algebraic Stacks (64) Cohomology of Algebraic Stacks (65) Introducing Algebraic Stacks (66) Examples (67) Exercises (68) Guide to Literature (69) Desirables (70) Coding Style (71) Obsolete (72) GNU Free Documentation License (73) Auto Generated Index

CHAPTER 11

Derived Categories 11.1. Introduction We first discuss triangulated categories and localization in triangulated categories. Next, we prove that the homotopy category of complexes in an additive category is a triangulated category. Once this is done we define the derived category of an abelian category as the localization of the of homotopy category with respect to quasi-isomorphisms. A good reference is Verdier’s thesis [Ver96]. 11.2. Triangulated categories Triangulated categories are a convenient tool to describe the type of structure inherent in the derived category of an abelian category. Some references are [Ver96], [KS06], and [Nee01]. 11.3. The definition of a triangulated category In this section we collect most of the definitions concerning triangulated and pretriangulated categories. Definition 11.3.1. Let D be an additive category. Let [n] : D → D, E 7→ E[n] be a collection of additive functors indexed by n ∈ Z such that [n] ◦ [m] = [n + m] and [0] = id (equality as functors). In this situation we call triangle a sixtuple (X, Y, Z, f, g, h) where X, Y, Z ∈ Ob(D) and f : X → Y , g : Y → Z and h : Z → X[1] are morphisms of D. A morphism of triangles (X, Y, Z, f, g, h) → (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) is given by morphisms a : X → X 0 , b : Y → Y 0 and c : Z → Z 0 of D such that b ◦ f = f 0 ◦ a, c ◦ g = g 0 ◦ b and a[1] ◦ h = h0 ◦ c. A morphism of triangles is visualized by the following commutative diagram X a

 X0

/Z

/Y b

 / Y0

c

 / Z0

/ X[1] 

a[1]

/ X 0 [1]

Here is the definition of a triangulated category as given in Verdier’s thesis. Definition 11.3.2. A triangulated category consists of a triple (D, {[n]}n∈Z , T ) where (1) D is an additive category, (2) [n] : D → D, E 7→ E[n] be a collection of additive functors indexed by n ∈ Z such that [n] ◦ [m] = [n + m] and [0] = id (equality as functors), and (3) T is a set of triangles called the distinguished triangles 751

752

11. DERIVED CATEGORIES

subject to the following conditions TR1 Any triangle isomorphic to a distinguished triangle is a distinguished triangle. Any triangle of the form (X, X, 0, id, 0, 0) is distinguished. For any morphism f : X → Y of D there exists a distinguished triangle of the form (X, Y, Z, f, g, h). TR2 The triangle (X, Y, Z, f, g, h) is distinguished if and only if the triangle (Y, Z, X[1], g, h, −f [1]) is. TR3 Given a solid commutative square /Y

X a

b

 / Y0

 X0

/Z

/ X[1]

 / Z0



a[1]

/ X 0 [1]

whose rows are distinguished triangles there exists a morphism c : Z → Z 0 such that (a, b, c) is a morphism of triangles. TR4 Given objects X, Y , Z of D, and morphisms f : X → Y , g : Y → Z, and distinguished triangles (X, Y, Q1 , f, p1 , d1 ), (X, Z, Q2 , g◦f, p2 , d2 ), and (Y, Z, Q3 , g, p3 , d3 ), there exist morphisms a : Q1 → Q2 and b : Q2 → Q3 such that (a) (Q1 , Q2 , Q3 , a, b, p1 [1] ◦ d3 ) is a distinguished triangle, (b) the triple (idX , g, a) is a morphism of triangles (X, Y, Q1 , f, p1 , d1 ) → (X, Z, Q2 , g ◦ f, p2 , d2 ), and (c) the triple (f, idZ , b) is a morphism of triangles (X, Z, Q2 , g◦f, p2 , d2 ) → (Y, Z, Q3 , g, p3 , d3 ). We will call (D, [ ], T ) a pre-triangulated category if TR1, TR2 and TR3 hold. The explanation of TR4 is that if you think of Q1 as Y /X, Q2 as Z/X and Q3 as Z/Y , then TR4(a) expresses the isomorphism (Z/X)/(Y /Z) ∼ = Z/Y and TR(b) and TR(c) express that we can compare the triangles X → Y → Q1 → X[1] etc with morphisms of triangles. For a more precise reformuation of this idea see the proof of Lemma 11.9.2. The sign in TR2 means that if (X, Y, Z, f, g, h) is a distinguished triangle then in the long sequence (11.3.2.1) −h[−1]

f

g

h

−f [1]

−g[1]

. . . → Z[−1] −−−−→ X − →Y − →Z− → X[1] −−−→ Y [1] −−−→ Z[1] → . . . each four term sequence gives a distinguished triangle. As usual we abuse notation and we simply speak of a (pre-)triangulated category D without explicitly introducing notation for the additional data. The notion of a pre-triangulated category is useful in finding statements equivalent to TR4. We have the following definition of a triangulated functor. Definition 11.3.3. Let D, D0 be pre-triangulated categories. An exact functor, or a triangulated functor from D to D0 is a functor F : D → D0 together with given functorial isomorphisms ξX : F (X[1]) → F (X)[1] such that for every distinguished triangle (X, Y, Z, f, g, h) of D the triangle (F (X), F (Y ), F (Z), F (f ), F (g), ξX ◦ F (h)) is a distinguished triangle of D0 .

11.3. THE DEFINITION OF A TRIANGULATED CATEGORY

753

An exact functor is additive, see Lemma 11.4.15. When we say two triangulated categories are equivalent we mean that they are equivalent in the 2-category of triangulated categories. A 2-morphism a : (F, ξ) → (F 0 , ξ 0 ) in this 2-category is simply a transformation of functors a : F → F 0 which is compatible with ξ and ξ 0 , i.e., / [1] ◦ F F ◦ [1] ξ

a?1

1?a

 F 0 ◦ [1]

ξ0

 / [1] ◦ F 0

commutes. Definition 11.3.4. Let (D, [ ], T ) be a pre-triangulated category. A pre-triangulated subcategory1 is a pair (D0 , T 0 ) such that (1) D0 is an additive subcategory of D which is preserved under [1] and [−1], (2) T 0 ⊂ T is a subset such that for every (X, Y, Z, f, g, h) ∈ T 0 we have X, Y, Z ∈ Ob(D0 ) and f, g, h ∈ Arrows(D0 ), and (3) (D0 , [ ], T 0 ) is a pre-triangulated category. If D is a triangulated category, then we say (D0 , T 0 ) is a triangulated subcategory if it is a pre-triangulated subcategory and (D0 , [ ], T 0 ) is a triangulated category. In this situation the inclusion functor D0 → D is an exact functor with ξX : X[1] → X[1] given by the identity on X[1]. We will see in Lemma 11.4.1 that for a distinguished triangle (X, Y, Z, f, g, h) in a pre-triangulated category the composition g ◦ f : X → Z is zero. Thus the sequence (11.3.2.1) is a complex. A homological functor is one that turns this complex into a long exact sequence. Definition 11.3.5. Let D be a pre-triangulated category. Let A be an abelian category. An additive functor H : D → A is called homological if for every distinguished triangle (X, Y, Z, f, g, h) the sequence H(X) → H(Y ) → H(Z) is exact in the abelian category A. An additive functor H : Dopp → A is called cohomological if the corresponding functor D → Aopp is homological. If H : D → A is a homological functor we often write H n (X) = H(X[n]) so that H(X) = H 0 (X). Our discussion of TR2 above implies that says that a distinguished triangle (X, Y, Z, f, g, h) determines a long exact sequence (11.3.5.1)

H −1 (Z)

h[−1]

/ H 0 (X)

f

/ H 0 (Y )

g

/ H 0 (Z)

h

/ H 1 (X)

This will be called the long exact sequence associated to the distinguished triangle and the homological functor. As indicated we will not use any signs for the morphisms in the long exact sequence. This has the side effect that maps in the long exact sequence associated to the rotation (TR2) of a distinguished triangle differ from the maps in the sequence above by some signs. 1This definition may be nonstandard. If D 0 is a full subcategory then T 0 is the intersection of the set of triangles in D0 with T , see Lemma 11.4.14. In this case we drop T 0 from the notation.

754

11. DERIVED CATEGORIES

Definition 11.3.6. Let A be an abelian category. Let D be a triangulated category. A δ-functor from A to D is given by a functor G : A → D and a rule which assigns to every short exact sequence a

b

0→A− →B→ − C→0 a morphism δ = δA→B→C : G(C) → G(A)[1] such that (1) the triangle (G(A), G(B), G(C), G(a), G(b), δA→B→C ) is a distinguished triangle of D for any short exact sequence as above, and (2) for every morphism (A → B → C) → (A0 → B 0 → C 0 ) of short exact sequences the diagram G(C)

δA→B→C

 G(C 0 )

δA0 →B 0 →C 0

/ G(A)[1]  / G(A0 )[1]

is commutative. In this situation we call (G(A), G(B), G(C), G(a), G(b), δA→B→C ) the image of the short exact sequence under the given δ-functor. Note how a δ-functor comes equipped with additional structure. Strictly speaking it does not make sense to say that a given functor A → D is a δ-functor, but we will often do so anyway. 11.4. Elementary results on triangulated categories Most of the results in this section are proved for pre-triangulated categories and a fortiori hold in any triangulated category. Lemma 11.4.1. Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) be a distinguished triangle. Then g ◦ f = 0, h ◦ g = 0 and f [1] ◦ h = 0. Proof. By TR1 we know (X, X, 0, 1, 0, 0) is a distinguished triangle. Apply TR3 to /X /0 / X[1] X f

1

 X

f

 /Y

1[1]

g

 /Z

h

 / X[1]

Of course the dotted arrow is the zero map. Hence the commutativity of the diagram implies that g ◦ f = 0. For the other cases rotate the triangle, i.e., apply TR2.  Lemma 11.4.2. Let D be a pre-triangulated category. For any object W of D the functor HomD (W, −) is homological, and the functor HomD (−, W ) is cohomological. Proof. Consider a distinguished triangle (X, Y, Z, f, g, h). We have already seen that g ◦ f = 0, see Lemma 11.4.1. Suppose a : W → Y is a morphism such that

11.4. ELEMENTARY RESULTS ON TRIANGULATED CATEGORIES

755

g ◦ a = 0. Then we get a commutative diagram W

/W

1

/0

a

b

 X

/ W [1] b[1]

0

 /Y

 / X[1]

 /Z

Both rows are distinguished triangles (use TR1 for the top row). Hence we can fill the dotted arrow b (first rotate using TR2, then apply TR3, and then rotate back). This proves the lemma.  Lemma 11.4.3. Let D be a pre-triangulated category. Let (a, b, c) : (X, Y, Z, f, g, h) → (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) be a morphism of distinguished triangles. If two among a, b, c are isomorphisms so is the third. Proof. Assume that a and c are isomorphisms. For any object W of D write HW (−) = HomD (W, −). Then we get a commutative diagram of abelian groups HW (Z[−1])

/ HW (X)

/ HW (Y )

/ HW (Z)

/ HW (X[1])

 HW (Z 0 [−1])

 / HW (X 0 )

 / HW (Y 0 )

 / HW (Z 0 )

 / HW (X 0 [1])

By assumption the right two and left two vertical arrows are bijective. As HW is homological by Lemma 11.4.2 and the five lemma (Homology, Lemma 10.3.25) it follows that the middle vertical arrow is an isomorphism. Hence by Yoneda’s lemma, see Categories, Lemma 4.3.5 we see that b is an isomorphism. This implies the other cases by rotating (using TR2).  Lemma 11.4.4. Let D be a pre-triangulated category. Let (0, b, 0), (0, b0 , 0) : (X, Y, Z, f, g, h) → (X, Y, Z, f, g, h) be endomorphisms of a distinguished triangle. Then bb0 = 0. Proof. Picture

/Y

X 0

 X

α

 /Y

/Z b,b0

β

/ X[1]

0

 /Z

0

 / X[1]

Applying Lemma 11.4.3 we find dotted arrows α and β such that b0 = f ◦ α and b = β ◦ g. Then bb0 = β ◦ g ◦ f ◦ α = 0 as g ◦ f = 0 by Lemma 11.4.1.  Lemma 11.4.5. Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) be a distinguished triangle. If / X[1] Z f

c

 Z

a[1]

f

 / X[1]

is commutative and a2 = a, c2 = c, then there exists a morphism b : Y → Y with b2 = b such that (a, b, c) is an endomorphism of the triangle (X, Y, Z, f, g, h).

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11. DERIVED CATEGORIES

Proof. By TR3 there exists a morphism b0 such that (a, b0 , c) is an endormorphism of (X, Y, Z, f, g, h). Then (0, (b0 )2 − b0 , 0) is also an endomorphism. By Lemma 11.4.4 we see that (b0 )2 − b0 has square zero. Set b = b0 − (2b0 − 1)((b0 )2 − b0 ) = 3(b0 )2 − 2(b0 )3 . A computation shows that (a, b, c) is an endomorphism and that b2 − b = (4(b0 )2 − 4b0 − 3)((b0 )2 − b0 )2 = 0.  Lemma 11.4.6. Let D be a pre-triangulated category. Let f : X → Y be a morphism of D. There exists a distinguished triangle (X, Y, Z, f, g, h) which is unique up to (nonunique) isomorphism of triangles. More precisely, given a second such distinguished triangle (X, Y, Z 0 , f, g 0 , h0 ) there exists an isomorphism (1, 1, c) : (X, Y, Z, f, g, h) −→ (X, Y, Z 0 , f, g 0 , h0 ) Proof. Existence by TR1. Uniqueness up to isomorphism by TR3 and Lemma 11.4.3.  Lemma 11.4.7. Let D be a pre-triangulated category. Let f : X → Y be a morphism of D. The following are equivalent (1) f is an isomorphism, (2) (X, Y, 0, f, 0, 0) is a distinguished triangle, and (3) for any distinguished triangle (X, Y, Z, f, g, h) we have Z = 0. Proof. Immediate from Lemma 11.4.6 and TR1.



Lemma 11.4.8. Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) and (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) be triangles. The following are equivalent (1) (X ⊕ X 0 , Y ⊕ Y 0 , Z ⊕ Z 0 , f ⊕ f 0 , g ⊕ g 0 , h ⊕ h0 ) is a distinguished triangle, (2) both (X, Y, Z, f, g, h) and (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) are distinguished triangles. Proof. Assume (2). By TR1 we may choose a distinguished triangle (X ⊕ X 0 , Y ⊕ Y 0 , Q, f ⊕ f 0 , g 00 , h00 ). By TR3 we can find morphisms of distinguished triangles (X, Y, Z, f, g, h) → (X ⊕ X 0 , Y ⊕ Y 0 , Q, f ⊕ f 0 , g 00 , h00 ) and (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) → (X ⊕ X 0 , Y ⊕ Y 0 , Q, f ⊕ f 0 , g 00 , h00 ). Taking the direct sum of these morphisms we obtain a morphism of triangles (X ⊕ X 0 , Y ⊕ Y 0 , Z ⊕ Z 0 , f ⊕ f 0 , g ⊕ g 0 , h ⊕ h0 ) 

(1,1,c)

(X ⊕ X 0 , Y ⊕ Y 0 , Q, f ⊕ f 0 , g 00 , h00 ). Let W be any object in D and apply the functor HW = HomD (W, −) to this diagram. By Lemma 11.4.2 (applied three times) we deduce that HW (c) : HW (Z ⊕ Z 0 ) → HW (Q) is an isomorphism. Hence c is an isomorphism and we conclude that (1) holds. Assume (1). We will show that (X, Y, Z, f, g, h) is a distinguished triangle. Let W be any object in D and set HW = HomD (W, −). By Lemma 11.4.2 we see that HW (X) → HW (Y ) → HW (Z) → HW (Z[1]) is exact as it is a direct summand of the exact sequence associated to the distinguished triangle (X ⊕ X 0 , Y ⊕ Y 0 , Z ⊕ Z 0 , f ⊕ f 0 , g ⊕ g 0 , h ⊕ h0 ). Using TR1 let (X, Y, Q, f, g 00 , h00 ) be a distinguished triangle. By TR3 there exists a morphism of distinguished triangles (X ⊕ X 0 , Y ⊕

11.4. ELEMENTARY RESULTS ON TRIANGULATED CATEGORIES

757

Y 0 , Z ⊕ Z 0 , f ⊕ f 0 , g ⊕ g 0 , h ⊕ h0 ) → (X, Y, Q, f, g 00 , h00 ). Composing this with the inclusion map we get a morphism of triangles (1, 1, c) : (X, Y, Z, f, g, h) −→ (X, Y, Q, f, g 00 , h00 ) Applying HW and using the above we once again see that HW (c) : HW (Z) → HW (Q) is an isomorphism and we conclude that c is an isomorphism. Hence we win.  Lemma 11.4.9. Let D be a pre-triangulated category. Let (X, Y, Z, f, g, h) be a distinguished triangle. (1) If h = 0, then there exists a left inverse s : Z → Y to g. (2) For any left inverse s : Z → Y of g the map f ⊕ s : X ⊕ Z → Y is an isomorphism. (3) For any objects X 0 , Z 0 of D the triangle (X 0 , X 0 ⊕ Z 0 , Z 0 , (1, 0), (0, 1), 0) is distinguished. Proof. To see (1) use that HomD (Z, Y ) → HomD (Z, Z) → HomD (Z, X[1]) is exact by Lemma 11.4.2. By the same token, if s is as in (2), then h = 0 and the sequence 0 → HomD (W, X) → HomD (W, Y ) → HomD (W, Z) → 0 is split exact (split by s : Z → Y ). Hence by Yoneda’s lemma we see that X⊕Z → Y is an isomorphism. The last assertion follows from TR1 and Lemma 11.4.8.  Lemma 11.4.10. Let D be a pre-triangulated category. Let f : X → Y be a morphism of D. The following are equivalent (1) f has a kernel, (2) f has a cokernel, (3) f is isomorphic to a map K ⊕ Z → Z ⊕ Q induced by idZ . Proof. Any morphism isomorphic to a map of the form X 0 ⊕ Z → Z ⊕ Y 0 has both a kernel and a cokernel. Hence (3) ⇒ (1), (2). Next we prove (1) ⇒ (3). Suppose first that f : X → Y is a monomorphism, i.e., its kernel is zero. By TR1 there exists a distinguished triangle (X, Y, Z, f, g, h) and by Lemma 11.4.2 we see that h = 0. Then Lemma 11.4.9 implies that Y = X ⊕ Z, i.e., we see that (3) holds. Next, assume f has a kernel K. As K → X is a monomorphism we conclude X = K ⊕ X 0 and f |X 0 : X 0 → Y is a monomorphism. Hence Y = X 0 ⊕ Y 0 and we win. The implication (2) ⇒ (3) is dual to this.  Let D be an additive category. Let e : A → A be an idempotent endomorphism of an object of D. If Ker(e) and Ker(1 − e) exist, then A = Ker(e) ⊕ Ker(1 − e) and moreover Ker(e) = Coker(1 − e). Dually, if Coker(e) and Coker(1 − e) exist, then A = Coker(e) ⊕ Coker(1 − e) and moreover Ker(e) = Coker(1 − e). Lemma 11.4.11. Let D be an additive category. (1) If D has countable products and kernels of maps which have a right inverse, then D has kernels of idempotents. (2) If D has countable coproducts and cokernels of maps which have a left inverse, then D has cokernels of idempotents. Proof. Let X be an object of D and let e : X → X be an idempotent. The functor e

W 7−→ Ker(MorD (W, X) − → MorD (W, X))

758

11. DERIVED CATEGORIES

if representable if and only if e has a kernel. Note that for any abelian group A and idempotent endomorphism e : A → A we have Y Y Ker(e : A → A) = Ker(Φ : A→ A) n∈N

n∈N

where Φ(a1 , a2 , a3 , . . .) = (ea1 + (1 − e)a2 , ea2 + (1 − e)a3 , . . .) Moreover, Φ has the right inverse Ψ(a1 , a2 , a3 , . . .) = (a1 , (1 − e)a1 + ea2 , (1 − e)a2 + ea3 , . . .). Hence (1) holds. The proof of (2) is dual.



Lemma 11.4.12. Let D be a pre-triangulated category. If D has countable products, then D has kernels of idempotents. If D has countable coproducts, then D has kernels of idempotents. Proof. Assume D has countable products. By Lemma 11.4.11 it suffices to check that morphisms which have a right inverse have kernels. Any morphism which has a right inverse is an epimorphism, hence has a kernel by Lemma 11.4.10. The second statement is dual to the first (see also remark preceding Lemma 11.4.11).  The following lemma makes it slightly easier to prove that a pre-triangulated category is triangulated. Lemma 11.4.13. Let D be a pre-triangulated category. In order to prove TR4 it suffices to show that given any pair of composable morphisms f : X → Y and g : Y → Z there exist (1) isomorphisms i : X 0 → X, j : Y 0 → Y and k : Z 0 → Z, and then setting f 0 = j −1 f i : X 0 → Y 0 and g 0 = k −1 gj : Y 0 → Z 0 there exist (2) distinguished triangles (X 0 , Y 0 , Q1 , f 0 , p1 , d1 ), (X 0 , Z 0 , Q2 , g 0 ◦f 0 , p2 , d2 ) and (Y 0 , Z 0 , Q3 , g 0 , p3 , d3 ), such that the assertion of TR4 holds. Proof. The replacement of X, Y, Z by X 0 , Y 0 , Z 0 is harmless by our definition of distinguished triangles and their isomorphisms. The lemma follows from the fact that the distinguished triangles (X 0 , Y 0 , Q1 , f 0 , p1 , d1 ), (X 0 , Z 0 , Q2 , g 0 ◦f 0 , p2 , d2 ) and (Y 0 , Z 0 , Q3 , g 0 , p3 , d3 ) are unique up to isomorphism by Lemma 11.4.6.  Lemma 11.4.14. Let D be a pre-triangulated category. Assume that D0 is an additive full subcategory of D. The following are equivalent (1) there exists a set of triangles T 0 such that (D0 , T 0 ) is a pre-triangulated subcategory of D, (2) D0 is preserved under [1], [−1] and given any morphism f : X → Y in D0 there exists a distinguished triangle (X, Y, Z, f, g, h) in D such that Z is isomorphic to an object of D0 . In this case T 0 is the set of distinguished triangles (X, Y, Z, f, g, h) of D such that X, Y, Z ∈ Ob(D0 ) and f, g, h ∈ Arrows(D0 ). Finally, if D is a triangulated category, then (1) and (2) are also equivalent to (3) D0 is a triangulated subcategory. Proof. Omitted. Lemma 11.4.15. An exact functor of pre-triangulated categories is additive.



11.4. ELEMENTARY RESULTS ON TRIANGULATED CATEGORIES

759

Proof. Let F : D → D0 be an exact functor of pre-triangulated categories. Since (0, 0, 0, 10 , 10 , 0) is a distinguished triangle of D the triangle (F (0), F (0), F (0), 1F (0) , 1F (0) , F (0)) 0

is distinguished in D . This implies that 1F (0) ◦ 1F (0) is zero, see Lemma 11.4.1. Hence F (0) is the zero object of D0 . This also implies that F applied to any zero morphism is zero (since a morphism in an additive category is zero if and only if it factors through the zero objet). Next, using that (X, X ⊕ Y, Y, (1, 0), (0, 1), 0) is a distinguished triangle, we see that (F (X), F (X ⊕ Y ), F (Y ), F (1, 0), F (0, 1), 0) is one too. This implies that the map F (1, 0) ⊕ F (0, 1) : F (X) ⊕ F (Y ) → F (X ⊕ Y ) is an isomorphism, see Lemma 11.4.9. We omit the rest of the argument.  Lemma 11.4.16. Let F : D → D0 be a fully faithful exact functor of pre-triangulated categories. Then a triangle (X, Y, Z, f, g, h) of D is distinguished if and only if (F (X), F (Y ), F (Z), F (f ), F (g), F (h)) is distinguished in D0 . Proof. The “if” part is clear. Assume (F (X), F (Y ), F (Z)) is distinguished in D0 . Pick a distinguished triangle (X, Y, Z 0 , f, g 0 , h0 ) in D. By Lemma 11.4.6 there exists an isomorphism of triangles (1, 1, c0 ) : (F (X), F (Y ), F (Z)) −→ (F (X), F (Y ), F (Z 0 )). Since F is fully faithful, there exists a morphism c : Z → Z 0 such that F (c) = c0 . Then (1, 1, c) is an isomorphism between (X, Y, Z) and (X, Y, Z 0 ). Hence (X, Y, Z) is distinguished by TR1.  Lemma 11.4.17. Let D, D0 , D00 be pre-triangulated categories. Let F : D → D0 and F 0 : D0 → D00 be exact functors. Then F 0 ◦ F is an exact functor. Proof. Omitted.



Lemma 11.4.18. Let D be a pre-triangulated category. Let A be an abelian category. Let H : D → A be a homological functor. (1) Let D0 be a pre-triangulated category. Let F : D0 → D be an exact functor. Then the composition G ◦ F is a homological functor as well. (2) Let A0 be an abelian category. Hence G : A → A0 be an exact functor. Hence G ◦ H is a homological functor as well. Proof. Omitted.



Lemma 11.4.19. Let D be a triangulated category. Let A be an abelian category. Let G : A → D be a δ-functor. (1) Let D0 be a triangulated category. Let F : D → D0 be an exact functor. Then the composition F ◦ G is a δ-functor as well. (2) Let A0 be an abelian category. Hence H : A0 → A be an exact functor. Hence G ◦ H is a δ-functor as well. Proof. Omitted.



Lemma 11.4.20. Let D be a triangulated category. Let A be an abelian category. Let G : A → D be a δ-functor. Let H : D → B be a homological functor. Assume that H −1 (G(A)) = 0 for all A in A. Then the collection {H n ◦ G, H n (δA→B→C )}n≥0 is a δ-functor from A → B, see Homology, Definition 10.9.1.

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b

Proof. The notation signifies the following. If 0 → A − →B → − C → 0 is a short exact sequence in A, then δ = δA→B→C : G(C) → G(A)[1] is a morphism in D such that (G(A), G(B), G(C), a, b, δ) is a distinguished triangle, see Definition 11.3.6. Then H n (δ) : H n (G(C)) → H n (G(A)[1]) = H n+1 (G(A)) is clearly functorial in the short exact sequence. Finally, the long exact cohomology sequence (11.3.5.1) combined with the vanishing of H −1 (G(C)) gives a long exact sequence H 0 (δ)

0 → H 0 (G(A)) → H 0 (G(B)) → H 0 (G(C)) −−−−→ H 1 (G(A)) → . . . in B as desired.



The proof of the following result uses TR4. Proposition 11.4.21. Let D be a triangulated category. Any commutative diagram /Y X  X0

 / Y0

can be extended to a diagram X

/Y

/Z

/ X[1]

 X0

 / Y0

 / Z0

 / X 0 [1]

 X 00

 / Y 00

 / Z 00

 / X 00 [1]

 X[1]

 / Y [1]

 / Z[1]

 / X[2]

where all the squares are commutative, except for the lower right square which is anticommutative. Moreover, each of the rows and columns are distinguished triangles. Finally, the morphisms on the bottom row (resp. right column) are obtained from the morphisms of the top row (resp. left column) by applying [1]. Proof. During this proof we avoid writing the arrows in order to make the proof legible. Choose distinguished triangles (X, Y, Z), (X 0 , Y 0 , Z 0 ), (X, X 0 , X 00 ), (Y, Y 0 , Y 00 ), and and (X, Y 0 , A). Note that the morphism X → Y 0 is both equal to the composition X → Y → Y 0 and equal to the composition X → X 0 → Y 0 . Hence, we can find morphisms (1) a : Z → A and b : A → Y 00 , and (2) a0 : X 00 → A and b0 : A → Z 0 as in TR4. Denote c : Y 00 → Z[1] the composition Y 00 → Y [1] → Z[1] and denote c0 : Z 0 → X 00 [1] the composition Z 0 → X 0 [1] → X 00 [1]. The conclusion of our application TR4 are that (1) (Z, A, Y 00 , a, b, c), (X 00 , A, Z 0 , a0 , b0 , c0 ) are distinguished triangles,

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761

(2) (X, Y, Z) → (X, Y 0 , A), (X, Y 0 , A) → (Y, Y 0 , Y 00 ), (X, X 0 , X 00 ) → (X, Y 0 , A), (X, Y 0 , A) → (X 0 , Y 0 , Z 0 ) are morphisms of triangles. First using that (X, X 0 , X 00 ) → (X, Y 0 , A) and (X, Y 0 , A) → (Y, Y 0 , Y 00 ). are morphisms of triangles we see the first of the commutative diagrams / Y0

X0  X 00  X[1]

b◦a0



/ Y 00

Y

/Z

 Y0

 / Z0

b0 ◦a

/ X[1]  / X 0 [1]

 / Y [1]

is commutative. The second is commutative too using that (X, Y, Z) → (X, Y 0 , A) and (X, Y 0 , A) → (X 0 , Y 0 , Z 0 ) are morphisms of triangles. At this point we choose a distinguished triangle (X 00 , Y 00 , Z 00 ) starting with the map b ◦ a0 : X 00 → Y 00 . Next we apply TR4 one more time to the morphisms X 00 → A → Y 00 and the triangles (X 00 , A, Z 0 , a0 , b0 , c0 ), (X 00 , Y 00 , Z 00 ), and (A, Y 00 , Z[1], b, c, −a[1]) to get morphisms a00 : Z 0 → Z 00 and b00 : Z 00 → Z[1]. Then (Z 0 , Z 00 , Z[1], a00 , b00 , −b0 [1] ◦ a[1]) is a distinguished triangle, hence also (Z, Z 0 , Z 00 , −b0 ◦ a, a00 , −b00 ) and hence also (Z, Z 0 , Z 00 , b0 ◦a, a00 , b00 ). Moreover, (X 00 , A, Z 0 ) → (X 00 , Y 00 , Z 00 ) and (X 00 , Y 00 , Z 00 ) → (A, Y 00 , Z[1], b, c, −a[1]) are morphisms of triangles. At this point we have defined all the distinguished triangles and all the morphisms, and all that’s left is to verify some commutativity relations. To see that the middle square in the diagram commutes, note that the arrow Y 0 → Z 0 factors as Y 0 → A → Z 0 because (X, Y 0 , A) → (X 0 , Y 0 , Z 0 ) is a morphism of triangles. Similarly, the morphism Y 0 → Y 00 factors as Y 0 → A → Y 00 because (X, Y 0 , A) → (Y, Y 0 , Y 00 ) is a morphism of triangles. Hence the middle square commutes because the square with sides (A, Z 0 , Z 00 , Y 00 ) commutes as (X 00 , A, Z 0 ) → (X 00 , Y 00 , Z 00 ) is a morphism of triangles (by TR4). The square with sides (Y 00 , Z 00 , Y [1], Z[1]) commutes because (X 00 , Y 00 , Z 00 ) → (A, Y 00 , Z[1], b, c, −a[1]) is a morphism of triangles and c : Y 00 → Z[1] is the composition Y 00 → Y [1] → Z[1]. The square with sides (Z 0 , X 0 [1], X 00 [1], Z 00 ) is commutative because (X 00 , A, Z 0 ) → (X 00 , Y 00 , Z 00 ) is a morphism of triangles and c0 : Z 0 → X 00 [1] is the composition Z 0 → X 0 [1] → X 00 [1]. Finally, we have to show that the square with sides (Z 00 , X 00 [1], Z[1], X[2]) anticommutes. This holds because (X 00 , Y 00 , Z 00 ) → (A, Y 00 , Z[1], b, c, −a[1]) is a morphism of triangles and we’re done.  11.5. Localization of triangulated categories In order to construct the derived category starting from the homotopy category of compexes, we will use a localization process. Definition 11.5.1. Let D be a pre-triangulated category. We say a multiplicative system S is compatible with the triangulated structure if the following two conditions hold: MS5 For s ∈ S we have s[n] ∈ S for all n ∈ Z.

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MS6 Given a solid commutative square /Y

X

s0

s

 X0

 / Y0

/Z

/ X[1]

 / Z0



s[1]

/ X 0 [1]

whose rows are distinguished triangles with s, s0 ∈ S there exists a morphism s00 : Z → Z 0 in S such that (s, s0 , s00 ) is a morphism of triangles. It turns out that these axioms are not independent of the axioms defining multiplicative systems. Lemma 11.5.2. Let D be a pre-triangulated category. Let S be a set of morphisms of D and assume that axioms MS1, MS5, MS6 hold (see Categories, Definition 4.24.1 and Definition 11.5.1). Then MS2 holds. Proof. Suppose that f : X → Y is a morphism of D and t : X → X 0 an element of S. Choose a distinguished triangle (X, Y, Z, f, g, h). Next, choose a distinguished triangle (X 0 , Y 0 , Z, f 0 , g 0 , t[1] ◦ h) (here we use TR1 and TR2). By MS5, MS6 (and TR2 to rotate) we can find the dotted arrow in the commutative diagram /Y

X

/Z s0

t

1

 / Y0

 X0

/ X[1]

 /Z



t[1]

/ X 0 [1]

with moreover s0 ∈ S. This proves LMS2. The proof of RMS2 is dual.



0

Lemma 11.5.3. Let F : D → D be an exact functor of pre-triangulated categories. Let S = {f ∈ Arrows(D) | F (f ) is an isomorphism} Then S is a saturated (see Categories, Definition 4.24.17) multiplicative system compatible with the triangulated structure on D. Proof. We have to prove axioms MS1 – MS6, see Categories, Definitions 4.24.1 and 4.24.17 and Definition 11.5.1. MS1, MS4, and MS5 are direct from the definitions. MS6 follows from TR3 and Lemma 11.4.3. By Lemma 11.5.2 we conclude that MS2 holds. To finish the proof we have to show that MS3 holds. To do this let f, g : X → Y be morphisms of D, and let t : Z → X be an element of S such that f ◦t = g ◦t. As D is additive this simply means that a◦t = 0 with a = f −g. Choose a distinguished triangle (Z, X, Q, t, g, h) using TR1 and TR2. Since a ◦ t = 0 we see by Lemma 11.4.2 there exists a morphism i : Q → Y such that i ◦ g = a. Finally, using TR1 again we can choose a triangle (Q, Y, W, i, j, k). Here is a picture Z

t

/X

g

/Q

a

 /Y

1

 X

i

j

 W

/ Z[1]

11.5. LOCALIZATION OF TRIANGULATED CATEGORIES

763

OK, and now we apply the functor F to this diagram. Since t ∈ S we see that F (Q) = 0, see Lemma 11.4.7. Hence F (j) is an isomorphism by the same lemma, i.e., j ∈ S. Finally, j ◦ a = j ◦ i ◦ g = 0 as j ◦ i = 0. Thus j ◦ f = j ◦ g and we see that LMS3 holds. The proof of RMS3 is dual.  Lemma 11.5.4. Let H : D → A be a homological functor between a pre-triangulated category and an abelian category. Let S = {f ∈ Arrows(D) | H i (f ) is an isomorphism for all i ∈ Z} Then S is a saturated (see Categories, Definition 4.24.17) multiplicative system compatible with the triangulated structure on D. Proof. We have to prove axioms MS1 – MS6, see Categories, Definitions 4.24.1 and 4.24.17 and Definition 11.5.1. MS1, MS4, and MS5 are direct from the definitions. MS6 follows from TR3 and the long exact cohomology sequence (11.3.5.1). By Lemma 11.5.2 we conclude that MS2 holds. To finish the proof we have to show that MS3 holds. To do this let f, g : X → Y be morphisms of D, and let t : Z → X be an element of S such that f ◦ t = g ◦ t. As D is additive this simply means that a ◦ t = 0 with a = f − g. Choose a distinguished triangle (Z, X, Q, t, g, h) using TR1 and TR2. Since a ◦ t = 0 we see by Lemma 11.4.2 there exists a morphism i : Q → Y such that i ◦ g = a. Finally, using TR1 again we can choose a triangle (Q, Y, W, i, j, k). Here is a picture Z

t

/X

g

/Q

a

 /Y

1

 X

/ Z[1]

i

j

 W

OK, and now we apply the functors H i to this diagram. Since t ∈ S we see that H i (Q) = 0 by the long exact cohomology sequence (11.3.5.1). Hence H i (j) is an isomorphism for all i by the same argument, i.e., j ∈ S. Finally, j ◦ a = j ◦ i ◦ g = 0 as j ◦ i = 0. Thus j ◦ f = j ◦ g and we see that LMS3 holds. The proof of RMS3 is dual.  Proposition 11.5.5. Let D be a pre-triangulated category. Let S be a multiplicative system compatible with the triangulated structure. Then there exists a unique structure of a pre-triangulated category on S −1 D such that the localization functor Q : D → S −1 D is exact. Moreover, if D is a triangulated category, so is S −1 D. Proof. We have seen that S −1 D is an additive category and that the localization functor Q is additive in Homology, Lemma 10.6.2, It is clear that we may define Q(X)[n] = Q(X[n]) since S is preserved under the shift functors [n] by MS5. Finally, we say a triangle of S −1 D is distinguished if it is isomorphic to the image of a distinguished triangle under the localization functor Q. Proof of TR1. The only thing to prove here is that if a : Q(X) → Q(Y ) is a morphism of S −1 D, then a fits into a distinguish triangle. Write a = Q(s)−1 ◦ Q(f ) for some s : Y → Y 0 in S and f : X → Y 0 . Choose a distinguished triangle

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11. DERIVED CATEGORIES

(X, Y 0 , Z, f, g, h) in D. Then we see that (Q(X), Q(Y ), Q(Z), a, Q(g) ◦ Q(s), Q(h)) is a distinguished triangle of S −1 D. Proof of TR2. This is immediate from the definitions. Proof of TR3. Note that the existence of the dotted arrow which is required to exist may be proven after replacing the two triangles by isomorphic triangles. Hence we may assume given distinguished triangles (X, Y, Z, f, g, h) and (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) of D and a commutative diagram Q(X)

Q(f )

a

 Q(X 0 )

/ Q(Y ) b

Q(f 0 )

 / Q(Y 0 )

in S −1 D. Now we apply Categories, Lemma 4.24.8 to find a morphism f 00 : X 00 → Y 00 in D and a commutative diagram X

k

f

 Y

l

/ X 00 o 

s

f 00

X0 f0

/ Y 00 o

t

 Y0

in D with s, t ∈ S and a = s−1 k, b = t−1 l. At this point we can use TR3 for D and MS6 to find a commutative diagram X k

/Y 

/Z l

m

 XO 00

/ Y 00 O

 / Z 00 O

s

t

r

X0

/ Y0

/ Z0

/ X[1] 

g[1]

/ X 00 [1] O s[1]

/ X 0 [1]

with r ∈ S. It follows that setting c = Q(r)−1 Q(m) we obtain the desired morphism of triangles (Q(X), Q(Y ), Q(Z), Q(f ), Q(g), Q(h)) (a,b,c)

 (Q(X 0 ), Q(Y 0 ), Q(Z 0 ), Q(f 0 ), Q(g 0 ), Q(h0 )) This proves the first statement of the lemma. If D is also a triangulated category, then we still have to prove TR4 in order to show that S −1 D is triangulated as well. To do this we reduce by Lemma 11.4.13 to the following statement: Given composable morphisms a : Q(X) → Q(Y ) and b : Q(Y ) → Q(Z) we have to produce an octahedron after possibly replacing Q(X), Q(Y ), Q(Z) by isomorphic objects. To do this we may first replace Y by an object such that a = Q(f ) for some morphism f : X → Y in D. (More precisely, write a = s−1 f with s : Y → Y 0 in S and f : X → Y 0 . Then replace Y by Y 0 .) After this we similarly replace Z by an object such that b = Q(g) for some morphism g : Y → Z. Now we can find distinguished triangles (X, Y, Q1 , f, p1 , d1 ), (X, Z, Q2 , g ◦ f, p2 , d2 ), and

11.5. LOCALIZATION OF TRIANGULATED CATEGORIES

765

(Y, Z, Q3 , g, p3 , d3 ) in D (by TR1), and morphisms a : Q1 → Q2 and b : Q2 → Q3 as in TR4. Then it is immediately verified that applying the functor Q to all these data gives a corresponding structure in S −1 D  The universal property of the localization of a triangulated category is as follows (we formulate this for pre-triangulated categories, hence it holds a fortiori for triangulated categories). Lemma 11.5.6. Let D be a pre-triangulated category. Let S be a multiplicative system compatible with the triangulated category. Let Q : D → S −1 D be the localization functor, see Proposition 11.5.5. (1) If H : D → A is a homological functor into an abelian category A such that H(s) is an isomorphism for all s ∈ S, then the unique factorization H 0 : S −1 D → A such that H = H 0 ◦ Q (see Categories, Lemma 4.24.6) is a homological functor too. (2) If F : D → D0 is an exact functor into a pre-triangulated category D0 such that F (s) is an isomorphism for all s ∈ S, then the unique factorization F 0 : S −1 D → D0 such that F = F 0 ◦ Q (see Categories, Lemma 4.24.6) is an exact functor too. Proof. This lemma proves itself. Details omitted.



The following lemma describes the kernel (see Definition 11.6.5) of the localization functor. Lemma 11.5.7. Let D be a pre-triangulated category. Let S be a multiplicative system compatible with the triangulated structure. Let Z be an object of D. The following are equivalent (1) (2) (3) (4)

Q(Z) = 0 in S −1 D, there exists Z 0 ∈ Ob(D) such that 0 : Z → Z 0 is an element of S, there exists Z 0 ∈ Ob(D) such that 0 : Z 0 → Z is an element of S, and there exists an object Z 0 and a distinguished triangle (X, Y, Z ⊕ Z 0 , f, g, h) such that f ∈ S.

If S is saturated, then these are also equivalent to (4) the morphism 0 → Z is an element of S, (5) the morphism Z → 0 is an element of S, (6) there exists a distinguished triangle (X, Y, Z, f, g, h) such that f ∈ S. Proof. The equivalence of (1), (2), and (3) is Homology, Lemma 10.6.3. If (2) holds, then (Z 0 [−1], Z 0 [−1] ⊕ Z, Z, (1, 0), (0, 1), 0) is a distinguised triangle (see Lemma 11.4.9) with “0 ∈ S”. By rotating we conclude that (4) holds. If (X, Y, Z ⊕ Z 0 , f, g, h) is a distinguished triangle with f ∈ S then Q(f ) is an isomorphism hence Q(Z ⊕ Z 0 ) = 0 hence Q(Z) = 0. Thus (1) – (4) are all equivalent. Next, assume that S is saturated. Note that each of (4), (5), (6) implies one of the equivalent conditions (1) – (4). Suppose that Q(Z) = 0. Then 0 → Z is a morphism of D which becomes an isomorphism in S −1 D. According to Categories, Lemma 4.24.18 the fact that S is saturated implies that 0 → Z is in S. Hence (1) ⇒ (4). Dually (1) ⇒ (5). Finally, if 0 → Z is in S, then the triangle (0, Z, Z, 0, idZ , 0) is distinguished by TR1 and TR2 and is a triangle as in (4). 

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Lemma 11.5.8. Let D be a triangulated category. Let S be a saturated multiplicative system in D. Let (X, Y, Z, f, g, h) be a distinguished triangle in D. Consider the category of morphisms of triangles I = {(s, s0 , s00 ) : (X, Y, Z, f, g, h) → (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) | (s, s0 , s00 ) ∈ S} Then I is a filtered category and the functors I → X/S, I → Y /S, and I → Z/S are surjective on objects. Proof. We strongly suggest the reader skip the proof of this lemma and instead works it out on a napkin. The category I is nonempty as the identity provides an object. This proves the first condition of the definition of a filtered category, see Categories, Definition 4.17.1. Note that if s : X → X 0 is a morphism of S, then using MS2 we can find s0 : Y → Y 0 and f 0 : X 0 → Y 0 such that f 0 ◦ s = s0 ◦ f , whereupon we can use MS6 to complete this into an object of I. Hence certainly the surjectivity statement is correct. Next we check condition (3) of Categories, Definition 4.17.1. Suppose (s1 , s01 , s001 ) : (X, Y, Z) → (X1 , Y1 , Z1 ) and (s2 , s02 , s002 ) : (X, Y, Z) → (X2 , Y2 , Z2 ) are objects of I, and suppose (a, b, c), (a0 , b0 , c0 ) are two morphisms between them. Since a ◦ s1 = a0 ◦ s1 there exists a morphism s3 : X2 → X3 such that s3 ◦ a = s3 ◦ a0 . Using the surjectivity statement we can complete this to a morphism of triangles (s3 , s03 , s003 ) : (X2 , Y2 , Z2 ) → (X3 , Y3 , Z3 ) with s3 , s03 , s003 ∈ S. Thus (s3 ◦ s2 , s03 ◦ s02 , s003 ◦ s002 ) : (X, Y, Z) → (X3 , Y3 , Z3 ) is also an object of I and after composing the maps (a, b, c), (a0 , b0 , c0 ) with (s3 , s03 , s003 ) we obtain a = a0 . By rotating we may do the same to get b = b0 and c = c0 . Finally, we check condition (2) of Categories, Definition 4.17.1. Suppose we are given two objects (s1 , s01 , s001 ) : (X, Y, Z) → (X1 , Y1 , Z1 ) and (s2 , s02 , s002 ) : (X, Y, Z) → (X2 , Y2 , Z2 ) of I. Pick a morphism s3 : X → X3 in S such that there exist morphisms a : X1 → X3 and a0 : X2 → X3 with s3 = a ◦ s1 and s3 = a0 ◦ s2 . Because S is a saturated multiplicative system we see that a0 ∈ S (because S is the set of arrows of D which are turned into isomorphisms in S −1 D, see Categories, Lemma 4.24.18). Hence, by the essential surjectivity above, we can find a morphism (a, b, c) : (X2 , Y2 , Z2 ) → (X3 , Y3 , Z3 ) with a, b, c ∈ S such that X → X3 factors through X → X1 . Replacing (X2 , Y2 , Z2 ) by (X3 , Y3 , Z3 ) and repeating this argument twice more, we may assume that s2 = a ◦ s1 , s02 = b ◦ s01 , and s002 = c ◦ s001 . The problem is that it may not be the case that b ◦ f1 = f2 ◦ a, etc, i.e., we don’t know that (a, b, c) is a morphism of triangles from (X1 , Y1 , Z1 ) to (X2 , Y2 , Z2 ). On the other hand, we do know that (a, b, c) ◦ (s1 , s01 , s001 ) is a morphism of triangles. Using MS3 one more time this means there exist morphisms s3 : X2 → X3 , s03 : Y2 → Y3 , and s003 : Z2 → Z3 in S such that the required equalities hold after post-composing with them, e.g., s03 ◦ b ◦ f1 = s03 ◦ f2 ◦ a, etc. Using the essential surjectivity above once more we see that we may find a morphism of triangles (s4 , s04 , s004 ) : (X2 , Y2 , Z2 ) → (X4 , Y4 , Z4 ) with s4 , s04 , s004 ∈ S such that s4 factors through s3 , s04 factors through s03 , and s004 factors through s003 . We conclude that (s4 ◦ a, s04 ◦ b, s004 ◦ c) is a morphism of triangles from (X1 , Y1 , Z1 ) to (X4 , Y4 , Z4 ).  11.6. Quotients of triangulated categories Given a triangulated category and a triangulated subcategory we can construct another triangulated category by taking the “quotient”. The construction uses

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767

a localization. This is similar to the quotient of an abelian category by a Serre subcategory, see Homology, Section 10.7. Before we do the actual construction we briefly discuss kernels of exact functors. Definition 11.6.1. Let D be a pre-triangulated category. We say a full pretriangulated subcategory D0 of D is saturated if whenever X ⊕ Y is isomorphic to an object of D0 then both X and Y are isomorphic to objects of D0 . Lemma 11.6.2. Let F : D → D0 be an exact functor of pre-triangulated categories. Let D00 be the full subcategory of D with objects Ob(D00 ) = {X ∈ Ob(D) | F (X) = 0} Then D00 is a strictly full saturated pre-triangulated subcategory of D. If D is a triangulated category, then D00 is a triangulated subcategory. Proof. It is clear that D00 is preserved under [1] and [−1]. If (X, Y, Z, f, g, h) is a distinguished triangle of D and F (X) = F (Y ) = 0, then also F (Z) = 0 as (F (X), F (Y ), F (Z), F (f ), F (g), F (h)) is distinguished. Hence we may apply Lemma 11.4.14 to see that D00 is a pre-triangulated subcategory (respectively a triangulated subcategory if D is a triangulated category). The final assertion of being saturated follows from F (X) ⊕ F (Y ) = 0 ⇒ F (X) = F (Y ) = 0.  Lemma 11.6.3. Let H : D → A be a homological functor of a pre-triangulated category into an abelian category. Let D0 be the full subcategory of D with objects Ob(D0 ) = {X ∈ Ob(D) | H(X[n]) = 0 for all n ∈ Z} Then D0 is a strictly full saturated pre-triangulated subcategory of D. If D is a triangulated category, then D0 is a triangulated subcategory. Proof. It is clear that D0 is preserved under [1] and [−1]. If (X, Y, Z, f, g, h) is a distinguished triangle of D and H(X[n]) = H(Y [n]) = 0 for all n, then also H(Z[n]) = 0 for all n by the long exact sequence (11.3.5.1). Hence we may apply Lemma 11.4.14 to see that D0 is a pre-triangulated subcategory (respectively a triangulated subcategory if D is a triangulated category). The assertion of being saturated follows from H((X ⊕ Y )[n]) = 0 ⇒ H(X[n] ⊕ Y [n]) = 0 ⇒ H(X[n]) ⊕ H(Y [n]) = 0 ⇒ H(X[n]) = H(Y [n]) = 0 for all n ∈ Z.



Lemma 11.6.4. Let H : D → A be a homological functor of a pre-triangulated + − b category into an abelian category. Let DH , DH , DH be the full subcategory of D with objects + Ob(DH ) = {X ∈ Ob(D) | H(X[n]) = 0 for all n  0} − Ob(DH ) = {X ∈ Ob(D) | H(X[n]) = 0 for all n  0} b Ob(DH ) = {X ∈ Ob(D) | H(X[n]) = 0 for all |n|  0}

Each of these is a strictly full saturated pre-triangulated subcategory of D. If D is a triangulated category, then each is a triangulated subcategory.

768

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+ Proof. Let us prove this for DH . It is clear that it is preserved under [1] and [−1]. If (X, Y, Z, f, g, h) is a distinguished triangle of D and H(X[n]) = H(Y [n]) = 0 for all n  0, then also H(Z[n]) = 0 for all n  0 by the long exact sequence (11.3.5.1). + Hence we may apply Lemma 11.4.14 to see that DH is a pre-triangulated subcategory (respectively a triangulated subcategory if D is a triangulated category). The assertion of being saturated follows from

H((X ⊕ Y )[n]) = 0 ⇒ H(X[n] ⊕ Y [n]) = 0 ⇒ H(X[n]) ⊕ H(Y [n]) = 0 ⇒ H(X[n]) = H(Y [n]) = 0 for all n ∈ Z.



Definition 11.6.5. Let D be a (pre-)triangulated category. (1) Let F : D → D0 be an exact functor. The kernel of F is the strictly full saturated (pre-)triangulated subcategory described in Lemma 11.6.2. (2) Let H : D → A be a homological functor. The kernel of H is the strictly full saturated (pre-)triangulated subcategory described in Lemma 11.6.3. These are sometimes denoted Ker(F ) or Ker(H). The proof of the following lemma uses TR4. Lemma 11.6.6. Let D be a triangulated category. Let D0 ⊂ D be a full triangulated subcategory. Set   f ∈ Arrows(D) such that there exists a distinguished triangle (11.6.6.1) S = (X, Y, Z, f, g, h) of D with Z isomorphic to an object of D0 Then S is a multiplicative system compatible with the triangulated structure on D. In this situation the following are equivalent (1) S is a saturated multiplicative system, (2) D0 is a saturated triangulated subcategory. Proof. To prove the first assertion we have to prove that MS1, MS2, MS3 and MS5, MS6 hold. Proof of MS1. It is clear that identities are in S because (X, X, 0, 1, 0, 0) is distinguished for every object X of D and because 0 is an object of D0 . Let f : X → Y and g : Y → Z be composable morphisms contained in S. Choose distinguished triangles (X, Y, Q1 , f, p1 , d1 ), (X, Z, Q2 , g ◦ f, p2 , d2 ), and (Y, Z, Q3 , g, p3 , d3 ). By assumption we know that Q1 and Q3 are isomorphic to objects of D0 . By TR4 we know there exists a distinguished triangle (Q1 , Q2 , Q3 , a, b, c). Since D0 is a triangulated subcategory we conclude that Q2 is isomorphic to an object of D0 . Hence g ◦ f ∈ S. Proof of MS3. Let a : X → Y be a morphism and let t : Z → X be an element of S such that a ◦ t = 0. To prove LMS3 we have to find a s ∈ S such that s ◦ a = 0. Choose a distinguished triangle (Z, X, Q, t, g, h) using TR1 and TR2. Since a◦t = 0 we see by Lemma 11.4.2 there exists a morphism i : Q → Y such that i ◦ g = a.

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Finally, using TR1 again we can choose a triangle (Q, Y, W, i, j, k). Here is a picture Z

/X

t

g

/Q

a

 /Y

1

 X

/ Z[1]

i

j

 W

Since t ∈ S we see that Q is isomorphic to an object of D0 . Hence j ∈ S. Finally, j ◦ a = j ◦ i ◦ g = 0 as j ◦ i = 0. Thus j ◦ f = j ◦ g and we see that LMS3 holds. The proof of RMS3 is dual. Proof of MS5. Follows as distinguished triangles and D0 are stable under translations Proof of MS6. Suppose given a commutative diagram /Y

X

s0

s

 X0

 / Y0

with s, s0 ∈ S. By Proposition 11.4.21 we can extend this to a nine square diagram. As s, s0 are elments of S we see that X 00 , Y 00 are isomorphic to objects of D0 . Since D0 is a full triangulated subcategory we see that Z 00 is also an object of D0 . Whence the morphism Z 0 → Z 00 is an element of S. This proves MS6. MS2 is a formal consquence of MS1, MS5, and MS6, see Lemma 11.5.2. This finishes the proof of the first assertion of the lemma. Let’s assume that S is saturated. (In the following we will use rotation of distinguished triangles without further mention.) Let X⊕Y be an object isomorphic to an object of D0 . Consider the morphism f : 0 → X. The composition 0 → X → X ⊕Y is an element of S as (0, X ⊕ Y, X ⊕ Y, 0, 1, 0) is a distinguished triangle. The composition Y [−1] → 0 → X is an element of S as (X, X ⊕ Y, Y, (1, 0), (0, 1), 0) is a distinguished triangle, see Lemma 11.4.9. Hence 0 → X is an element of S (as S is saturated). Thus X is isomorphic to an object of D0 as desired. Finally, assume D0 satisfies condition (2) of the lemma. Let h

g

f

W − →X− →Y − →Z be composable morphisms of D such that f g, gh ∈ S. We will build up a picture of objects as in the diagram below.

+1

+1

W

~

} Q1 a o

Q12 a o

+1 +1

+1 +1

/X

}

} Q2 a o

Q23 a +1 +1

/Y

}

Q3 ` /Z

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11. DERIVED CATEGORIES

First choose distinguished triangles (W, X, Q1 ), (X, Y, Q2 ), (Y, Z, Q3 ) (W, Y, Q12 ), and (X, Z, Q23 ). Denote s : Q2 → Q1 [1] the composition Q2 → X[1] → Q1 [1]. Denote t : Q3 → Q2 [1] the composition Q3 → Y [1] → Q2 [1]. By TR4 applied to the composition W → X → Y and the composition X → Y → Z there exist a distinguished triangles (Q1 , Q12 , Q2 ) and (Q2 , Q23 , Q3 ) which use the morphisms s and t. The objects Q12 and Q23 are isomorphic to objects of D0 as W → Y and X → Z are assumed in S. Hence also s[1]t is an element of S as S is closed under compositions and shifts. Note that s[1]t = 0 as Y [1] → Q2 [1] → X[2] is zero, see Lemma 11.4.1. Hence Q3 ⊕ Q1 [2] is isomorphic to an object of D0 , see Lemma 11.4.9. By assumption on D0 we conclude that Q3 , Q1 are isomorphic to objects of D0 . Looking at the distinguished triangle (Q1 , Q12 , Q2 ) we conclude that Q2 is also isomorphic to an object of D0 . Looking at the distinguished triangle (X, Y, Q2 ) we finally conclude that g ∈ S. (It is also follows that h, f ∈ S, but we don’t need this.)  Definition 11.6.7. Let D be a triangulated category. Let B be a full triangulated subcategory. We define the quotient category D/B by the formula D/B = S −1 D, where S is the multiplicative system of D associated to B via Lemma 11.6.6. The localization functor Q : D → D/B is called the quotient functor in this case. Note that the quotient functor Q : D → D/B is an exact functor of triangulated categories, see Proposition 11.5.5. The universal property of this construction is the following. Lemma 11.6.8. Let D be a triangulated category. Let B be a full triangulated subcategory of D. Let Q : D → D/B be the quotient functor. (1) If H : D → A is a homological functor into an abelian category A such that B ⊂ Ker(H) then there exists a unique factorization H 0 : D/B → A such that H = H 0 ◦ Q and H 0 is a homological functor too. (2) If F : D → D0 is an exact functor into a pre-triangulated category D0 such that B ⊂ Ker(F ) then there exists a unique factorization F 0 : D/B → D0 such that F = F 0 ◦ Q and F 0 is an exact functor too. Proof. This lemma follows from Lemma 11.5.6. Namely, if f : X → Y is a morphism of D such that for some distinguished triangle (X, Y, Z, f, g, h) the object Z is isomorphic to an object of B, then H(f ), resp. F (f ) is an isomorphism under the assumptions of (1), resp. (2). Details omitted.  The kernel of the quotient functor can be described as follows. Lemma 11.6.9. Let D be a triangulated category. Let B be a full triangulated subcategory. The kernel of the quotient functor Q : D → D/B is the strictly full subcategory of D whose objects are   Z ∈ Ob(D) such that there exists a Z 0 ∈ Ob(D) Ob(Ker(Q)) = such that Z ⊕ Z 0 is isomorphic to an object of B In other words it is the smallest strictly full saturated triangulated subcategory of D containing B. Proof. First note that the kernel is automatically a strictly full triangulated subcategory stable containing summands of any of its objects, see Lemma 11.6.2. The description of its objects follows from the definitions and Lemma 11.5.7 part (4). 

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771

Let D be a triangulated category. At this point we have constructions which induce order preserving maps between (1) the partially ordered set of multiplicative systems S in D compatible with the triangulated structure, and (2) the partially ordered set of full triangulated subcategories B ⊂ D. Namely, the constructions are given by S 7→ B(S) = Ker(Q : D → S −1 D) and B 7→ S(B) where S(B) is the multiplicative set of (11.6.6.1), i.e.,   f ∈ Arrows(D) such that there exists a distinguished triangle S(B) = (X, Y, Z, f, g, h) of D with Z isomorphic to an object of B Note that it is not the case that these operations are mutually inverse. Lemma 11.6.10. Let D be a triangulated category. The operations described above have the following properties (1) S(B(S)) is the “saturation” of S, i.e., it is the smallest saturated multiplicative system in D containing S, and (2) B(S(B)) is the “saturation” of B, i.e., it is the smallest strictly full saturated triangulated subcategory of D containing B. In particular, the constructions define mutually inverse maps between the (partially ordered) set of saturated multiplicative systems in D compatible with the triangulated structure on D and the (partially ordered) set of strictly full saturated triangulated subcategories of D. Proof. First, let’s start with a full triangulated subcategory B. Then B(S(B)) = Ker(Q : D → D/B) and hence (2) is the content of Lemma 11.6.9. Next, suppose that S is multiplicative system in D compatible with the triangulation on D. Then B(S) = Ker(Q : D → S −1 D). Hence (using Lemma 11.4.7 in the localized category)   f ∈ Arrows(D) such that there exists a distinguished S(B(S)) = triangle (X, Y, Z, f, g, h) of D with Q(Z) = 0 = {f ∈ Arrows(D) | Q(f ) is an isomorphism} = Sˆ = S 0 in the notation of Categories, Lemma 4.24.18. The final statement of that lemma finishes the proof.  Lemma 11.6.11. Let H : D → A be a homological functor from a triangulated category D to an abelian category A, see Definition 11.3.5. The subcategory Ker(H) of D is a strictly full saturated triangulated subcategory of D whose corresponding saturated multiplicative system (see Lemma 11.6.10) is the set S = {f | H i (f ) is an isomorphism for all i ∈ Z}. The functor H factors through the quotient functor Q : D → D/Ker(H). Proof. The category Ker(H) is a strictly full saturated triangulated subcategory of D by Lemma 11.6.3. The set S is a saturated multiplicative system compatible with the triangulated structure by Lemma 11.5.4. Recall that the multiplicative system corresponding to Ker(H) is the set   f ∈ Arrows(K(A)) such that there exists a distinguished triangle (X, Y, Z, f, g, h) with H i (Z) = 0 for all i

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By the long exact cohomology sequence, see (11.3.5.1), it is clear that f is an element of this set if and only if f is an element of S. Finally, the factorization of H through Q is a consequence of Lemma 11.6.8.  It is clear that in the lemma above the factorization of H through D/Ker(H) is the universal factorization. Namely, if F : D → D0 is an exact functor of triangulated categories and if there exists a homological functor H 0 : D0 → A such that H ∼ = H 0 ◦ F , then F factors through the quotient functor Q : D → D/Ker(H). 11.7. The homotopy category Let A be an additive category. The homotopy category K(A) of A is the the category of complexes of A with morphisms given by morphisms of complexes up to homotopy. Here is the formal definition. Definition 11.7.1. Let A be an additive category. (1) We set Comp(A) = CoCh(A) be the category of (cochain) complexes. (2) A complex K • is said to be bounded below if K n = 0 for all n  0. (3) A complex K • is said to be bounded above if K n = 0 for all n  0. (4) A complex K • is said to be bounded if K n = 0 for all |n|  0. (5) We let Comp+ (A), Comp− (A), resp. Compb (A) be the full subcategory of Comp(A) whose objects are the complexes which are bounded below, bounded above, resp. bounded. (6) We let K(A) be the category with the same objects as Comp(A) but as morphisms homotopy classes of maps of complexes (see Homology, Lemma 10.10.7). (7) We let K + (A), K − (A), resp. K b (A) be the full subcategory of K(A) whose objects are bounded below, bounded above, resp. bounded complexes of A. It will turn out that the categories K(A), K + (A), K − (A), and K b (A) are triangulated categories. To prove this we first develop some machinery related to cones and split exact sequences. 11.8. Cones and termwise split sequences Let A be an additive category, and let K(A) denote the category of complexes of A with morphisms given by morphisms of complexes up to homotopy. Note that the shift functors [n] on complexes, see Homology, Definition 10.12.7, give rise to functors [n] : K(A) → K(A) such that [n] ◦ [m] = [n + m] and [0] = id. Definition 11.8.1. Let A be an additive category. Let f : K • → L• be a morphism of complexes of A. The cone of f is the complex C(f )• given by C(f )n = Ln ⊕K n+1 and differential  n  dL f n+1 dnC(f ) = 0 −dn+1 K It comes equipped with canonical morphisms of complexes i : L• → C(f )• and p : C(f )• → K • [1] induced by the obvious maps Ln → C(f )n → K n+1 . In other words (K, L, C(f ), f, i, p) forms a triangle: K • → L• → C(f )• → K • [1] The formation of this triangle is functorial in the following sense.

11.8. CONES AND TERMWISE SPLIT SEQUENCES

773

Lemma 11.8.2. Suppose that K1•

f1

a

 K2•

/ L•1 b

f2

 / L•2

is a diagram of morphisms of complexes which is commutative up to homotopy. Then there exists a morphism c : C(f1 )• → C(f2 )• which gives rise to a morphism of triangles (a, b, c) : (K1• , L•1 , C(f1 )• , f1 , i1 , p1 ) → (K1• , L•1 , C(f1 )• , f2 , i2 , p2 ) of K(A). Proof. Let hn : K1n → Ln−1 be a family of morphisms such that f2 ◦ a − b ◦ f1 = 2 d ◦ h + h ◦ d. Define cn by the matrix  n  a hn+1 cn = : Ln1 ⊕ K1n+1 → Ln2 ⊕ K2n+1 0 bn A matrix computation show that c is a morphism of complexes. It is trivial that c ◦ i1 = i2 ◦ b, and it is trivial also to check that p2 ◦ c = a ◦ p1 .  Note that the morphism c : C(f1 )• → C(f2 )• constructed in the proof of Lemma 11.8.2 in general depends on the chosen homotopy h between f2 ◦ a and b ◦ f1 . Definition 11.8.3. Let A be an additive category. A termwise split injection α : A• → B • is a morphism of complexes such that each An → B n is isomorphic to the inclusion of a direct summand. A termwise split surjection β : B • → C • is a morphism of complexes such that each B n → C n is isomorphic to the projection onto a direct summand. Lemma 11.8.4. Let A be an additive category. Let / B• A• f

a

 C•

b

g

 / D•

be a diagram of morphisms of complexes commuting up to homotopy. If f is a split injection, then b is homotopic to a morphism which makes the diagram commute. If g is a split surjection, then a is homotopic to a morphism which makes the diagram commute. Proof. Let hn : An → Dn−1 be a collection of morphisms such that bf − ga = dh + hd. Suppose that π n : B n → An are morphisms splitting the morphisms f n . Take b0 = b + dhπ + hπd. Suppose sn : Dn → C n are morphisms splitting the morphisms g n : C n → Dn . Take a0 = a + dsh + shd. Computations omitted.  The following lemma can be used to replace an morphism of complexes by a morphism where in each degree the map is the injection of a direct summand. Lemma 11.8.5. Let A be an additive category. Let α : K • → L• be a morphism of complexes of A. There exists a factorization K•

α ˜

˜• /L α

such that

π

/6 L•

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11. DERIVED CATEGORIES

(1) α ˜ is a termwise split injection (see Definition 11.8.3), ˜ • such that π ◦ s = idL• and such (2) there is a map of complexes s : L• → L that s ◦ π is homotopic to idL˜ • . ˜•. Moreover, if both K • and L• are in K + (A), K − (A), or K b (A), then so is L Proof. We set ˜ n = Ln ⊕ K n ⊕ K n+1 L and we define dnL dnL˜ =  0 0 

0 dnK 0

0



idK n+1  −dn+1 K

Moreover, we set  α α ˜ = idK n  0 

which is clearly a split injection. It is also clear that it defines a morphism of complexes. We define
π = idLn 0 0 so that clearly π ◦ α ˜ = α. We set   idLn s= 0  0 ˜n → L ˜ n−1 be the map which maps the so that π ◦ s = idL• . Finally, let hn : L n n ˜ ˜ n−1 . Then summand K of L via the identity morphism to the summand K n of L it is a trivial matter (see computations in remark below) to prove that idL˜ • − s ◦ π = d ◦ h + h ◦ d which finishes the proof of the lemma.



Remark 11.8.6. To see the last displayed equality in the proof above we can argue with elements as follows. We have sπ(l, k, k + ) = (l, 0, 0). Hence the morphism of the left hand side maps (l, k, k + ) to (0, k, k + ). On the other hand h(l, k, k + ) = (0, 0, k) and d(l, k, k + ) = (dl, dk + k + , −dk + ). Hence (dh + hd)(l, k, k + ) = d(0, 0, k) + h(dl, dk + k + , −dk + ) = (0, k, −dk) + (0, 0, dk + k + ) = (0, k, k + ) as desired. Lemma 11.8.7. Let A be an additive category. Let α : K • → L• be a morphism of complexes of A. There exists a factorization K•

i

˜• /K

α ˜

/6 L•

α

such that (1) α ˜ is a termwise split surjection (see Definition 11.8.3), ˜ • → K • such that s ◦ i = idK • and such (2) there is a map of complexes s : K that i ◦ s is homotopic to idK˜ • . ˜ •. Moreover, if both K • and L• are in K + (A), K − (A), or K b (A), then so is K

11.8. CONES AND TERMWISE SPLIT SEQUENCES

775

Proof. Dual to Lemma 11.8.5. Take ˜ n = K n ⊕ Ln ⊕ Ln+1 K and we define dnK˜

 n dK = 0 0

0 dnL 0

α ˜= α

idLn

0



idLn+1  −dn+1 L

Moreover, we set 0




which is clearly a split surjection. It is also clear that it defines a morphism of complexes. We define   idK n i= 0  0 so that clearly α ˜ ◦ i = α. We set
s = idK n 0 0 ˜n → K ˜ n−1 be the map which maps the so that s ◦ i = idK • . Finally, let hn : K n n ˜ ˜ n−1 . Then summand L of K via the identity morphism to the summand Ln of K it is a trivial matter to prove that idK˜ • − i ◦ s = d ◦ h + h ◦ d which finishes the proof of the lemma.



Definition 11.8.8. Let A be an additive category. A termwise split sequence of complexes of A is a complex of complexes α

β

0 → A• − → B• − → C• → 0 together with given direct sum decompositions B n = An ⊕ C n compatible with αn and β n . We often write sn : C n → B n and π n : B n → An for the maps induced by the direct sum decompositions. According to Homology, Lemma 10.12.10 we get an associated morphism of complexes δ : C • −→ A• [1] which in degree n is the map π n+1 ◦ dnC ◦ sn . In other words (A• , B • , C • , α, β, δ) forms a triangle A• → B • → C • → A• [1] This will be the triangle associated to the termwise split sequence of complexes. Lemma 11.8.9. Let A be an additive category. Let 0 → A• → B • → C • → 0 be termwise split exact sequences as in Definition 11.8.8. Let (π 0 )n , (s0 )n be a second collection of splittings. Denote δ 0 : C • −→ A• [1] the morphism associated to this second set of splittings. Then (1, 1, 1) : (A• , B • , C • , α, β, δ) −→ (A• , B • , C • , α, β, δ 0 ) is an isomorphism of triangles in K(A). Proof. The statement simply means that δ and δ 0 are homotopic maps of complexes. This is Homology, Lemma 10.12.12. 

776

11. DERIVED CATEGORIES

Lemma 11.8.10. Let A be an additive category. Let 0 → A•i → Bi• → Ci• → 0, i = 1, 2 be termwise split exact sequences. Suppose that a : A•1 → A•2 , b : B1• → B2• , and c : C1• → C2• are morphisms of complexes such that A•1 a

 A•2

/ B1• b

 / B2•

/ C1• c

 / C2•

commutes in K(A). Then there exists a morphism b0 : B1• → B2• which is homotopic to b such that the diagram above commutes in the category of complexes. Proof. Let f n : An1 → B2n−1 be a collection of morphisms such that b ◦ α1 − α2 ◦ a = d ◦ f + f ◦ d. Let g n : B1n → C2n−1 be a collection of morphisms such that c◦β1 −β2 ◦b = d◦g+g◦d. Suppose that π n : B1n → An1 (resp. sn : C2n → B2n ) are the morphisms splitting the morphisms α1n (resp. β2n ). Set hn = −f n ◦ π n + sn−1 ◦ g n . Take b0 = b + d ◦ h + h ◦ d. Computation omitted.  Lemma 11.8.11. Let A be an additive category. Let f1 : K1• → L•1 and f2 : K2• → L•2 be morphisms of complexes. Let (a, b, c) : (K1• , L•1 , C(f1 )• , f1 , i1 , p1 ) −→ (K1• , L•1 , C(f1 )• , f2 , i2 , p2 ) be any morphism of triangles of K(A). If a and b are homotopy equivalences then so is c. Proof. Let a−1 : K2• → K1• be a morphism of complexes which is inverse to a in K(A). Let b−1 : L•2 → L•1 be a morphism of complexes which is inverse to b in K(A). Let c0 : C(f2 )• → C(f1 )• be the morphism from Lemma 11.8.2 applied to f1 ◦ a−1 = b−1 ◦ f2 . If we can show that c ◦ c0 and c0 ◦ c are isomorphisms in K(A) then we win. Hence it suffices to prove the following: Given a morphism of triangles (1, 1, c) : (K • , L• , C(f )• , f, i, p) in K(A) the morphism c is an isomorphism in K(A). By assumption the two squares in the diagram L• 1

 L•

/ C(f )• c

 / C(f )•

/ K • [1] 1

 / K • [1]

commute up to homotopy. By construction of C(f )• the rows form termwise split sequences of complexes. By Lemma 11.8.10 we may replace c by a morphism homotopic to c such that the diagram commutes in the category of complexes. In this case each cn is an isomorphism (because an upper triangular matrix with 1’s on the diagonal is invertible).  Hence if a and b are homotopy equivalences then the resulting morphism of triangles is an isomorphism of triangles in K(A). It turns out that the collection of triangles of K(A) given by cones and the collection of triangles of K(A) given by termwise split sequences of complexes are the same up to isomorphisms, at least up to sign! Lemma 11.8.12. Let A be an additive category.

11.8. CONES AND TERMWISE SPLIT SEQUENCES

777

(1) Given a termwise split sequence of complexes (α : A• → B • , β : B • → C • , sn , π n ) there exists a homotopy equivalence C(α)• → C • such that the diagram A•

/ B•

/ C(α)•

 A•

 / B•

 / C•

−p

δ

/ A• [1]  / A• [1]

defines an isomorphism of triangles in K(A). (2) Given a morphism of complexes f : K • → L• there exists an isomorphism of triangles K•

˜• /L

/ M•

 K•

 / L•

 / C(f )•

δ

−p

/ K • [1]  / K • [1]

where the upper triangle is the triangle associated to a termwise split exact ˜• → M •. sequence K • → L Proof. Proof of (1). We have C(α)n = B n ⊕ An+1 and we simply define C(α)n → C n via the projection onto B n followed by β n . This defines a morphism of complexes because the compositions An+1 → B n+1 → B n → C n are zero. To get a homotopy inverse we take C • → C(α)• given by (sn , −δ n ) in degree n. This is a morphism of complexes because the morphism δ n can be characterized as the unique morphism C n → An+1 such that d ◦ sn − sn+1 ◦ d = α ◦ δ n , see proof of Homology, Lemma 10.12.10. The composition C • → C(f )• → C • is the identity. The composition C(f )• → C • → C(f )• is equal to the morphism  n  s ◦ βn 0 −δ n ◦ β n 0 To see that this is homotopic to the identity map use the homotopy hn : C(α)n → C(α)n−1 ) given by the matrix   0 0 : C(α)n = B n ⊕ An+1 → B n−1 ⊕ An = C(α)n−1 πn 0 It is trivial to verify that     
βn 1 0 d n n −δ − s = 0 1 0 0

αn+1 −d



0 πn

  0 0 + 0 π n+1

 0 d 0 0

αn+1 −d



To finish the proof of (1) we have to show that the morphisms −p : C(α)• → A• [1] (see Definition 11.8.1) and C(α)• → C • → A• [1] agree up to homotopy. This is clear from the above. Namely, we can use the homotopy inverse (s, −δ) : C • → C(α)• and check instead that the two maps C • → A• [1] agree. And note that p ◦ (s, −δ) = −δ as desired. ˜ • , s : L• → L ˜ • and π : L• → L• be as in Proof of (2). We let f˜ : K • → L Lemma 11.8.5. By Lemmas 11.8.2 and 11.8.11 the triangles (K • , L• , C(f ), i, p) ˜ • , C(f˜), ˜i, p˜) are isomorphic. Note that we can compose isomorphisms and (K • , L ˜ • and f by f˜. In other words we may of triangles. Thus we may replace L• by L

778

11. DERIVED CATEGORIES

assume that f is a termwise split injection. In this case the result follows from part (1).  Lemma 11.8.13. Let A be an additive category. Let A•1 → A•2 → . . . → A•n be a sequence of composable morphisms of complexes. There exists a commutative diagram / A•2 / ... / A•n A•1 O O O B1•

/ B2•

/ ...

/ Bn•

• is a split injection and each Bi• → A•i is a such that each morphism Bi• → Bi+1 homotopy equivalence. Moreover, if all A•i are in K + (A), K − (A), or K b (A), then so are the Bi• .

Proof. The case n = 1 is without content. Lemma 11.8.5 is the case n = 2. Suppose we have constructed the diagram except for Bn . Applying Lemma 11.8.5 to the composition Bn−1 → An−1 → An . The result is a factorization Bn−1 → ˜n → An as desired. B  Lemma 11.8.14. Let A be an additive category. Let (α : A• → B • , β : B • → C • , sn , π n ) be a termwise split sequence of complexes. Let (A• , B • , C • , α, β, δ) be the associated triangle. Then the triangle (C • [−1], A• , B • , δ[−1], α, β) is isomorphic to the triangle (C • [−1], A• , C(δ[−1])• , δ[−1], i, p). Proof. We write B n = An ⊕ C n and we identify αn and β n with the natural inclusion and projection maps. By construction of δ we have   n dA δ n dnB = 0 dnC On the other hand the cone of δ[−1] : C • [−1] → A• is given as C(δ[−1])n = An ⊕C n with differential identical with the matrix above! Whence the lemma.  Lemma 11.8.15. Let A be an additive category. Let f : K • → L• be a morphism of complexes. The triangle (L• , C(f )• , K • [1], i, p, f [1]) is the triangle associated to the termwise split sequence 0 → L• → C(f )• → K • [1] → 0 coming from the definition of the cone of f . Proof. Immediate from the definitions.



11.9. Distinguished triangles in the homotopy category Since we want our boundary maps in long exact sequences of cohomology to be given by the maps in the snake lemma without signs we define distinguished triangles in the homotopy category as follows. Definition 11.9.1. Let A be an additive category. A triangle (X, Y, Z, f, g, h) of K(A) is called a distinguished triangle of K(A) if it is isomorphic to the triangle associated to a termwise split exact sequence of complexes, see Definition 11.8.8. Same definition for K + (A), K − (A), and K b (A).

11.9. DISTINGUISHED TRIANGLES IN THE HOMOTOPY CATEGORY

779

Note that according to Lemma 11.8.12 a triangle of the form (K • , L• , C(f )• , f, i, −p) is a distinguished triangle. This does indeed lead to a triangulated category, see Lemma 11.8.12. Before we can prove the proposition we need one more lemma in order to be able to prove TR4. Lemma 11.9.2. Let A be an additive category. Suppose that α : A• → B • and β : B • → C • are split injections of complexes. Then there exist distinguished triangles (A• , B • , Q•1 , α, p1 , d1 ), (A• , C • , Q•2 , β ◦α, p2 , d2 ) and (B • , C • , Q•3 , β, p3 , d3 ) for which TR4 holds. Proof. Say π1n : B n → An , and π3n : C n → B n are the splittings. Then also A• → C • is a split injection with splittings π2n = π1n ◦ π3n . Let us write Q•1 , Q•2 and Q•3 for the “quotient” complexes. In other words, Qn1 = Ker(π1n ), Qn3 = Ker(π3n ) and Qn2 = Ker(π2n ). Note that the kernels exist. Then B n = An ⊕ Qn1 and Cn = B n ⊕ Qn3 , where we think of An as a subobject of B n and so on. This implies C n = An ⊕ Qn1 ⊕ Qn3 . Note that π2n = π1n ◦ π3n is zero on both Qn1 and Qn3 . Hence Qn2 = Qn1 ⊕ Qn3 . Consider the commutative diagram 0 0 0

A• ↓ → A• ↓ → B• →

→ B• ↓ → C• ↓ → C•

→ → →

Q•1 ↓ Q•2 ↓ Q•3

→

0

→

0

→

0

The rows of this diagram are termwise split exact sequences, and hence determine distinguished triangles by definition. Moreover downward arrows in the diagram above are compatible with the chosen splittings and hence define morphisms of triangles (A• → B • → Q•1 → A• [1]) −→ (A• → C • → Q•2 → A• [1]) and (A• → C • → Q•2 → A• [1]) −→ (B • → C • → Q•3 → B • [1]). Note that the splittings Qn3 → C n of the bottom split sequence in the diagram provides a splitting for the split sequence 0 → Q•1 → Q•2 → Q•3 → 0 upon composing with C n → Qn2 . It follows easily from this that the morphism δ : Q•3 → Q•1 [1] in the corresponding distinguished triangle (Q•1 → Q•2 → Q•3 → Q•1 [1]) is equal to the composition Q•3 → B • [1] → Q•1 [1]. Hence we get a structure as in the conclusion of axiom TR4.  Proposition 11.9.3. Let A be an additive category. The category K(A) of complexes up to homotopy with its natural translation functors and distinguished triangles as defined above is a triangulated category. Proof. Proof of TR1. By definition every triangle isomorphic to a distinguished one is distinguished. Also, any triangle (A• , A• , 0, 1, 0, 0) is distinguished since 0 → A• → A• → 0 → 0 is a termwise split sequence of complexes. Finally, given any morphism of complexes f : K • → L• the triangle (K, L, C(f ), f, i, −p) is distinguished by Lemma 11.8.12. Proof of TR2. Let (X, Y, Z, f, g, h) be a triangle. Assume (Y, Z, X[1], g, h, −f [1]) is distinguished. Then there exists a termwise split sequence of complexes A• →

780

11. DERIVED CATEGORIES

B • → C • such that the associated triangle (A• , B • , C • , α, β, δ) is isomorphic to (Y, Z, X[1], g, h, −f [1]). Rotating back we see that (X, Y, Z, f, g, h) is isomorphic to (C • [−1], A• , B • , −δ[−1], α, β). It follows from Lemma 11.8.14 that the triangle (C • [−1], A• , B • , δ[−1], α, β) is isomorphic to (C • [−1], A• , C(δ[−1])• , δ[−1], i, p). Precomposing the previous isomorphism of triangles with −1 on Y it follows that (X, Y, Z, f, g, h) is isomorphic to (C • [−1], A• , C(δ[−1])• , δ[−1], i, −p). Hence it is distinguished by Lemma 11.8.12. On the other hand, suppose that (X, Y, Z, f, g, h) is distinguished. By Lemma 11.8.12 this means that it is isomorphic to a triangle of the form (K • , L• , C(f ), f, i, −p) for some morphism of complexes f . Then the rotated triangle (Y, Z, X[1], g, h, −f [1]) is isomorphic to (L• , C(f ), K • [1], i, −p, −f [1]) which is isomorphic to the triangle (L• , C(f ), K • [1], i, p, f [1]). By Lemma 11.8.15 this triangle is distinguished. Hence (Y, Z, X[1], g, h, −f [1]) is distinguished as desired. Proof of TR3. Let (X, Y, Z, f, g, h) and (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) be distinguished triangles of K(A) and let a : X → X 0 and b : Y → Y 0 be morphisms such that f 0 ◦ a = b ◦ f . By Lemma 11.8.2 we may assume that (X, Y, Z, f, g, h) = (X, Y, C(f ), f, i, p) and (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) = (X 0 , Y 0 , C(f 0 ), f 0 , i0 , p0 ). At this point we simply apply Lemma 11.8.2 to the commutative diagram given by f, f 0 , a, b. Proof of TR4. At this point we know that K(A) is a pre-triangulated category. Hence we can use Lemma 11.4.13. Let A• → B • and B • → C • be composable morphisms of K(A). By Lemma 11.8.13 we may assume that A• → B • and B • → C • are split injective morphisms. In this case the result follows from Lemma 11.9.2.  Remark 11.9.4. Let A be an additive category. Exactly the same proof as the proof of Proposition 11.9.3 shows that the categories K + (A), K − (A), and K b (A) are triangulated categories. Namely, the cone of a morphisms between bounded (above, below) is bounded (above, below). But we prove below that these are triangulated subcategories of K(A) which gives another proof. Lemma 11.9.5. Let A be an additive subcategory. The categories K + (A), K − (A), and K b (A) are full triangulated subcategories of K(A). Proof. Each of the categories mentioned is a full additive subcategory. We use the criterion of Lemma 11.4.14 to show that they are triangulated subcategories. It is clear that each of the categories K + (A), K − (A), and K b (A) is preserved under the shift functors [1], [−1]. Finally, suppose that f : A• → B • is a morphism in K + (A), K − (A), or K b (A). Then (A• , B • , C(f )• , f, i, −p) is a distinguished triangle of K(A) with C(f )• ∈ K + (A), K − (A), or K b (A) as is clear from the construction of the cone. Thus the lemma is proved. (Alternatively, K • → L• is isomorphic to an termwise split injection of complexes in K + (A), K − (A), or K b (A), see Lemma 11.8.5 and then one can directly take the associated distinguished triangle.)  Lemma 11.9.6. Let A, B be additive categories. Let F : A → B be an additive functor. The induced functors F : K(A) −→ K(B) F : K + (A) −→ K + (B) F : K − (A) −→ K − (B) F : K b (A) −→ K b (B)

11.10. DERIVED CATEGORIES

781

are exact functors of triangulated categories. Proof. Suppose A• → B • → C • is a termwise split sequence of complexes of A with splittings (sn , π n ) and associated morphism δ : C • → A• [1], see Definition 11.8.8. Then F (A• ) → F (B • ) → F (C • ) is a termwise split sequence of complexes with splittings (F (sn ), F (π n )) and associated morphism F (δ) : F (C • ) → F (A• )[1]. Thus F transforms distinguished triangles into distinguished triangles.  11.10. Derived categories In this section we construct the derived category of an abelian category A by inverting the quasi-isomorphisms in K(A). Before we do this recall that the functors H i : Comp(A) → A factor through K(A), see Homology, Lemma 10.10.11. Moreover, in Homology, Definition 10.12.8 we have defined identifications H i (K • [n]) = H i+n (K • ). At this point it makes sense to redefine H i (K • ) = H 0 (K • [i]) in order to avoid confusion and possible sign errors. Lemma 11.10.1. Let A be an abelian category. The functor H 0 : K(A) −→ A is homological. Proof. Because H 0 is a functor, and by our definition of distinguished triangles it suffices to prove that given a termwise split short exact sequence of complexes 0 → A• → B • → C • → 0 the sequence H 0 (A• ) → H 0 (B • ) → H 0 (C • ) is exact. This follows from Homology, Lemma 10.10.12.  In particular, this lemma implies that a distinguished triangle (X, Y, Z, f, g, h) in K(A) gives rise to a long exact cohomology sequence (11.10.1.1) ...

/ H i (X)

H i (f )

/ H i (Y )

H i (g)

/ H i (Z)

H i (h)

/ H i+1 (X)

/ ...

see (11.3.5.1). Moreover, there is a compatibility with the long exact sequence of cohomology associated to a short exact sequence of complexes (insert future reference here). For example, if (A• , B • , C • , α, β, δ) is the distinguished triangle associated to a termwise split exact sequence of complexes (see Definition 11.8.8), then the cohomology sequence above agrees with the one defined using the snake lemma, see Homology, Lemma 10.10.12 and for agreement of sequences, see Homology, Lemma 10.12.11. Recall that a complex K • is acyclic if H i (K • ) = 0 for all i ∈ Z. Moreover, recall that a morphism of complexes f : K • → L• is a quasi-isomorphism if and only if H i (f ) is an isomorphism for all i. See Homology, Definition 10.10.10. Lemma 11.10.2. Let A be an abelian category. The full subcategory Ac(A) of K(A) consisting of acyclic complexes is a strictly full saturated triangulated subcategory of K(A). The corresponding saturated multiplicative system (see Lemma 11.6.10) of K(A) is the set Qis(A) of quasi-isomorphisms. In particular, the kernel of the localization functor Q : K(A) → Qis(A)−1 K(A) is Ac(A) and the functor H 0 factors through Q.

782

11. DERIVED CATEGORIES

Proof. We know that H 0 is a homological functor by Lemma 11.10.1. Thus this lemma is a special case of Lemma 11.6.11.  Definition 11.10.3. Let A be an abelian category. Let Ac(A) and Qis(A) be as in Lemma 11.10.2. The derived category of A is the triangulated category D(A) = K(A)/Ac(A) = Qis(A)−1 K(A). We denote H 0 : D(A) → A the unique functor whose composition with the quotient functor gives back the functor H 0 defined above. Using Lemma 11.6.4 we introduce the strictly full saturated triangulated subcategories D+ (A), D− (A), Db (A) whose sets of objects are Ob(D+ (A)) = {X ∈ Ob(D(A)) | H n (X) = 0 for all n  0} Ob(D− (A)) = {X ∈ Ob(D(A)) | H n (X) = 0 for all n  0} Ob(Db (A)) = {X ∈ Ob(D(A)) | H n (X) = 0 for all |n|  0} The category Db (A) is called the bounded derived category of A. Each of the variants D+ (A), D− (A), Db (A) can be constructed as a localization of the corresponding homotopy category. This relies on the following simple lemma. Lemma 11.10.4. Let A be an abelian category. Let K • be a complex. (1) If H n (K • ) = 0 for all n  0, then there exists a quasi-isomorphism K • → L• with L• bounded below. (2) If H n (K • ) = 0 for all n  0, then there exists a quasi-isomorphism M • → K • with M • bounded above. (3) If H n (K • ) = 0 for all |n|  0, then there exists a commutative diagram of morphisms of complexes KO •

/ L• O

M•

/ N•

where all the arrows are quasi-isomorphisms, L• bounded below, M • bounded above, and N • a bounded complex. Proof. Pick a  0  b and set M • = τ≤a K • , L• = K • /τ≤b K • , and N • = L• /M • . See Homology, Section 10.11 for the truncation functors.  To state the following lemma denote Ac+ (A), Ac− (A), resp. Acb (A) the intersection of K + (A), K − (A), resp. K b (A) with Ac(A). Denote Qis+ (A), Qis− (A), resp. Qisb (A) the intersection of K + (A), K − (A), resp. K b (A) with Qis(A). Lemma 11.10.5. Let A be an abelian category. The subcategories Ac+ (A), Ac− (A), resp. Acb (A) are strictly full saturated triangulated subcategories of K + (A), K − (A), resp. K b (A). The corresponding saturated multiplicative systems (see Lemma 11.6.10) are the sets Qis+ (A), Qis− (A), resp. Qisb (A). (1) The kernel of the functor K + (A) → D+ (A) is Ac+ (A) and this induces an equivalence of triangulated categories K + (A)/Ac+ (A) = Qis+ (A)−1 K + (A) −→ D+ (A)

11.11. THE CANONICAL DELTA-FUNCTOR

783

(2) The kernel of the functor K − (A) → D− (A) is Ac− (A) and this induces an equivalence of triangulated categories K − (A)/Ac− (A) = Qis− (A)−1 K − (A) −→ D− (A) (3) The kernel of the functor K b (A) → Db (A) is Acb (A) and this induces an equivalence of triangulated categories K b (A)/Acb (A) = Qisb (A)−1 K b (A) −→ Db (A) Proof. The initial statements follow from Lemma 11.6.11 by considering the restriction of the homological functor H 0 . The statement on kernels in (1), (2), (3) is a consequence of the definitions in each case. Each of the functors is essentially surjective by Lemma 11.10.4. To finish the proof we have to show the functors are fully faithful. We first do this for the bounded below version. Suppose that K • , L• are bounded above complexes. A morphism between these in D(A) is of the form s−1 f for a pair f : K • → (L0 )• , s : L• → (L0 )• where s is a quasi-isomorphism. This implies that (L0 )• has cohomology bounded below. Hence by Lemma 11.10.4 we can choose a quasi-isomorphism s0 : (L0 )• → (L00 )• with (L00 )• bounded below. Then the pair (s0 ◦ f, s0 ◦ s) defines a morphism in Qis+ (A)−1 K + (A). Hence the functor is “full”. Finally, suppose that the pair f : K • → (L0 )• , s : L• → (L0 )• defines a morphism in Qis+ (A)−1 K + (A) which is zero in D(A). This means that there exists a quasi-isomorphism s0 : (L0 )• → (L00 )• such that s0 ◦f = 0. Using Lemma 11.10.4 once more we obtain a quasi-isomorphism s00 : (L00 )• → (L000 )• with (L000 )• bounded below. Thus we see that s00 ◦ s0 ◦ f = 0 which implies that s−1 f is zero in Qis+ (A)−1 K + (A). This finishes the proof that the functor in (1) is an equivalence. The proof of (2) is dual to the proof of (1). To prove (3) we may use the result of (2). Hence it suffices to prove that the functor Qisb (A)−1 K b (A) → Qis− (A)−1 K − (A) is fully faithful. The argument given in the previous paragraph applies directly to show this where we consistently work with complexes which are already bounded above.  11.11. The canonical delta-functor The derived category should be the receptacle for the universal cohomology functor. In order to state the result we use the notion of a δ-functor from an abelian category into a triangulated category, see Definition 11.3.6. Consider the functor Comp(A) → K(A). This functor is not a δ-functor in general. The easiest way to see this is to consider a nonsplit short exact sequence 0 → A → B → C → 0 of objects of A. Since HomK(A) (C[0], A[1]) = 0 we see that any distinguished triangle arising from this short exact sequence would look like (A[0], B[0], C[0], a, b, 0). But the existence of such a distinguished triangle in K(A) implies that the extension is split. A contradiction. It turns out that the functor Comp(A) → D(A) is a δ-functor. In order to see this we have to define the morphisms δ associated to a short exact sequence a

b

0 → A• − → B• → − C• → 0

784

11. DERIVED CATEGORIES

of complexes in the abelian category A. Consider the cone C(a)• of the morphism a. We have C(a)n = B n ⊕ An+1 and we define q n : C(a)n → C n via the projection to B n followed by bn . Hence a morphism of complexes q : C(a)• −→ C • . It is clear that q ◦ i = b where i is as in Definition 11.8.1. Note that, as a• is injective in each degree, the kernel of q is identified with the cone of idA• which is acyclic. Hence we see that q is a quasi-isomorphism. According to Lemma 11.8.12 the triangle (A, B, C(a), a, i, −p) is a distinguished triangle in K(A). As the localization functor K(A) → D(A) is exact we see that (A, B, C(a), a, i, −p) is a distinguished triangle in D(A). Since q is a quasi-isomorphism we see that q is an isomorphism in D(A). Hence we deduce that (A, B, C, a, b, −p ◦ q −1 ) is a distinguished triangle of D(A). This suggests the following lemma. Lemma 11.11.1. Let A be an abelian category. The functor Comp(A) → D(A) defined has the natural structure of a δ-functor, with δA• →B • →C • = −p ◦ q −1 with p and q as explained above. The same construction turns the functors Comp+ (A) → D+ (A), Comp− (A) → D− (A), and Compb (A) → Db (A) into δ-functors. Proof. We have already seen that this choice leads to a distinguished triangle whenever given a short exact sequence of complexes. We have to show that given a commutative diagram / A•

0

a

g

f

 / (A0 )•

0

/ B•

a

0



/ (B 0 )•

b

/ C•

/0

h

0

b

 / (C 0 )•

/0

we get the desired commutative diagram of Definition 11.3.6 (2). By Lemma 11.8.2 the pair (f, g) induces a canonical morphism c : C(a)• → C(a0 )• . It is a simple computation to show that q 0 ◦ c = h ◦ q and f [1] ◦ p = p0 ◦ c. From this the result follows directly.  Lemma 11.11.2. Let A be an abelian category. Let 0

/ A•

/ B•

/ C•

/0

0

 / D•

 / E•

 / F•

/0

be a commutative diagram of morphisms of complexes such that the rows are short exact sequences of complexes, and the vertical arrows are quasi-isomorphisms. The δ-functor of Lemma 11.11.1 above maps the to short exact sequences 0 → A• → B • → C • → 0 and 0 → D• → E • → F • → 0 to isomorphic distinguished triangles. Proof. Trivial from the fact that K(A) → D(A) transforms quasi-isomorphisms into isomorphisms and that the associated distinguished triangles are functorial. 

11.12. TRIANGULATED SUBCATEGORIES OF THE DERIVED CATEGORY

785

Lemma 11.11.3. Let A be an abelian category. Let 0

/ A•

/ B•

/ C•

/0

be a short exact sequences of complexes. Assume this short exact sequence is termwise split. Let (A• , B • , C • , α, β, δ) be the distinguished triangle of K(A) associated to the sequence. The δ-functor of Lemma 11.11.1 above maps the short exact sequences 0 → A• → B • → C • → 0 to a triangle isomorphic to the distinguished triangle (A• , B • , C • , α, β, δ). Proof. Follows from Lemma 11.8.12.



11.12. Triangulated subcategories of the derived category Let A be an abelian category. In this section we are going to look for strictly full saturated triangulated subcategories D0 ⊂ D(A) and in the bounded versions. Here is a simple construction. Let B ⊂ A be a weak Serre subcategory, see Homology, Section 10.7. We let DB (A) the full subcategory of D(A) whose objects are Ob(DB (A)) = {X ∈ Ob(D(A)) | H n (X) is an object of B for all n} We also define DB+ (A) = D+ (A) ∩ DB (A) and similarly for the other bounded versions. Lemma 11.12.1. Let A be an abelian category. Let B ⊂ A be a weak Serre subcategory. The category DB (A) is a strictly full saturated abelian subcategory of D(A). Similarly for the bounded versions. Proof. It is clear that DB (A) is an additive subcategory preserved under the translation functors. If X ⊕ Y is in DB (A), then it is clear that both X and Y are in DB because H n (X ⊕ Y ) = H n (X) ⊕ H n (Y ), hence both H n (X) and H n (Y ) are kernels of maps between maps of an object of B, hence objects of B. By Lemma 11.4.14 it therefore suffices to show that given a distinguished triangle (X, Y, Z, f, g, h) such that X and Y are in DB (A) then Z is an object of DB (A). The long exact cohomology sequence (11.10.1.1) and the definition of a weak Serre subcategory (see Homology, Definition 10.7.1) show that H n (Z) is an object of B for all n. Thus Z is an object of DB (A).  An interesting feature of the situation of the lemma is that the functor D(B) → D(A) factors through a canonical exact functor (11.12.1.1)

D(B) −→ DB (A)

After all a complex made from objects of B certainly gives rise to an object of DB (A) and as distinguished triangles in DB (A) are exactly the distinguished triangles of D(A) whose vertices are in DB (A) we see that the functor is exact since D(B) → D(A) is exact. Similarly we obtain functors D+ (B) −→ DB+ (A) etc for the bounded versions. A key question in many cases is whether the displayed functor is an equivalence. Now, suppose that B is a Serre subcategory of A. In this case we have the quotient functor A → A/B, see Homology, Lemma 10.7.6. In this case DB (A) is the kernel

786

11. DERIVED CATEGORIES

of the functor D(A) → D(A/B). Thus we obtain a canonical functor D(A)/DB (A) −→ D(A/B) by Lemma 11.6.8. Similarly for the bounded versions. Lemma 11.12.2. Let A be an abelian category. Let B ⊂ A be a Serre subcategory. Then D(A) → D(A/B) is essentially surjective. Proof. We will use the description of the category A/B in the proof of Homology, Lemma 10.7.6. Let (X • , d• ) be a complex of A/B. For each i we have an object X i of A and di = (si , f i ) where si : Y i → X i is a morphism of A whose kernel and cokernel are in B and f i : Y i → X i+1 is an arbitrary morphism of A. Next, consider the complex . . . → X i ⊕ Y i ⊕ Y i+1 → X i+1 ⊕ Y i+1 ⊕ Y i+2 → . . . in A with differential given by  0 0 0

fi 0 0

 si+1 −idY i+1  . 0

This complex becomes quasi-isomorphic to the complex (X • , d• ) in A/B by the maps (idX i , si , 0) : X i ⊕ Y i ⊕ Y i+1 → X i Calculation omitted.  Lemma 11.12.3. Let A be an abelian category. Let B ⊂ A be a Serre subcategory. Suppose that the functor v : A → A/B has a left adjoint u : A/B → A such that vu ∼ = id. Then D(A)/DB (A) = D(A/B) and similarly for the bounded versions. Proof. The functor D(v) : D(A) → D(A/B) is essentially surjective by Lemma 11.12.2. For an object X of D(A) the adjunction mapping cX : uvX → X maps to an isomorphism in D(A/B) because vuv ∼ = v by the assumption that vu ∼ = id. Thus in a distinguished triangle (uvX, X, Z, cX , g, h) the object Z is an object of DB (A) as we see by looking at the long exact cohomology sequence. Hence cX is an element of the multiplicative sytem used to define the quotient category D(A)/DB (A). Thus uvX ∼ = X in D(A)/DB (A). For X, Y ∈ Ob(A)) the map HomD(A)/DB (A) (X, Y ) −→ HomD(A/B) (vX, vY ) is bijective because u gives an inverse (by the remarks above).



11.13. Filtered derived categories A reference for this section is [Ill72, I, Chapter V]. Let A be an abelian category. The goal is to define the filtered derived category DF (A) of A. In some sense this is the derived category of the category Filf (A) of objects with a finite filtration in A. We will slightly generalize Illusie’s discussion by allowing our filtered complexes to have infinitely many nonzero grp (K • ) but we retaining the requirement that each term has a finite filtration. The rationale for this generalization is that it is not harder and it allows us to apply the discussion to the spectral sequences of Lemma 11.20.3, see also Remark 11.20.4.

11.13. FILTERED DERIVED CATEGORIES

787

We will use the notation regarding filtered objects introduced in Homology, Section 10.13. The category of filtered objects of A is denoted Fil(A). All filtrations will be decreasing by fiat. Definition 11.13.1. Let A be an abelian category. The category of finite filtered objects of A is the category of filtered objects (A, F ) of A whose filtration F is finite. We denote it Filf (A). Thus Filf (A) is a full subcategory of Fil(A). For each p ∈ Z there is a functor L grp : Filf (A) → A. There is a functor gr = p∈Z grp : Filf (A) → A. Finally, there is a functor (forget F ) : Filf (A) −→ A which associates to the filtered object (A, F ) the underlying object of A. The category Filf (A) is an additive category, but not abelian in general, see Homology, Example 10.3.11. The construction in this section is a special case of a more general construction of the derived category of an “exact category”, see for example [B¨ uh10], [Kel90]. Because the functors grp , gr, (forget F ) are additive they induce exact functors of triangulated categories grp , gr, (forget F ) : K(Filf (A)) −→ K(A) by Lemma 11.9.6. By analogy with the case of the homotopy category of an abelian category we make the following definitions. Definition 11.13.2. Let A be an abelian category. (1) Let α : K • → L• be a morphism of K(Filf (A)). We say that α is a filtered quasi-isomorphism if the morphism gr(α) is a quasi-isomorphism. (2) Let K • be an object of K(Filf (A)). We say that K • is filtered acyclic if the complex gr(K • ) is acyclic. Note that α : K • → L• is a filtered quasi-isomorphism if and only if each grp (α) is a quasi-isomorphism. Similarly a complex K • is filtered acyclic if and only if each grp (K • ) is acyclic. Lemma (1) (2) (3)

11.13.3. Let A be an abelian category. The functor K(Filf (A)) −→ A, K • 7−→ H 0 (gr(K • )) is homological. The functor K(Filf (A)) → A, K • 7−→ H 0 (grp (K • )) is homological. The functor K(Filf (A)) −→ A, K • 7−→ H 0 ((forget F )K • ) is homological.

Proof. This follows from the fact that H 0 : K(A) → A is homological, see Lemma 11.10.1 and the fact that the functors gr, grp , (forget F ) are exact functors of triangulated categories. See Lemma 11.4.18.  Lemma 11.13.4. Let A be an abelian category. The full subcategory FAc(A) of K(Filf (A)) consisting of filtered acyclic complexes is a strictly full saturated triangulated subcategory of K(Filf (A)). The corresponding saturated multiplicative system (see Lemma 11.6.10) of K(Filf (A)) is the set FQis(A) of filtered quasiisomorphisms. In particular, the kernel of the localization functor Q : K(Filf (A)) −→ FQis(A)−1 K(Filf (A)) is FAc(A) and the functor H 0 ◦ gr factors through Q.

788

11. DERIVED CATEGORIES

Proof. We know that H 0 ◦ gr is a homological functor by Lemma 11.13.3. Thus this lemma is a special case of Lemma 11.6.11.  Definition 11.13.5. Let A be an abelian category. Let FAc(A) and FQis(A) be as in Lemma 11.13.4. The filtered derived category of A is the triangulated category DF (A) = K(Filf (A))/FAc(A) = FQis(A)−1 K(Filf (A)). Lemma 11.13.6. The functors grp , gr, (forget F ) induce canonical exact functors grp , gr, (forget F ) : DF (A) −→ D(A) which commute with the localization functors. Proof. This follows from the universal propery of localization, see Lemma 11.5.6, provided we can show that a filtered quasi-isomorphism is turned into a quasiisomorphism by each of the functors grp , gr, (forget F ). This is true by definition for the first two. For the last one the statement we have to do a little bit of work. Let f : K • → L• be a filtered quasi-isomorphism in K(Filf (A)). Choose a distinguished triangle (K • , L• , M • , f, g, h) which contains f . Then M • is filtered acyclic, see Lemma 11.13.4. Hence by the corresponding lemma for K(A) it suffices to show that a filtered acyclic complex is an acyclic complex if we forget the filtration. This follows from Homology, Lemma 10.13.17.  Definition 11.13.7. Let A be an abelian category. The bounded filtered derived category DF b (A) is the full subcategory of DF (A) with objects those X such that gr(X) ∈ Db (A). Similarly for the bounded below filtered derived category DF + (A) and the bounded above filtered derived category DF − (A). Lemma 11.13.8. Let A be an abelian category. Let K • ∈ K(Filf (A)). (1) If H n (gr(K • )) = 0 for all n < a, then there exists a filtered quasiisomorphism K • → L• with Ln = 0 for all n < a. (2) If H n (gr(K • )) = 0 for all n > b, then there exists a filtered quasiisomorphism M • → K • with M n = 0 for all n > b. (3) If H n (gr(K • )) = 0 for all |n|  0, then there exists a commutative diagram of morphisms of complexes KO •

/ L• O

M•

/ N•

where all the arrows are filtered quasi-isomorphisms, L• bounded below, M • bounded above, and N • a bounded complex. Proof. Suppose that H n (gr(K • )) = 0 for all n < a. By Homology, Lemma 10.13.17 the sequence da−2

da−1

K a−1 −−−→ K a−1 −−−→ K a is an exact sequence of objects of A and the morphisms da−2 and da−1 are strict. Hence Coim(da−1 ) = Im(da−1 ) in Filf (A) and the map gr(Im(da−1 )) → gr(K a ) is injective with image equal to the image of gr(K a−1 ) → gr(K a ), see Homology, Lemma 10.13.15. This means that the map K • → τ≥a K • into the truncation τ≥a K • = (. . . → 0 → K a /Im(da−1 ) → K a+1 → . . .)

11.14. DERIVED FUNCTORS IN GENERAL

789

is a filtered quasi-isomorphism. This proves (1). The proof of (2) is dual to the proof of (1). Part (3) follows formally from (1) and (2).  To state the following lemma denote FAc+ (A), FAc− (A), resp. FAcb (A) the intersection of K + (Filf A), K − (Filf A), resp. K b (Filf A) with FAc(A). Denote FQis+ (A), FQis− (A), resp. FQisb (A) the intersection of K + (Filf A), K − (Filf A), resp. K b (Filf A) with FQis(A). Lemma 11.13.9. Let A be an abelian category. The subcategories FAc+ (A), FAc− (A), resp. FAcb (A) are strictly full saturated triangulated subcategories of K + (Filf A), K − (Filf A), resp. K b (Filf A). The corresponding saturated multiplicative systems (see Lemma 11.6.10) are the sets FQis+ (A), FQis− (A), resp. FQisb (A). (1) The kernel of the functor K + (Filf A) → DF + (A) is FAc+ (A) and this induces an equivalence of triangulated categories K + (Filf A)/FAc+ (A) = FQis+ (A)−1 K + (Filf A) −→ DF + (A) (2) The kernel of the functor K − (Filf A) → DF − (A) is FAc− (A) and this induces an equivalence of triangulated categories K − (Filf A)/FAc− (A) = FQis− (A)−1 K − (Filf A) −→ DF − (A) (3) The kernel of the functor K b (Filf A) → DF b (A) is FAcb (A) and this induces an equivalence of triangulated categories K b (Filf A)/FAcb (A) = FQisb (A)−1 K b (Filf A) −→ DF b (A) Proof. This follows from the results above, in particular Lemma 11.13.8, by exactly the same arguments as used in the proof of Lemma 11.10.5.  11.14. Derived functors in general A reference for this section is Deligne’s expos´ee XVII in [AGV71]. A very general notion of right and left derived functors exists where we have an exact functor between triangulated categories, a multiplicative system in the source category and we want to find the “correct” extension of the exact functor to the localized category. Situation 11.14.1. Here F : D → D0 is an exact functor of triangulated categories and S is a saturated multiplicative system in D compatible with the structure of triangulated category on D. Let X ∈ Ob(D). Recall from Categories, Remark 4.24.5 the filtered category X/S of arrows s : X → X 0 in S with source X. Dually, in Categories, Remark 4.24.12 we defined the cofiltered category S/X of arrows s : X 0 → X in S with target X. Definition 11.14.2. Assumptions and notation as in Situation 11.14.1. Let X ∈ Ob(D). (1) we say the right derived functor RF is defined at X if the ind-object (X/S) −→ D0 ,

(s : X → X 0 ) 7−→ F (X 0 )

790

11. DERIVED CATEGORIES

is essentially constant2; in this case the value Y in D0 is called the value of RF at X. (2) we say the left derived functor LF is defined at X if the pro-object (S/X) −→ D0 ,

(s : X 0 → X) 7−→ F (X 0 )

is essentially constant; in this case the value Y in D0 is called the value of LF at X. By abuse of notation we often denote the values simply RF (X) or LF (X). It will turn out that the full subcategory of D consisting of objects where RF is defined is a triangulated subcategory, and RF will define functor on this subcategory which transforms morphisms of s into isomorphisms. Lemma 11.14.3. Assumptions and notation as in Situation 11.14.1. Let f : X → Y be a morphism of D. (1) If RF is defined at X and Y then there exists a unique morphism RF (f ) : RF (X) → RF (Y ) between the values such that for any commutative diagram / X0 X s

f0

f

 Y

 / Y0

s0

with s, s0 ∈ S the diagram F (X)

/ F (X 0 )

/ RF (X)

 F (Y )

 / F (Y 0 )

 / RF (Y )

commutes. (2) If LF is defined at X and Y then there exists a unique morphism LF (f ) : LF (X) → LF (Y ) between the values such that for any commutative diagram /X X0 s

f0

 Y0

0

s

 /Y

f

with s, s0 in S the diagram LF (X)

/ F (X 0 )

/ F (X)

 LF (Y )

 / F (Y 0 )

 / F (Y )

commutes. 2For a discussion of when an ind-object or pro-object of a category is essentially constant we refer to Categories, Section 4.20.

11.14. DERIVED FUNCTORS IN GENERAL

791

Proof. Part (1) holds if we only assume that the colimits RF (X) = colims:X 0 →X F (X 0 )

and RF (Y ) = colims0 :Y 0 →Y F (Y 0 )

exist. Namely, to give a morphism RF (X) → RF (Y ) between the colimits is the same thing as giving for each s : X → X 0 in Ob(X/S) a morphism F (X 0 ) → RF (Y ) compatible with morphisms in the category X/S. To get the morphism we choose a commutative diagram X s / X0 f0

f

 Y

s0

 / Y0

with s, s0 in S as is possible by MS2 and we set F (X 0 ) → RF (Y ) equal to the composition F (X 0 ) → F (Y 0 ) → RF (Y ). To see that this is independent of the choice of the diagram above use MS3. Details omitted. The proof of (2) is dual.  Lemma 11.14.4. Assumptions and notation as in Situation 11.14.1. Let s : X → Y be an element of S. (1) RF is defined at X if and only if it is defined at Y . In this case the map RF (s) : RF (X) → RF (Y ) between values is an isomorphism. (2) LF is defined at X if and only if it is defined at Y . In this case the map LF (s) : LF (X) → LF (Y ) between values is an isomorphism. Proof. Omitted.



Lemma 11.14.5. Assumptions and notation as in Situation 11.14.1. Let X be an object of D and n ∈ Z. (1) RF is defined at X if and only if it is defined at X[n]. In this case there is a canonical isomorphism RF (X)[n] = RF (X[n]) between values. (2) LF is defined at X if and only if it is defined at X[n]. In this case there is a canonical isomorphism LF (X)[n] → LF (X[n]) between values. Proof. Omitted.



Lemma 11.14.6. Assumptions and notation as in Situation 11.14.1. Let (X, Y, Z, f, g, h) be a distinguished triangle of D. If RF is defined at two out of three of X, Y, Z, then it is defined at the third. Moreover, in this case (RF (X), RF (Y ), RF (Z), RF (f ), RF (g), RF (h)) is a distinguished triangle in D0 . Similarly for LF . Proof. Say RF is defined at X, Y with values A, B. Let RF (f ) : A → B be the induced morphism, see Lemma 11.14.3. We may choose a distinguished triangle (A, B, C, RF (f ), b, c) in D0 . We claim that C is a value of RF at Z. To see this pick s : X → X 0 in S such that there exists a morphism α : A → F (X 0 ) as in Categories, Definition 4.20.1. We may choose a commutative diagram X

s

f0

f

 Y

/ X0

0

s

 / Y0

792

11. DERIVED CATEGORIES

with s0 ∈ S by MS2. Using that Y /S is filtered we can (after replacing s0 by some s00 : Y → Y 00 in S) assume that there exists a morphism β : B → F (Y 0 ) as in Categories, Definition 4.20.1. Picture A

α

F (f 0 )

RF (f )

 B

/ F (X 0 )

β

 / F (Y 0 )

/A RF (f )

 /B

It may not be true that the left square commutes, but the outer and right squares commute. The assumption that the ind-object {F (Y 0 )}s0 :Y 0 →Y is essentially constant means that there exists a s00 : Y → Y 00 in S and a morphism h : Y 0 → Y 00 with such that s00 = h ◦ s0 and F (h) equal to F (Y 0 ) → B → F (Y 0 ) → F (Y 00 ). Hence after replacing Y 0 by Y 00 and β by F (h) ◦ β the diagram will commute (by direct computation with arrows). Using MS6 choose a morphism of triangles (s, s0 , s00 ) : (X, Y, Z, f, g, h) −→ (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) with s00 ∈ S. By TR3 choose a morphism of triangles (α, β, γ) : (A, B, C, RF (f ), b, c) −→ (F (X 0 ), F (Y 0 ), F (Z 0 ), F (f 0 ), F (g 0 ), F (h0 )) By Lemma 11.14.4 it suffices to prove that RF (Z 0 ) is defined and has value C. Consider the category I of Lemma 11.5.8 of triangles I = {(t, t0 , t00 ) : (X 0 , Y 0 , Z 0 , f 0 , g 0 , h0 ) → (X 00 , Y 00 , Z 00 , f 00 , g 00 , h00 ) | (t, t0 , t00 ) ∈ S} To show that the system F (Z 00 ) is essentially constant over the category Z 0 /S is equivalent to showing that the system of F (Z 00 ) is essentially constant over I by using the surjectivity of the functor I → Z 0 /S. For any object W in D0 we can consider the diagram limI MorD0 (F (X 00 ), W ) O

/ MorD0 (A, W ) O

limI MorD0 (F (Y 00 ), W ) O

/ MorD0 (B, W ) O

limI MorD0 (F (Z 00 ), W ) O

/ MorD0 (C, W ) O

limI MorD0 (F (X 00 [1]), W ) O

/ MorD0 (A[1], W ) O

limI MorD0 (F (Y 00 [1]), W )

/ MorD0 (B[1], W )

which shows that the middle arrow is an isomorphism by the 5 lemma. In this way we conclude that C is the colimit colimI F (Z 00 ). To see that the ind-object is essentially constant it now suffices to show that for any object W in D0 the map colimI MorD0 (W, F (Z 00 )) −→ MorD0 (W, C)

11.14. DERIVED FUNCTORS IN GENERAL

793

is bijective, see Categories, Lemma 4.20.6. To see this we can use again the 5 lemma and the commutative diagram colimI MorD0 (W, F (X 00 )) O

/ MorD0 (W, A) O

colimI MorD0 (W, F (Y 00 )) O

/ MorD0 (W, B) O

colimI MorD0 (W, F (Z 00 )) O

/ MorD0 (W, C) O

colimI MorD0 (W, F (X 00 [1])) O

/ MorD0 (W, A[1]) O

colimI MorD0 (W, F (Y 00 [1]))

/ MorD0 (W, B[1])

and the fact that Categories, Lemma 4.20.6 guarantees that the other horizontal arrows are isomorphisms.  Lemma 11.14.7. Assumptions and notation as in Situation 11.14.1. Let X, Y be objects of D. If RF is defined at X ⊕ Y , then it is defined at X and Y . Moreover, in this case RF (X ⊕ Y ) = RF (X) ⊕ RF (Y ). Similarly for LF . Proof. Since S is a saturated system for any s : X → X 0 and s0 : Y → Y 0 in S the morphism s ⊕ s0 : X ⊕ Y → X 0 ⊕ Y 0 is an element of S (as S is the set of arrows which become invertible under the additive localization functor Q : D → S −1 D, see Categories, Lemma 4.24.18). To prove the lemma for RF it suffices to show that these arrows s ⊕ s0 are cofinal in the filtered category (X ⊕ Y )/S. To do this pick any t : X ⊕ Y → Z in S. Using MS2 we can find morphisms Z → X 0 , Z → Y 0 and s : X → X 0 , s0 : Y → Y 0 in S such that Xo s

/Y

X ⊕Y

 X0 o

 Z

s0

 / Y0

commutes. Hence the desired result. The proof for LF is dual.



Proposition 11.14.8. Assumptions and notation as in Situation 11.14.1. The full subcategory E of D consisting of objects at which RF is defined is a strictly full saturated triangulated subcategory of D. Choosing values using the axiom of choice gives rise to an exact functor RF : E −→ D0 of triangulated categories. Elements of S with either source or target in E are morphisms of E. Any element of SE = Arrows(E) ∩ S is transformed into an isomorphism by RF . Hence an exact functor RF : SE−1 E −→ D0 . A similar result holds for LF .

794

11. DERIVED CATEGORIES

Proof. This is just a summary of the results obtained in Lemmas 11.14.3, 11.14.4, 11.14.5, 11.14.6, and 11.14.7.  Definition 11.14.9. In Situation 11.14.1. We say F is right deriveable, or that RF everywhere defined if RF is defined at every object of D. We say F is left deriveable, or that LF everywhere defined if LF is defined at every object of D. In this case we obtain a right (resp. left) derived functor RF : S −1 D −→ D0 ,

(11.14.9.1)

(resp. LF : S −1 D −→ D0 ),

see Proposition 11.14.8. In most interesting situations it is not the case that RF ◦Q is equal to F . In fact, it might happen that the canonical map F (X) → RF (X) is never an isomorphism. In practice this does not happen, because in practice we only know how to prove F is right deriveable by showing that RF can be computed by evaluating F at judiciously chosen objects of the triangulated category D. This warrents a definition. Definition 11.14.10. In Situation 11.14.1. (1) An object X of D computes RF if RF is defined at X and the canonical map F (X) → RF (X) is an isomorphism. (2) An object X of D computes LF if LF is defined at X and the canonical map LF (X) → F (X) is an isomorphism. Lemma 11.14.11. Assumptions and notation as in Situation 11.14.1. Let X be an object of D and n ∈ Z. (1) X computes RF if and only if X[n] computes RF . (2) X computes LF if and only if X[n] computes LF . Proof. Omitted.



Lemma 11.14.12. Assumptions and notation as in Situation 11.14.1. Let (X, Y, Z, f, g, h) be a distinguished triangle of D. If X, Y compute RF then so does Z. Similar for LF . Proof. By Lemma 11.14.6 we know that RF is defined at Z and that RF applied to the triangle produces a distinguished triangle. Consider the morphism of distinguished triangles (F (X), F (Y ), F (Z), F (f ), F (g), F (h))  (RF (X), RF (Y ), RF (Z), RF (f ), RF (g), RF (h)) Two out of three maps are isomorphisms, hence so is the third.



Lemma 11.14.13. Assumptions and notation as in Situation 11.14.1. Let X, Y be objects of D. If X ⊕ Y computes RF , then X and Y compute RF . Similarly for LF . Proof. By Lemma 11.14.7 we know that RF is defined at X and Y and that RF (X ⊕ Y ) = RF (X) ⊕ RF (Y ). Since the map F (X) ⊕ F (Y ) = F (X ⊕ Y ) −→ RF (X ⊕ Y ) = RF (X) ⊕ RF (Y ) is compatible with direct sum decompositions we win.



11.14. DERIVED FUNCTORS IN GENERAL

795

Lemma 11.14.14. Assumptions and notation as in Situation 11.14.1. (1) If for every object X ∈ Ob(D) there exists an arrow s : X → X 0 in S such that X 0 computes RF , then RF is everywhere defined. (2) If for every object X ∈ Ob(D) there exists an arrow s : X 0 → X in S such that X 0 computes LF , then LF is everywhere defined. Proof. This is clear from the definitions.



Lemma 11.14.15. Assumptions and notation as in Situation 11.14.1. If there exists a subset I ⊂ Ob(D) such that (1) for all X ∈ Ob(D) there exists s : X → X 0 in S with X 0 ∈ I, and (2) for every arrow s : X → X 0 in S with X, X 0 ∈ I the map F (s) : F (X) → F (X 0 ) is an isomorphism, then RF is everywhere defined and every X ∈ I computes RF . Dually, if there exists a subset P ⊂ Ob(D) such that (1) for all X ∈ Ob(D) there exists s : X 0 → X in S with X 0 ∈ P, and (2) for every arrow s : X → X 0 in S with X, X 0 ∈ P the map F (s) : F (X) → F (X 0 ) is an isomorphism, then LF is everywhere defined and every X ∈ P computes LF . Proof. Let X be an object of D. Assumption (1) implies that the arrows s : X → X 0 in S with X 0 ∈ I are cofinal in the category X/S. Assumption (2) implies that F is constant on this cofinal subcategory. Clearly this implies that F : (X/S) → D0 is essentially constant with value F (X 0 ) for any s : X → X 0 in S with X 0 ∈ I.  Lemma 11.14.16. Let A, B, C be triangulated categories. Let S, resp. S 0 be a saturated multiplicative system in A, resp. B compatible with the triangulated structure. Let F : A → B and G : B → C be exact functors. Denote F 0 : A → (S 0 )−1 B the composition of F with the localization functor. (1) If RF 0 , RG, R(G ◦ F ) are everywhere defined, then there is a canonical transformation of functors t : R(G ◦ F ) −→ RG ◦ RF 0 . (2) If LF 0 , LG, L(G ◦ F ) are everywhere defined, then there is a canonical transformation of functors t : LG ◦ LF 0 → L(G ◦ F ). Proof. In this proof we try to be careful. Hence let us think of the derived functors as the functors RF 0 : S −1 A → (S 0 )−1 B,

R(G ◦ F ) : S −1 A → C,

RG : (S 0 )−1 B → C.

Let us denote QA : A → S −1 A and QB : B → (S 0 )−1 B the localization functors. Then F 0 = QB ◦ F . Note that for every object Y of B there is a canonical map G(Y ) −→ RG(QB (Y )) in other words, there is a transformation of functors t0 : G → RG ◦ QB . Let X be an object of A. We have R(G ◦ F )(QA (X)) = colims:X→X 0 ∈S G(F (X 0 )) t0

− → colims:X→X 0 ∈S RG(QB (F (X 0 ))) = colims:X→X 0 ∈S RG(F 0 (X 0 )) = RG(colims:X→X 0 ∈S F 0 (X 0 )) = RG(RF 0 (X)).

796

11. DERIVED CATEGORIES

The system F 0 (X 0 ) is essentially constant in the category (S 0 )−1 B. Hence we may pull the colimit inside the functor RG in the third equality of the diagram above, see Categories, Lemma 4.20.5 and its proof. We omit the proof this this defines a transformation of functors. The case of left derived functors is similar.  11.15. Derived functors on derived categories In practice derived functors come about most often when given an additive functor between abelian categories. Situation 11.15.1. Here F : A → B is an additive functor between abelian categories. This induces exact functors F : K(A) → K(B),

K + (A) → K + (B),

K − (A) → K − (B).

We also denote F the composition K(A) → D(B), K + (A) → D+ (B), and K − (A) → D(B) of F with the localization functor K(B) → D(B), etc. This situation leads to four derived functors we will consider in the following. (1) The right derived functor of F : K(A) → D(B) relative to the multiplicative system Qis(A). (2) The right derived functor of F : K + (A) → D+ (B) relative to the multiplicative system Qis+ (A). (3) The left derived functor of F : K(A) → D(B) relative to the multiplicative system Qis(A). (4) The left derived functor of F : K − (A) → D− (B) relative to the multiplicative system Qis(A). Each of these cases is an example of Situation 11.14.1. Some of the ambiguity that may arise is alleviated by the following. Lemma 11.15.2. In Situation 11.15.1. (1) Let X be an object of K + (A). The right derived functor of K(A) → D(B) is defined at X if and only if the right derived functor of K + (A) → D+ (B) is defined at X. Moreover, the values are canonically isomorphic. (2) Let X be an object of K + (A). Then X computes the right derived functor of K(A) → D(B) if and only if X computes the right derived functor of K + (A) → D+ (B). (3) Let X be an object of K − (A). The left derived functor of K(A) → D(B) is defined at X if and only if the left derived functor of K − (A) → D− (B) is defined at X. Moreover, the values are canonically isomorphic. (4) Let X be an object of K − (A). Then X computes the left derived functor of K(A) → D(B) if and only if X computes the left derived functor of K − (A) → D− (B). Proof. Let X be an object of K + (A). Consider a quasi-isomorphism s : X → X 0 in K(A). By Lemma 11.10.4 there exists quasi-isomorphism X 0 → X 00 with X 00 bounded below. Hence we see that X/Qis+ (A) is cofinal in X/Qis(A). Thus it is clear that (1) holds. Part (2) follows directly from part (1). Parts (3) and (4) are dual to parts (1) and (2).  Given an object A of an abelian category A we get a complex A[0] = (. . . → 0 → A → 0 → . . .)

11.15. DERIVED FUNCTORS ON DERIVED CATEGORIES

797

where A is placed in degree zero. Hence a functor A → K(A), A 7→ A[0]. Let us temporarily say that a partial functor is one that is defined on a subcategory. Definition 11.15.3. In Situation 11.15.1. (1) The right derived functors of F are the partial functors RF associated to cases (1) and (2) of Situation 11.15.1. (2) The left derived functors of F are the partial functors LF associated to cases (3) and (4) of Situation 11.15.1. (3) An object A of A is said to be right acyclic for F , or acyclic for RF if A[0] computes RF . (4) An object A of A is said to be left acyclic for F , or acyclic for LF if A[0] computes RF . The following few lemmas give some criteria for the existence of enough acyclics. Lemma 11.15.4. Let A be an abelian category. Let I ⊂ Ob(A) be a subset such that every object of A is a subobject of an element of I. For every K • with K n = 0 for n < a there exists a quasi-isomorphism K • → I • with I n = 0 for n < a, each I n ∈ I, and each K n → I n injective. Proof. Consider the following induction hypothesis IHn : There are I j ∈ I, j ≤ n almost all zero, maps dj : I j → I j+1 for j < n and injective maps αj : K j → I j for j ≤ n such that the diagram ...

/ K n−1 

...

α

/ I n−1

/ Kn 

/ K n+1

/ ...

α

/ In

is commutative, such that dj ◦ dj−1 = 0 for j < n and such that α induces isomorphisms H j (K • ) → Ker(dj )/Im(dj−1 ) for j < n. Note that this implies (11.15.4.1)

n n−1 α(Im(dn−1 ) ⊂ α(K n ) ∩ Im(dn−1 ). K )) ⊂ α(Ker(dK )) ∩ Im(d

If these inclusions are not equalities, then choose an injection I n ⊕ K n /Im(dn−1 K ) −→ I with I ∈ I. Denote α0 : K n → I the map obtained by composing α⊕1 : K n → I n ⊕ 0 n−1 K n /Im(dn−1 → I the composition K ) with the displayed injection. Denote d : I n−1 n n−1 I → I → I of d by the inclusion of the first summand. Then α0 (K n ) ∩ 0 n Im(d0 ) = α0 (Im(dn−1 K )) simply because the intersection of α (K ) with the first n−1 n−1 n n 0 summand of I ⊕ K /Im(dK ) is equal to α (Im(dK )). Hence, after replacing I n by I, α by α0 and dn−1 by d0 we may assume that we have equality in Equation (11.15.4.1). Once this is the case consider the solid diagram K n /Ker(dnK )

/ K n+1

 I n /(Im(dn−1 ) + α(Ker(dnK )))

 /M

The horizontal arrow is injective by fiat and the vertical arrow is injective as we have equality in (11.15.4.1). Hence the push-out M of this diagram contains both K n+1 and I n /(Im(dn−1 ) + α(Ker(dnK ))) as subobjects. Choose an injection M → I n+1

798

11. DERIVED CATEGORIES

with I n+1 ∈ I. By construction we get dn : I n → I n+1 and an injective map αn+1 : K n+1 → I n+1 . The equality in Equation (11.15.4.1) and the construction of dn guarantee that α : H n (K • ) → Ker(dn )/Im(dn−1 ) is an isomorphism. In other words IHn+1 holds. We finish the proof of by the following observations. First we note that IHn is true for n = a since we can just take I j = 0 for j < a and K a → I a an injection of K a into an element of A. Next, we note that in the proof of IHn ⇒ IHn+1 we only modified the object I n , the map dn−1 and the map αn . Hence we see that proceeding by induction we produce a complex I • with I n = 0 for n < a consisting of objects from I, and a termwise injective quasi-isomorphism α : K • → I • as desired.  Lemma 11.15.5. Let A be an abelian category. Let P ⊂ Ob(A) be a subset such that every object of A is a quotient of an element of P. Then for every bounded above complex K • there exists a quasi-isomorphism P • → K • with P • bounded above and each P n ∈ P. Proof. This lemma is dual to Lemma 11.15.4.



Lemma 11.15.6. In Situation 11.15.1. Let I ⊂ Ob(A) be a subset with the following properties: (1) every object of A is a subobject of an element of I, (2) for any short exact sequence 0 → P → Q → R → 0 of A with P, Q ∈ I, then R ∈ I, and 0 → F (P ) → F (Q) → F (R) → 0 is exact. Then every object of I is acyclic for RF . Proof. Pick A ∈ I. Let A[0] → K • be a quasi-isomorphism with L• bounded below. Then we can find a quasi-isomorphism K • → I • with I • bounded below and each I n ∈ I, see Lemma 11.15.4. Hence we see that these resolutions are cofinal in the category A[0]/Qis+ (A). To finish the proof it therefore suffices to show that for any quasi-isomorphism A[0] → I • with I • bounded above and I n ∈ I we have F (A)[0] → F (I • ) is a quasi-isomorphism. To see this suppose that I n = 0 for n < n0 . Of course we may assume that n0 < 0. Starting with n = n0 we prove inductively that Im(dn−1 ) = Ker(dn ) and Im(d−1 ) are elements of I using propert (2) and the exact sequences 0 → Ker(dn ) → I n → Im(dn ) → 0. Moreover, property (2) also guarantees that the complex 0 → F (I n0 ) → F (I n0 +1 ) → . . . → F (I −1 ) → F (Im(d−1 )) → 0 is exact. The exact sequence 0 → Im(d−1 ) → I 0 → I 0 /Im(d−1 ) → 0 implies that I 0 /Im(d−1 ) is an element of I. The exact sequence 0 → A → I 0 /Im(d−1 ) → Im(d0 ) → 0 then implies that Im(d0 ) = Ker(d1 ) is an elements of I and from then on one continues as before to show that Im(dn−1 ) = Ker(dn ) is an element of I for all n > 0. Applying F to each of the short exact sequences mentioned above and using (2) we observe that F (A)[0] → F (I • ) is an isomorphism as desired.  Lemma 11.15.7. In Situation 11.15.1. Let P ⊂ Ob(A) be a subset with the following properties: (1) every object of A is a quotient of an element of P,

11.16. HIGHER DERIVED FUNCTORS

799

(2) for any short exact sequence 0 → P → Q → R → 0 of A with Q, R ∈ P, then P ∈ P, and 0 → F (P ) → F (Q) → F (R) → 0 is exact. Then every object of P is acyclic for LF . Proof. Dual to the proof of Lemma 11.15.6.



Proposition 11.15.8. In Situation 11.15.1. (1) If every object of A injects into an object acyclic for RF , then RF is defined on all of K + (A) and we obtain an exact functor RF : D+ (B) −→ D+ (A) see (11.14.9.1). Moreover, any bounded below complex K • whose terms are acyclic for RF computes RF . (2) If every object of A is quotient of an object acyclic for LF , then LF is defined on all of K − (A) and we obtain an exact functor LF : D− (B) −→ D− (A) see (11.14.9.1). Moreover, any bounded above complex K • whose terms are acyclic for LF computes LF . Proof. Suppose every object of A injects into an object acyclic for RF . Let I be the set of objects acyclic for RF . Let K • be a bounded below complex with K n ∈ I. By Lemma 11.15.4 the quasi-isomorphisms α : K • → I • with I • bounded below and I n ∈ I are cofinal in the category K • /Qis+ (A). Hence in order to show that K • computes RF it suffices to show that F (K • ) → F (I • ) is an isomorphism. Note that C(α)• is an acyclic bounded below complex all of whose terms are in I. Hence it suffices to show: given an acyclic bounded below complex I • all of whose terms are in I the complex F (I • ) is acyclic. Say I n = 0 for n < n0 . Then we break I • into short exact sequences 0 → Im(dn ) → I n+1 → Im(dn+1 ) → 0 for n ≥ n0 . These sequences induce distinguished triangles (Im(dn ), I n+1 , Im(dn+1 )) by Lemma 11.11.1. This implies inductively that each Im(dn ) is acyclic for RF by Lemma 11.14.12. Moreover, the long exact cohomology sequences (11.10.1.1) associated to the distinguished triangles (F (Im(dn )), F (I n+1 ), F (Im(dn+1 ))) of D+ (B) imply that 0 → F (Im(dn )) → F (I n+1 ) → F (Im(dn+1 )) → 0 is short exact, and this in turn proves that F (I • ) is exact. Finally, since by Lemma 11.15.4 every object of K + (A) is quasi-isomorphic to such a bounded below complex with terms in I we see that RF is everywhere defined, see Lemma 11.14.14. The proof in the case of LF is dual.  11.16. Higher derived functors The following simple lemma shows that right derived functors “move to the right”. Lemma 11.16.1. Let F : A → B be an additive functor between abelian categories. Let K • ∈ K + (A) and a ∈ Z such that H i (K • ) = 0 for all i < a. If RF is defined at K • , then H i (RF (K • )) = 0 for all i < a.

800

11. DERIVED CATEGORIES

Proof. Let K • → L• be any quasi-isomorphism. Then it is also true that K • → τ≥a L• is a quasi-isomorphism by our assumption on K • . Hence in the category K • /Qis+ (A) the quasi-isomorphisms s : K • → L• with Ln = 0 for n < a are cofinal. Thus RF is the value of the essentially constant ind-object F (L• ) for these s it follows that H i (RF (K • )) = 0 for i < 0.  Definition 11.16.2. Let F : A → B be an additive functor between abelian categories. Assume RF : D+ (A) → D+ (B) is everywhere defined. Let i ∈ Z. The ith right derived functor Ri F of F is the functor Ri F = H i ◦ RF : A −→ B The following lemma shows that it really does not make a lot of sense to take the right derived functor unless the functor is left exact. Lemma 11.16.3. Let F : A → B be an additive functor between abelian categories and assume RF : D+ (A) → D+ (B) is everywhere defined. (1) We have Ri F = 0 for i < 0, (2) R0 F is left exact, (3) the map F → R0 F is an isomorphism if and only if F is left exact. Proof. Let A be an object of A. Let A[0] → K • be any quasi-isomorphism. Then it is also true that A[0] → τ≥0 K • is a quasi-isomorphism. Hence in the category A[0]/Qis+ (A) the quasi-isomorphisms s : A[0] → K • with K n = 0 for n < 0 are cofinal. Thus it is clear that H i (RF (A[0])) = 0 for i < 0. Moreover, for such an s the sequence 0 → A → K0 → K1 is exact. Hence if F is left exact, then 0 → F (A) → F (K 0 ) → F (K 1 ) is exact as well, and we see that F (A) → H 0 (F (K • )) is an isomorphism for every s : A[0] → K • as above which implies that H 0 (RF (A[0])) = F (A). Let 0 → A → B → C → 0 be a short exact sequence of A. By Lemma 11.11.1 we obtain a distinguished triangle (A[0], B[0], C[0], a, b, c) in K + (A). From the long exact cohomology sequence (and the vanishing for i < 0 proved above) we deduce that 0 → R0 F (A) → R0 F (B) → R0 F (C) is exact. Hence R0 F is left exact. Of course this also proves that if F → R0 F is an isomorphism, then F is left exact.  Lemma 11.16.4. Let F : A → B be an additive functor between abelian categories and assume RF : D+ (A) → D+ (B) is everywhere defined. Let A be an object of A. (1) A is right acyclic for F if and only if F (A) → R0 F (A) is an isomorphism and Ri F (A) = 0 for all i > 0, (2) if F is left exact, then A is right acyclic for F if and only if Ri F (A) = 0 for all i > 0. Proof. If A is right acyclic for F , then RF (A[0]) = F (A)[0] and in particular F (A) → R0 F (A) is an isomorphism and Ri F (A) = 0 for i 6= 0. Conversely, if F (A) → R0 F (A) is an isomorphism and Ri F (A) = 0 for all i > 0 then F (A[0]) → RF (A[0]) is a quasi-isomorphism by Lemma 11.16.3 part (1) and hence A is acyclic. If F is left exact then F = R0 F , see Lemma 11.16.3.  Lemma 11.16.5. Let F : A → B be a left exact functor between abelian categories and assume RF : D+ (A) → D+ (B) is everywhere defined. Let 0 → A → B → C → 0 be a short exact sequence of A.

11.16. HIGHER DERIVED FUNCTORS

801

(1) If A and C are right acyclic for F then so is B. (2) If A and B are right acyclic for F then so is C. (3) If B and C are right acyclic for F and F (B) → F (C) is surjective then A is right acyclic for F . In each of the three cases 0 → F (A) → F (B) → F (C) → 0 is a short exact sequence of B. Proof. By Lemma 11.11.1 we obtain a distinguished triangle (A[0], B[0], C[0], a, b, c) in K + (A). As RF is an exact functor and since Ri F = 0 for i < 0 and R0 F = F (Lemma 11.16.3) we obtain an exact cohomology sequence 0 → F (A) → F (B) → F (C) → R1 F (A) → . . . in the abelian category B. Thus the lemma follows from the characterization of acyclic objects in Lemma 11.16.4.  Lemma 11.16.6. Let F : A → B be an additive functor between abelian categories and assume RF : D+ (A) → D+ (B) is everywhere defined. (1) The functors Ri F , i ≥ 0 come equipped with a canonical structure of a δ-functor from A → B, see Homology, Definition 10.9.1. (2) If every object of A is a subobject of a right acyclic object for F , then {Ri F, δ}i≥0 is a universal δ-functor, see Homology, Definition 10.9.3. Proof. The functor A → Comp+ (A), A 7→ A[0] is exact. The functor Comp+ (A) → D+ (A) is a δ-functor, see Lemma 11.11.1. The functor RF : D+ (A) → D+ (B) is exact. Finally, the functor H 0 : D+ (B) → B is a homological functor, see Definition 11.10.3. Hence we get the structure of a δ-functor from Lemma 11.4.20 and Lemma 11.4.19. Part (2) follows from Homology, Lemma 10.9.4 and the description of acyclics in Lemma 11.16.4.  Lemma 11.16.7 (Leray’s acyclicity lemma). Let F : A → B be an additive functor between abelian categories and assume RF : D+ (A) → D+ (B) is everywhere defined. Let A• be a bounded below complex of F -acyclic objects. The canonical map F (A• ) −→ RF (A• ) is an isomorphism in D+ (B), i.e., A• computes RF . Proof. First we claim the lemma holds for a bounded complex of acyclic objects. Namely, it holds for complexes with at most one nonzero object by definition. Suppose that A• is a complex with An = 0 for n 6∈ [a, b]. Using the “stupid” truncations we obtain a termwise split short exact sequence of complexes 0 → σ≥a+1 A• → A• → σ≤a A• → 0 see Homology, Section 10.11. Thus a distinguished triangle (σ≥a+1 A• , A• , σ≤a A• ). By induction hypothesis the two outer complexes compute RF . Then the middle one does too by Lemma 11.14.12. Suppose that A• is a bounded below complex of acyclic objects. To show that F (A) → RF (A) is an isomorphism in D+ (B) it suffices to show that H i (F (A)) →

802

11. DERIVED CATEGORIES

H i (RF (A)) is an isomorphism for all i. Pick i. Consider the termwise split short exact sequence of complexes 0 → σ≥i+2 A• → A• → σ≤i+1 A• → 0. Note that this induces a termwise split short exact sequence 0 → σ≥i+2 F (A• ) → F (A• ) → σ≤i+1 F (A• ) → 0. Hence we get distinguished triangles (σ≥i+2 A• , A• , σ≤i+1 A• ) (σ≥a+1 F (A• ), F (A• ), σ≤a F (A• )) (RF (σ≥a+1 A• ), RF (A• ), RF (σ≤a A• )) Using the last two we obtain a map of exact sequences H i (σ≥i+2 F (A• ))

/ H i (F (A• ))



 / Ri F (A• )

Ri F (σ≥i+2 A• )

α

/ H i (σ≤i+1 F (A• )) 

β

/ Ri F (σ≤i+1 A• )

/ H i+1 (σ≥i+2 F (A• ))  / Ri+1 F (σ≥i+2 A• )

By the results of the first paragraph the map β is an isomorphism. By inspection the objects on the upper left and the upper right are zero. Hence to finish the proof we have to show that Ri F (σ≥i+2 A• ) = 0 and Ri+1 F (σ≥i+2 A• ) = 0. This follows immediately from Lemma 11.16.1.  11.16.8. Let F : A → B be an exact functor of abelian categories. Then every object of A is right acyclic for F , RF : D+ (A) → D+ (A) is everywhere defined, RF : D(A) → D(A) is everywhere defined, every complex computes RF , in other words, the canonical map F (K • ) → RF (K • ) is an isomorphism for all complexes, and (5) Ri F = 0 for i 6= 0.

Lemma (1) (2) (3) (4)

Proof. This is true because F transforms acyclic complexes into acyclic complexes and quasi-isomorphisms into quasi-isomorphisms. Details omitted.  11.17. Injective resolutions In this section we prove some lemmas regarding the existence of injective resolutions in abelian categories having enough injectives. Definition 11.17.1. Let A be an abelian category. Let A ∈ Ob(A). An injective resolution of A is a complex I • together with a map A → I 0 such that: (1) We have I n = 0 for n < 0. (2) Each I n is an injective object of A. (3) The map A → I 0 is an isomorphism onto Ker(d0 ). (4) We have H i (I • ) = 0 for i > 0. Hence A[0] → I • is a quasi-isomorphism. In other words the complex . . . → 0 → A → I0 → I1 → . . . is acyclic. Let K • be a complex in A. An injective resolution of K • is a complex I • together with a map α : K • → I • of complexes such that (1) We have I n = 0 for n  0, i.e., I • is bounded below.

11.17. INJECTIVE RESOLUTIONS

803

(2) Each I n is an injective object of A. (3) The map α : K • → I • is a quasi-isomorphism. In other words an injective resolution K • → I • gives rise to a diagram ...

/ K n−1

/ Kn

/ K n+1

/ ...

...

 / I n−1

 / In

 / I n+1

/ ...

which induces an isomorphism on cohomology objects in each degree. An injective resolution of an object A of A is almost the same thing as an injective resolution of the complex A[0]. Lemma 11.17.2. Let A be an abelian category. Let K • be a complex of A. (1) If K • has an injective resolution then H n (K • ) = 0 for n  0. (2) If H n (K • ) = 0 for all n  0 then there exists a quasi-isomorphism K • → L• with L• bounded below. Proof. Omitted. For the second statement use L• = K • /τ≤n K • for some n  0. See Homology, Section 10.11 for the definition of the truncation τ≤n .  Lemma (1) (2) (3)

11.17.3. Let A be an abelian category. Assume A has enough injectives. Any object of A has an injective resolution. If H n (K • ) = 0 for all n  0 then K • has an injective resolution. If K • is a complex with K n = 0 for n < a, then there exists an injective resolution α : K • → I • with I n = 0 for n < a such that each αn : K n → I n is injective.

Proof. Proof of (1). First choose an injection A → I 0 of A into an injective object of A. Next, choose an injection I0 /A → I 1 into an injective object of A. Denote d0 the induced map I 0 → I 1 . Next, choose an injection I 1 /Im(d0 ) → I 2 into an injective object of A. Denote d1 the induced map I 1 → I 2 . And so on. By Lemma 11.17.2 part (2) follows from part (3). Part (3) is a special case of Lemma 11.15.4.  Lemma 11.17.4. Let A be an abelian category. Let K • be an acyclic complex. Let I • be bounded below and consisting of injective objects. Any morphism K • → I • is homotopic to zero. Proof. Let α : K • → I • be a morphism of complexes. Assume that αj = 0 for j < n. We will show that there exists a morphism h : K n+1 → I n such that αn = h ◦ d. Thus α will be homotopic to the morphism of complexes β defined by  0 if j≤n  β j = αn+1 − d ◦ h if j = n + 1  if j > n + 1 αj This will clearly prove the lemma (by induction). To prove the existence of h note that αn |dn−1 (K n−1 ) = 0 since αn−1 = 0. Since K • is acyclic we have dn−1 (K n−1 ) = Ker(K n → K n+1 ). Hence we can think of αn as a map into I n defined on the subobject Im(K n → K n+1 ) of K n+1 . By injectivity of the object I n we can extend this to a map h : K n+1 → I n as desired. 

804

11. DERIVED CATEGORIES

Remark 11.17.5. Let A be an abelian category. Using the fact that K(A) is a triangulated category we may use Lemma 11.17.4 to obtain proofs of some of the lemmas below which are usually proved by chasing through diagrams. Namely, suppose that α : K • → L• is a quasi-isomorphism of complexes. Then (K • , L• , C(α)• , α, i, −p) is a distinguished triangle in K(A) (Lemma 11.8.12) and C(f )• is an acyclic complex (Lemma 11.10.2). Next, let I • be a bounded below complex of injective objects. Then HomK(A) (C(α)• , I • )

/ HomK(A) (L• , I • )

/ HomK(A) (K • , I • )

r HomK(A) (C(α)• [−1], I • ) is an exact sequence of abelian groups, see Lemma 11.4.2. At this point Lemma 11.17.4 guarantees that the outer two groups are zero and hence HomK(A) (L• , I • ) = HomK(A) (K • , I • ). Lemma 11.17.6. Let A be an abelian category. Consider a solid diagram K• γ

I

 }

α

/ L•

β

•

where I • is bounded below and consists of injective objects, and α is a quasiisomorphism. (1) There exists a map of complexes β making the diagram commute up to homotopy. (2) If α is injective in every degree then we can find a β which makes the diagram commute. Proof. The “correct” proof of part (1) is explained in Remark 11.17.5. We also give a direct proof here. ˜ • , π, s be as in Lemma We first show that (2) implies (1). Namely, let α ˜:K→L ˜ • → I • such that 11.8.5. Since α ˜ is injective by (2) there exists a morphism β˜ : L γ = β˜ ◦ α ˜ . Set β = β˜ ◦ s. Then we have β ◦ α = β˜ ◦ s ◦ π ◦ α ˜ ∼ β˜ ◦ α ˜=γ as desired. Assume that α : K • → L• is injective. Suppose we have already defined β in all degrees ≤ n − 1 compatible with differentials and such that γ j = β j ◦ αj for all

11.17. INJECTIVE RESOLUTIONS

805

j ≤ n − 1. Consider the commutative solid diagram / Kn

K n−1 

γ

α

α

 / Ln

Ln−1 β

  I n−1

γ

  / In

Thus we see that the dotted arrow is prescribed on the subobjects α(K n ) and dn−1 (Ln−1 ). Moreover, these two arrows agree on α(dn−1 (K n−1 )). Hence if (11.17.6.1)

α(dn−1 (K n−1 )) = α(K n ) ∩ dn−1 (Ln−1 )

then these morphisms glue to a morphism α(K n ) + dn−1 (Ln−1 ) → I n and, using the injectivity of I n , we can extend this to a morphism from all of Ln into I n . After this by induction we get the morphism β for all n simultaneously (note that we can set β n = 0 for all n  0 since I • is bounded below – in this way starting the induction). It remains to prove the equality (11.17.6.1). The reader is encouraged to argue this for themselves with a suitable diagram chase. Nonetheless here is our argument. Note that the inclusion α(dn−1 (K n−1 )) ⊂ α(K n ) ∩ dn−1 (Ln−1 ) is obvious. Take an object T of A and a morphism x : T → Ln whose image is contained in the subobject α(K n ) ∩ dn−1 (Ln−1 ). Since α is injective we see that x = α ◦ x0 for some x0 : T → K n . Moreover, since x lies in dn−1 (Ln−1 ) we see that dn ◦ x = 0. Hence using injectivity of α again we see that dn ◦ x0 = 0. Thus x0 gives a morphism [x0 ] : T → H n (K • ). On the other hand the corresponding map [x] : T → H n (L• ) induced by x is zero by assumption. Since α is a quasi-isomorphism we conclude that [x0 ] = 0. This of course means exactly that the image of x0 is contained in dn−1 (K n−1 ) and we win.  Lemma 11.17.7. Let A be an abelian category. Consider a solid diagram K•

α

/ L•

γ

 } I•

βi

where I • is bounded below and consists of injective objects, and α is a quasiisomorphism. Any two morphisms β1 , β2 making the diagram commute up to homotopy are homotopic. Proof. This follows from Remark 11.17.5. We also give a direct argument here. ˜ • , π, s be as in Lemma 11.8.5. If we can show that β1 ◦π is homotopic Let α ˜:K→L to β2 ◦ π, then we deduce that β1 ∼ β2 because π ◦ s is the identity. Hence we may assume αn : K n → Ln is the inclusion of a direct summand for all n. Thus we get a short exact sequence of complexes 0 → K • → L• → M • → 0 which is termwise split and such that M • is acyclic. We choose splittings Ln = K n ⊕ M n , so we have βin : K n ⊕ M n → I n and γ n : K n → I n . In this case the

806

11. DERIVED CATEGORIES

condition on βi is that there are morphisms hni : K n → I n−1 such that γ n − βin |K n = d ◦ hni + hn+1 ◦d i Thus we see that β1n |K n − β2n |K n = d ◦ (hn1 − hn2 ) + (hn+1 − hn+1 )◦d 1 2 Consider the map hn : K n ⊕M n → I n−1 which equals hn1 −hn2 on the first summand and zero on the second. Then we see that β1n − β2n − (d ◦ hn + hn+1 ) ◦ d) is a morphism of complexes L• → I • which is identically zero on the subcomplex K • . Hence it factors as L• → M • → I • . Thus the result of the lemma follows from Lemma 11.17.4.  Lemma 11.17.8. Let A be an abelian category. Let I • be bounded below complex consisting of injective objects. Let L• ∈ K(A). Then MorK(A) (L• , I • ) = MorD(A) (L• , I • ). Proof. Let a be an element of the right hand side. We may represent a = γα−1 where α : K • → L• is a quasi-isomorphism and γ : K • → I • is a map of complexes. By Lemma 11.17.6 we can find a morphism β : L• → I • such that β ◦α is homotopic to γ. This proves that the map is surjective. Let b be an element of the left hand side which maps to zero in the right hand side. Then b is the homotopy class of a morphism β : L• → I • such that there exists a quasi-iomorphism α : K • → L• with β ◦ α homotopic to zero. Then Lemma 11.17.7 shows that β is homotopic to zero also, i.e., b = 0.  Lemma 11.17.9. Let A be an abelian category. Assume A has enough injectives. For any short exact sequence 0 → A• → B • → C • → 0 of Comp+ (A) there exists a commutative diagram in Comp+ (A) 0

/ A•

/ B•

/ C•

/0

0

 / I1•

 / I2•

 / I3•

/0

where the vertical arrows are injective resolutions and the rows are short exact sequences of complexes. In fact, given any injective resolution A• → I • we may assume I1• = I • . Proof. Step 1. Choose an injective resolution A• → I • (see Lemma 11.17.3) or use the given one. Recall that Comp+ (A) is an abelian category, see Homology, Lemma 10.10.9. Hence we may form the pushout along the injective map A• → I • to get / A• / B• / C• /0 0    / I• / E• / C• /0 0 Note that the lower short exact sequence is termwise split, see Homology, Lemma 10.20.2. Hence it suffices to prove the lemma when 0 → A• → B • → C • → 0 is termwise split.

11.18. PROJECTIVE RESOLUTIONS

807

Step 2. Choose splittings. In other words, write B n = An ⊕ C n . Denote δ : C • → A• [1] the morphism as in Homology, Lemma 10.12.10. Choose injective resolutions f1 : A• → I1• and f3 : C • → I3• . (If A• is a complex of injectives, then use I1• = A• .) We may assume f3 is injective in every degree. By Lemma 11.17.6 we may find a morphism δ 0 : I3• → I1• [1] such that δ 0 ◦ f3 = f1 [1] ◦ δ (equality of morphisms of complexes). Set I2n = I1n ⊕ I3n . Define  n  dI1 (δ 0 )n n dI2 = 0 dnI3 and define the maps B n → I2n to be given as the sum of the maps An → I1n and C n → I3n . Everything is clear.  11.18. Projective resolutions This section is dual to Section 11.17. We give definitions and state results, but we do not reprove the lemmas. Definition 11.18.1. Let A be an abelian category. Let A ∈ Ob(A). An projective resolution of A is a complex P • together with a map P 0 → A such that: (1) We have P n = 0 for n > 0. (2) Each P n is an projective object of A. (3) The map P 0 → A induces an isomorphism Coker(d−1 ) → A. (4) We have H i (P • ) = 0 for i < 0. Hence P • → A[0] is a quasi-isomorphism. In other words the complex . . . → P −1 → P 0 → A → 0 → . . . is acyclic. Let K • be a complex in A. An projective resolution of K • is a complex P • together with a map α : P • → K • of complexes such that (1) We have P n = 0 for n  0, i.e., P • is bounded above. (2) Each P n is an projective object of A. (3) The map α : P • → K • is a quasi-isomorphism. Lemma 11.18.2. Let A be an abelian category. Let K • be a complex of A. (1) If K • has a projective resolution then H n (K • ) = 0 for n  0. (2) If H n (K • ) = 0 for n  0 then there exists a quasi-isomorphism L• → K • with L• bounded above. Proof. Dual to Lemma 11.17.2. Lemma (1) (2) (3)



11.18.3. Let A be an abelian category. Assume A has enough projectives. Any object of A has a projective resolution. If H n (K • ) = 0 for all n  0 then K • has a projective resolution. If K • is a complex with K n = 0 for n > a, then there exists a projective resolution α : P • → K • with P n = 0 for n > a such that each αn : P n → K n is surjective.

Proof. Dual to Lemma 11.17.3.



Lemma 11.18.4. Let A be an abelian category. Let K • be an acyclic complex. Let P • be bounded above and consisting of projective objects. Any morphism P • → K • is homotopic to zero. Proof. Dual to Lemma 11.17.4.



808

11. DERIVED CATEGORIES

Remark 11.18.5. Let A be an abelian category. Suppose that α : K • → L• is a quasi-isomorphism of complexes. Let P • be a bounded above complex of projectives. Then HomK(A) (P • , K • ) −→ HomK(A) (P • , L• ) is an isomorphism. This is dual to Remark 11.17.5. Lemma 11.18.6. Let A be an abelian category. Consider a solid diagram KO • o α = L• β

P• where P • is bounded above and consists of projective objects, and α is a quasiisomorphism. (1) There exists a map of complexes β making the diagram commute up to homotopy. (2) If α is surjective in every degree then we can find a β which makes the diagram commute. Proof. Dual to Lemma 11.17.6.



Lemma 11.18.7. Let A be an abelian category. Consider a solid diagram KO • o α = L• βi

P• where P • is bounded above and consists of projective objects, and α is a quasiisomorphism. Any two morphisms β1 , β2 making the diagram commute up to homotopy are homotopic. Proof. Dual to Lemma 11.17.7.



Lemma 11.18.8. Let A be an abelian category. Let P • be bounded above complex consisting of projective objects. Let L• ∈ K(A). Then MorK(A) (P • , L• ) = MorD(A) (P • , L• ). Proof. Dual to Lemma 11.17.8.



Lemma 11.18.9. Let A be an abelian category. Assume A has enough projectives. For any short exact sequence 0 → A• → B • → C • → 0 of Comp+ (A) there exists a commutative diagram in Comp+ (A) 0

/ P1•

/ P2•

/ P3•

/0

0

 / A•

 / B•

 / C•

/0

where the vertical arrows are projective resolutions and the rows are short exact sequences of complexes. In fact, given any projective resolution P • → C • we may assume P3• = P • . Proof. Dual to Lemma 11.17.9.



11.19. RIGHT DERIVED FUNCTORS AND INJECTIVE RESOLUTIONS

809

Lemma 11.18.10. Let A be an abelian category. Let P • , K • be complexes. Let n ∈ Z. Assume that (1) P • is a bounded complex consisting of projective objects, (2) P i = 0 for i < n, and (3) H i (K • ) = 0 for i ≥ n. Then HomK(A) (P • , K • ) = HomD(A) (P • , K • ) = 0. Proof. The first equality follows from Lemma 11.18.8. Note that there is a distinguished triangle (τ≤n−1 K • , K • , τ≥n K • , f, g, h) because the complex K • /τ≤n−1 K • is quasi-isomorphic to τ≥n K • . Hence, by Lemma 11.4.2 it suffices to prove HomK(A) (P • , τ≤n−1 K • ) = 0 and HomK(A) (P • , τ≥n K • ) = 0. The first vanishing is trivial and the second is Lemma 11.18.4.  Lemma 11.18.11. Let A be an abelian category. Let β : P • → L• and α : E • → L• be maps of complexes. Let n ∈ Z. Assume (1) P • is a bounded complex of projectives and P i = 0 for i < n, (2) H i (α) is an isomorphism for i > n and surjective for i = n. Then there exists a map of complexes γ : P • → E • such that α ◦ γ and β are homotopic. Proof. Consider the cone C • = C(α)• with map i : L• → C • . Note that i ◦ β is zero by Lemma 11.18.10. Hence we can lift β to E • by Lemma 11.4.2.  11.19. Right derived functors and injective resolutions At this point we can use the material above to define the right derived functors of an additive functor between an abelian category having enough injectives and a general abelian category. Lemma 11.19.1. Let A be an abelian category. Let I ∈ Ob(A) be an injective object. Let I • be a bounded below complex of injectives in A. (1) I • computes RF relative to Qis+ (A) for any exact functor F : K + (A) → D into any triangulated category D. (2) I is right acyclic for any additive functor F : A → B into any abelian category B. Proof. Part (2) is a direct consequences of part (1) and Definition 11.15.3. To prove (1) let α : I • → K • be a quasi-isomorphism into a complex. By Lemma 11.17.7 we see that α has a left inverse. Hence the category I • /Qis+ (A) is essentially constant with value id : I • → I • . Thus also the ind-object I • /Qis+ (A) −→ D,

(I • → K • ) 7−→ F (K • )

is essentially constant with value F (I • ). This proves (1), see Definitions 11.14.2 and 11.14.10.  Lemma 11.19.2. Let A be an abelian category with enough injectives. (1) For any exact functor F : K + (A) → D into a triangulated category D the right derived functor RF : D+ (A) −→ D is everywhere defined.

810

11. DERIVED CATEGORIES

(2) For any additive functor F : A → B into an abelian category B the right derived functor RF : D+ (A) −→ D+ (B) is everywhere defined. Proof. Combine Lemma 11.19.1 and Proposition 11.15.8 for the second assertion. To see the first assertion combine Lemma 11.17.3, Lemma 11.19.1, Lemma 11.14.14, and Equation (11.14.9.1).  Lemma 11.19.3. Let A be an abelian category with enough injectives. Let F : A → B be an additive functor. (1) (2) (3) (4)

The The The The

functor functor functor functor

RF RF RF RF

is an exact functor D+ (A) → D+ (B). induces an exact functor K + (A) → D+ (B). induces a δ-functor Comp+ (A) → D+ (B). induces a δ-functor A → D+ (B).

Proof. This lemma simply reviews some of the results obtained sofar. Note that by Lemma 11.19.2 RF is everywhere defined. Here are some references: (1) The derived functor is exact: This boils down to Lemma 11.14.6. (2) This is true because K + (A) → D+ (A) is exact and compositions of exact functors are exact. (3) This is true because Comp+ (A) → D+ (A) is a δ-functor, see Lemma 11.11.1. (4) This is true because A → Comp+ (A) is exact and precomposing a δfunctor by an exact functor gives a δ-functor.  Lemma 11.19.4. Let A be an abelian category with enough injectives. Let F : A → B be a left exact functor. (1) For any short exact sequence 0 → A• → B • → C • → 0 of complexes in Comp+ (A) there is an associated long exact sequence . . . → H i (RF (A• )) → H i (RF (B • )) → H i (RF (C • )) → H i+1 (RF (A• )) → . . . (2) The functors Ri F : A → B are zero for i < 0. Also R0 F = F : A → B. (3) We have Ri F (I) = 0 for i > 0 and I injective. (4) The sequence (Ri F, δ) forms a universal δ-functor (see Homology, Definition 10.9.3) from A to B. Proof. This lemma simply reviews some of the results obtained sofar. Note that by Lemma 11.19.2 RF is everywhere defined. Here are some references: (1) This follows from Lemma 11.19.3 part (3) combined with the long exact cohomology sequence (11.10.1.1) for D+ (B). (2) This is Lemma 11.16.3. (3) This is the fact that injective objects are acyclic. (4) This is Lemma 11.16.6. 

11.20. CARTAN-EILENBERG RESOLUTIONS

811

11.20. Cartan-Eilenberg resolutions This section can be expanded. The material can be generalized and applied in more cases. Resolutions need not use injectives and the method also works in the unbounded case in some situations. Definition 11.20.1. Let A be an abelian category. Let K • be a bounded below complex. A Cartan-Eilenberg resolution of K • is given by a double complex I •,• and a morphism of complexes  : K • → I •,0 with the following properties: (1) There exists a i  0 such that I p,q = 0 for all p < i and all q. (2) We have I p,q = 0 if q < 0. (3) The complex I p,• is an injective resolution of K p . (4) The complex Ker(d1p,• ) is an injective resolution of Ker(dpK ). p (5) The complex Im(dp,• 1 ) is an injective resolution of Im(dK ). p •,• (6) The complex HI (I ) is an injective resolution of H p (K • ). Lemma 11.20.2. Let A be an abelian category with enough injectives. Let K • be a bounded below complex. There exists a Cartan-Eilenberg resolution of K • . Proof. Suppose that K p = 0 for p < n. Decompose K • into short exact sequences as follows: Set Z p = Ker(dp ), B p = Im(dp−1 ), H p = Z p /B p , and consider 0 → Z n → K n → B n+1 → 0 0 → B n+1 → Z n+1 → H n+1 → 0 0 → Z n+1 → K n+1 → B n+2 → 0 0 → B n+2 → Z n+2 → H n+2 → 0 ... Set I p,q = 0 for p < n. Inductively we choose injective resolutions as follows: (1) Choose an injective resolution Z n → JZn,• . (2) Using Lemma 11.17.9 choose injective resolutions K n → I n,• , B n+1 → n+1,• n+1,• JB , and an exact sequence of complexes 0 → JZn,• → I n,• → JB → 0 compatible with the short exact sequence 0 → Z n → K n → B n+1 → 0. (3) Using Lemma 11.17.9 choose injective resolutions Z n+1 → JZn+1,• , H n+1 → n+1,• n+1,• JH , and an exact sequence of complexes 0 → JB → JZn+1,• → n+1,• JH → 0 compatible with the short exact sequence 0 → B n+1 → n+1 Z → H n+1 → 0. (4) Etc. p+1,• Taking as maps d•1 : I p,• → I p+1,• the compositions I p,• → JB → JZp+1,• → p+1,• I everything is clear. 

Lemma 11.20.3. Let F : A → B be a left exact functor of abelian categories. Let K • be a bounded below complex of A. Let I •,• be a Cartan-Eilenberg resolution for K • . The spectral sequences (0 Er , 0 dr )r≥0 and (00 Er , 00 dr )r≥0 associated to the double complex F (I •,• ) satisfy the relations 0

E2p,q = H p (Rq F (K • ))3

and

00

E2p,q = Rp F (H q (K • ))

Moreover, these spectral sequences converge to H p+q (RF (K • )) and the associated induced filtrations on H p+q (RF (K • )) are finite. 3This notation sucks! It really means the pth cohomology group of the complex with terms Rq F (K n ). Not the pth cohomology of the qth derived functor of F applied to K • ...

812

11. DERIVED CATEGORIES

Proof. We will use the following remarks without further mention: (1) As I p,• is an injective resolution of K p we see that RF is defined at K p [0] with value F (I p,• ). (2) As HIp (I •,• ) is an injective resolution of H p (K • ) the derived functor RF is defined at H p (K • )[0] with value F (HIp (I •,• )). (3) By Homology, Lemma 10.19.6 the total complex sI • is an injective resolution of K • . Hence RF is defined at K • with value F (sI • ). Consider the spectral sequences associated to the double complex K •,• = F (I •,• ), see Homology, Lemma 10.19.3. These both converge, see Homology, Lemma 10.19.5, to the cohomology groups of the associated total complex s(F (I •,• ) = F (sI • ) which computes H n (RF (K • )). Computation of the first spectral sequence. We have 0 E1p,q = H q (K p,• ) in other words 0 p,q E1 = H q (F (I p,• )) = Rq F (K p ) and the maps 0 E1p,q → 0 E p+1,q are the maps Rq F (K p ) → Rq F (K p+1 ) as desired. Computation of the second spectral sequence. We have 00 E1p,q = H q (K •,p ) = H q (F (I •,p )). Note that the complex I •,p is bounded below, consists of injectives, and moreover each kernel, image, and cohomology group of the differentials is an injective object of A. Hence we can split the differentials, i.e., each differential is a split surjection onto a direct summand. It follows that the same is true after applying F . Hence 00 E1p,q = F (H q (I •,p )) = F (HIq (I •,p )). The differentials on this are (−1)q times F applied to the differential of the complex HIp (I •,• ) which is an injective resolution of H p (K • ). Hence the description of the E2 terms.  Remark 11.20.4. The spectral sequences of Lemma 11.20.3 are functorial in the complex K • . This follows from functoriality properties of Cartan-Eilenberg resolutions. On the other hand, they are both examples of a more general spectral sequence which may be associated to a filtered complex of A. The functoriality will follow from its construction. We will return to this in the section on the filtered derived category, see Remark 11.25.15. 11.21. Composition of right derived functors Sometimes we can compute the right derived functor of a composition. Suppose that A, B, C be abelian categories. Let F : A → B and G : B → C be left exact functors. Assume that the right derived functors RF : D+ (A) → D+ (B), RG : D+ (B) → D+ (C), and R(G ◦ F ) : D+ (A) → D+ (C) are everywhere defined. Then there exists a canonical transformation t : R(G ◦ F ) −→ RG ◦ RF, see Lemma 11.14.16. This transformation need not always be an isomorphism. Lemma 11.21.1. Let A, B, C be abelian categories. Let F : A → B and G : B → C be left exact functors. Assume A, B have enough injectives. If F (I) is right acyclic for G for each injective object I of A, then we have an isomorphism of functors t : R(G ◦ F ) −→ RG ◦ RF.

11.22. RESOLUTION FUNCTORS

813

Proof. Let A• be a bounded below complex of A. Choose an injective resolution A• → I • . The map t is given (see proof of Lemma 11.14.16) by the maps R(G ◦ F )(A• ) = (G ◦ F )(I • ) = G(F (I • ))) → RG(F (I • )) = RG(RF (A• )) where the arrow is an isomorphism by Lemma 11.16.7.



Lemma 11.21.2 (Grothendieck spectral sequence). With assumptions as in Lemma 11.21.1. Let A be an object of A. There exists a spectral sequence (Erp,q , dp,q r )r≥0 associated to a filtered complex with E2p,q = Rp G(Rq F (A)) converging to Rp+q (G ◦ F )(A). Moreover, the induced filtration on each Rn (G ◦ F )(A) is finite. Proof. Choose an injective resolution A → I • . Choose a Cartan-Eilenberg resolution F (I • ) → I •,• using Lemma 11.20.2. Apply Lemma 11.20.3 (use the second spectral sequence). Details omitted.  11.22. Resolution functors Let A be an abelian category with enough injectives. Denote I the full additive subcategory of A whose objects are the injective objects of A. It turns out that K + (I) and D+ (A) are equivalent in this case (see Proposition 11.22.1). For many purposes it therefore makes sense to think of D+ (A) as the (easier to grok) category K + (I) in this case. Proposition 11.22.1. Let A be an abelian category. Assume A has enough injectives. Denote I ⊂ A the strictly full additive subcategory whose objects are the injective objects of A. The functor K + (I) −→ D+ (A) is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories. Proof. It is clear that the functor is exact. It is essentially surjective by Lemma 11.17.3. Fully faithfulness is a consequence of Lemma 11.17.8.  Proposition 11.22.1 implies that we can find resolution functors. It turns out that we can prove resolution functors exist even in some cases where the abelian category A is a “big” category, i.e., has a class of objects. Definition 11.22.2. Let A be an abelian category with enough injectives. A resolution functor4 for A is given by the following data: (1) for all K • ∈ Ob(K + (A)) a bounded below complex of injectives j(K • ), and (2) for all K • ∈ Ob(K + (A)) a quasi-isomorphism iK • : K • → j(K • ). 4This is likely nonstandard terminology.

814

11. DERIVED CATEGORIES

Lemma 11.22.3. Let A be an abelian category with enough injectives. Given a resolution functor (j, i) there is a unique way to turn j into a functor and i into a 2-isomorphism producing a 2-commutative diagram K + (A)

/ K + (I)

j

$ z D+ (A) where I is the full additive subcategory of A consisting of injective objects. Proof. For every morphism α : K • → L• of K + (A) there is a unique morphism j(α) : j(K • ) → j(L• ) in K + (I) such that K•

α

iK •

 j(K • )

/ L• iL•

j(α)

 / j(L• )

is commutative in K + (A). To see this either use Lemmas 11.17.6 and 11.17.7 or the equivalent Lemma 11.17.8. The uniqueness implies that j is a functor, and the commutativity of the diagram implies that i gives a 2-morphism which witnesses the 2-commutativity of the diagram of categories in the statement of the lemma.  Lemma 11.22.4. Let A be an abelian category. Assume A has enough injectives. Then a resolution functor j exists and is unique up to unique isomorphism of functors. Proof. Consider the set of all objects K • of K + (A). (Recall that by our conventions any category has a set of objects unless mentioned otherwise.) By Lemma 11.17.3 every object has an injective resolution. By the axiom of choice we can choose for each K • an injective resolution iK • : K • → j(K • ).  Lemma 11.22.5. Let A be an abelian category with enough injectives. Any resolution functor j : K + (A) → K + (I) is exact. Proof. Denote iK • : K • → j(K • ) the canonical maps of Definition 11.22.2. First we discuss the existence of the functorial isomorphism j(K • [1]) → j(K • )[1]. Consider the diagram K • [1] 

K • [1]

iK • [1]

j(K • [1])

ξK •



iK • [1]

/ j(K • )[1]

By Lemmas 11.17.6 and 11.17.7 there exists a unique dotted arrow ξK • in K + (I) making the diagram commute in K + (A). We omit the verification that this gives a functorial isomorphism. (Hint: use Lemma 11.17.7 again.) Let (K • , L• , M • , f, g, h) be a distinguished triangle of K + (A). We have to show that (j(K • ), j(L• ), j(M • ), j(f ), j(g), ξK • ◦j(h)) is a distinguished triangle of K + (I).

11.23. FUNCTORIAL INJECTIVE EMBEDDINGS AND RESOLUTION FUNCTORS

815

Note that we have a commutative diagram K•  j(K • )

f

j(f )

/ L•  / j(L• )

g

j(g)

/ M•  / j(M • )

h

ξK • ◦j(h)

/ K • [1]  / j(K • )[1]

in K + (A) whose vertical arrows are the quasi-isomorphisms iK , iL , iM . Hence we see that the image of (j(K • ), j(L• ), j(M • ), j(f ), j(g), ξK • ◦ j(h)) in D+ (A) is isomorphic to a distinguished triangle and hence a distinguished triangle by TR1. Thus we see from Lemma 11.4.16 that (j(K • ), j(L• ), j(M • ), j(f ), j(g), ξK • ◦ j(h)) is a distinguished triangle in K + (I).  Lemma 11.22.6. Let A be an abelian category which has enough injectives. Let j be a resolution functor. Write Q : K + (A) → D+ (A) for the natural functor. Then j = j 0 ◦ Q for a unique functor j 0 : D+ (A) → K + (I) which is quasi-inverse to the canonical functor K + (I) → D+ (A). Proof. By Lemma 11.10.5 Q is a localization functor. To prove the existence of j 0 it suffices to show that any element of Qis+ (A) is mapped to an isomorphism under the functor j, see Lemma 11.5.6. This is true by the remarks following Definition 11.22.2.  Remark 11.22.7. Suppose that A is a “big” abelian category with enough injectives such as the category of abelian groups. In this case we have to be slightly more careful in constructing our resolution functor since we cannot use the axiom of choice with a quantifier ranging over a class. But note that the proof of the lemma does show that any two localization functors are canonically isomorphic. Namely, given quasi-isomorphisms i : K • → I • and i0 : K • → J • of a bounded below complex K • into bounded below complexes of injectives there exists a unique(!) morphism a : I • → J • in K + (I) such that i0 = i ◦ a as morphisms in K + (I). Hence the only issue is existence, and we will see how to deal with this in the next section. 11.23. Functorial injective embeddings and resolution functors In this section we redo the construction of a resolution functor K + (A) → K + (I) in case the category A has functorial injective embeddings. There are two reasons for this: (1) the proof is easier and (2) the construction also works if A is a “big” abelian category. See Remark 11.23.3 below. Let A be an abelian category. As before denote I the additive full subcategory of A consisting of injective objects. Consider the category InjRes(A) of arrows α : K • → I • where K • is a bounded below complex of A, I • is a bounded below complex of injectives of A and α is a quasi-isomorphism. In other words, α is an injective resolution and K • is bounded below. There is an obvious functor s : InjRes(A) −→ Comp+ (A) defined by (α : K • → I • ) 7→ K • . There is also a functor t : InjRes(A) −→ K + (I) defined by (α : K • → I • ) 7→ I • .

816

11. DERIVED CATEGORIES

Lemma 11.23.1. Let A be an abelian category. Assume A has functorial injective embeddings, see Homology, Definition 10.20.5. (1) There exists a functor inj : Comp+ (A) → InjRes(A) such that s◦inj = id. (2) For any functor inj : Comp+ (A) → InjRes(A) such that s ◦ inj = id we obtain a resolution functor, see Definition 11.22.2. Proof. Let A 7→ (A → J(A)) be a functorial injective embedding, see Homology, Definition 10.20.5. We first note that we may assume J(0) = 0. Namely, if not then for any object A we have 0 → A → 0 which gives a direct sum decomposition J(A) = J(0) ⊕ Ker(J(A) → J(0)). Note that the functorial morphism A → J(A) has to map into the second summand. Hence we can replace our functor by J 0 (A) = Ker(J(A) → J(0)) if needed. Let K • be a bounded below complex of A. Say K p = 0 if p < B. We are going to construct a double complex I •,• of injectives, together with a map α : K • → I •,0 such that α induces a quasi-isomorphism of K • with the associated total complex of I •,• . First we set I p,q = 0 whenever q < 0. Next, we set I p,0 = J(K p ) and αp : K p → I p,0 the functorial embedding. Since J is a functor we see that I •,0 is a complex and that α is a morphism of complexes. Each αp is injective. And I p,0 = 0 for p < B because J(0) = 0. Next, we set I p,1 = J(Coker(K p → I p,0 )). Again by functoriality we see that I •,1 is a complex. And again we get that I p,1 = 0 for p < B. It is also clear that K p maps isomorphically onto Ker(I p,0 → I p,1 ). As our third step we take I p,2 = J(Coker(I p,0 → I p,1 )). And so on and so forth. At this point we can apply Homology, Lemma 10.19.6 to get that the map α : K • → sI • is a quasi-isomorphism. To prove we get a functor inj it rests to show that the construction above is functorial. This verification is omitted. Suppose we have a functor inj such that s ◦ inj = id. For every object K • of Comp+ (A) we can write inj(K • ) = (iK • : K • → j(K • )) This provides us with a resolution functor as in Definition 11.22.2.



Remark 11.23.2. Suppose inj is a functor such that s ◦ inj = id as in part (2) of Lemma 11.23.1. Write inj(K • ) = (iK • : K • → j(K • )) as in the proof of that lemma. Suppose α : K • → L• is a map of bounded below complexes. Consider the map inj(α) in the category InjRes(A). It induces a commutative diagram K•

α

/ L•

inj(α)

 / j(L)•

iK

 j(K)•

iL

of morphisms of complexes. Hence, looking at the proof of Lemma 11.22.3 we see that the functor j : K + (A) → K + (I) is given by the rule j(α up to homotopy) = inj(α) up to homotpy ∈ HomK + (I) (j(K • ), j(L• ))

11.25. FILTERED DERIVED CATEGORY AND INJECTIVE RESOLUTIONS

817

Hence we see that j matches t ◦ inj in this case, i.e., the diagram Comp+ (A)

/ K + (I) :

t◦inj

& K + (A)

j

is commutative. Remark 11.23.3. Let Mod(OX ) be the category of OX -modules on a ringed space (X, OX ) (or more generally on a ringed site). We will see later that Mod(OX ) has enough injectives and in fact functorial injective embeddings, see Injectives, Theorem 17.12.4. Note that the proof of Lemma 11.22.4 does not apply to Mod(OX ). But the proof of Lemma 11.23.1 does apply to Mod(OX ). Thus we obtain j : K + (Mod(OX )) −→ K + (I) which is a resolution functor where I is the additive category of injective OX modules. This argument also works in the following cases: (1) The category ModR of R-modules over a ring R. (2) The category PMod(O) of presheaves of O-modules on a site endowed with a presheaf of rings. (3) The category Mod(O) of sheaves of O-modules on a ringed site. (4) Add more here as needed. 11.24. Right derived functors via resolution functors The content of the following lemma is that we can simply define RF (K • ) = F (j(K • ) if we are given a resolution functor j. Lemma 11.24.1. Let A be an abelian category with enough injectives Let F : A → B be an additive functor into an abelian category. Let (i, j) be a resolution functor, see Definition 11.22.2. The right derived functor RF of F fits into the following 2-commutative diagram j0

D+ (A) RF

$ z D+ (B)

/ K + (I) F

where j 0 is the functor from Lemma 11.22.6. Proof. By Lemma 11.19.1 we have RF (K • ) = F (j(K • )).



Remark 11.24.2. In the situation of Lemma 11.24.1 we see that we have actually lifted the right derived functor to an exact functor F ◦ j 0 : D+ (A) → K + (B). It is occasionally useful to use such a factorization. 11.25. Filtered derived category and injective resolutions If the underlying abelian category A has enough injectives then the category Filf (A) has enough right acyclic objects relative to any left exact functor. Definition 11.25.1. Let A be an abelian category. We say an object I of Filf (A) is filtered injective if each grp (I) is an injective object of A.

818

11. DERIVED CATEGORIES

This category is an example of an exact category, see Injectives, Remark 17.13.6. A special role is played by the strict morphisms, see Homology, Definition 10.13.3, i.e., the morphisms f such that Coim(f ) = Im(f ). We will say that a complex A → B → C in Filf (A) is exact if the sequence gr(A) → gr(B) → gr(C) is exact in A. This implies that A → B and B → C are strict morphisms, see Homology, Lemma 10.13.17. Lemma 11.25.2. Let A be an abelian category. An object I of Filf (A) is filtered injective if and only L if there exist a ≤ b, injectiveL objects In , a ≤ n ≤ b of A and an isomorphism I ∼ = a≤n≤b In such that F p I = n≥p In . Proof. Follows from the fact that any injection J → M of A is split if J is an injective object. Details omitted.  Lemma 11.25.3. Let A be an abelian category. Any strict monomorphism u : I → A of Filf (A) where I is a filtered injective object is a split injection. Proof. Let p be the largest integer such that F p I 6= 0. In particular grp (I) = F p I. Let I 0 be the object of Filf (A) whose underlying object of A is F p I and with filtration given by F n I 0 = 0 for n > p and F n I 0 = I 0 = F p I for n ≤ p. Note that I 0 → I is a strict monomorphism too. The fact that u is a strict monomorphism implies that F p I → A/F p+1 (A) is injective, see Homology, Lemma 10.13.15. Choose a splitting s : A/F p+1 A → F p I in A. The induced morphism s0 : A → I 0 is a strict morphism of filtered objects splitting the composition I 0 → I → A. Hence we can write A = I 0 ⊕ Ker(s0 ) and I = I 0 ⊕ Ker(s0 |I ). Note that ker(s0 |I ) → ker(s0 ) is a strict monomorphism and that ker(s0 |I ) is a filtered injective object. By induction on the length of the filtration on I the map ker(s0 |I ) → ker(s0 ) is a split injection. Thus we win.  Lemma 11.25.4. Let A be an abelian category. Let u : A → B be a strict monomorphism of Filf (A) and f : A → I a morphism from A into a filtered injective object in Filf (A). Then there exists a morphism g : B → I such that f = g ◦ u. Proof. The pushout f 0 : I → I qA B of f by u is a strict monomorphism, see Homology, Lemma 10.13.10. Hence the result follows formally from Lemma 11.25.3.  Lemma 11.25.5. Let A be an abelian category with enough injectives. For any object A of Filf (A) there exists a strict monomorphism A → I where I is a filtered injective object. Proof. Pick a ≤ b such that grp (A) = 0 unless p ∈ {a, a + 1, . . . , b}. For each n n ∈ {a, a + 1, . .L . , b} choose an injection un : A/F L A → In with In and injective p object. Set I = a≤n≤b Ip with filtration F I = n≥p In and set u : A → I equal to the direct sum of the maps un .  Lemma 11.25.6. Let A be an abelian category with enough injectives. For any object A of Filf (A) there exists a filtered quasi-isomorphism A[0] → I • where I • is a complex of filtered injective objects with I n = 0 for n < 0. Proof. First choose a strict monomorphism u0 : A → I 0 of A into a filtered injective object, see Lemma 11.25.5. Next, choose a strict monomorphism u1 :

11.25. FILTERED DERIVED CATEGORY AND INJECTIVE RESOLUTIONS

819

Coker(u0 ) → I 1 into a filtered injective object of A. Denote d0 the induced map I 0 → I 1 . Next, choose a strict monomorphism u2 : Coker(u1 ) → I 2 into a filtered injective object of A. Denote d1 the induced map I 1 → I 2 . And so on. This works because each of the sequences 0 → Coker(un ) → I n+1 → Coker(un+1 ) → 0 is short exact, i.e., induces a short exact sequence on applying gr. To see this use Homology, Lemma 10.13.15.  Lemma 11.25.7. Let A be an abelian category with enough injectives. Let f : A → B be a morphism of Filf (A). Given filtered quasi-isomorphisms A[0] → I • and B[0] → J • where I • , J • are complexes of filtered injective objects with I n = J n = 0 for n < 0, then there exists a commutative diagram A[0]

/ B[0]

 I•

 / J•

Proof. As A[0] → I • and C[0] → J • are filtered quasi-isomorphisms we conclude that a : A → I 0 , b : B → J 0 and all the morphisms dnI , dnJ are strict, see Homology, Lemma 11.13.4. We will inductively construct the maps f n in the following commutative diagram A

a

f0

f

 B

/ I0

b

 / J0

/ I1 f1

 / J1

/ I2

/ ...

f2

 / J2

/ ...

Because A → I 0 is a strict monomorphism and because J 0 is filtered injective, we can find a morphism f 0 : I 0 → J 0 such that f 0 ◦ a = b ◦ f , see Lemma 11.25.4. The composition d0J ◦ b ◦ f is zero, hence d0J ◦ f 0 ◦ a = 0, hence d0J ◦ f 0 factors through a unique morphism Coker(a) = Coim(d0I ) = Im(d0I ) −→ J 1 . As Im(d0I ) → I 1 is a strict monomorphism we can extend the displayed arrow to a morphism f 1 : I 1 → J 1 by Lemma 11.25.4 again. And so on.  Lemma 11.25.8. Let A be an abelian category with enough injectives. Let 0 → A → B → C → 0 be a short exact sequence in Filf (A). Given filtered quasiisomorphisms A[0] → I • and C[0] → J • where I • , J • are complexes of filtered injective objects with I n = J n = 0 for n < 0, then there exists a commutative diagram / A[0] / B[0] / C[0] /0 0    / I• / M• / J• /0 0 where the lower row is a termwise split sequence of complexes. Proof. As A[0] → I • and C[0] → J • are filtered quasi-isomorphisms we conclude that a : A → I 0 , c : C → J 0 and all the morphisms dnI , dnJ are strict, see Homology,

820

11. DERIVED CATEGORIES

Lemma 11.13.4. We are going to step by step construct the south-east and the south arrows in the following commutative diagram BO

/C

β b

α

A

a

/ J0

c b



/ I0



/ J1

δ0



/ I1

/ ...

δ1

/ I2

/ ...

As A → B is a strict monomorphism, we can find a morphism b : B → I 0 such that b ◦ α = a, see Lemma 11.25.4. As A is the kernel of the strict morphism I 0 → I 1 and β = Coker(α) we obtain a unique morphism b : C → I 1 fitting into the diagram. As c is a strict monomorphism and I 1 is filtered injective we can find δ 0 : J 0 → I 1 , see Lemma 11.25.4. Because B → C is a strict epimorphism and because B → I 0 → I 1 → I 2 is zero, we see that C → I 1 → I 2 is zero. Hence d1I ◦ δ 0 is zero on C ∼ = Im(c). Hence d1I ◦ δ 0 factors through a unique morphism Coker(c) = Coim(d0J ) = Im(d0J ) −→ I 2 . As I 2 is filtered injective and Im(d0J ) → J 1 is a strict monomorphism we can extend the displayed morphism to a morphism δ 1 : J 1 → I 2 , see Lemma 11.25.4. And so on. We set M • = I • ⊕ J • with differential  n  dI (−1)n+1 δ n dnM = 0 dnJ Finally, the map B[0] → M • is given by b ⊕ c ◦ β : M → I 0 ⊕ J 0 .



Lemma 11.25.9. Let A be an abelian category with enough injectives. For every K • ∈ K + (Filf (A)) there exists a filtered quasi-isomorphism K • → I • with I • bounded below, each I n a filtered injective object, and each K n → I n a strict monomorphism. Proof. After replacing K • by a shift (which is harmless for the proof) we may assume that K n = 0 for n < 0. Consider the short exact sequences 0 → ker(d0K ) → K 0 → Coim(d0K ) → 0 0 → ker(d1K ) → K 1 → Coim(d1K ) → 0 0 → ker(d2K ) → K 2 → Coim(d2K ) → 0 ... of the exact category Filf (A) and the maps ui : Coim(diK ) → Ker(di+1 K ). For each i ≥ 0 we may choose filtered quasi-isomorphisms • ker(diK )[0] → Iker,i i • Coim(dK )[0] → Icoim,i n n with Iker,i , Icoim,i filtered injective and zero for n < 0, see Lemma 11.25.6. By • • Lemma 11.25.7 we may lift ui to a morphism of complexes u•i : Icoim,i → Iker,i+1 . Finally, for each i ≥ 0 we may complete the diagrams

0

/ ker(di )[0] K

0

 / I• ker,i

/ K i [0]

αi

 / I• i

βi

/ Coim(di )[0] K

/0

 / I• coim,i

/0

11.25. FILTERED DERIVED CATEGORY AND INJECTIVE RESOLUTIONS

821

with the lower sequence a termwise split exact sequence, see Lemma 11.25.8. For • i ≥ 0 set di : Ii• → Ii+1 equal to di = αi+1 ◦ u•i ◦ βi . Note that di ◦ di−1 = 0 because βi ◦ αi = 0. Hence we have constructed a commutative diagram I0• O

/ I1• O

/ I2• O

/ ...

K 0 [0]

/ K 1 [0]

/ K 2 [0]

/ ...

Here the vertical arrows are filtered quasi-isomorphisms. The upper row is a complex of complexes and each complex consists of filtered injective objects with no nonzero objects in degree < 0. Thus we obtain a double complex by setting I a,b = Iab and using a,b b da,b = Iab → Ia+1 = I a+1,b 1 :I the map dba and using for a,b da,b = Iab → Iab+1 = I a,b+1 2 :I

the map dbIa . Denote Tot(I •,• ) the total complex associated to this double complex, see Homology, Definition 10.19.2. Observe that the maps K n [0] → In• come from maps K n → I n,0 which give rise to a map of complexes K • −→ Tot(I •,• ) We claim this is a filtered quasi-isomorphism. As gr(−) is an additive functor, we see that gr(Tot(I •,• )) = Tot(gr(I •,• )). Thus we can use Homology, Lemma 10.19.6 to conclude that gr(K • ) → gr(Tot(I •,• )) is a quasi-isomorphism as desired.  Lemma 11.25.10. Let A be an abelian category. Let K • , I • ∈ K(Filf (A)). Assume K • is filtered acyclic and I • bounded below and consisting of filtered injective objects. Any morphism K • → I • is homotopic to zero: HomK(Filf (A)) (K • , I • ) = 0. Proof. Let α : K • → I • be a morphism of complexes. Assume that αj = 0 for j < n. We will show that there exists a morphism h : K n+1 → I n such that αn = h ◦ d. Thus α will be homotopic to the morphism of complexes β defined by  0 if j≤n  β j = αn+1 − d ◦ h if j = n + 1  αj if j > n + 1 This will clearly prove the lemma (by induction). To prove the existence of h note that αn ◦ dn−1 = 0 since αn−1 = 0. Since K • is filtered acyclic we see that dn−1 K K n and dK are strict and that n n 0 → Im(dn−1 K ) → K → Im(dK ) → 0

is an exact sequence of the exact category Filf (A), see Homology, Lemma 10.13.17. Hence we can think of αn as a map into I n defined on Im(dnK ). Using that Im(dnK ) → K n+1 is a strict monomorphism and that I n is filtered injective we may lift this map to a map h : K n+1 → I n as desired, see Lemma 11.25.4.  Lemma 11.25.11. Let A be an abelian category. Let I • ∈ K(Filf (A)) be a bounded below complex consisting of filtered injective objects.

822

11. DERIVED CATEGORIES

(1) Let α : K • → L• in K(Filf (A)) be a filtered quasi-isomorphism. Then the map HomK(Filf (A)) (L• , I • ) → HomK(Filf (A)) (K • , I • ) is bijective. (2) Let L• ∈ K(A). Then HomK(Filf (A)) (L• , I • ) = HomDF (A) (L• , I • ). Proof. Proof of (1). Note that (K • , L• , C(α)• , α, i, −p) is a distinguished triangle in K(Filf (A)) (Lemma 11.8.12) and C(f )• is a filtered acyclic complex (Lemma 11.13.4). Then / HomK(Filf (A)) (K • , I • )

/ HomK(Filf (A)) (L• , I • )

HomK(Filf (A)) (C(α)• , I • ) q HomK(Filf (A)) (C(α)• [−1], I • )

is an exact sequence of abelian groups, see Lemma 11.4.2. At this point Lemma 11.25.10 guarantees that the outer two groups are zero and hence HomK(A) (L• , I • ) = HomK(A) (K • , I • ). Proof of (2). Let a be an element of the right hand side. We may represent a = γα−1 where α : K • → L• is a filtered quasi-isomorphism and γ : K • → I • is a map of complexes. By part (1) we can find a morphism β : L• → I • such that β ◦ α is homotopic to γ. This proves that the map is surjective. Let b be an element of the left hand side which maps to zero in the right hand side. Then b is the homotopy class of a morphism β : L• → I • such that there exists a filtered quasi-iomorphism α : K • → L• with β ◦ α homotopic to zero. Then part (1) shows that β is homotopic to zero also, i.e., b = 0.  Lemma 11.25.12. Let A be an abelian category. Let I f ⊂ Filf (A) denote the strictly full additive subcategory whose objects are the filtered injective objects. The canonical functor K + (I f ) −→ DF + (A) is exact, fully faithful and essentially surjective, i.e., an equivalence of triangulated categories. Furthermore the diagrams K + (I f ) grp



K + (I)

/ DF + (A) grp

 / D+ (A)

K + (I f ) 

forget F

K + (I)

/ DF + (A) forget F

 / D+ (A)

are commutative, where I ⊂ A is the strictly full additive subcategory whose objects are the injective objects. Proof. The functor K + (I f ) → DF + (A) is essentially surjective by Lemma 11.25.9. It is fully faithful by Lemma 11.25.11. It is an exact functor by our definitions regarding distinguished triangles. The commutativity of the squares is immediate. 

11.25. FILTERED DERIVED CATEGORY AND INJECTIVE RESOLUTIONS

823

Remark 11.25.13. We can invert the arrow of the lemma only if A is a category in our sense, namely if it has a set of objects. However, suppose given a big abelian category A with enough injectives, such as Mod(OX ) for example. Then for any given set of objects {Ai }i∈I there is an abelian subcategory A0 ⊂ A containing all of them and having enough injectives, see Sets, Lemma 3.12.1. Thus we may use the lemma above for A0 . This essentially means that if we use a set worth of diagrams, etc then we will never run into trouble using the lemma. Let A, B be abelian categories. Let T : A → B be a left exact functor. (We cannot use the letter F for the functor since this would conflict too much with our use of the letter F to indicate filtrations.) Note that T induces an additive functor T : Filf (A) → Filf (B) by the rule T (A, F ) = (T (A), F ) where F p T (A) = T (F p A) which makes sense as T is left exact. (Warning: It may not be the case that gr(T (A)) = T (gr(A)).) This induces functors of triangulated categories (11.25.13.1)

T : K + (Filf (A)) −→ K + (Filf (B))

The filtered right derived functor of T are is the right derived functor of Definition 11.14.2 for this exact functor composed with the exact functor K + (Filf (B)) → DF + (B) and the multiplicative set FQis+ (A). Assume A has enough injectives. At this point we can redo the discussion of Section 11.19 to define the filtered right derived functors (11.25.13.2)

RT : DF + (A) −→ DF + (B)

of our functor T . However, instead we will proceed as in Section 11.24, and it will turn out that we can define RT even if T is just additive. Namely, we first choose a quasi-inverse j 0 : DF + (A) → K + (I f ) of the equivalence of Lemma 11.25.12. By Lemma 11.4.16 we see that j 0 is an exact functor of triangulated categories. Next, we note that for a filtered injective object I we have a (noncanonical) decomposition M M (11.25.13.3) I∼ Ip , with F p I = Iq = p∈Z

q≥p

by Lemma 11.25.2. Hence if T is any additive functor T : A → B then we get an additive functor Text : I f → Filf (B) L L by setting Text (I) = T (Ip ) with F p Text (I) = q≥p T (Iq ). Note that we have the property gr(Text (I)) = T (gr(I)) by construction. Hence we obtain a functor

(11.25.13.4)

(11.25.13.5)

Text : K + (I f ) → K + (Filf (B))

which commutes with gr. Then we define (11.25.13.2) by the composition (11.25.13.6)

RT = Text ◦ j 0 .

Since RT : D+ (A) → D+ (B) is computed by injective resolutions as well, see Lemmas 11.19.1, the commutation of T with gr, and the commutative diagrams of Lemma 11.25.12 imply that (11.25.13.7) grp ◦ RT ∼ = RT ◦ grp

824

11. DERIVED CATEGORIES

and (11.25.13.8)

(forget F ) ◦ RT ∼ = RT ◦ (forget F )

as functors DF + (A) → D+ (B). The filtered derived functor RT (11.25.13.2) induces functors RT : Filf (A) → DF + (B), RT : Comp+ (Filf (A)) → DF + (B), RT : KF + (A) → DF + (B). Note that since Filf (A), and Comp+ (Filf (A)) are no longer abelian it does not make sense to say that RT restricts to a δ-functor on them. (This can be repaired by thinking of these categories as exact categories and formulating the notion of a δ-functor from an exact category into a triangulated category.) But it does make sense, and it is true by construction, that RT is an exact functor on the triangulated category KF + (A). Lemma 11.25.14. Let A, B be abelian categories. Let T : A → B be a left exact functor. Assume A has enough injectives. Let (K • , F ) be an object of Comp+ (Filf (A)). There exists a spectral sequence (Erp,q , dr )r≥0 which is the spectral sequence associated to an object of Comp+ (Filf (B)) with E1p,q = Rp+q T (grp (K • )) which converges to Rp+q T (K • ) inducing a finite filtration on each Rn T (K • ). Moreover the construction of this spectral sequence is functorial in the object K • of Comp+ (Filf (A)). In fact the terms (Er , dr ) for r ≥ 2 do not depend on any choices. Proof. Choose a filtered quasi-isomorphism K • → I • with I • a bounded below complex of filtered injective objects, see Lemma 11.25.9. Consider the complex RT (K • ) = Text (I • ), see (11.25.13.6). Thus we can consider the spectral sequence (Er , dr )r≥0 associated to this as a filtered complex in B, see Homology, Section 10.18. By Homology, Lemma 10.18.2 we have E1p,q = H p+q (grp (T (I • ))). By Equation (11.25.13.3) we have E1p,q = H p+q (T (grp (I • ))), and by definition of a filtered injective resolution the map grp (K • ) → grp (I • ) is an injective resolution. Hence E1p,q = Rp+q T (grp (K • )). On the other hand, each I n has a finite filtration and hence each T (I n ) has a finite filtration. Thus we may apply Homology, Lemma 10.18.9 to conclude that the spectral sequence converges to H n (T (I • )) = Rn T (K • ) moreover inducing finite filtrations on each of the terms. Suppose that K • → L• is a morphism of Comp+ (Filf (A)). Choose a filtered quasi-isomorphism L• → J • with J • a bounded below complex of filtered injective objects, see Lemma 11.25.9. By our results above, for example Lemma 11.25.11, there exists a diagram / L• K•   / J• I• which commutes up to homotopy. Hence we get a morphism of filtered complexes T (I • ) → T (J • ) which gives rise to the morphism of spectral sequences, see Homology, Lemma 10.18.4. The last statement follows from this. 

11.26. EXT GROUPS

825

Remark 11.25.15. As promised in Remark 11.20.4 we discuss the connection of the lemma above with the constructions using Cartan-Eilenberg resolutions. Namely, assume the notations of Lemma 11.20.3. In particular K • is a bounded below complex of A and T : A → B is a left exact functor. We give an alternative construction of the spectral sequences 0 E and 00 E First spectral sequence. Consider the “stupid” filtration on K • obtained by setting F p (K • ) = σ≥p (K • ), see Homology, Section 10.11. Note that this stupid in the sense that d(F p (K • )) ⊂ F p+1 (K • ), compare Homology, Lemma 10.18.3. Note that grp (K • ) = K p [p] with this filtration. According to the above there is a spectral sequence with E1 term E1p,q = Rp+q T (K p [p]). Then the E2 term is clearly E2p,q = H p (Rp+q T (K • )) as in the spectral sequence 0 Er . Second spectral sequence. Consider the filtration on the complex K • obtained by setting F p (K • ) = τ≤−p (K • ), see Homology, Section 10.11. The minus sign is necessary to get a decreasing filtration. Note that grp (K • ) is quasi-isomorphic to H −p (K • )[−p] with this filtration. According to the above there is a spectral sequence with E1 term E1p,q = Rp+q T (H −p (K • )[−p]) = R2p+q T (H −p (K • )) = 00 E2i,j with i = 2p+q and j = −p. (This looks unnatural, but note that we could just have well developped the whole theory of filtered complexes using increasing filtrations, with the end result that this then looks natural, but the other one doesn’t.) We leave it to the reader to see that the differentials match up. Actually, given a Cartan-Eilenberg resolution K • → I •,• the induced morphism K • → sI • into the associated simple complex will be a filtered injective resolution for either filtration using suitable filtrations on sI • . This can be used to match up the spectral sequences exactly. 11.26. Ext groups In this section we start describing the ext groups of objects of an abelian category. First we have the following very general definition. Definition 11.26.1. Let A be an abelian category. Let i ∈ Z. Let X, Y be objects of D(A). The ith extension group of X by Y is the group ExtiA (X, Y ) = HomD(A) (X, Y [i]) = HomD(A) (X[−i], Y ). If A, B ∈ Ob(A) we set ExtiA (A, B) = ExtiA (A[0], B[0]). Since HomD(A) (X, −), resp. HomD(A) (−, Y ) is a homological, resp. cohomological functor, see Lemma 11.4.2, we see that a distinguished triangle (Y, Y 0 , Y 00 ), resp. (X, X 0 , X 00 ) leads to a long exact sequence . . . → ExtiA (X, Y ) → ExtiA (X, Y 0 ) → ExtiA (X, Y 00 ) → Exti+1 A (X, Y ) → . . . respectively 00 . . . → ExtiA (X 00 , Y ) → ExtiA (X 0 , Y ) → ExtiA (X, Y ) → Exti+1 A (X , Y ) → . . .

Note that since D+ (A), D− (A), Db (A) are full subcategories we may compute the ext groups by Hom groups in these categories provided X, Y are contained in them. In case the category A has enough injectives or enough projectives we can compute the Ext groups using injective or projective resolutions. To avoid confusion, recall

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that having an injective (resp. projective) resolution implies vanishing of homology in all low (resp. high) degrees, see Lemmas 11.17.2 and 11.18.2. Lemma 11.26.2. Let A be an abelian category. Let X • , Y • ∈ Ob(K(A)). (1) Let Y • → I • be an injective resolution (Definition 11.17.1). Then ExtiA (X • , Y • ) = HomK(A) (X • , I • [i]). (2) Let P • → X • be a projective resolution (Definition 11.18.1). Then ExtiA (X • , Y • ) = HomK(A) (P • [−i], Y • ). Proof. Follows immediately from Lemma 11.17.8 and Lemma 11.18.8.



In the rest of this section we discuss extensions of objects of the abelian category itself. First we observe the following. Lemma 11.26.3. Let A be an abelian category and let A, B ∈ Ob(A). For i < 0 we have ExtiA (B, A) = 0. We have Ext0A (B, A) = HomA (B, A). Proof. Let L• → B[0] be any quasi-isomorphism. Then it is also true that τ≤0 L• → B[0] is a quasi-isomorphism. Hence a morphism B[0] → A[i] in D(A) can be represented as f s−1 where s : L• → B[0] is a quasi-isomorphism, f : L• → A[i] a morphism, and Ln = 0 for n > 0. Thus f = 0 if i < 0. If i = 0, then f corresponds exactly to a morphism B = Coker(L−1 → L0 ) → A.  Let A be an abelian category. Suppose that 0 → A → A0 → A00 → 0 is a short exact sequence of objects of A. Then 0 → A[0] → A0 [0] → A00 [0] → 0 leads to a distiguished triangle in D(A) (see Lemma 11.11.1) hence a long exact sequence of Ext groups 0 → Ext0A (B, A) → Ext0A (B, A0 ) → Ext0A (B, A00 ) → Ext1A (B, A) → . . . Similarly, given a short exact sequence 0 → B → B 0 → B 00 → 0 we obtain a long exact sequence of Ext groups 0 → Ext0A (B 00 , A) → Ext0A (B 0 , A) → Ext0A (B, A) → Ext1A (B 00 , A) → . . . We may view these Ext groups as an application of the construction of the derived category. It shows one can define Ext groups and construct the long exact sequence of Ext groups without needing the existence of enough injectives or projectives. There is an alternative construction of the Ext groups due to Yoneda which avoids the use of the derived category, see [Yon60]. Definition 11.26.4. Let A be an abelian category. Let A, B ∈ Ob(A). A degree i Yoneda extension of B by A is an exact sequence E : 0 → A → Zi−1 → Zi−2 → . . . → Z0 → B → 0

11.26. EXT GROUPS

827

in A. We say two Yoneda extensions E and E 0 of the same degree are equivalent if there exists a commutative diagram 0

/A O

/ Zi−1 O

/ ...

/ Z0 O

/B O

/0

0

/A

/ Z 00 i−1

/ ...

/ Z000

/B

/0

0

 /A

 / Z0 i−1

/ ...

 / Z00

 /B

/0

where the middle row is a Yoneda extension as well. It is not immediately clear that the equivalence of the definition is an equivalence relation. Although it is instructive to prove this directly this will also follow from Lemma 11.26.5 below. Let A be an abelian category with objects A, B. Given a Yoneda extension E : 0 → A → Zi−1 → Zi−2 → . . . → Z0 → B → 0 we define an associated element δ(E) ∈ Exti (B, A) as the morphism δ(E) = f s−1 : B[0] → A[i] where s is the quasi-isomorphism (. . . → 0 → A → Zi−1 → . . . → Z0 → 0 → . . .) −→ B[0] and f is the morphism of complexes (. . . → 0 → A → Zi−1 → . . . → Z0 → 0 → . . .) −→ A[i] We call δ(E) = f s−1 the class of the Yoneda extension. It turns out that this class characterizes the equivalence class of the Yoneda extension. Lemma 11.26.5. Let A be an abelian category with objects A, B. Any element in ExtiA (B, A) is δ(E) for some degree i Yoneda extension of B by A. Given two Yoneda extensions E, E 0 of the same degree then E is equivalent to E 0 if and only if δ(E) = δ(E 0 ). Proof. Let ξ : B[0] → A[i] be an element of ExtiA (B, A). We may write ξ = f s−1 for some quasi-isomorphism s : L• → B[0] and map f : L• → A[i]. After replacing L• by τ≤0 L• we may assume that Li = 0 for i > 0. Picture L−i−1

/ L−i

/ ...

/ L0

/B

/0

 A Then setting Zi−1 = (L−i+1 ⊕ A)/L−i and Zj = L−j for j = i − 2, . . . , 0 we see that we obtain a degree i extension E of B by A whose class δ(E) equals ξ. It is immediate from the definitions that equivalent Yoneda extensions have the same class. Suppose that E : 0 → A → Zi−1 → Zi−2 → . . . → Z0 → B → 0 and 0 0 E 0 : 0 → A → Zi−1 → Zi−2 → . . . → Z00 → B → 0 are Yoneda extensions with the same class. By construction of D(A) as the localization of K(A) at the set of quasi-isomorphisms, this means there exists a complex L• and quasi-isomorphisms t : L• → (. . . → 0 → A → Zi−1 → . . . → Z0 → 0 → . . .)

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and 0 t0 : L• → (. . . → 0 → A → Zi−1 → . . . → Z00 → 0 → . . .) such that s ◦ t = s0 ◦ t0 and f ◦ t = f 0 ◦ t0 , see Categories, Section 4.24. Let E 00 be the degree i extension of B by A constructed from the pair L• → B[0] and L• → A[i] in the first paragraph of the proof. Then the reader sees readily that there exists “morphisms” of degree i Yoneda extensions E 00 → E and E 00 → E 0 as in the definition of equivalent Yoneda extensions (details omitted). This finishes the proof. 

Lemma 11.26.6. Let A be an abelian category. Let A, B be objects of A. Then Ext1A (B, A) is the group ExtA (B, A) constructed in Homology, Definition 10.4.2. Proof. This is the case i = 1 of Lemma 11.26.5.



11.27. Unbounded complexes A reference for the material in this section is [Spa88]. The following lemma is useful to find “good” left resolutions of unbounded complexes. Lemma 11.27.1. Let A be an abelian category. Let P ⊂ Ob(A) be a subset. Assume that every object of A is a quotient of an element of P. Let K • be a complex. There exists a commutative diagram P1•

/ P2•

/ ...

 τ≤1 K •

 / τ≤2 K •

/ ...

in the category of complexes such that (1) the vertical arrows are quasi-isomorphisms, (2) P1• is a bounded above complex with terms in P, • are termwise split injections and each cokernel (3) the arrows Pn• → Pn+1 i i Pn+1 /Pn is an element of P. Proof. By Lemma 11.15.5 any bounded above complex has a resolution by a bounded above complex whose terms are in P. Thus we obtain the first com• and the plex P1• . By induction it suffices, given P1• , . . . , Pn• to construct Pn+1 • • • • • maps Pn → Pn+1 and Pn → τ≤n+1 K . Consider the cone C1 of the composition Pn• → τ≤n K • → τ≤n+1 K • . This fits into the distinguished triangle Pn• → τ≤n+1 K • → C1• → Pn• [1] Note that C1• is bounded above, hence we can choose a quasi-isomorphism Q• → C1• where Q• is a bounded above complex whose terms are elements of P. Take the cone C2• of the map of complexes Q• → Pn• [1] to get the distinguished triangle Q• → Pn• [1] → C2• → Q• [1] By the axioms of triangulated categories we obtain a map of distinguished triangles Pn•

/ C2• [−1]

/ Q•

/ Pn• [1]

 Pn•

 / τ≤n+1 K •

 / C1•

 / Pn• [1]

11.27. UNBOUNDED COMPLEXES

829

• in the triangulated category K(A). Set Pn+1 = C2• [−1]. Note that (3) holds by • construction. Choose an actual morphism of complexes f : Pn+1 → τ≤n+1 K • . The • left square of the diagram above commutes up to homotopy, but as Pn• → Pn+1 is a termwise split injection we can lift the homotopy and modify our choice of f to make it commute. Finally, f is a quasi-isomorphism, because both Pn• → Pn• and Q• → C1• are. 

In some cases we can use the lemma above to show that a left derived functor is everywhere defined. Proposition 11.27.2. Let F : A → B be a right exact functor of abelian categories. Let P ⊂ Ob(A) be a subset. Assume (1) every object of A is a quotient of an element of P, (2) for any bounded above acyclic complex P • of A with P n ∈ P for all n the complex F (P • ) is exact, (3) A and B have colimits of systems over N, (4) colimits over N are exact in both A and B, and (5) F commutes with colimits over N. Then LF is defined on all of D(A). Proof. By (1) and Lemma 11.15.5 for any bounded above complex K • there exists a quasi-isomorphism P • → K • with P • bounded above and P n ∈ P for all n. Suppose that s : P • → (P 0 )• is a quasi-isomorphism of bounded above complexes consisting of objects of P. Then F (P • ) → F ((P 0 )• ) is a quasi-isomorphism because F (C(s)• ) is acyclic by assumption (2). This already shows that LF is defined on D− (A) and that a bounded above complex consisting of objects of P computes LF , see Lemma 11.14.15. Next, let K • be an arbitrary complex of A. Choose a diagram P1•

/ P2•

/ ...

 τ≤1 K •

 / τ≤2 K •

/ ...

as in Lemma 11.27.1. Note that the map colim Pn• → K • is a quasi-isomorphism because colimits over N in A are exact and H i (Pn• ) = H i (K • ) for n > i. We claim that F (colim Pn• ) = colim F (Pn• ) (termwise colimits) is LF (K • ), i.e., that colim Pn• computes LF . To see this, by Lemma 11.14.15, it suffices to prove the following claim. Suppose that α

colim Q•n = Q• −−→ P • = colim Pn• is a quasi-isomorphism of complexes, such that each Pn• , Q•n is a bounded above complex whose terms are in P and the maps Pn• → τ≤n P • and Q•n → τ≤n Q• are quasi-isomorphisma. Claim: F (α) is a quasi-isomorphism. The problem is that we do not assume that α is given as a colimit of maps between the complexes Pn• and Q•n . However, for each n we know that the solid arrows in

830

11. DERIVED CATEGORIES

the diagram R•  L•

Pn• o  τ≤n P •

/ Q•n  / τ≤n Q•

τ≤n α

are quasi-isomorphisms. Because quasi-isomorphisms form a multiplicative system in K(A) (see Lemma 11.10.2) we can find a quasi-isomorphism L• → Pn• and map of complexes L• → Q•n such that the diagram above commutes up to homotopy. Then τ≤n L• → L• is a quasi-isomorphism. Hence (by the first part of the proof) we can find a bounded above complex R• whose terms are in P and a quasiisomorphism R• → L• (as indicated in the diagram). Using the result of the first paragraph of the proof we see that F (R• ) → F (Pn• ) and F (R• ) → F (Q•n ) are quasiisomorphisms. Thus we obtain a isomorphisms H i (F (Pn• )) → H i (F (Q•n )) fitting into the commutative diagram H i (F (Pn• ))

/ H i (F (Q•n ))

 H i (F (P • ))

 / H i (F (Q• ))

The exact same argument shows that these maps are also compatible as n varies. Since by (4) and (5) we have H i (F (P • )) = H i (F (colim Pn• )) = H i (colim F (Pn• )) = colim H i (F (Pn• )) and similarly for Q• we conclude that H i (α) : H i (F (P • ) → H i (F (Q• ) is an isomorphism and the claim follows.  Lemma 11.27.3. Let A be an abelian category. Let I ⊂ Ob(A) be a subset. Assume that every object of A is a subobject of an element of I. Let K • be a complex. There exists a commutative diagram ...

/ τ≥−2 K •

/ τ≥−1 K •

...

 / I2•

 / I1•

in the category of complexes such that (1) the vertical arrows are quasi-isomorphisms, (2) I1• is a bounded above complex with terms in I, • i (3) the arrows In+1 → In• are termwise split surjections and Ker(In+1 → Ini ) is an element of I. Proof. This lemma is dual to Lemma 11.27.1.



11.28. K-INJECTIVE COMPLEXES

831

11.28. K-injective complexes The following types of complexes can be used to compute right derived functors on the unbounded derived category. Definition 11.28.1. Let A be an abelian category. A complex I • is K-injective if for every acyclic complex M • we have HomK(A) (M • , I • ) = 0. In the situation of the definition we have in fact HomK(A) (M • [i], I • ) = 0 for all i as the translate of an acyclic complex is acyclic. Lemma 11.28.2. Let A be an abelian category. Let I • be a complex. The following are equivalent (1) I • is K-injective, (2) for every quasi-isomorphism M • → N • the map HomK(A) (N • , I • ) → HomK(A) (M • , I • ) is bijective, and (3) for every complex N • the map HomK(A) (N • , I • ) → HomD(A) (N • , I • ) is an isomorphism. Proof. Assume (1). Then (2) holds because the functor HomK(A) (−, I • ) is cohomological and the cone on a quasi-isomorphism is acyclic. Assume (2). A morphism N • → I • in D(A) is of the form f s−1 : N • → I • where s : M • → N • is a quasi-isomorphism and f : M • → I • is a map. By (2) this corresponds to a unique morphism N • → I • in K(A), i.e., (3) holds. Assume (3). If M • is acyclic then M • is isomorphic to the zero complex in D(A) hence HomD(A) (N • , I • ) = 0, whence HomK(A) (N • , I • ) = 0 by (3), i.e., (1) holds.  Lemma 11.28.3. Let A be an abelian category. A bounded below complex of injectives is K-injective. Proof. Follows from Lemmas 11.28.2 and 11.17.8.



Lemma 11.28.4. Let A be an abelian category. Let F : K(A) → D0 be an exact functor of triangulated categories. Then RF is defined at every complex in K(A) which is quasi-isomorphic to a K-injective complex. In fact, every K-injective complex computes RF . Proof. By Lemma 11.14.4 it suffices to show that RF is defined at a K-injective complex, i.e., it suffices to show a K-injective complex I • computes RF . Any quasiisomorphism I • → N • is a homotopy equivalence as it has an inverse by Lemma 11.28.2. Thus I • → I • is a final object of I • /Qis(A) and we win.  Lemma 11.28.5. Let A be an abelian category. Assume every complex has a quasiisomorphism towards a K-injective complex. Then any exact functor F : K(A) → D0 of triangulated categories has a right derived functor RF : D(A) −→ D0 and RF (I • ) = I • for K-injective complexes I • .

832

11. DERIVED CATEGORIES

Proof. To see this we apply Lemma 11.14.15 with I the collection of K-injective complexes. Since (1) holds by assumption, it suffices to prove that if I • → J • is a quasi-isomorphism of K-injective complexes, then F (I • ) → F (J • ) is an isomorphism. This is clear because I • → J • is a homotopy equivalence, i.e., an isomorphism in K(A), by Lemma 11.28.2.  The following lemma can be generalized to limits over bigger ordinals. Lemma 11.28.6. Let A be an abelian category. Let . . . → I3• → I2• → I1• be an inverse system of K-injective complexes. Assume (1) each In• is K-injective, m (2) each map In+1 → Inm is a split surjection, m (3) the limits I = lim Inm exist. Then the complex I • is K-injective. Proof. Let M • be an acyclic complex. Let us abbreviate Hn (a, b) = HomA (M a , Inb ). With this notation HomK(A) (M • , I • ) is the cohomology of the complex Y Y Y Y lim Hn (m, m−2) → lim Hn (m, m−1) → lim Hn (m, m) → lim Hn (m, m+1) m

n

m

n

m

n

m

n

Q

in the third spot from the left. We may exchange the order of and lim and each of the complexes Y Y Y Y Hn (m, m − 2) → Hn (m, m − 1) → Hn (m, m) → Hn (m, m + 1) m

m

m

m

is exact by assumption (1). By assumption (2) the maps in the systems Y Y Y ... → H3 (m, m − 2) → H2 (m, m − 2) → H1 (m, m − 2) m

m

m

are surjective. Thus the lemma follows from Homology, Lemma 10.23.4.



Remark 11.28.7. It appears that a combination of Lemmas 11.27.3, 11.28.5, and 11.28.6 produces “enough K-injectives” for any abelian category with enough injectives and countable limits. Actually, this may not work! Namely, suppose that K • is a complex and In• is the system of bounded above complexes of injectives produced by Lemma 11.27.3. Let I • = lim In• be the termwise limit which is Kinjective by Lemma 11.28.6. The problem is that the map K • → I • may not be a quasi-isomorphism. Namely, if limn is not exact in A then there is no reason to think that it is a quasi-isomorphism in general. 11.29. Bounded cohomological dimension There is another case where the unbounded derived functor exists. Namely, when the functor has bounded cohomological dimension. Lemma 11.29.1. Let A be an abelian category. Let d : Ob(A) → {0, 1, 2, . . . , ∞} be a function. Assume that (1) every object of A is a subobject of an object A with d(A) = 0, (2) if 0 → A → B → C → 0 is a short exact sequence then d(C) ≤ max{d(A) − 1, d(B)}.

11.29. BOUNDED COHOMOLOGICAL DIMENSION

833

Let K • be a complex such that n + d(K n ) tends to −∞ as n → −∞. Then there exists a quasi-isomorphism K • → L• with d(Ln ) = 0 for all n ∈ Z. Proof. By Lemma 11.15.4 we can find a quasi-isomorphism σ≥0 K • → M • with M n = 0 for n < 0 and d(M n ) = 0 for n ≥ 0. Then K • is quasi-isomorphic to the complex . . . → K −2 → K −1 → M 0 → M 1 → . . . Hence we may assume that d(K n ) = 0 for n  0. Note that the condition n + d(K n ) → −∞ as n → −∞ is not violated by this replacement. We are going to improve K • by an (infinite) sequence of elementary replacements. An elementary replacement is the following. Choose an index n such that d(K n ) > 0. Choose an injection K n → M where d(M ) = 0. Set M 0 = Coker(K n → M ⊕ K n+1 ). Consider the map of complexes K• :

K n−1

/ Kn

/ K n+1

/ K n+2

 (K 0 )• :

 K n−1

 /M

 / M0

 / K n+2

It is clear that K • → (K 0 )• is a quasi-isomorphism. Moreover, it is clear that d((K 0 )n ) = 0 and d((K 0 )n+1 ) ≤ max{d(K n+1 ), d(K n ) − 1} and the other values are unchanged. To finish the proof we carefully choose the order in which to do the elementary replacements so that for every integer m the complex σ≥m K • is changed only a finite number of times. To do this set ξ(K • ) = max{n + d(K n ) | d(K n ) > 0} and I = {n ∈ Z | ξ(K • ) = n + d(K n ) ∧ d(K n ) > 0} Our assumption that n + d(K n ) tends to −∞ as n → −∞ and the fact that d(K n ) = 0 for n >> 0 implies ξ(K • ) < +∞ and that I is a finite set. It is clear that ξ((K 0 )• ) ≤ ξ(K • ) for an elementary transformation as above. An elementary transformation changes the complex in degrees ≤ ξ(K • ) + 1. Hence if we can find finite sequence of elementary transformations which decrease ξ(K • ), then we win. However, note that if we do an elementary transformation starting with the smallest element n ∈ I, then we either decrease the size of I, or we increase min I. Since every element of I is ≤ ξ(K • ) we see that we win after a finite number of steps.  Lemma 11.29.2. Let F : A → B be a left exact functor of abelian categories. If (1) every object of A is a subobject of an object which is right acyclic for F , (2) there exists an integer n such that Rn F = 0, then RF : D(A) → D(B) exists. Any complex consisting of right acyclic objects for F computes RF and any complex is the source of a quasi-isomorphism into such a complex. Proof. Note that the first condition implies that RF : D+ (A) → D+ (B) exists, see Proposition 11.15.8. Let A be an object of A. Choose an injection A → A0 with A0 acyclic. Then we see that Rn+1 F (A) = Rn F (A0 /A) = 0 by the long exact cohomology sequence. Hence we conclude that Rn+1 F = 0. Continuing like this using induction we find that Rm F = 0 for all m ≥ n.

834

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We are going to use Lemma 11.29.1 with the function d : Ob(A) → {0, 1, 2, . . .} given by d(A) = min{0} ∪ {i | Ri F (A) 6= 0}. The first assumption of Lemma 11.29.1 is our assumption (3) and the second assumption of Lemma 11.29.1 follows from the long exact cohomology sequence. Hence for every complex K • there exists a quasi-isomorphism K • → L• with Ln right acyclic for F . We claim that if L• → M • is a quasi-isomorphism of complexes of right acyclic objects for F , then F (L• ) → F (M • ) is a quasi-isomorphism. If we prove this claim then we are done by Lemma 11.14.15. To prove the claim pick an integer i ∈ Z. Consider the distinguished triangle σ≥i−n−1 L• → σ≥i−n−1 M • → Q• , i.e., let Q• be the cone of the first map. Note that Q• is bounded below and that H j (Q• ) is zero except possibly for j = i − n − 1 or j = i − n − 2. We may apply RF to Q• . Using the spectral sequence of Lemma 11.20.3 and the assumed vanishing of cohomology (2) we conclude that Rj F (Q• ) is zero except possibly for j ∈ {i − n − 2, . . . , i − 1}. Hence we see that RF (σ≥i−n−1 L• ) → RF (σ≥i−n−1 L• ) induces an isomorphism of cohomology objects in degrees ≥ i. By Proposition 11.15.8 we know that RF (σ≥i−n−1 L• ) = σ≥i−n−1 F (L• ) and RF (σ≥i−n−1 M • ) = σ≥i−n−1 F (M • ). We conclude that F (L• ) → F (M • ) is an isomorphism in degree i as desired.  Lemma 11.29.3. Let F : A → B be a right exact functor of abelian categories. If (1) every object of A is a quotient of an object which is left acyclic for F , (2) there exists an integer n such that Ln F = 0, then LF : D(A) → D(B) exists. Any complex consisting of left acyclic objects for F computes LF and any complex is the target of a quasi-isomorphism into such a complex. Proof. This is dual to Lemma 11.29.2.



11.30. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves

(19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36)

Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes

11.30. OTHER CHAPTERS

(37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55)

´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap

(56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

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Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 12

More on Algebra 12.1. Introduction In this chapter we prove some results in commutative algebra which are less elementary than those in the first chapter on commutative algebra, see Algebra, Section 7.1. A reference is [Mat70]. 12.2. A comment on the Artin-Rees property Some of this material is taken from [CdJ02]. A general discussion with additional references can be found in [EH05, Section 1]. Let A be a Noetherian ring and let I ⊂ A be an ideal. Given a homomorphism f : M → N of finite A-modules there exists a c ≥ 0 such that f (M ) ∩ I n N ⊂ f (I n−c M ) for all n ≥ c, see Algebra, Lemma 7.48.5. In this situation we will say c works for f in the Artin-Rees lemma. Lemma 12.2.1. Let A be a Noetherian ring. Let I ⊂ A be an ideal contained in the Jacobson radical of A. Let f

g

S:L− →M − →N

and

f0

g0

S 0 : L −→ M −→ N

be two complexes of finite A-modules as shown. Assume that (1) c works in the Artin-Rees lemma for f and g, (2) the complex S is exact, and (3) f 0 = f mod I c+1 M and g 0 = g mod I c+1 N . Then c works in the Artin-Rees lemma for g 0 and the complex S 0 is exact. Proof. We first show that g 0 (L) ∩ I n M ⊂ g 0 (I n−c L) for n ≥ c. Let a be an element of M such that g 0 (a) ∈ I n N . We want to adjust a by an element of f 0 (L), i.e, without changing g 0 (a), so that a ∈ I n−c M . Assume that a ∈ I r M , where r < n − c. Then g(a) = g 0 (a) + (g − g 0 )(a) ∈ I n N + I r+c+1 N = I r+c+1 N. By Artin-Rees for g we have g(a) ∈ g(I r+1 M ). Say g(a) = g(a1 ) with a1 ∈ I r+1 M . Since the sequence S is exact, a − a1 ∈ f (L). Accordingly, we write a = f (b) + a1 for some b ∈ L. Then f (b) = a − a1 ∈ I r M . Artin-Rees for f shows that if r ≥ c, we may replace b by an element of I r−c L. Then in all cases, a = f 0 (b) + a2 , where a2 = (f − f 0 )(b) + a1 ∈ I r+1 M . (Namely, either c ≥ r and (f − f 0 )(b) ∈ I r+1 M by assumption, or c < r and b ∈ I r−c , whence again (f − f 0 )(b) ∈ I c+1 I r−c M = I r+1 M .) So we can adjust a by the element f 0 (b) ∈ f 0 (L) to increase r by 1. 837

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In fact, the argument above shows that (g 0 )−1 (I n M ) ⊂ f 0 (L)+I n−c M for all n ≥ c. Hence S 0 is exact because \ \ (g 0 )−1 (0) = (g 0 )−1 ( I n N ) ⊂ f 0 (L) + I n−c M = f 0 (L) as I ⊂ rad(A), see Algebra, Lemma 7.48.7.



Given an ideal I ⊂ A of a ring A and an A-module M we set M GrI (M ) = I n M/I n+1 M. We think of this as a graded GrI (A)-module. Lemma 12.2.2. Assumptions as in Lemma 12.2.1. Let Q = Coker(g) and Q0 = Coker(g 0 ). Then GrI (Q) ∼ = GrI (Q0 ) as graded GrI (A)-modules. Proof. In degree n we have GrI (Q)n = I n N/(I n+1 N + g(M ) ∩ I n N ) and similarly for Q0 . We claim that g(M ) ∩ I n N ⊂ I n+1 N + g 0 (M ) ∩ I n N. By symmetry (the proof of the claim will only use that c works for g which also holds for g 0 by the lemma) this will imply that I n+1 N + g(M ) ∩ I n N = I n+1 N + g 0 (M ) ∩ I n N whence GrI (Q)n and GrI (Q0 )n agree as subquotients of N , implying the lemma. Observe that the claim is clear for n ≤ c as f = f 0 mod I c+1 N . If n > c, then suppose b ∈ g(M ) ∩ I n N . Write b = g(a) for a ∈ I n−c M . Set b0 = g 0 (a). We have b − b0 = (g − g 0 )(a) ∈ I n+1 N as desired.  Lemma 12.2.3. Let A → B be a flat map of Noetherian rings. Let I ⊂ A be an ideal. Let f : M → N be a homomorphism of finite A-modules. Assume that c works for f in the Artin-Rees lemma. Then c works for f ⊗ 1 : M ⊗A B → N ⊗A B in the Artin-Rees lemma for the ideal IB. Proof. Note that (f ⊗ 1)(M ) ∩ I n N ⊗A B = (f ⊗ 1) (f ⊗ 1)−1 (I n N ⊗A B)




On the other hand, (f ⊗ 1)−1 (I n N ⊗A B) = Ker(M ⊗A B → N ⊗A B/(I n N ⊗A B)) = Ker(M ⊗A B → (N/I n N ) ⊗A B) As A → B is flat taking kernels and cokernels commutes with tensoring with B, whence this is equal to f −1 (I n N ) ⊗A B. By assumption f −1 (I n N ) is contained in Ker(f ) + I n−c M . Thus the lemma holds.  12.3. Fitting ideals The fitting ideals of a finite module are the ideals determined by the construction of Lemma 12.3.2. Lemma 12.3.1. Let R be a ring. Let A be an n × m matrix with coefficients in R. Let Ir (A) be the ideal generated by the r × r-minors of A with the convention that I0 (A) = R and Ir (A) = 0 if r > min(n, m). Then (1) I0 (A) ⊃ I1 (A) ⊃ I2 (A) . . .,

12.3. FITTING IDEALS

839

(2) if B is an (n + n0 ) × m matrix, and A is the first n rows of B, then Ir+n0 (B) ⊂ Ir (A), (3) if C is an n × n matrix then Ir (CA) ⊂ Ir (A). (4) If A is a block matrix   A1 0 0 A2 P then Ir (A) = r1 +r2 =r Ir1 (A1 )Ir2 (A2 ). (5) Add more here. Proof. Omitted. (Hint: Use that a determinant can be computed by expanding along a column or a row.)  Lemma 12.3.2. Let R be a ring. Let M be a finite R-module. Choose a presentation M R −→ R⊕n −→ M −→ 0. j∈J L ⊕n of M . Let A = (aij )i=1,...,n,j∈J be the matrix of the map . The j∈J R → R ideal Fitk (M ) generated by the (n − k) × (n − k) minors of A is independent of the choice of the presentation. Proof. Let K ⊂ R⊕n be the kernel of the surjection R⊕n → M . Pick z1 , . . . , zn−k ∈ K and write zj = (z1j , . . . , znj ). Another description of the ideal Fitk (M ) is that it is the ideal generated by the (n − k) × (n − k) minors of all the matrices (zij ) we obtain in this way. 0

Suppose we change the surjection into the surjection R⊕n+n → M with kernel K 0 0 where we use the original map on the first n standard basis elements of R⊕n+n and 0 on the last n0 basis vectors. Then the corresponding ideals are the same. Namely, if z1 , . . . , zn−k ∈ K as above, let zj0 = (z1j , . . . , znj , 0, . . . , 0) ∈ K 0 for 0 0 j = 1, . . . , n − k and zn+j 0 = (0, . . . , 0, 1, 0, . . . , 0) ∈ K . Then we see that the ideal of (n − k) × (n − k) minors of (zij ) agrees with the ideal of (n + n0 − k) × (n + n0 − k) 0 0 minors of (zij ). This gives one of the inclusions. Conversely, given z10 , . . . , zn+n 0 −k 0 ⊕n in K we can project these to R to get z1 , . . . , zn+n0 −k in K. By Lemma 12.3.1 0 we see that the ideal generated by the (n + n0 − k) × (n + n0 − k) minors of (zij ) is contained in the ideal generated by the (n − k) × (n − k) minors of (zij ). This gives the other inclusion. Let R⊕m → M be another surjection with kernel L. By the previous paragraph we may assume m = n. By Algebra, Lemma 7.5.2 we can choose a map R⊕n → R⊕m commuting with the surjections to M . Let C = (cli ) be the matrix of this map (it is a square matrix as n = m). Then given z1 , . . . , zn−k ∈ K as above we get Cz1 , . . . , Czn−k ∈ L. By Lemma 12.3.1 we get one of the inclusions. By symmetry we get the other.  Definition 12.3.3. Let R be a ring. Let M be a finite R-module. Let k ≥ 0. The kth fitting ideal of M is the ideal Fitk (M ) constructed in Lemma 12.3.1. Since the fitting ideals are the ideals of minors of a big matrix (numbered in reverse ordering from the ordering in Lemma 12.3.1) we see that Fit0 (M ) ⊂ Fit1 (M ) ⊂ . . . ⊂ Fitt (M ) = R for some t  0. Here are some basic properties of fitting ideals.

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Lemma 12.3.4. Let R be a ring. Let M be a finite R-module. (1) If M can be generated by n elemements, then Fitn (M ) = R. (2) Given a second finite R-module M 0 we have X Fitk (M ⊕ M 0 ) = Fitk (M )Fitk0 (M 0 ) 0 k+k =l

(3) If R → R0 is a ring map, then Fitk (M ⊗R R0 ) is the ideal of R0 generated by the image of Fitk (M ). (4) If M is an R-module of finite presentation, then Fitk (M ) is a finitely generated ideal. (5) If M → M 0 is a surjection, then Fitk (M ) ⊂ Fitk (M 0 ). (6) Add more here. Proof. Part (1) follows from the fact that I0 (A) = R in Lemma 12.3.1. part (2) follows form the corresponding statement in Lemma 12.3.1. Part (3) follows from the fact that ⊗R R0 is right exact, so the base change of a presentation of M is A a presentation of M ⊗R R0 . Proof of (4). Let R⊕m − → R⊕n → M → 0 be a presentation. Then Fitk (M ) is the ideal generated by the n − k × n − k minors of the matrix A. Part (5) is immediate from the definition.  Example 12.3.5. Let R be a ring. The fitting ideals of the finite free module M = R⊕n are are Fitk (M ) = 0 for k < n and Fitk (M ) = R for k ≥ n. Lemma 12.3.6. Let R be a ring. Let M be a finite R-module. Let k ≥ 0. Let p be a prime ideal with Fitk (M ) 6⊂ p. Then there exists an f ∈ R, f 6∈ p such that Mf can be generated by k elements over Rf . Proof. By Nakayama’s lemma (Algebra, Lemma 7.18.1) we see that Mf can be generated by k elements over Rf for some f ∈ R, f 6∈ p if M ⊗R κ(p) can be generated by k elements. This reduces the problem to the case where R is a field and p = (0). In this case the result follows from Example 12.3.5.  Lemma 12.3.7. Let R be a ring. Let M be a finite R-module. Let r ≥ 0. The following are equivalent (1) M is finite locally free of rank k (Algebra, Definition 7.73.1), (2) Fitr−1 (M ) = 0 and Fitr (M ) = R, and (3) Fitk (M ) = 0 for k < r and Fitk (M ) = R for k ≥ r. Proof. It is immediate that (2) is equivalent to (3) because the fitting ideals form an increasing sequence of ideals. Since the formation of Fitk (M ) commutes with base change (Lemma 12.3.4) we see that (1) implies (2) by Example 12.3.5 and glueing results (Algebra, Section 7.22). Conversely, assume (2). By Lemma L 12.3.6 we may assume that M is generated by r elements. Thus a presentation j∈J R → R⊕r → M → 0. But now the assumption that Fitr−1 (M ) = 0 implies that all L entries of the matrix of the map j∈J R → R⊕r are zero. Thus M is free.  12.4. Computing Tor Let R be a ring. We denote D(R) the derived category of the abelian category ModR of R-modules. Note that ModR has enough projectives as every free Rmodule is projective. Thus we can define the left derived functors of any additive functor from ModR to any abelian category.

12.5. DERIVED TENSOR PRODUCT

841

This implies in particular to the functor − ⊗R M : ModR → ModR whose right derived functors are the Tor functors TorR i (−, M ), see Algebra, Section 7.70. There is also a total right derived functor − − − ⊗L R M : D (R) −→ D (R)

(12.4.0.1)

which is denoted − ⊗L R M . Its satellites are the Tor modules, i.e., we have R H −p (N ⊗L R M ) = Torp (N, M ).

A special situation occurs when we consider the tensor product with an R-algebra A. In this case we think of − ⊗R A as a functor from ModR to ModA . Hence the total right derived functor − − − ⊗L R A : D (R) −→ D (A)

(12.4.0.2)

which is denoted − ⊗L R A. Its satellites are the tor groups, i.e., we have R H −p (N ⊗L R A) = Torp (N, A).

In particular these Tor groups naturally have the structure of A-modules. 12.5. Derived tensor product We can construct the derived tensor product in greater generality. In fact, it turns out that the boundedness assumptions are not necessary, provided we choose K-flat resolutions. Lemma 12.5.1. Let R be a ring. Let P • be a complex of R-modules. Let α, β : L• → M • be homotopy equivalent maps of complexes. Then α and β induce homotopy equivalent maps Tot(α ⊗ idP ), Tot(β ⊗ idP ) : Tot(L• ⊗R P • ) −→ Tot(M • ⊗R P • ). In particular the construction L• 7→ Tot(L• ⊗R P • ) defines an endo-functor of the homotopy category of complexes. Proof. Say α = β + dh + hd for some homotopy h defined by hn : Ln → M n−1 . Set M M M Hn = ha ⊗ idP b : La ⊗R P b −→ M a−1 ⊗R P b a+b=n

a+b=n

a+b=n

Then a straightforward computation shows that Tot(α ⊗ idP ) = Tot(β ⊗ idP ) + dH + Hd •

as maps Tot(L ⊗R P • ) → Tot(M • ⊗R P • ).



Lemma 12.5.2. Let R be a ring. Let P • be a complex of R-modules. The functor K(ModR ) −→ K(ModR ),

L• 7−→ Tot(L• ⊗R P • )

is an exact functor of triangulated categories. Proof. By our definition of the triangulated structure on K(ModR ) we have to check that our functor maps a termwise split short exact sequence of complexes to a termwise split short exact sequence of complexes. As the terms of Tot(L• ⊗R P • ) are direct sums of the tensor products La ⊗R P b this is clear.  The following definition will allow us to think intelligently about derived tensor products of unbounded complexes.

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Definition 12.5.3. Let R be a ring. A complex K • is called K-flat if for every acyclic complex M • the total complex Tot(M • ⊗R K • ) is acyclic. Lemma 12.5.4. Let R be a ring. Let K • be a K-flat complex. Then the functor K(ModR ) −→ K(ModR ),

L• 7−→ Tot(L• ⊗R K • )

transforms quasi-isomorphisms into quasi-isomorphisms. Proof. Follows from Lemma 12.5.2 and the fact that quasi-isomorphisms in K(ModR ) and K(ModA ) are characterized by having acyclic cones.  Lemma 12.5.5. Let R → R0 be a ring map. If K • is a K-flat complex of Rmodules, then K • ⊗R R0 is a K-flat complex of R0 -modules. Proof. Follows from the definitions and the fact that (K • ⊗R R0 )⊗R0 L• = K • ⊗R L• for any complex L• of R0 -modules.  Lemma 12.5.6. Let R be a ring. If K • , L• are K-flat complexes of R-modules, then Tot(K • ⊗R L• ) is a K-flat complex of R-modules. Proof. Follows from the isomorphism Tot(M • ⊗R Tot(K • ⊗R L• )) = Tot(Tot(M • ⊗R K • ) ⊗R L• ) and the definition.



Lemma 12.5.7. Let R be a ring. Let (K1• , K2• , K3• ) be a distinguished triangle in K(ModR ). If two out of three of Ki• are K-flat, so is the third. Proof. Follows from Lemma 12.5.2 and the fact that in a distinguished triangle in K(ModA ) if two out of three are acyclic, so is the third.  Lemma 12.5.8. Let R be a ring. Let P • be a bounded above complex of flat R-modules. Then P • is K-flat. Proof. Let L• be an acyclic complex of R-modules. Let ξ ∈ H n (Tot(L• ⊗R P • )). We have to show that ξ = 0. Since Totn (L• ⊗R P • ) is a direct sum with terms La ⊗R P b we see that ξ comes from an element in H n (Tot(τ≤m L• ⊗R P • )) for some m ∈ Z. Since τ≤m L• is also acyclic we may replace L• by τ≤m L• . Hence we may assume that L• is bounded above. In this case the spectral sequence of Homology, Lemma 10.19.5 has 0 p,q E1 = H p (L• ⊗R P q ) q which is zero as P is flat and L• acyclic. Hence H ∗ (Tot(L• ⊗R P • )) = 0.  In the following lemma by a colimit of a system of complexes we mean the termwise colimit. Lemma 12.5.9. Let R be a ring. Let K1• → K2• → . . . be a system of K-flat complexes. Then colimi Ki• is K-flat. Proof. Because we are taking termwise colimits it is clear that colimi Tot(M • ⊗R Ki• ) = Tot(M • ⊗R colimi Ki• ) Hence the lemma follows from the fact that filtered colimits are exact. •



Lemma 12.5.10. Let R be a ring. For any complex M there exists a K-flat complex K • and a quasi-isomorphism K • → M • . Moreover each K n is a flat R-module.

12.5. DERIVED TENSOR PRODUCT

843

Proof. Let P ⊂ Ob(ModR ) be the class of flat R-modules. By Derived Categories, Lemma 11.27.1 there exists a system K1• → K2• → . . . and a diagram K1•

/ K2•

/ ...

 τ≤1 M •

 / τ≤2 M •

/ ...

with the properties (1), (2), (3) listed in that lemma. These properties imply each complex Ki• is a bounded above complex of flat modules. Hence Ki• is Kflat by Lemma 12.5.8. The induced map colimi Ki• → M • is a quasi-isomorphism by construction. The complex colimi Ki• is K-flat by Lemma 12.5.9. The final assertion of the lemma is true because the colimit of a system of flat modules is flat, see Algebra, Lemma 7.36.2.  Lemma 12.5.11. Let R be a ring. Let α : P • → Q• be a quasi-isomorphism of K-flat complexes of R-modules. For every complex L• of R-modules the induced map Tot(idL ⊗ α) : Tot(L• ⊗R P • ) −→ Tot(L• ⊗R Q• ) is a quasi-isomorphism. Proof. Choose a quasi-isomorphism K • → L• with K • a K-flat complex, see Lemma 12.5.10. Consider the commutative diagram Tot(K • ⊗R P • )

/ Tot(K • ⊗R Q• )

 Tot(L• ⊗R P • )

 / Tot(L• ⊗R Q• )

The result follows as by Lemma 12.5.4 the vertical arrows and the top horizontal arrow are quasi-isomorphisms.  Let R be a ring. Let M • be an object of D(R). Choose a K-flat resolution K • → M • , see Lemma 12.5.10. By Lemmas 12.5.1 and 12.5.2 we obtain an exact functor of triangulated categories K(ModR ) −→ K(ModR ),

L• 7−→ Tot(L• ⊗R K • )

By Lemma 12.5.4 this functor induces a functor D(R) → D(R) simply because D(R) is the localization of K(ModR ) at quasi-isomorphism. By Lemma 12.5.11 the resulting functor (up to isomorphism) does not depend on the choice of the K-flat resolution. Definition 12.5.12. Let R be a ring. Let M • be an object of D(R). The derived tensor product • − ⊗L R M : D(R) −→ D(R) is the exact functor of triangulated categories described above. This functor extends the functor (12.4.0.1). It is clear from our explicit constructions that there is a canonical isomorphism • ∼ • L • M • ⊗L R L = L ⊗R M

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• whenever both L• and M • are in D(R). Hence when we write M • ⊗L R L we will usually be agnostic about which variable we are using to define the derived tensor product with.

12.6. Derived change of rings Let R → A be a ring map. We can also use K-flat resolutions to define a derived base change functor − ⊗L R A : D(R) → D(A) extending the functor (12.4.0.2). Namely, for every complex of R-modules M • we • can choose a K-flat resolution K • → M • and set M • ⊗L R A = K ⊗R A. You can use Lemmas 12.5.10 and 12.5.11 to see that this is well defined. However, to cross all the t’s and dot all the i’s it is perhaps more convenient to use some general theory. Lemma 12.6.1. The construction above is independent of choices and defines an exact functor of triangulated categories D(R) → D(A). Proof. To see this we use the general theory developed in Derived Categories, Section 11.14. Set D = K(ModR ) and D0 = D(A). Let us write F : D → D0 the exact functor of triangulated categories defined by the rule F (M • ) = M • ⊗R A. We let S be the set of quasi-isomorphisms in D = K(ModR ). This gives a situation as in Derived Categories, Situation 11.14.1 so that Derived Categories, Definition 11.14.2 applies. We claim that LF is everywhere defined. This follows from Derived Categories, Lemma 11.14.15 with P ⊂ Ob(D) the collection of K-flat complexes: (1) follows from Lemma 12.5.10 and (2) follows from Lemma 12.5.11. Thus we obtain a derived functor LF : D(R) = S −1 D −→ D0 = D(A) see Derived Categories, Equation (11.14.9.1). Finally, Derived Categories, Lemma 11.14.15 also guarantees that LF (K • ) = F (K • ) = K • ⊗R A when K • is K-flat, i.e., LF is indeed computed in the way described above.  12.7. Tor independence We often encounter the following situation. Suppose that AO

/ A0 O

R

/ R0

is a “base change” diagram of rings, i.e., A0 = A ⊗R R0 . In this situation, for any A-module M we have M ⊗A A0 = M ⊗R R0 . Thus − ⊗R R0 is equal to − ⊗A A0 as a functor ModA → ModA0 . In general this equality does not extend to derived tensor products. Let K • ∈ D− (A). We have 0 • 0 K • ⊗L A A = P ⊗A A

where P • → K • is a projective resolution in the category of A-modules. Pick a projective resolution E • → P • in the category of R-modules. Then it is also the case that E • → K • is a projective resolution in the category of R-modules. Hence 0 • 0 K • ⊗L R R = E ⊗R R

12.8. SPECTRAL SEQUENCES FOR TOR

845

The map E • → P • and the map R0 → A0 combined determine a comparison map (12.7.0.1)

0 • 0 • 0 • L 0 K • ⊗L R R = E ⊗R R −→ P ⊗A A = K ⊗A A

A simple example where this is not an isomorphism is to take R = k[x], A = R0 = A0 = k[x]/(x) = k and K • = A[0]. Clearly, a necessary condition is that 0 TorR p (A, R ) = 0 for all p > 0. Definition 12.7.1. Let R be a ring. Let A, B be R-algebras. We say A and B are Tor independent over R if TorR p (A, B) = 0 for all p > 0. Lemma 12.7.2. The comparison map (12.7.0.1) is an isomorphism if A and R0 are Tor independent over R. Proof. To prove this we choose a free resolution F • → R0 of R0 as an R-module. Because A and R0 are Tor independent over R we see that F • ⊗R A is a free Amodule resolution of A0 over A. By our general construction of the derived tensor product above we see that P • ⊗ A A0 ∼ = Tot(P • ⊗A (F • ⊗R A)) = Tot(P • ⊗R F • ) ∼ = Tot(E • ⊗R F • ) ∼ = E • ⊗R R0 as desired.

 12.8. Spectral sequences for Tor

In this section we collect various spectral sequences that come up when considering the Tor functors. Example 12.8.1. Let R be a ring. Let K• be a bounded above chain complex of R-modules. Let M be an R-module. Then there is a spectral sequence with E2 -page L TorR i (Hj (K• ), M ) ⇒ Hi+j (K• ⊗R M ) and another spectral sequence with E1 -page L TorR i (Kj , M ) ⇒ Hi+j (K• ⊗R M )

This follows from the dual to Derived Categories, Lemma 11.20.3. Example 12.8.2. Let R → S be a ring map. Let M be an R-module and let N be an S-module. Then there is a spectral sequence R TorSn (TorR m (M, S), N ) ⇒ Torn+m (M, N ).

To construct it choose a R-free resolution P• of M . Then we have • • M ⊗L R N = P ⊗R N = (P ⊗R S) ⊗S N

and then apply the first spectral sequence of Example 12.8.1. Example 12.8.3. Consider a commutative diagram BO

/ B 0 = B ⊗ A A0 O

A

/ A0

and B-modules M, N . Set M 0 = M ⊗A A0 = M ⊗B B 0 and N 0 = N ⊗A A0 = N ⊗B B 0 . Assume that A → B is flat and that M and N are A-flat. Then there is a spectral sequence B B0 0 0 0 TorA i (Torj (M, N ), A ) ⇒ Tori+j (M , N )

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The reason is as follows. Choose free resolution F• → M as a B-module. As B and M are A-flat we see that F• ⊗A A0 is a free B 0 -resolution of M 0 . Hence we see that 0 0 0 the groups TorB n (M , N ) are computed by the complex 0 (F• ⊗A A0 ) ⊗B 0 N 0 = (F• ⊗B N ) ⊗A A0 = (F• ⊗B N ) ⊗L AA

the last equality because F• ⊗B N is a complex of flat A-modules as N is flat over A. Hence we obtain the spectral sequence by applying the spectral sequence of Example 12.8.1. Example 12.8.4. Let K • , L• be objects of D− (R). Then there are spectral sequences q • p+q • H p (K • ⊗L (K • ⊗L R H (L )) ⇒ H RL ) and • p+q • H q (H p (K • ) ⊗L (K • ⊗L RL )⇒H RL )

After replacing K • and L• by bounded above complexes of projectives, these spectral sequences are simply the two spectral sequences for computing the cohomology of Tot(K • ⊗ L• ) discussed in Homology, Section 10.19. 12.9. Products and Tor The simplest example of the product maps comes from the following situation. Suppose that K • , L• ∈ D(R) with one of them contained in D− (R). Then there are maps (12.9.0.1)

• H i (K • ) ⊗R H j (L• ) −→ H i+j (K • ⊗L RL )

Namely, to define these maps we may assume that one of K • , L• is a bounded • above complex of projective R-modules. In that case K • ⊗L R L is represented by • • the complex Tot(K ⊗R L ), see Section 12.4. Next, suppose that ξ ∈ H i (K • ) and ζ ∈ H j (L• ). Choose k ∈ Ker(K i → K i+1 ) and l ∈ Ker(Lj → Lj+1 ) representing ξ and ζ. Then we set ξ ∪ ζ = class of k ⊗ l in H i+j (Tot(K • ⊗R L• )). This make sense because the formula (see Homology, Definition 10.19.2) for the differential d on the total complex shows that k ⊗ l is a cocycle. Moreover, if k 0 = dK (k 00 ) for some k 00 ∈ K i−1 , then k 0 ⊗ l = d(k 00 ⊗ l) because l is a cocycle. Similarly, altering the choice of l representing ζ does not change the class of k ⊗l. It is equally clear that ∪ is bilinear, and hence to a general element of H i (K • ) ⊗R H j (L• ) we assign X X ξi ⊗ ζi 7−→ ξi ∪ ζi in H i+j (Tot(K • ⊗R L• )). Let R → A be a ring map. Let K • , L• ∈ D− (R). Then we have a canonical identification (12.9.0.2)

L • L • L • L (K • ⊗L R A) ⊗A (L ⊗R A) = (K ⊗R L ) ⊗R A

in D(A). It is constructed as follows. First, choose projective resolutions P • → K • and Q• → L• over R. Then the left hand side is represented by the complex Tot((P • ⊗R A) ⊗A (Q• ⊗R A)) and the right hand side by the complex Tot(P • ⊗R

12.9. PRODUCTS AND TOR

847

Q• ) ⊗R A. These complexes are canonically isomorphic. Thus the construction above induces products R R • • • • TorR n (K , A) ⊗A Torm (L , A) −→ Torn+m (K ⊗R L , A)

which are occasionally usefull. Let M , N be R-modules. Using the general construction above and functoriality of Tor we obtain canonical maps (12.9.0.3)

R R TorR n (M, A) ⊗A Torm (N, A) −→ Torn+m (M ⊗R N, A)

Here is a direct construction using projective resolutions. First, choose projective resolutions P• → M,

Q• → N,

T • → M ⊗R N

over R. We have H0 (Tot(P• ⊗R Q• )) = M ⊗R N by right exactness of ⊗R . Hence Derived Categories, Lemmas 11.18.6 and 11.18.7 guarantee the existence and uniqueness of a map of complexes µ : Tot(P• ⊗R Q• ) → T• such that H0 (µ) = idM ⊗R N . This induces a canonical map L L (M ⊗L R A) ⊗A (N ⊗R A) = Tot((P• ⊗R A) ⊗A (Q• ⊗R A))

= Tot(P• ⊗R Q• ) ⊗R A → T• ⊗R A = (M ⊗R N ) ⊗L RA in D(A). Hence the products (12.9.0.3) above are constructed using (12.9.0.1) over A to construct R −n−m L L ((M ⊗L TorR R A) ⊗A (N ⊗R A)) n (M, A) ⊗A Torm (N, A) → H

and then composing by the displayed map above to end up in TorR n+m (M ⊗R N, A). An interesting special case of the above occurs when M = N = B where B is an R-algebra. In this case we obtain maps R R R TorR n (B, A) ⊗A Torm (B, A) −→ Torn (B ⊗R B, A) −→ Torn (B, A)

the second arrow being induced by the multiplication map B ⊗R B → B via functoriality for Tor. In other words we obtain an A-algebra structure on TorR ? (B, A). This algebra structure has many intriguing properties (associativity, graded commutative, B-algebra structure, divided powers in some case, etc) which we will discuss elsewhere (insert future reference here). Lemma 12.9.1. Let R be a ring. Let A, B, C be R-algebras and let B → C be an R-algebra map. Then the induced map R TorR ? (B, A) −→ Tor? (C, A)

is an A-algebra homomorphism. Proof. Omitted. Hint: You can prove this by working through the definitions, writing all the complexes explicitly. 

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12.10. Formal glueing of module categories Fix a noetherian scheme X, and a closed subscheme Z with complement U . Our goal is to explain a result of Artin that describes how coherent sheaves on X can be constructed (uniquely) from coherent sheaves on the formal completion of X along Z, and those on U with a suitable compatibility on the overlap. Definition 12.10.1. Let R be a ring. Let M be an R-module. (1) Let I ⊂ R be an ideal. We say M is an I-power torsion module if for every m ∈ M there exists an n > 0 such that I n m = 0. (2) Let f ∈ R. We say M is an f -power torsion module if for each m ∈ M , there exists an n > 0 such that f n m = 0. Thus an f -power torsion module is the same thing as a I-power torsion module for I = (fS ). We sometimes use the notation M [I n ] = {m ∈ M | I n m = 0} and ∞ M [I ] = M [I n ] for an R-module S M . Thus M is I-power torsion if and only if M = M [I ∞ ] if and only if M = M [I n ]. Lemma 12.10.2. Let ϕ : R → S be a ring map. Let I ⊂ R be an ideal. The following are equivalent (1) ϕ is flat and R/I → S/IS is faithfully flat, (2) ϕ is flat, and the map Spec(S/IS) → Spec(R/I) is surjective. (3) ϕ is flat, and the base change functor M 7→ M ⊗R S is faithful on modules annihilated by I, and (4) ϕ is flat, and the base change functor M 7→ M ⊗R S is faithful on I-power torsion modules. Proof. If R → S is flat, then R/I n → S/I n S is flat for every n, see Algebra, Lemma 7.36.6. Hence (1) and (2) are equivalent by Algebra, Lemma 7.36.15. The equivalence of (1) with (3) follows by identifying I-torsion R-modules with R/Imodules, using that M ⊗R S = M ⊗R/I S/IS for R-modules M annihilated by I, and Algebra, Lemma 7.36.13. The implication (4) ⇒ (3) is immediate. Assume (3). We have seen above that R/I n → S/I n S is flat, and by assumption it induces a surjection on spectra, as Spec(R/I n ) = Spec(R/I) and similarly for S. Hence the base change functor is faithful on modules S annihilated by I n . Since any I-power torsion module M is the union M = Mn where Mn is annihilated by I n we see that the base change functor is faithful on the category of all I-power torsion modules (as tensor product commutes with colimits).  Lemma 12.10.3. Let R be a ring. Let I be an ideal of R. Let M be an I-power torsion module. Then M admits a resolution . . . → K2 → K1 → K0 → M → 0 with each Ki a direct sum of copies of R/I n for n variable. Proof. There is a canonical surjection ⊕m∈M R/I nm → M → 0 where nm is the smallest positive integer such that I nm · m = 0. The kernel of the preceding surjection is also an I-power torsion module. Proceeding inductively, we construct the desired resolution of M . 

12.10. FORMAL GLUEING OF MODULE CATEGORIES

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Lemma 12.10.4. Assume (ϕ : R → S, I) satisfies the equivalent conditions of Lemma 12.10.2. The following are equivalent (1) for any I-power torsion module M , the natural map M → M ⊗R S is an isomorphism, and (2) R/I → S/IS is an isomorphism. Proof. The implication (1) ⇒ (2) is immediate. Assume (2). First assume that M is annihilated by I. In this case, M is an R/I-module. Hence, we have an isomorphism M ⊗R S = M ⊗R/I S/IS = M ⊗R/I R/I = M proving the claim. Next we prove by induction that M → M ⊗R S is an isomorphism for any module M is annihilated by I n . Assume the induction hypothesis holds for n and assume M is annihilated by I n+1 . Then we have a short exact sequence 0 → I n M → M → M/I n M → 0 and as R → S is flat this gives rise to a short exact sequence 0 → I n M ⊗R S → M ⊗R S → M/I n M ⊗R S → 0 Using that the canonical map is an isomorphism for M 0 = I n M and M 00 = M/I n M (by induction hypothesis) we conclude the same thing is true S for M . Finally, suppose that M is a general I-power torsion module. Then M = Mn where Mn is annihilated by I n and we conclude using that tensor products commute with colimits.  Lemma 12.10.5. Let R be a ring. Let I be an ideal of R. For any R-module M set M [I n ] = {m ∈ M | I n m = 0}. If I is finitely generated then the following are equivalent (1) M [I] = 0, (2) M [I n ] = 0 for all n ≥ 1, and L (3) if I = (f1 , . . . , ft ), then the map M → Mfi is injective. Proof. This follows from Algebra, Lemma 7.21.4.



Lemma 12.10.6. Let R be a ring. Let I be an ideal of R. For any R-module M S set M [I ∞ ] = n≥1 M [I n ]. If I is finitely generated, then (M/M [I ∞ ])[I] = 0. Proof. Let m ∈ M . If m maps to an element of (M/M [I ∞ ])[I] then Im ⊂ M [I ∞ ]. Write I = (f1 , . . . , ft ). Then we see that fi m P ∈ M [I ∞ ], i.e., I ni fi m = 0 for some N ni > 0. Thus we see that I m = 0 with N = ni + 2. Hence m maps to zero in (M/M [I ∞ ]) which proves the lemma.  Lemma 12.10.7. Assume ϕ : R → S is a flat ring map and I ⊂ R is a finitely generated ideal such that R/I → S/IS is an isomorphism. Then (1) for any R-module M the map M → M ⊗R S induces an isomorphism M [I ∞ ] → (M ⊗R S)[(IS)∞ ] of I-power torsion submodules, (2) the natural map HomR (M, N ) −→ HomS (M ⊗R S, N ⊗R S) is an isomorphism if either M or N is I-power torsion, and (3) the base change functor M 7→ M ⊗R S defines an equivalence of categories between I-power torsion modules and IS-power torsion modules.

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Proof. Note that the equivalent conditions of both Lemma 12.10.2 and Lemma 12.10.4 are satisfied. We will use these without further mention. We first prove (1). Let M be any R-module. Set M 0 = M/M [I ∞ ] and consider the exact sequence 0 → M [I ∞ ] → M → M 0 → 0 As M [I ∞ ] = M [I ∞ ]⊗R S we see that it suffices to show that (M 0 ⊗R S)[(IS)∞ ] = 0. Write I = (f1 , . . . , ft ). By Lemma 12.10.6 we see that M 0 [I ∞ ] = 0. Hence for every n > 0 the map M M 0 −→ M 0 , x 7−→ (f1n x, . . . , ftn x) i=1,...t L is injective. As S is flat over R also the corresponding map M 0 ⊗R S → i=1,...t M 0 ⊗R S is injective. This means that (M 0 ⊗R S)[I n ] = 0 as desired. Next we prove (2). If N is I-power torsion, then N ⊗R S = N and the displayed map of (2) is an isomorphism by Algebra, Lemma 7.11.17. If M is I-power torsion, then the image of any map M → N factors through M [I ∞ ] and the image of any map M ⊗R S → N ⊗R S factors through (N ⊗R S)[(IS)∞ ]. Hence in this case part (1) guarantees that we may replace N by N [I ∞ ] and the result follows from the case where N is I-power torsion we just discussed. Next we prove (3). The functor is fully faithful by (2). For essential surjectivity, we simply note that for any IS-power torsion S-module N , the natural map N ⊗R S → N is an isomorphism.  Lemma 12.10.8. Let R be a ring. Let I = (f1 , . . . , fn ) be a finitely generated ideal of R. Let M be the R-module generated by elements e1 , . . . , en subject to the relations fi ej − fj ei = 0. There exists a short exact sequence 0→K→M →I→0 such that K is annihilated by I. Proof. This is just a truncation of the Koszul complex, see (insert future P reference here). The map M → I isP is determined by the rule e → 7 f . If m = ai ei is in i i P P the kernel of M → I, i.e., ai fi = 0, then fj m = fj ai ei = ( fi ai )ej = 0.  Lemma 12.10.9. Let R be a ring. Let I = (f1 , . . . , fn ) be a finitely generated ideal of R. For any R-module N set {(x1 , . . . , xn ) ∈ N ⊕n | fi xj = fj xi } {f1 x, . . . , fn x) | x ∈ N } For any R-module N there exists a canonical short exact sequence H1 (N, f• ) =

0 → ExtR (R/I, N ) → H1 (N, f• ) → HomR (K, N ) where K is as in Lemma 12.10.8. Proof. The notation above indicates the Ext-groups in ModR as defined in Homology, Section 10.4. These are denoted ExtR (M, N ). Using the long exact sequence of Homology, Lemma 10.4.4 associated to the short exact sequence 0 → I → R → R/I → 0 and the fact that ExtR (R, N ) = 0 we see that ExtR (R/I, N ) = Coker(N −→ Hom(I, N )) Using the short exact sequence of Lemma 12.10.8 we see that we get a complex N → Hom(M, N ) → HomR (K, N )

12.10. FORMAL GLUEING OF MODULE CATEGORIES

851

whose homology in the middle is canonically isomorphic to ExtR (R/I, N ). The proof of the lemma is now complete as the cokernel of the first map is canonically isomorphic to H1 (N, f• ).  Lemma 12.10.10. Let R be a ring. Let I = (f1 , . . . , fn ) be a finitely generated ideal of R. For any R-module N the Koszul homology group H1 (N, f• ) defined in Lemma 12.10.9 is annihilated by I. Proof. Let (x1 , . . . , xn ) ∈ N ⊕n with fi xj = fj xi . Then we have fi (x1 , . . . , xn ) = (fi xi , . . . , fi xn ). In other words fi annihilates H1 (N, f• ).  We can improve on the full faithfulness of Lemma 12.10.7 by showing that Extgroups whose source is I-power torsion are insensitive to passing to S as well. See Remark 12.10.12 below for a more highbrow version of the following lemma. Lemma 12.10.11. Assume ϕ : R → S is a flat ring map and I ⊂ R is a finitely generated ideal such that R/I → S/IS is an isomorphism. Let M , N be R-modules. Assume M is I-power torsion. Given an short exact sequence ˜ → M ⊗R S → 0 0 → N ⊗R S → E there exists a commutative diagram /N 0

0

 / N ⊗R S

/E

/M

/0

 ˜ /E

 / M ⊗R S

/0

with exact rows. Proof. As M is I-power torsion we see that M ⊗R S = M , see Lemma 12.10.4. We will use this identification without further mention. As R → S is flat, the base change functor is exact and we obtain a functorial map of Ext-groups ExtR (M, N ) −→ ExtS (M ⊗R S, N ⊗R S), see Homology, Lemma 10.5.2. The claim of the lemma is that this map is surjective when M is I-power torsion. In fact we will show that it is an isomorphism. By Lemma 12.10.3 we can find a surjection M 0 → M with M 0 a direct sum of modules of the form R/I n . Using the long exact sequence of Homology, Lemma 10.4.4 we see that it suffices to prove the lemma for M 0 . Using compatibility of Ext with direct sums (details omitted) we reduce to the case where M = R/I n for some n. Let f1 , . . . , ft be generators for I n . By Lemma 12.10.9 we have a commutative diagram 0

/ ExtR (R/I n , N )

/ H1 (N, f• )

/ HomR (K, N )

0

 / ExtS (S/I n S, N ⊗ S)

 / H1 (N ⊗ S, f• )

 / HomS (K ⊗ S, N ⊗ S)

with exact rows where K is as in Lemma 12.10.8. Hence it suffices to prove that the two right vertical arrows are isomorphisms. Since K is annihilated by I n we see that HomR (K, N ) = HomS (K ⊗R S, N ⊗R S) by Lemma 12.10.7. As R → S is flat we have H1 (N, f• ) ⊗R S = H1 (N ⊗R S, f• ). As H1 (N, f• ) is annihilated by I n , see Lemma 12.10.10 we have H1 (N, f• ) ⊗R S = H1 (N, f• ) by Lemma 12.10.4. 

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Remark 12.10.12. Assume ϕ : R → S is a flat ring map and I ⊂ R is a finitely generated ideal such that R/I → S/IS is an isomorphism. Let M , N be R-modules and assume M is I-power torsion. Then the canonical map ExtiR (M, N ) −→ ExtiS (M ⊗R S, N ⊗R S) is an isomorphism for all i. We sketch a proof of this strengthening of Lemma 12.10.11. Consider the Koszul complex K• = K• (R, f• ) which is the complex 0 → ∧n Rn → ∧n−1 Rn → . . . → ∧i Rn → . . . → Rn → R → 0 where the last term R is placed in degree 0 with maps given by Xi ej1 ∧ . . . ∧ eji 7−→ (−1)i+1 fja ej1 ∧ . . . ∧ eˆja ∧ . . . ∧ eji a=1

Then H0 (K• ) = R/I and every homology module Hi (K• ) is annihilated by I. Having said this, we prove the statement on Ext-groups by induction on i. The case i = 0 is Lemma 12.10.7. Assume that the result holds for all i ≤ i0 and all modules N , M with M being I-power torsion. Pick a pair of modules N and M with M being I-power torsion and let’s prove that the map ExtiR0 +1 (M, N ) → ExtiS0 +1 (M ⊗R S, N ⊗R S) is an isomorphism. By Lemma 12.10.3 and the long exact sequence of Ext-groups and compatibility of Ext with direct sums we reduce to the case that M = R/I n . Since I n is finitely generated we can choose finitely many generators f1 , . . . , ft ∈ I n and consider the Koszul complex K• = K• (R, f• ). Note that K• ⊗R S = K• (S, f• ). As K• is a finite complex of finite free R-modules we see that the map (12.10.12.1)

HomR (K• , N ) ⊗R S −→ HomS (K• ⊗R S, N ⊗R S)

is an isomorphism of complexes. As R → S is flat and using Lemmas 12.10.7 we see that Hb (K• ) = Hb (K• ) ⊗R S = Hb (K• ⊗R S). Below we will use the spectral sequences a a+b (HomR (K• , N )), E(R)a,b 2 = ExtR (Hb (K• ), N ) ⇒ H a a+b (HomR (K• ⊗R S, N ⊗R S)) E(S)a,b 2 = ExtR (Hb (K• ⊗R S), N ⊗R S) ⇒ H

see (insert future reference here). The first one combined with the fact that each Hb (K• ) is annihilated by I n implies that H c (HomR (K• , N )) is annihilated by I n(t+1) . Hence using Lemma 12.10.7 once more we see that H c (HomR (K• , N )) = H c (HomR (K• , N )) ⊗R S = H c (HomS (K• ⊗R S, N ⊗R S)) because (12.10.12.1) is an isomorphism and R → S is flat. Combined we see that the map E(R)a,b → E(S)a,b of spectral sequences is an isomorphism for r = 2 r r and a ≤ i0 (induction hypothesis) and an isomorphism on abutments in all degrees. Then a formal argument on spectral sequences (insert future reference here) implies that E(R)i20 +1,0 → E(R)i20 +1,0 is an isomorphism as well, which is the result we wanted to prove. This ends the sketch of the proof of the result on Ext-groups; if we ever need to use this result in the stacks project we will put in a detailed proof. Let R → S be a ring map. Let f1 , . . . , ft ∈ R and I = (f1 , . . . , ft ). Then for any R-module M we can define a complex Y Y β Y α (12.10.12.2) 0→M − → M ⊗R S × Mfi − → (M ⊗R S)fi × Mfi fj

12.10. FORMAL GLUEING OF MODULE CATEGORIES

853

where α(m) = (m ⊗ 1, m/1, . . . , m/1) and β(m0 , m1 , . . . , mt ) = ((m0 /1−m1 ⊗1, . . . , m0 /1−mt ⊗1), (m1 −m2 , . . . , mt−1 −mt ). We would like to know when this complex is exact. Lemma 12.10.13. Assume ϕ : R → S is a flat ring map and I = (f1 , . . . , ft ) ⊂ R is an ideal such that R/I → S/IS is an isomorphism. Let M be an R-module. Then the complex (12.10.12.2) is exact. Proof. Let m ∈ M . If α(m) = 0, then m ∈ M [I ∞ ], see Lemma 12.10.5. Pick n such that I n m = 0 and consider the map ϕ : R/I n → M . If m ⊗ 1 = 0, then ϕ ⊗ 1S = 0, hence ϕ = 0 (see Lemma 12.10.7) hence m = 0. In this way we see that α is injective. Let (m0 , m01 , . . . , m0t ) ∈ Ker(β). Write m0i = mi /fin for some n > 0 and mi ∈ M . We may, after possibly enlarging n assume that fin m0 = mi ⊗ 1 in M ⊗R S and fjn mi − fin mj = 0 in M . In particular we see that (m1 , . . . , mt ) defines an element ξ of H1 (M, (f1n , . . . , ftn )). Since H1 (M, (f1n , . . . , ftn )) is annihilated by I tn+1 (see Lemma 12.10.10) and since R → S is flat we see that H1 (M, (f1n , . . . , ftn )) = H1 (M, (f1n , . . . , ftn )) ⊗R S = H1 (M ⊗R S, (f1n , . . . , ftn )) by Lemma 12.10.4 The existence of m0 implies that ξ maps to zero in the last group, i.e., the element ξ is zero. Thus there exists an m ∈ M such that mi = fin m. Then (m0 , m01 , . . . , m0t ) − α(m) = (m00 , 0, . . . , 0) for some m00 ∈ (M ⊗R S)[(IS)∞ ]. By Lemma 12.10.7 we conclude that m00 ∈ M [I ∞ ] and we win.  Remark 12.10.14. In this remark we define a category of glueing data. Let R → S be a ring map. Let f1 , . . . , ft ∈ R and I = (f1 , . . . , ft ). Consider the category Glue(R → S, f1 , . . . , ft ) as the category whose (1) objects are systems (M 0 , Mi , αi , αij ), where M 0 is an S-module, Mi is an Rfi -module, αi : (M 0 )fi → Mi ⊗R S is an isomorphism, and αij : (Mi )fj → (Mj )fi are isomorphisms such that (a) αij ◦ αi = αj as maps (M 0 )fi fj → (Mj )fi , and (b) αjk ◦ αij = αik as maps (Mi )fj fk → (Mk )fi fj (cocycle condition). (2) morphisms (M 0 , Mi , αi , αij ) → (N 0 , Ni , βi , βij ) are given by maps ϕ0 : M 0 → N 0 and ϕi : Mi → Ni compatible with the given maps αi , βi , αij , βij . There is a canonical functor Can : ModR −→ Glue(R → S, f1 , . . . , ft ),

M 7−→ (M ⊗R S, Mfi , cani , canij )

where cani : (M ⊗R S)fi → Mfi ⊗R S and canij : (Mfi )fj → (Mfj )fi are the canonical isomorphisms. For any object M = (M 0 , Mi , αi , αij ) of the category Glue(R → S, f1 , . . . , ft ) we define H 0 (M) = {(m0 , mi ) | αi (m0 ) = mi ⊗ 1, αij (mi ) = mj } in other words defined by the exact sequence Y Y Y 0 → H 0 (M) → M 0 × Mi → Mf0 i × (Mi )fj similar to (12.10.12.2). We think of H 0 (M) as an R-module. Thus we also get a functor H 0 : Glue(R → S, f1 , . . . , ft ) −→ ModR

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Our next goal is to show that the functors Can and H 0 are sometimes quasi-inverse to each other. Lemma 12.10.15. Assume ϕ : R → S is a flat ring map and I = (f1 , . . . , ft ) ⊂ R is an ideal such that R/I → S/IS is an isomorphism. Then the functor H 0 is a left quasi-inverse to the functor Can of Remark 12.10.14. Proof. This is a reformulation of Lemma 12.10.13.



Lemma 12.10.16. Assume ϕ : R → S is a flat ring map and let I = (f1 , . . . , ft ) ⊂ R be an ideal. Then Glue(R → S, f1 , . . . , ft ) is an abelian category, and the functor Can is exact and commutes with arbitrary colimits. Proof. Given a morphism (ϕ0 , ϕi ) : (M 0 , Mi , αi , αij ) → (N 0 , Ni , βi , βij ) of the category Glue(R → S, f1 , . . . , ft ) we see that its kernel exists and is equal to the object (Ker(ϕ0 ), Ker(ϕi ), αi , αij ) and its cokernel exists and is equal to the object (Coker(ϕ0 ), Coker(ϕi ), βi , βij ). This works because R → S is flat, hence taking kernels/cokernels commutes with − ⊗R S. Details omitted. The exactness follows from the R-flatness of Rfi and S, while commuting with colimits follows as tensor products commute with colimits.  Lemma 12.10.17. Let ϕ : R → S be a flat ring map and (f1 , . . . , ft ) = R. Then Can and H 0 are quasi-inverse equivalences of categories ModR = Glue(R → S, f1 , . . . , ft ) Proof. Consider an object M = (M 0 , Mi , αi , αij ) of Glue(R → S, f1 , . . . , ft ). By Algebra, Lemma 7.22.4 there exists a unique module M and isomorphisms Mfi → Mi which recover the glueing data αij . Then both M 0 and M ⊗R S are S-modules which recover the modules Mi ⊗R S upon localizing at fi . Whence there is a canonical isomorphism M ⊗R S → M 0 . This shows that M is in the essential image of Can. Combined with Lemma 12.10.15 the lemma follows.  Lemma 12.10.18. Let ϕ : R → S be a flat ring map and I = (f1 , . . . , ft ) and ideal. Let R → R0 be a flat ring map, and set S 0 = S ⊗R R0 . Then we obtain a commutative diagram of categories and functors ModR

Can

−⊗R R0

 ModR0

/ Glue(R → S, f1 , . . . , ft )

H0

−⊗R R0

Can

 / Glue(R0 → S 0 , f1 , . . . , ft )

/ ModR −⊗R R0

H0

 / ModR0

Proof. Omitted.



Proposition 12.10.19. Assume ϕ : R → S is a flat ring map and I = (f1 , . . . , ft ) ⊂ R is an ideal such that R/I → S/IS is an isomorphism. Then Can and H 0 are quasi-inverse equivalences of categories ModR = Glue(R → S, f1 , . . . , ft ) Proof. We have already seen that H 0 ◦ Can is isomorphic to the identity functor, see Lemma 12.10.15. Consider an object M = (M 0 , Mi , αi , αij ) of Glue(R → S, f1 , . . . , ft ). We get a natural morphism Ψ : (H 0 (M) ⊗R S, H 0 (M)fi , cani , canij ) −→ (M 0 , Mi , αi , αij ).

12.10. FORMAL GLUEING OF MODULE CATEGORIES

855

Namely, by definition H 0 (M) comes equipped with compatible R-module maps H 0 (M) → M 0 and H 0 (M) → Mi . We have to show that this map is an isomorphism. Pick an index i and set R0 = Rfi . Combining Lemmas 12.10.18 and 12.10.17 we see that Ψ ⊗R R0 is an isomorphism. Hence the kernel, resp. cokernel of Ψ is a system of the form (K, 0, 0, 0), resp. (Q, 0, 0, 0). Note that H 0 ((K, 0, 0, 0)) = K, that H 0 is left exact, and that by construction H 0 (Ψ) is bijective. Hence we see K = 0, i.e., the kernel of Ψ is zero. The conclusion of the above is that we obtain a short exact sequence 0 → H 0 (M) ⊗R S → M 0 → Q → 0 and that Mi = H 0 (M)fi . Note that we may think of Q as an R-module which is I-power torsion so that Q = Q ⊗R S. By Lemma 12.10.11 we see that there exists a commutative diagram 0

/ H 0 (M)

/E

/Q

/0

0

 / H 0 (M) ⊗R S

 / M0

 /Q

/0

with exact rows. This clearly determines an isomorphism Can(E) → (M 0 , Mi , αi , αij ) in the category Glue(R → S, f1 , . . . , ft ) and we win. (Of course, a posteriori we have Q = 0.)  Next, we specialize this very general proposition to get something more useable. Namely, if I = (f ) is a principal ideal then the objects of Glue(R → S, f ) are simply triples (M 0 , M1 , α1 ) and there is no cocycle condition to check! Theorem 12.10.20. Let R be a ring, and let f ∈ R. Let ϕ : R → S be a flat ring map inducing an isomorphism R/f R → S/f S. Then the functor ModR −→ ModS ×ModSf ModRf ,

M 7−→ (M ⊗R S, Mf , can)

is an equivalence. Proof. The category appearing on the right side of the arrow is the category of triples (M 0 , M1 , α1 ) where M 0 is an S-module, M1 is a Rf -module, and α1 : Mf0 → M1 ⊗R S is a Sf -isomorphism, see Categories, Example 4.28.3. Hence this theorem is a special case of Proposition 12.10.19.  A useful special case of Theorem 12.10.20 is when R is noetherian, and S is a completion of R at an element f . The completion R → S is flat, and the functor M 7→ M ⊗R S can be identified with the f -adic completion functor when M is finitely generated. To state this more precisely, let Modf g (R) denote the category of finitely generated R-modules. Proposition 12.10.21. Let R be a noetherian ring. Let f ∈ R be an element. Let R∧ be the f -adic completion of R. Then the functor M 7→ (M ∧ , Mf , can) defines an equivalence Modf g (R) −→ Modf g (R∧ ) ×Modf g (Rf∧ ) Modf g (Rf )

856

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Proof. The ring map R → R∧ is flat by Algebra, Lemma 7.91.3. It is clear that R/f R = R∧ /f R∧ . By Algebra, Lemma 7.91.2 the completion of a finite R-module M is equal to M ⊗R R∧ . Hence the displayed functor of the proposition is equal to the functor occuring in Theorem 12.10.20. In particular it is fully faithful. Let (M1 , M2 , ψ) be an object of the right hand side. By Theorem 12.10.20 there exists an R-module M such that M1 = M ⊗R R∧ and M2 = Mf . As R → R∧ × Rf is faithfully flat we conclude from Algebra, Lemma 7.22.2 that M is finitely generated, i.e., M ∈ Modf g (R). This proves the proposition.  Remark 12.10.22. The equivalences of Proposition 12.10.19, Theorem 12.10.20, and Proposition 12.10.21 preserve the ⊗-structures on either side. Thus, it defines equivalences of various categories built out of the pair (ModR , ⊗), such as the category of R-algebras. Remark 12.10.23. Given a differential manifold X with a compact closed submanifold Z having complement U , specifying a sheaf on X is the same as specifying a sheaf on U , a sheaf on an unspecified tubular neighbourhood T of Z in X, and an isomorphism between the two resulting sheaves along T ∩ U . Tubular neighbourhoods do not exist in algebraic geometry as such, but results such as Proposition 12.10.19, Theorem 12.10.20, and Proposition 12.10.21 allow us to work with formal neighbourhoods instead. 12.11. Lifting In this section we collection some lemmas concerning lifting statements of the following kind: If A is a ring and I ⊂ A is an ideal, and ξ is some kind of structure over A/I, then we can lift ξ to a similar kind of structure ξ over A or over some ´etale extension of A. Here are some types of structure for which we have already proved some results: (1) idempotents, see Algebra, Lemmas 7.50.5 and 7.50.6, (2) projective modules, see Algebra, Lemma 7.72.4, (3) basis elements, see Algebra, Lemmas 7.94.1 and 7.94.3, (4) ring maps, i.e., proving certain algebras are formally smooth, see Algebra, Lemma 7.128.4, Proposition 7.128.13, and Lemma 7.128.16, (5) syntomic ring maps, see Algebra, Lemma 7.126.19, (6) smooth ring maps, see Algebra, Lemma 7.127.19, (7) ´etale ring maps, see Algebra, Lemma 7.133.10, (8) factoring polynomials, see Algebra, Lemma 7.133.19, and (9) Algebra, Section 7.140 discusses henselian local rings. The interested reader will find more results of this nature in Smoothing Ring Maps, Section 13.4 in particular Smoothing Ring Maps, Proposition 13.4.2. Let A be a ring and let I ⊂ A be an ideal. Let ξ be some kind of structure over A/I. In the following lemmas we look for ´etale ring maps A → A0 which induce isomorphisms A/I → A0 /IA0 and objects ξ 0 over A0 lifting ξ. A general remark is that given ´etale ring maps A → A0 → A00 such that A/I ∼ = A0 /IA0 and 0 0 ∼ 00 00 00 A /IA = A /IA the composition A → A is also ´etale (Algebra, Lemma 7.133.3) and also satisfies A/I ∼ = A00 /IA00 . We will frequently use this in the following lemmas without further mention. Here is a trivial example of the type of result we are looking for.

12.11. LIFTING

857

Lemma 12.11.1. Let A be a ring, let I ⊂ A be an ideal, let u ∈ A/I be an invertible element. There exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and an invertible element u0 ∈ A0 lifting u. Proof. Choose any lift f ∈ A of u and set A0 = Af and u the image of f in A0 .  Lemma 12.11.2. Let A be a ring, let I ⊂ A be an ideal, let e ∈ A/I be an idempotent. There exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and an idempotent e0 ∈ A0 lifting e. Proof. Choose any lift x ∈ A of e. Set  1 A = A[t]/(t − t) . t−1+x 0

2



The ring map A → A0 is ´etale because (2t − 1)dt = 0 and (2t − 1)(2t − 1) = 1 which 1 ]∼ is invertible. We have A0 /IA0 = A/I[t]/(t2 − t)[ t−1+e = A/I the last map sending 2 t to e which works as e is a root of t − t. This also shows that setting e0 equal to the class of t in A0 works.  ` Lemma 12.11.3. Let A be a ring, let I ⊂ A be an ideal. Let Spec(A/I) = j∈J U j be a finite disjoint open covering. Then there exists an ´etale ring map A → A0 which induces A/I → A0 /IA0 and a finite disjoint open covering ` an isomorphism 0 0 Spec(A ) = j∈J Uj lifting the given covering. Proof. This follows from Lemma 12.11.2 and the fact that open and closed subsets of Spectra correspond to idempotents, see Algebra, Lemma 7.19.3.  Lemma 12.11.4. Let A → B be a ring map and J ⊂ B an ideal. If A → B is ´etale at every prime of V (J), then there exists a g ∈ B mapping to an invertible element of B/J such that A0 = Bg is ´etale over A. Proof. The set of points of Spec(B) where A → B is not ´etale is a closed subset of Spec(B), see Algebra, Definition 7.133.1. Write this as V (J 0 ) for some ideal J 0 ⊂ B. Then V (J 0 ) ∩ V (J) = ∅ hence J + J 0 = B by Algebra, Lemma 7.16.2. Write 1 = f + g with f ∈ J and g ∈ J 0 . Then g works.  The assumption on the leading coefficient in the following lemma will be removed in Lemma 12.11.6. Lemma 12.11.5. Let A be a ring, let I ⊂ A be an ideal. Let f ∈ A[x] be a monic polynomial. Let f = gh be a factorization of f in A/I[x] and assume (1) the leading coefficient of g is an invertible element of A/I, and (2) g, h generate the unit ideal in A/I[x]. Then there exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and a factorization f = g 0 h0 in A0 [x] lifting the given factorization over A/I. Proof. Applying Lemma 12.11.1 we may assume that the leading coefficient of g is the reduction of an invertible element u ∈ A. Then we may replace g by u−1 g and h by uh. Thus we may assume that g is monic. Since f is monic we conclude that h is monic too. P Say deg(g) = n and deg(h) = m so that deg(f ) = n + m. Write f = xn+m + αi xn+m−i for some α1 , . . . , αn+m ∈ A. Consider the ring map R = Z[a1 , . . . , an+m ] −→ S = Z[b1 , . . . , bn , c1 , . . . , cm ]

858

12. MORE ON ALGEBRA

of Algebra, Example 7.133.12. Let R → A be the ring map which sends ai to αi . Set B = A ⊗R S P By construction the image of f in B[x] factors. Write g = xn + β i xn−i and P m m−i h = x + γix . The A-algebra map B −→ A/I,

1 ⊗ bi 7→ β i ,

1 ⊗ ci 7→ γ i

maps the factorization of f over B to the given factorization over A/I. The displayed map is surjective; denote J ⊂ B its kernel. From the discussion in Algebra, Example 7.133.12 it is clear that A → B is etale at all points of V (J) ⊂ Spec(B). Choose g ∈ B as in Lemma 12.11.4 and set A0 = Bg .  Lemma 12.11.6. Let A be a ring, let I ⊂ A be an ideal. Let f ∈ A[x] be a monic polynomial. Let f = gh be a factorization of f in A/I[x] and assume that g, h generate the unit ideal in A/I[x]. Then there exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and a factorization f = g 0 h0 in A0 [x] lifting the given factorization over A/I. P Proof. Say f = xd + a1 xd−1 + . . . + ad has degree d. Write g = bj xj and P P j h= cj x . Then we see that 1 = bj cd−j . It follows that Spec(A/I) is covered by the standard opens D(bj cd−j ). However, each point p of Spec(A/I) is contained in at most one of these as by looking at the induced factorization of f over the field κ(p) we see that deg(g mod p) + deg(h mod p) = d. Hence our open covering is a disjoint open covering. Applying Lemma 12.11.3 (and replacing A by A0 ) we see that we may assume there is a corresponding disjoint open covering of Spec(A). This disjoint open covering corresponds to a product decomposition of A, see Algebra, Lemma 7.21.3. It follows that A = A0 × . . . × Ad ,

I = I0 × . . . × Id ,

where the image of g, resp. h in Aj /Ij has degree j, resp. d−j with invertible leading coefficient. Clearly, it suffices to prove the result for each factor Aj separatedly. Hence the lemma follows from Lemma 12.11.5.  Lemma 12.11.7. Let R → S be a ring map. Let I ⊂ R be an ideal of R and let J ⊂ S be an ideal of S. If the closure of the image of V (J) in Spec(R) is disjoint from V (I), then there exists an element f ∈ R which maps to 1 in R/I and to an element of J in S. Proof. Let I 0 ⊂ R be an ideal such that V (I 0 ) is the closure of the image of V (J). Then V (I) ∩ V (I 0 ) = ∅ by assumption and hence I + I 0 = R by Algebra, Lemma 7.16.2. Write 1 = g + f with g ∈ I and f ∈ I 0 . We have V (f 0 ) ⊃ V (J) where f 0 is the image of f in S. Hence (f 0 )n ∈ J for some n, see Algebra, Lemma 7.16.2. Replacing f by f n we win.  Lemma 12.11.8. Let A be a ring, let I ⊂ A be an ideal. Let A → B be an integral ring map. Let e ∈ B/IB be an idempotent. Then there exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and an idempotent e0 ∈ B ⊗A A0 lifting e. Proof. Choose an element y ∈ B lifting e. Then z = y 2 − y is an element P of IB. By Algebra, Lemma 7.35.4 there exist a monic polynomial g(x) = xd + aj xj of degree d with aj ∈ I such that g(z) = 0 in B. Hence f (x) = g(x2 − x) ∈ A[x] is

12.11. LIFTING

859

a monic polynomial such that f (x) ≡ xd (x − 1)d mod I and such that f (y) = 0 in B. By Lemma 12.11.5 we can find an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and such that f = gh in A[x] with g(x) = xd mod IA0 and h(x) = (x − 1)d mod IA0 . After replacing A by A0 we may assume that the factorization is defined over A. In that case we see that b1 = g(y) ∈ B is a lift of ed = e and b2 = h(y) ∈ B is a lift of (e − 1)d = (−1)d (1 − e)d = (−1)d (1 − e) and moreover b1 b2 = 0. Thus (b1 , b2 )B/IB = B/IB and V (b1 , b2 ) ⊂ Spec(B) is disjoint from V (IB). Since Spec(B) → Spec(A) is closed (see Algebra, Lemmas 7.33.20 and 7.37.6) we can find an a ∈ A which maps to an invertible element of A/I whose image in B lies in (b1 , b2 ), see Lemma 12.11.7. After replacing A by the localization Aa we get that (b1 , b2 ) = B. Then Spec(B) = D(b1 ) q D(b2 ); disjoint union because b1 b2 = 0. Let e ∈ B be the idempotent corresponding to the open and closed subset D(b1 ), see Algebra, Lemma 7.19.3. Since b1 is a lift of e and b2 is a lift of ±(1 − e) we conclude that e is a lift of e by the uniqueness statement in Algebra, Lemma 7.19.3.  Lemma 12.11.9. Let A be a ring, let I ⊂ A be an ideal. Let P be finite projective A/I-module. Then there exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and a finite projective A0 -module P 0 lifting P . Proof. We can choose an integer n and a direct sum decomposition (A/I)⊕n = P ⊕ K for some R/I-module K. Choose a lift ϕ : A⊕n → A⊕n of the projector p associated to the direct summand P . Let f ∈ A[x] be the characteristic polynomial of ϕ. Set B = A[x]/(f ). By Cayley-Hamilton (Algebra, Lemma 7.15.1) there is a map B → EndA (A⊕n ) mapping x to ϕ. For every prime p ⊃ I the image of f in κ(p) is (x − 1)r xn−r where r is the dimension of P ⊗A/I κ(p). Hence (x − 1)n xn maps to zero in B ⊗A κ(p) for all p ⊃ I. Hence the image of (x − 1)n xn in B is contained in [ [ √ pB = ( p)B = IB p⊃I

p⊃I

the first equality because B is a free A-module and the second by Algebra, Lemma 7.16.2. Thus (x−1)N xN is contained in IB for some N . It follows that xN +(1−x)N is a unit in B/IB and that e = image of

xN in B/IB xN + (1 − x)N

is an idempotent as both assertions hold in Z[x]/(xn (x − 1)N ). The image of e in EndA/I ((A/I)⊕n ) is pN =p p + (1 − p)N N

as p is an idempotent. After replacing A by an ´etale extension A0 as in the lemma, we may assume there exists an idempotent e ∈ B which maps to e in B/IB, see Lemma 12.11.8. Then the image of e under the map B = A[x]/(f ) −→ EndA (A⊕n ). is an idempotent element p which lifts p. Setting P = Im(p) we win.



Lemma 12.11.10. Let A be a ring. Let 0 → K → A⊕m → M → 0 be a sequence of A-modules. Consider the A-algebra C = Sym∗A (M ) with its presentation α :

860

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A[y1 , . . . , ym ] → C coming from the surjection A⊕m → M . Then M N L(α) = (K ⊗A C → Cdyj ) j=1,...,m

(see Algebra, Section 7.124) in particular ΩC/A = M ⊗A C. Proof. Let J = Ker(α). The lemma asserts that J/J 2 ∼ = K ⊗A C. Note that α is a homomorphism of graded algebras. We will prove that in degree d we have (J/J 2 )d = K ⊗A Cd−1 . Note that d−1 Jd = Ker(SymdA (A⊕m ) → SymdA (M )) = Im(K ⊗A SymA (A⊕m ) → SymdA (A⊕m )), P see Algebra, Lemma 7.12.2. It follows that (J 2 )d = a+b=d Ja · Jb is the image of d−2 K ⊗A K ⊗A SymA (A⊗m ) → SymdA (A⊕m ). d−2 d−1 d−1 The cokernel of the map K ⊗A SymA (A⊗m ) → SymA (A⊕m ) is SymA (M ) by the lemma referenced above. Hence it is clear that (J/J 2 )d = Jd /(J 2 )d is equal to d−1 d−1 ⊗m Coker(K ⊗A K ⊗A Symd−2 ) → K ⊗A SymA (A⊗m )) = K ⊗A SymA (M ) A (A

= K ⊗A Cd−1 as desired.



Lemma 12.11.11. Let A be a ring. Let M be an A-module. Then C = Sym∗A (M ) is smooth over A if and only if M is a finite projective A-module. Proof. Let σ : C → A be the projection onto the degree 0 part of C. Then J = Ker(σ) is the part of degree > 0 and we see that J/J 2 = M as an A-module. Hence if A → C is smooth then M is a finite projective A-module by Algebra, Lemma 7.129.4. Conversely, assume that M is finite projective and choose a surjection A⊕n → M with kernel K. Of course the sequence 0 → K → A⊕n → M → 0 is split as M is projective. In particular we see that K is a finite A-module and hence C is of finite presentation over A as C is a quotient of A[x1 , . . . , xn ] by the ideal generated by L K⊂ Axi . The computation of Lemma 12.11.10 shows that N LC/A is homotopy equivalent to (K → M ) ⊗A C. Hence N LC/A is quasi-isomorphic to C ⊗A M placed in degree 0 which means that C is smooth over A by Algebra, Definition 7.127.1.  Lemma 12.11.12. Let A be a ring, let I ⊂ A be an ideal. Consider a commutative diagram BO A

! / A/I

where B is a smooth A-algebra. Then there exists an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and an A-algebra map B → A0 lifting the ring map B → A/I. Proof. Let J ⊂ B be the kernel of B → A/I so that B/J = A/I. By Algebra, Lemma 7.129.3 the sequence 0 → I/I 2 → J/J 2 → ΩB/A ⊗B B/J → 0

12.12. AUTO-ASSOCIATED RINGS

861

is split exact. Thus P = J/(J 2 + IB) = ΩB/A ⊗B B/J is a finite projective A/Imodule. Choose an integer n and a direct sum decomposition A/I ⊕n = P ⊕ K. By Lemma 12.11.9 we can find an ´etale ring map A → A0 which induces an isomorphism A/I → A0 /IA0 and a finite projective A-module K which lifts K. We may and do replace A by A0 . Set B 0 = B ⊗A Sym∗A (K). Since A → Sym∗A (K) is smooth by Lemma 12.11.11 we see that B → B 0 is smooth which in turn implies that A → B 0 is smooth (see Algebra, Lemmas 7.127.4 and 7.127.13). Moreover the section Sym∗A (K) → A determines a section B 0 → B and we let B 0 → A/I be the composition B 0 → B → A/I. Let J 0 ⊂ B 0 be the kernel of B 0 → A/I. We have JB 0 ⊂ J 0 and B ⊗A K ⊂ J 0 . These maps combine to give an isomorphism (A/I)⊕n ∼ = J/J 2 ⊕ K −→ J 0 /((J 0 )2 + IB 0 ) Thus, after replacing B by B 0 we may assume that J/(J 2 + IB) = ΩB/A ⊗B B/J is a free A/I-module of rank n. In this case, choose f1 , . . . , fn ∈ J which map to a basis of J/(J 2 + IB). Consider the finitely presented A-algebra C = B/(f1 , . . . , fn ). Note that we have an exact sequence 0 → H1 (LC/A ) → (f1 , . . . , fn )/(f1 , . . . , fn )2 → ΩB/A ⊗B C → ΩC/A → 0 see Algebra, Lemma 7.124.3 (note that H1 (LB/A ) = 0 and that ΩB/A is finite projective, in particular flat so the Tor group vanishes). For any prime q ⊃ J of B the module ΩB/A,q is free of rank n because ΩB/A is finite projective and because ΩB/A ⊗B B/J is free of rank n. By our choice of f1 , . . . , fn the map
(f1 , . . . , fn )/(f1 , . . . , fn )2 q → ΩB/A,q is surjective modulo I. Hence we see that this map of modules over the local ring Cq has to be an isomorphism. Thus H1 (LC/A )q = 0 and ΩC/A,q = 0. By Algebra, Lemma 7.127.12 we see that A → C is smooth at the prime q of C corresponding to q. Since ΩC/A,q = 0 it is actually ´etale at q. Thus A → C is ´etale at all primes of C containing JC. By Lemma 12.11.4 we can find an f ∈ C mapping to an invertible element of C/JC such that A → Cf is ´etale. By our choice of f it is still true that Cf /JCf = A/I. The map Cf /ICf → A/I is surjective and ´etale by Algebra, Lemma 7.133.8. Hence A/I is isomorphic to the localization of Cf /ICf at some element g ∈ C, see Algebra, Lemma 7.133.9. Set A0 = Cf g to conclude the proof.  12.12. Auto-associated rings Some of this material is in [Laz69]. Definition 12.12.1. A ring R is said to be auto-associated if R is local and its maximal ideal m is weakly associated to R. Lemma 12.12.2. An auto-associated ring R has the following property: (P) Every proper finitely generated ideal I ⊂ R has a nonzero annihilator. Proof. By assumption there exists a nonzero element x ∈ R such that forLevery f ∈ m we have f n x = 0. Say I = (f1 , . . . , fr ). Then x is in the kernel of R → Rfi . Hence we see that there exists a nonzero y ∈ R such that fi y = 0 for all i, see Algebra, Lemma 7.21.4. As y ∈ AnnR (I) we win. 

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Lemma 12.12.3. Let R be a ring having property (P) of Lemma 12.12.2. Let u : N → M be a homomorphism of projective R-modules. Then u is universally injective if and only if u is injective. Proof. Assume u is injective. Our goal is to show u is universally injective. First we choose a module Q such that N ⊕Q is free. On considering the map N ⊕Q → M ⊕Q we see that it suffices to prove the lemma in case N is free. In this case N is a directed colimit of finite free R-modules. Thus we reduce to the case that N is a finite free R-module, say N = R⊕n . We prove the lemma by induction on n. The case n = 0 is trivial. Let u : R⊕n → M be an injective module map with M projective. Choose an R-module Q such that M ⊕ Q is free. After replacing u by the composition R⊕n → M → M ⊕ Q we see that we may assume that M is free. Then we can find a direct summand R⊕m ⊂ M such that u(R⊕n ) ⊂ R⊕m . Hence we ⊕m may assume that M = R P . In this Pcase u is given by a matrix A = (aij ) so that u(x1 , . . . , xn ) = ( xi ai1 , . . . , xi aim ). As u is injective, in particular u(x, 0, . . . , 0) = (xa11 , xa12 , . . . , xa1m ) 6= 0 if x 6= 0, and as R has property (P) we see that a11 R + a12 R + . . . + a1m R = R. Hence see that R(a11 , . . . , a1m ) ⊂ R⊕m is a direct summand of R⊕m , in particular R⊕m /R(a11 , . . . , a1m ) is a projective R-module. We get a commutative diagram 0

/R

0

 /R

/ R⊕n

1 (a11 ,...,a1m )



u

/ R⊕m

/ R⊕n−1

/0

 / R⊕m /R(a11 , . . . , a1m )

/0

with split exact rows. Thus the right vertical arrow is injective and we may apply the induction hypothesis to conclude that the right verical arrow is universally injective. It follows that the middle vertical arrow is universally injective.  Lemma 12.12.4. Let R be a ring. The following are equivalent (1) R has property (P) of Lemma 12.12.2, (2) any injective map of projective R-modules is universally injective, (3) if u : N → M is injective and N , M are finite projective R-modules then Coker(u) is a finite projective R-module, (4) if N ⊂ M and N , M are finite projective as R-modules, then N is a direct summand of M , and (5) any injective map R → R⊕n is a split injection. Proof. The implication (1) ⇒ (2) is Lemma 12.12.3. It is clear that (3) and (4) are equivalent. We have (2) ⇒ (3), (4) by Algebra, Lemma 7.77.4. Part (5) is a special case of (4). Assume (5). Let I = (a1 , . . . , an ) be a proper finitely generated ideal of R. As I 6= R we see that R → R⊕n , x 7→ (xa1 , . . . , xan ) is not a split injection. Hence it has a nonzero kernel and we conclude that AnnR (I) 6= 0. Thus (1) holds.  Example 12.12.5. If the equivalent conditions of Lemma 12.12.4 hold, then it is not always the case that every injective map of free R-modules is a split injection. For example suppose that R = k[x1 , x2 , x3 , . . .]/(x2i ). This is an auto-associated

12.13. FLATTENING STRATIFICATION

863

ring. Consider the map of free R-modules M M u: Rei −→ Rfi , ei 7−→ fi − xi fi+1 . i≥1 i≥1 L For any integer n the restriction of u to i=1,...,n Rei is injective as the images u(e1 ), . . . , u(en ) are R-linearly independent. Hence u is injective and hence universally injective by the lemma. Since u ⊗ idk is bijective we see that if u were a split injection then u would be surjective. But u is not surjective because the inverse image of f1 would be the element X x1 . . . xi ei+1 = e1 + x1 e2 + x1 x2 e3 + . . . i≥0

which is not an element of the direct sum. A side remark is that Coker(u) is a flat (because u is universally injective), countably generated R-module which is not projective (as u is not split), hence not Mittag-Leffler (see Algebra, Lemma 7.88.1). 12.13. Flattening stratification Let R → S be a ring map and let N be an S-module. For any R-algebra R0 we can consider the base changes S 0 = S ⊗R R0 and M 0 = M ⊗R R0 . We say R → R0 flattens M if the module M 0 is flat over R0 . We would like to understand the structure of the collection of ring maps R → R0 which flatten M . In particular we would like to know if there exists a universal flattening R → Runiv of M , i.e., a ring map R → Runiv which flattens M and has the property that any ring map R → R0 which flattens M factors through R → Runiv . It turns out that such a universal solution usually does not exist. We will discuss universal flattenings and flattening stratifications in a scheme theoretic setting F/X/S in More on Flatness, Section 34.21. If the universal flattening R → Runiv exists then the morphism of schemes Spec(Runiv ) → Spec(R) is the f on Spec(S). universal flattening of the quasi-coherent module M In this and the next few sections we prove some basic algebra facts related to this. The most basic result is perhaps the following. Lemma 12.13.1. Let R be a ring. Let M be an R-module. Let I1 , I2 be ideals of R. If M/I1 M is flat over R/I1 and M/I2 M is flat over R/I2 , then M/(I1 ∩ I2 )M is flat over R/(I1 ∩ I2 ). Proof. By replacing R with R/(I1 ∩ I2 ) and M by M/(I1 ∩ I2 )M we may assume that I1 ∩ I2 = 0. Let J ⊂ R be an ideal. To prove that M is flat over R we have to show that J ⊗R M → M is injective, see Algebra, Lemma 7.36.4. By flatness of M/I1 M over R/I1 the map J/(J ∩ I1 ) ⊗R M = (J + I1 )/I1 ⊗R/I1 M/I1 M −→ M/I1 M is injective. As 0 → (J ∩ I1 ) → J → J/(J ∩ I1 ) → 0 is exact we obtain a diagram (J ∩ I1 ) ⊗R M

/ J ⊗R M

/ J/(J ∩ I1 ) ⊗R M

 M

 M

 / M/I1 M

/0

hence it suffices to show that (J ∩ I1 ) ⊗R M → M is injective. Since I1 ∩ I2 = 0 the ideal J ∩ I1 maps isomorphically to an ideal J 0 ⊂ R/I2 and we see that

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(J ∩ I1 ) ⊗R M = J 0 ⊗R/I2 M/I2 M . By flatness of M/I2 M over R/I2 the map J 0 ⊗R/I2 M/I2 M → M/I2 M is injective, which clearly implies that (J ∩I1 )⊗R M → M is injective.  12.14. Flattening over an Artinian ring A universal flattening exists when the base ring is an Artinian local ring. It exists for an arbitrary module. Hence, as we will see later, a flatting stratification exists when the base scheme is the spectrum of an Artinian local ring. Lemma 12.14.1. Let R be an Artinian ring. Let M be an R-module. Then there exists a smallest ideal I ⊂ R such that M/IM is flat over R/I. Proof. This follows directly from Lemma 12.13.1 and the Artinian property.



This ideal has the following universal property. Lemma 12.14.2. Let R be an Artinian ring. Let M be an R-module. Let I ⊂ R be the smallest ideal I ⊂ R such that M/IM is flat over R/I. Then I has the following universal property: For every ring map ϕ : R → R0 we have R0 ⊗R M is flat over R0 ⇔ we have ϕ(I) = 0. Proof. Note that I exists by Lemma 12.14.1. The implication ⇒ follows from Algebra, Lemma 7.36.6. Let ϕ : R → R0 be such that M ⊗R R0 is flat over R0 . Let J = Ker(ϕ). By Algebra, Lemma 7.94.7 and as R0 ⊗R M = R0 ⊗R/J M/JM is flat over R0 we conclude that M/JM is flat over R/J. Hence I ⊂ J as desired.  12.15. Flattening over a closed subset of the base Let R → S be a ring map. Let I ⊂ R be an ideal. Let M be an S-module. In the following we will consider the following condition (12.15.0.1)

∀q ∈ V (IS) ⊂ Spec(S) : Mq is flat over R.

Geometrically, this means that M is flat over R along the inverse image of V (I) in Spec(S). If R and S are Noetherian rings and M is a finite S-module, then (12.15.0.1) is equivalent to the condition that M/I n M is flat over R/I n for all n ≥ 1, see Algebra, Lemma 7.92.10. Lemma 12.15.1. Let R → S be a ring map. Let I ⊂ R be an ideal. Let M be an S-module. Let R → R0 be a ring map and IR0 ⊂ I 0 ⊂ R0 an ideal. If (12.15.0.1) holds for (R → S, I, M ), then (12.15.0.1) holds for (R0 → S ⊗R R0 , I 0 , M ⊗R R0 ). Proof. Assume (12.15.0.1) holds for (R → S, I ⊂ R, M ). Let I 0 (S ⊗R R0 ) ⊂ q0 be a prime of S ⊗R R0 . Let q ⊂ S be the corresponding prime of S. Then IS ⊂ q. Note that (M ⊗R R0 )q0 is a localization of the base change Mq ⊗R R0 . Hence (M ⊗R R0 )q0 is flat over R0 as a localization of a flat module, see Algebra, Lemmas 7.36.6 and 7.36.19.  Lemma 12.15.2. Let R → S be a ring map. Let I ⊂ R be an ideal. Let M be an S-module. Let R → R0 be a ring map and IR0 ⊂ I 0 ⊂ R0 an ideal such that (1) the map V (I 0 ) → V (I) induced by Spec(R0 ) → Spec(R) is surjective, and (2) Rp0 0 is flat over R for all primes p0 ∈ V (I 0 ). If (12.15.0.1) holds for (R0 → S ⊗R R0 , I 0 , M ⊗R R0 ), then (12.15.0.1) holds for (R → S, I, M ).

12.16. FLATTENING OVER A CLOSED SUBSETS OF SOURCE AND BASE

865

Proof. Assume (12.15.0.1) holds for (R0 → S ⊗R R0 , IR0 , M ⊗R R0 ). Pick a prime IS ⊂ q ⊂ S. Let I ⊂ p ⊂ R be the corresponding prime of R. By assumption there exists a prime p0 ∈ V (I 0 ) of R0 lying over p and Rp → Rp0 0 is flat. Choose a prime q0 ⊂ κ(q) ⊗κ(p) κ(p0 ) which corresponds to a prime q0 ⊂ S ⊗R R0 which lies over q and over p0 . Note that (S ⊗R R0 )q0 is a localization of Sq ⊗Rp Rp0 0 . By assumption the module (M ⊗R R0 )q0 is flat over Rp0 0 . Hence Algebra, Lemma 7.93.1 implies that Mq is flat over Rp which is what we wanted to prove.  Lemma 12.15.3. Let R → S be a ring map of finite presentation. Let M be an S-module of finite presentation. Let R0 = colimλ∈Λ Rλ be a directed colimit of R-algebras. Let Iλ ⊂ Rλ be ideals such that Iλ Rµ ⊂ Iµ for all µ ≥ λ and set I 0 = colimλ Iλ . If (12.15.0.1) holds for (R0 → S ⊗R R0 , I 0 , M ⊗R R0 ), then there exists a λ ∈ Λ such that (12.15.0.1) holds for (Rλ → S ⊗R Rλ , Iλ , M ⊗R Rλ ). Proof. We are going to write Sλ = S ⊗R Rλ , S 0 = S ⊗R R0 , Mλ = M ⊗R Rλ , and M 0 = M ⊗R R0 . The base change S 0 is of finite presentation over R0 and M 0 is of finite presentation over S 0 and similarly for the versions with subscript λ, see Algebra, Lemma 7.13.2. By Algebra, Theorem 7.121.4 the set U 0 = {q0 ∈ Spec(S 0 ) | Mq0 0 is flat over R0 } is open in Spec(S 0 ). Note that V (I 0 S 0 ) is a quasi-compact space which is contained in U 0 by assumption. Hence there exist finitely many gj0 ∈ S 0 , j = 1, . . . , m such S 0 0 0 0 0 that D(gj ) ⊂ U and such that V (I S ) ⊂ D(gj ). Note that in particular (M 0 )gj0 is a flat module over R0 . We are going to pick increasingly large elements λ ∈ Λ. First we pick it largeSenough 0 0 so that we can find gj,λ ∈ Sλ mapping to gj0 . The inclusion V (IP S) ⊂ P D(gj0 ) 0 0 0 0 0 zs hs + fj gj0 means that I S + (g1 , . . . , gm ) = S which can be expressed as 1 = for some zs ∈ I 0 , hs , fj ∈ S 0 . After increasing λ weSmay assume such an equation holds in Sλ . Hence we may assume that V (Iλ Sλ ) ⊂ D(gj,λ ). By Algebra, Lemma 7.151.1 we see that for some sufficiently large λ the modules (Mλ )gj,λ are flat over Rλ . In particular the module Mλ is flat over Rλ at all the primes lying over the ideal Iλ .  12.16. Flattening over a closed subsets of source and base In this section we slightly generalize the discussion in Section 12.15. We strongly suggest the reader first read and understand that section. Situation 12.16.1. Let R → S be a ring map. Let J ⊂ S be an ideal. Let M be an S-module. In this situation, given an R-algebra R0 and an ideal I 0 ⊂ R0 we set S 0 = S ⊗R R0 and M 0 = M ⊗R R0 . We will consider the condition (12.16.1.1)

∀q0 ∈ V (I 0 S 0 + JS 0 ) ⊂ Spec(S 0 ) : Mq0 0 is flat over R0 .

Geometrically, this means that M 0 is flat over R0 along the intersection of the inverse image of V (I 0 ) with the inverse image of V (J). Since (R → S, J, M ) are fixed, condition (12.16.1.1) only depends on the pair (R0 , I 0 ) where R0 is viewed as an R-algebra.

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Lemma 12.16.2. In Situation 12.16.1 let R0 → R00 be an R-algebra map. Let I 0 ⊂ R0 and I 0 R00 ⊂ I 00 ⊂ R00 be ideals. If (12.16.1.1) holds for (R0 , I 0 ), then (12.16.1.1) holds for (R00 , I 00 ). Proof. Assume (12.16.1.1) holds for (R0 , I 0 ). Let I 00 S 00 + JS 00 ⊂ q00 be a prime of S 00 . Let q0 ⊂ S 0 be the corresponding prime of S 0 . Then both I 0 S 0 ⊂ q0 and JS 0 ⊂ q0 because the corresponding conditions hold for q00 . Note that (M 00 )q00 is a localization of the base change Mq0 0 ⊗R R00 . Hence (M 00 )q00 is flat over R00 as a localization of a flat module, see Algebra, Lemmas 7.36.6 and 7.36.19.  Lemma 12.16.3. In Situation 12.16.1 let R0 → R00 be an R-algebra map. Let I 0 ⊂ R0 and I 0 R00 ⊂ I 00 ⊂ R00 be ideals. Assume (1) the map V (I 00 ) → V (I 0 ) induced by Spec(R00 ) → Spec(R0 ) is surjective, and (2) Rp0000 is flat over R0 for all primes p00 ∈ V (I 00 ). If (12.16.1.1) holds for (R00 , I 00 ), then (12.16.1.1) holds for (R0 , I 0 ). Proof. Assume (12.16.1.1) holds for (R00 , I 00 ). Pick a prime I 0 S 0 + JS 0 ⊂ q0 ⊂ S 0 . Let I 0 ⊂ p0 ⊂ R0 be the corresponding prime of R0 . By assumption there exists a prime p00 ∈ V (I 00 ) of R00 lying over p0 and Rp0 0 → Rp0000 is flat. Choose a prime q00 ⊂ κ(q0 )⊗κ(p0 ) κ(p00 ). This corresponds to a prime q00 ⊂ S 00 = S 0 ⊗R0 R00 which lies over q0 and over p00 . In particular we see that I 00 S 00 ⊂ q00 and that JS 00 ⊂ q00 . Note that (S 0 ⊗R0 R00 )q00 is a localization of Sq0 0 ⊗Rp0 0 Rp0000 . By assumption the module (M 0 ⊗R0 R00 )q00 is flat over Rp0000 . Hence Algebra, Lemma 7.93.1 implies that Mq0 0 is flat over Rp0 0 which is what we wanted to prove.  Lemma 12.16.4. In Situation 12.16.1 assume R → S is essentially of finite presentation and M is an S-module of finite presentation. Let R0 = colimλ∈Λ Rλ be a directed colimit of R-algebras. Let Iλ ⊂ Rλ be ideals such that Iλ Rµ ⊂ Iµ for all µ ≥ λ and set I 0 = colimλ Iλ . If (12.16.1.1) holds for (R0 , I 0 ), then there exists a λ ∈ Λ such that (12.16.1.1) holds for (Rλ , Iλ ). Proof. We first prove the lemma in case R → S is of finite presentation and then we explain what needs to be changed in the general case. We are going to write Sλ = S ⊗R Rλ , S 0 = S ⊗R R0 , Mλ = M ⊗R Rλ , and M 0 = M ⊗R R0 . The base change S 0 is of finite presentation over R0 and M 0 is of finite presentation over S 0 and similarly for the versions with subscript λ, see Algebra, Lemma 7.13.2. By Algebra, Theorem 7.121.4 the set U 0 = {q0 ∈ Spec(S 0 ) | Mq0 0 is flat over R0 } is open in Spec(S 0 ). Note that V (I 0 S 0 + JS 0 ) is a quasi-compact space which is contained in U 0 by assumption. Hence there exist finitely many gj0 ∈ S 0 , j = S 0 0 0 0 0 1, . . . , m such that D(gj ) ⊂ U and such that V (I S + JS ) ⊂ D(gj0 ). Note that in particular (M 0 )gj0 is a flat module over R0 . We are going to pick increasingly large elements λ ∈ Λ. First we pick it largeSenough so that we can find gj,λ ∈ Sλ mapping to gj0 . The inclusion V (I 0 S 0 +JS 0 ) ⊂ D(gj0 ) 0 means that I 0 S 0 + JS 0 + (g10 , . . . , gm ) = S 0 which can be expressed as X X X 1= yt kt + zs hs + fj gj0

12.17. FLATTENING OVER A NOETHERIAN COMPLETE LOCAL RING

867

for some zs ∈ I 0 , yt ∈ J, kt , hs , fj ∈ S 0 . After increasing λ we may assume such S an equation holds in Sλ . Hence we may assume that V (Iλ Sλ + JSλ ) ⊂ D(gj,λ ). By Algebra, Lemma 7.151.1 we see that for some sufficiently large λ the modules (Mλ )gj,λ are flat over Rλ . In particular the module Mλ is flat over Rλ at all the primes corresponding to points of V (Iλ Sλ + JSλ ). In the case that S is essentially of finite presentation, we can write S = Σ−1 C where R → C is of finite presentation and Σ ⊂ C is a multiplicative subset. We can also write M = Σ−1 N for some finitely presented C-module N , see Algebra, Lemma 7.118.3. At this point we introduce Cλ , C 0 , Nλ , N 0 . Then in the discussion above we obtain an open U 0 ⊂ Spec(C 0 ) over which N 0 is flat over R0 . The assumption that (12.16.1.1) is true means that V (I 0 S 0 + JS 0 ) maps into U 0 , because for a prime q0 ⊂ S 0 , corresponding to a prime r0 ⊂ C 0 we have Mq0 0 = Nr00 . Thus we can find S gj0 ∈ C 0 such that D(gj0 ) contains the image of V (I 0 S 0 + JS 0 ). The rest of the proof is exactly the same as before.  12.16.5. In Situation 12.16.1. Let I ⊂ R be an ideal. Assume R is a Noetherian ring, S is a Noetherian ring, M is a finite S-module, and for each n ≥ 1 and any prime q ∈ V (J + IS) the module (M/I n M )q is flat over R/I n . Then (12.16.1.1) holds for (R, I), i.e., for every prime q ∈ V (J +IS) the localization Mq is flat over R. Lemma (1) (2) (3) (4)

Proof. Let q ∈ V (J + IS). Then Algebra, Lemma 7.92.10 applied to R → Sq and Mq implies that Mq is flat over R.  12.17. Flattening over a Noetherian complete local ring The following three lemmas give a completely algebraic proof of the existence of the “local” flattening stratification when the base is a complete local Noetherian ring R and the given module is finite over a finite type R-algebra S. Lemma 12.17.1. Let R → S be a ring map. Let M be an S-module. Assume (1) (R, m) is a complete local Noetherian ring, (2) S is a Noetherian ring, and (3) M is finite over S. Then there exists an ideal I ⊂ m such that (1) (M/IM )q is flat over R/I for all primes q of S/IS lying over m, and (2) if J ⊂ R is an ideal such that (M/JM )q is flat over R/J for all primes q lying over m, then I ⊂ J. In other words, I is the smallest ideal of R such that (12.15.0.1) holds for (R → S, m, M ) where R = R/I, S = S/IS, m = m/I and M = M/IM . Proof. Let J ⊂ R be an ideal. Apply Algebra, Lemma 7.92.10 to the module M/JM over the ring R/J. Then we see that (M/JM )q is flat over R/J for all primes q of S/JS if and only if M/(J + mn )M is flat over R/(J + mn ) for all n ≥ 1. We will use this remark below. For every n ≥ 1 the local ring R/mn is Artinian. Hence, by Lemma 12.14.1 there exists a smallest ideal In ⊃ mn such that M/In M is flat over R/In . It is clear that

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In+1 + mn is contains In and applying Lemma 12.13.1 we see that In = In+1 + mn . Since R = limn R/mn we see that I = limn In /mn is an ideal in R such that In = I + mn for all n ≥ 1. By the initial remarks of the proof we see that I verifies (1) and (2). Some details omitted.  Lemma 12.17.2. With notation R → S, M , and I and assumptions as in Lemma 12.17.1. Consider a local homomorphism of local rings ϕ : (R, m) → (R0 , m0 ) such that R0 is Noetherian. Then the following are equivalent (1) condition (12.15.0.1) holds for (R0 → S ⊗R R0 , m0 , M ⊗R R0 ), and (2) ϕ(I) = 0. Proof. The implication (2) ⇒ (1) follows from Lemma 12.15.1. Let ϕ : R → R0 be as in the lemma satisfying (1). We have to show that ϕ(I) = 0. This is equivalent to the condition that ϕ(I)R0 = 0. By Artin-Rees in the Noetherian local ring R0 (see Algebra, Lemma 7.48.6) this is equivalent to the condition that ϕ(I)R0 + (m0 )n = (m0 )n for all n > 0. Hence this is equivalent to the condition that the composition ϕn : R → R0 → R0 /(m0 )n annihilates I for each n. Now assumption (1) for ϕ implies assumption (1) for ϕn by Lemma 12.15.1. This reduces us to the case where R0 is Artinian local. Assume R0 Artinian. Let J = Ker(ϕ). We have to show that I ⊂ J. By the construction of I in Lemma 12.17.1 it suffices to show that (M/JM )q is flat over R/J for every prime q of S/JS lying over m. As R0 is Artinian, condition (1) signifies that M ⊗R R0 is flat over R0 . As R0 is Artinian and R/J → R0 is a local injective ring map, it follows that R/J is Artinian too. Hence the flatness of M ⊗R R0 = M/JM ⊗R/J R0 over R0 implies that M/JM is flat over R/J by Algebra, Lemma 7.94.7. This concludes the proof.  Lemma 12.17.3. With notation R → S, M , and I and assumptions as in Lemma 12.17.1. In addition assume that R → S is of finite type. Then for any local homomorphism of local rings ϕ : (R, m) → (R0 , m0 ) the following are equivalent (1) condition (12.15.0.1) holds for (R0 → S ⊗R R0 , m0 , M ⊗R R0 ), and (2) ϕ(I) = 0. Proof. The implication (2) ⇒ (1) follows from Lemma 12.15.1. Let ϕ : R → R0 be as in the lemma satisfying (1). As R is Noetherian we see that R → S is of finite presentation and M is an S-module of finite presentation. Write R0 = colimλ Rλ as a directed colimit of local R-subalgebras Rλ ⊂ R0 , with maximal ideals mλ = Rλ ∩m0 such that each Rλ is essentially of finite type over R. By Lemma 12.15.3 we see that condition (12.15.0.1) holds for (Rλ → S ⊗R Rλ , mλ , M ⊗R Rλ ) for some λ. Hence Lemma 12.17.2 applies to the ring map R → Rλ and we see that I maps to zero in Rλ , a fortiori it maps to zero in R0 .  12.18. Descent flatness along integral maps First a few simple lemmas. Lemma 12.18.1. Let R be a ring. Let P (T ) be a monic polynomial with coefficients in R. If there exists an α ∈ R such that P (α) = 0, then P (T ) = (T − α)Q(T ) for some monic polynomial Q(T ) ∈ R[T ].

12.18. DESCENT FLATNESS ALONG INTEGRAL MAPS

869

Proof. By induction on the degree of P . If deg(P ) = 1, then P (T ) = T − α and the result is true. If deg(P ) > 1, then we can write P (T ) = (T − α)Q(T ) + r for some polynomial Q ∈ R[T ] of degree < deg(P ) and some r ∈ R by long division. By assumption 0 = P (α) = (α − α)Q(α) + r = r and we conclude that r = 0 as desired.  Lemma 12.18.2. Let R be a ring. Let P (T ) be a monic polynomial with coefficients in R. There exists a finite free ring map R → R0 such that P (T ) = (T − α)Q(T ) for some α ∈ R0 and some monic polynomial Q(T ) ∈ R0 [T ]. Proof. Write P (T ) = T d +a1 T d−1 +. . .+a0 . Set R0 = R[x]/(xd +a1 xd−1 +. . .+a0 ). Set α equal to the congruence class of x. Then it is clear that P (α) = 0. Thus we win by Lemma 12.18.1.  Lemma 12.18.3. Let R → S be a finite ring map. There exists a finite free ring extension R ⊂ R0 such that S ⊗R R0 is a quotient of a ring of the form R0 [T1 , . . . , Tn ]/(P1 (T1 ), . . . , Pn (Tn )) with Pi (T ) =

Q

j=1,...,di (T

− αij ) for some αij ∈ R0 .

Proof. Let x1 , . . . , xn ∈ S be generators of S over R. For each i we can choose a monic polynomial Pi (T ) ∈ R[T ] such that P (xi ) = 0 P in S, see Algebra, Lemma 7.33.3. Say deg(Pi ) = di . By Lemma 12.18.2 (applied di times) there exists a finite free ring extension R ⊂ R0 such that each Pi splits completely: Y Pi (T ) = (T − αij ) j=1,...,di

for certain αik ∈ R0 . Let R0 [T1 , . . . , Tn ] → S ⊗R R0 be the R0 -algebra map which maps Ti to xi ⊗ 1. As this maps Pi (Ti ) to zero, this induces the desired surjection.  Lemma 12.18.4. Let R be a ring. Let S = QR[T1 , . . . , Tn ]/J. Assume J contains elements of the form Pi (Ti ) with Pi (T ) = j=1,...,di (T − αij ) for some αij ∈ R. For k = (k1 , . . . , kn ) with 1 ≤ ki ≤ di consider the ring map Φk : R[T1 , . . . , Tn ] → R,

Ti 7−→ αiki

T Set Jk = Φk (J). Then the image of Spec(S) → Spec(R) is equal to V ( Jk ). S T Proof. This lemma proves itself. Hint: V ( Jk ) = V (Jk ).



The following result is due to Ferrand, see [Fer69]. Lemma 12.18.5. Let R → S be a finite injective homomorphism of Noetherian rings. Let M be an R-module. If M ⊗R S is a flat S-module, then M is a flat R-module. Proof. Let M be an R-module such that M ⊗R S is flat over S. By Algebra, Lemma 7.36.7 in order to prove that M is flat we may replace R by any faithfully flat ring extension. By Lemma 12.18.3 we can find a finite locally free ring extension 0 R ⊂ R0 such that S 0 = S ⊗R R0 = R0 [T1 , . . . , Tn ]/J for some ideal Q J ⊂ R [T1 , . . . , Tn ] which contains the elements of the form Pi (Ti ) with Pi (T ) = j=1,...,di (T − αij ) for some αij ∈ R0 . Note that R0 is Noetherian and that R0 ⊂ S 0 is a finite extension of rings. Hence we may replace R by R0 and assume that S has a presentation as in Lemma 12.18.4. Note that Spec(S) → Spec(R) is surjective, see Algebra, Lemma

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T 7.33.15. Thus, using Lemma 12.18.4 we conclude that I = Jk is an ideal such p that V (I) = Spec(R). This means that I ⊂ (0), and since R is Noetherian that I is nilpotent. The maps Φk induce commutative diagrams / R/Jk =

S^

R from which we conclude that M/Jk M is flat over R/Jk . By Lemma 12.13.1 we see that M/IM is flat over R/I. Finally, applying Algebra, Lemma 7.94.5 we conclude that M is flat over R.  Lemma 12.18.6. Let R → S be an injective integral ring map. Let M be a finitely presented module over R[x1 , . . . , xn ]. If M ⊗R S is flat over S, then M is flat over R. Proof. Choose a presentation R[x1 , . . . , xn ]⊕t → R[x1 , . . . , xn ]⊕r → M → 0. Let’s say that the first map by the r × t-matrix T = (fij ) with fij ∈ Pis given R[x1 , . . . , xn ]. Write fij = fij,I xI with fij,I ∈ R (multi-index notation). Consider diagrams /S RO O Rλ

/ Sλ

where Rλ is a finitely generated Z-subalgebra of R containing all fij,I and Sλ is a finite Rλ -subalgebra of S. Let Mλ be the finite Rλ [x1 , . . . , xn ]-module defined by a presentation as above, using the same matrix T but now viewed as a matrix over Rλ [x1 , . . . , xn ]. Note that S is the directed colimit of the Sλ (details omitted). By Algebra, Lemma 7.151.1 we see that for some λ the module Mλ ⊗Rλ Sλ is flat over Sλ . By Lemma 12.18.5 we conclude that Mλ is flat over Rλ . Since M = Mλ ⊗Rλ R we win by Algebra, Lemma 7.36.6.  12.19. Torsion and flatness In this section we discuss the relationship between torsion and flatness. Definition 12.19.1. Let R be a domain. Let M be an R-module. (1) We say an element x ∈ M is torsion if there exists a nonzero f ∈ R such that f x = 0. (2) We say M is torsion free if the only torsion element of M is 0. Lemma 12.19.2. Let R be a domain. Let M be an R-module. The set of torsion elements of M forms a submodule Mtors ⊂ M . The quotient module M/Mtors is torsion free. Proof. Omitted. Lemma 12.19.3. Let R be a domain. Any flat R-module is torsion free.



12.20. FLATNESS AND FINITENESS CONDITIONS

871

Proof. If x ∈ R is nonzero, then x : R → R is injective, and hence if M is flat over R, then x : M → M is injective. Thus if M is flat over R, then M is torsion free.  Lemma 12.19.4. Let A be a valuation ring. An A-module M is flat over A if and only if M is torsion free. Proof. The implication “flat ⇒ torsion free” is Lemma 12.19.3. For the converse, assume M is torsion free. By the equational criterion of flatness (see Algebra, Lemma 7.36.10) P we have to show that every relation in M is trivial. To do this assume that i=1,...,n ai xi = 0 with xi ∈ M and fi ∈ A. After renumbering we may assume that v(a1 ) ≤ v(ai ) for all i. Hence we can write ai P = a0i a1 for some 0 0 ai ∈ A. Note that a1 = 1. As A is torsion free we see that x1 = − i≥2 a0i xi . Thus, if we choose yi = xi , i = 2, . . . , n then X x1 = −a0j yj , xi = yi , (i ≥ 2) 0 = a1 · (−a0j ) + aj · 1(j ≥ 2) j≥2

shows that the relation was trivial (to be explicit the elements aij are defined by setting a1j = −a0j and aij = δij for i, j ≥ 2).  12.20. Flatness and finiteness conditions In this section we discuss some implications of the type “flat + finite type ⇒ finite presentation”. We will revisit this result in the chapter on flatness, see More on Flatness, Section 34.1. A first result of this type was proved in Algebra, Lemma 7.101.6. Lemma 12.20.1. Let R be a ring. Let S = R[x1 , . . . , xn ] be a polynomial ring over R. Let M be an S-module. Assume (1) there exist finitely many primes p1 , . . . , pm of R such that the map R → Q Rpj is injective, (2) M is a finite S-module, (3) M flat over R, and (4) for every prime p of R the module Mp is of finite presentation over Sp . Then M is of finite presentation over S. Proof. Choose a presentation 0 → K → S ⊕r → M → 0 of M as an S-module. Let q be a prime ideal of S lying over a prime p of R. By assumption there exist finitely many elements k1 , . . . , kt ∈ K such that if we P set K 0 = Skj ⊂ K then Kp0 = Kp and Kp0 j = Kpj for j = 1, . . . , m. Setting M 0 = S ⊕r /K 0 we deduce that in particular Mq0 = Mq . By openness of flatness, see Algebra, Theorem 7.121.4 we conclude that there exists a g ∈ S, g 6∈ q such that Mg0 is flat over R. Thus Mg0 → Mg is a surjective map of flat R-modules. Consider the commutative diagram / Mg

Mg0 Q  0 (Mg )pj /

Q  (Mg )pj

872

12. MORE ON ALGEBRA

The bottom arrow is an isomorphism by choice of k1 , . . . , kt . The left vertical arrow Q is an injective map as R → Rpj is injective and Mg0 is flat over R. Hence the top horizontal arrow is injective, hence an isomorphism. This proves that Mg is of finite presentation over Sg . We conclude by applying Algebra, Lemma 7.22.2.  Lemma 12.20.2. Let R → S be a ring homomorphism. Assume (1) there exist finitely many primes p1 , . . . , pm of R such that the map R → Q Rpj is injective, (2) R → S is of finite type, (3) S flat over R, and (4) for every prime p of R the ring Sp is of finite presentation over Rp . Then S is of finite presentation over R. Proof. By assumption S is a quotient of a polynomial ring over R. Thus the result follows directly from Lemma 12.20.1.  Lemma 12.20.3. Let R be a ring. Let S = R[x1 , . . . , xn ] be a graded polynomial algebra over R, i.e., deg(xi ) > 0 but not necessarily equal to 1. Let M be a graded S-module. Assume (1) R is a local ring, (2) M is a finite S-module, and (3) M is flat over R. Then M is finitely presented as an S-module. L Proof. Let M = Md be the grading on M . Pick homogeneous generators m1 , . . . , mr ∈ M of M . Say deg(mi ) = di ∈ Z. This gives us a presentation M 0→K→ S(−di ) → M → 0 i=1,...,r

which in each degree d leads to the short exact sequence M 0 → Kd → Sd−di → Md → 0. i=1,...,r

By assumption each Md is a finite flat R-module. By Algebra, Lemma 7.73.4 this implies each Md is a finite free R-module. Hence we see each Kd is a finite R-module. Also each Kd is flat over R by Algebra, Lemma 7.36.12. Hence we conclude that each Kd is finite free by Algebra, Lemma 7.73.4 again. Let m be the maximal ideal of R. By the flatness of M over R the short exact sequences above remain short exact after tensoring with κ = κ(m). As the ring S ⊗R κ is Noetherian we see that there exist homogeneous elements k1 , . . . , kt ∈ K such that the images k j generate K ⊗R κ over S ⊗R κ. Say deg(kj ) = ej . Thus for any d the map M Sd−ej −→ Kd j=1,...,t

becomes surjective after tensoring with κ. By Nakayama’s lemma (Algebra, Lemma 7.18.1) this implies the map is surjective over R. Hence K is generated by k1 , . . . , kt over S and we win.  L Lemma 12.20.4. Let R be a ring. Let S = n≥0 Sn be a graded R-algebra. L Let M = M be a graded S-module. Assume S is finitely generated as an d d∈Z R-algebra, assume S0 is a finite R-algebra, and assume there exist finitely many Q primes pj , i = 1, . . . , m such that R → Rpj is injective.

12.20. FLATNESS AND FINITENESS CONDITIONS

873

(1) If S is flat over R, then S is a finitely presented R-algebra. (2) If M is flat as an R-module and finite as an S-module, then M is finitely presented as an S-module. Proof. As S is finitely generated as an R-algebra, it is finitely generated as an S0 algebra, say by homogeneous elements t1 , . . . , tn ∈ S of degrees d1 , . . . , dn > 0. Set P = R[x1 , . . . , xn ] with deg(xi ) = di . The ring map P → S, xi → ti is finite as S0 is a finite R-module. To prove (1) it suffices to prove that S is a finitely presented P -module. To prove (2) it suffices to prove that M is a finitely presented P -module. Thus it suffices to prove that if S = P is a graded polynomial ring and M is a finite S-module flat over R, then M is finitely presented as an S-module. By Lemma 12.20.3 we see Mp is a finitely presented Sp -module for every prime p of R. Thus the result follows from Lemma 12.20.1.  Remark 12.20.5. Let R be a ring. When does R satisfy the condition mentioned in Lemmas 12.20.1, 12.20.2, and 12.20.4? This holds if (1) R is local, (2) R is Noetherian, (3) R is a domain, (4) R is a reduced ring with finitely many minimal primes, or (5) R has finitely many weakly associated primes, see Algebra, Lemma 7.64.16. Thus these lemmas hold in all cases listed above. The following lemma wil be improved below, see Proposition 12.20.8. Lemma 12.20.6. Let A be a valuation ring. Let A → B be a ring map of finite type. Let M be a finite B-module. (1) If B is flat over A, then B is a finitely presented A-algebra. (2) If M is flat as an A-module, then M is finitely presented as a B-module. Proof. We are going to use that an A-module is flat if and only if it is torsion free, see Lemma 12.19.4. By Algebra, Lemma 7.54.10 we can find a graded A-algebra S with S0 = A and generated by finitely many elements in degree 1, an element f ∈ S1 and a finite graded S-module N such that B ∼ = S(f ) and M ∼ = N(f ) . If M is torsion free, then we can take N torsion free by replacing it by N/Ntors , see Lemma 12.19.2. Similarly, if B is torsion free, then we can take S torsion free by replacing it by S/Stors . Hence in case (1), we may apply Lemma 12.20.4 to see that S is a finitely presented A-algebra, which implies that B = S(f ) is a finitely presented A-algebra. To see (2) we may first replace S by a graded polynomial ring, and then we may apply Lemma 12.20.3 to conclude.  Lemma 12.20.7. Let R be a domain with fraction field K. Let S = R[x1 , . . . , xn ] be a polynomial ring over R. Let M be a finite S-module. Assume that M is flat over R. If for every subring R ⊂ R0 ⊂ K, R 6= R0 the module M ⊗R R0 is finitely presented over S ⊗R R0 , then M is finitely presented over S. Proof. Suppose that f1 , . . . , fn ∈ R are elements which generate the unit ideal. If R 6= Rfi for each i = 1, . . . , n, then we conclude that Mfi is finitely presented over Sfi for each i, and hence M is finitely presented over S by Algebra, Lemma 7.22.2. Thus we are done if such a sequence of elements exists. Assume this is not the case. In particular, for every x ∈ R we have either x ∈ R∗ , or 1 − x ∈ R∗ . This implies that R is local, see Algebra, Lemma 7.17.2.

874

12. MORE ON ALGEBRA

Choose a presentation 0 → K → R[x1 , . . . , xn ]⊕r → M → 0. Throughout the rest of the proof we will use that this sequence stays exact after tensoring with any R-algebra, see Algebra, Lemma 7.36.11. Let R0 be the integral closure of R in its fraction field. If R 6= R0 , then we see that M ⊗R R0 is finitely presented over R0 [x1 , . . . , xn ]. In particular, the module K ⊗R R0 is finitely generated. Thus we kt ⊗ 1 generate K ⊗R R0 . Set P may pick k1 , . . . , kt ∈ K such0 that k1 ⊗ 1, . . . ,⊕t 0 K = R[x1 , . . . , xn ]ki ⊂ K. Set M = R[x1 , . . . , xn ] /K 0 . Then M 0 is a finitely presented module over R[x1 , . . . , xn ] such that M 0 ⊗R R0 ∼ = M ⊗R R0 is flat over 0 0 R . By Lemma 12.18.6 we conclude that M is flat over R. Hence the surjective map M 0 → M is also injective as M 0 is torsion free, see Lemma 12.19.3. In other words, M 0 ∼ = M and we conclude that M is finitely presented. Thus we are done if R is not a normal domain. Assume this is not the case. This reduces us to the case where R is a normal local domain. Pick any pair of nonzero elements x, y ∈ R. Consider the inclusions R ⊂ R[x/y] and R[y/x]. As R is a normal domain we get a short exact sequence (−1,1)

(1,1)

0 → R −−−−→ R[x/y] ⊕ R[y/x] −−−→ R[x/y, y/x] → 0 see Algebra, Lemma 7.33.21. If R 6= R[x/y] and R 6= R[y/x] then we see that K ⊗R R[x/y] and K ⊗R R[y/x] are finitely generated as R[x/y][x1 , . . . , xn ] and R[y/x][x1 , . . . , xn ] modules. Thus we can find k1 , . . . , kt ∈ K such that the elements ki ⊗ 1 generate K ⊗R R[x/y] and P K ⊗R R[y/x] as R[x/y][x1 , . . . , xn ] and R[y/x][x1 , . . . , xn ] modules. Set K 0 = R[x1 , . . . , xn ]ki ⊂ K. Tensoring the sequence above with K 0 ⊂ K we get the diagram

0

K0

/ K 0 ⊗R R[x/y] ⊕ K 0 ⊗R R[y/x]

/ K 0 ⊗R R[x/y, y/x]

/0

 /K

 / K ⊗R R[x/y] ⊕ K ⊗R R[y/x]

 / K ⊗R R[x/y, y/x]

/0

Now we know that the vertical arrows in the middle and on the right are isomorphisms. The lower row is exact as K is flat over R. Hence the left vertical arrow is surjective, i.e., an isomorphism. Thus we win if there exists a pair of nonzero elements such that neither x/y nor y/x is an element of R. Assume this is not the case. Then we know that R ⊂ f.f.(R) is a normal local domain such that for every x ∈ f.f.(R) either x ∈ R, or x−1 ∈ R. In other words, R is a valuation ring, see Algebra, Lemma 7.47.4. In this case M is finitely presented by Lemma 12.20.6 and we win.  The following result is a special case of results in [GR71] which we discuss in great detail in More on Flatness, Section 34.1. Proposition 12.20.8. Let R be a domain. Let R → S be a ring map of finite type. Let M be a finite S-module. (1) If S is flat over R, then S is a finitely presented R-algebra. (2) If M is flat as an R-module, then M is finitely presented as a S-module. Proof. It suffices to prove part (2) in case S = R[x1 , . . . , xn ]. Choose a presentation 0 → K → R[x1 , . . . , xn ]⊕r → M → 0.

12.21. BLOWING UP AND FLATNESS

875

Throughout the rest of the proof we will use that this sequence stays exact after tensoring with any R-algebra, see Algebra, Lemma 7.36.11. Let L be the fraction field of R. Consider the set R = {R0 | R ⊂ R0 ⊂ L and M ⊗R R0 not of finite presentation over S ⊗R R0 } We order RSby inclusion. Suppose that {Ri }i∈I is a totally ordered subset of R. Set R∞ = i∈I Ri . We claim that R∞ ∈ R. Namely, if M ⊗R R∞ is finitely presented over S ⊗R R∞ , then K ⊗R R∞ is finitely generated, say by k1 , . . . , kt . Then for some i P ∈ I we have k1 , . . . , kt ∈ K ⊗R Ri . For any i0 ≥ i set Mi0 = ⊕r Ri [x1 , . . . , xn ] / Ri [x1 , . . . , xn ]ki . By Algebra, Lemma 7.151.1 we see that Mi0 is flat over Ri for some sufficiently large i0 ∈ I. For such an i0 the surjective map Mi0 → M ⊗R Ri is also injective as Mi0 is torsion free. Hence we conclude that M ⊗R Ri is finitely presented which is a contradiction. In other words R∞ ∈ R. This shows that Zorn’s lemma applies to R if R is not empty. But Lemma 12.20.7 shows that R does not have any maximal elements and the proposition is proved.  12.21. Blowing up and flatness In this section we begin our discussion of results of the form: “After a blow up the strict transform becomes flat”. More results of this type may be found in More on Flatness, Section 34.27. Definition 12.21.1. Let R be a domain. Let M be an R-module. Let R ⊂ R0 be an extension of domains. The strict transform of M along R → R01 is the torsion free R0 -module M 0 = (M ⊗R R0 )/(M ⊗R R0 )tors . The following is a very weak version of flattening by blowing up, but it is already sometimes a useful result. Lemma 12.21.2. Let (R, m) be a local domain with fraction field K. Let S be a finite type R-algebra. Let M be a finite S-module. For every valuation ring A ⊂ K dominating R there exists an ideal I ⊂ m and a nonzero element a ∈ I such that (1) I is finitely generated, (2) A has center on R[ aI ], (3) the fibre ring of R → R[ aI ] at m is not zero, and (4) the strict transform SI,a of S along R → R[ aI ] is flat and of finite presentation over R, and the strict transform MI,a of M along R → R[ aI ] is flat over R and finitely presented over SI,a . Proof. Note that the assertion makes sense as R[ aI ] is a domain, and R → R[ aI ] is injective, see Algebra, Lemmas 7.55.4 and 7.55.6. Before we start the proof of the Lemma, note that there is no loss in generality assuming that S = R[x1 , . . . , xn ] is a polynomial ring over R. We also fix a presentation 0 → K → S ⊕r → M → 0. Let MA be the strict transform of M along R → A. It is a finite module over SA = A[x1 , . . . , xn ]. By Lemma 12.19.4 we see that MA is flat over A. By Lemma 12.20.6 we see that MA is finitely presented. Hence there exist finitely many elements ⊕r ⊕r k1 , . . . , kt ∈ SA which generate the kernel of the presentation SA → MA as an 1This is somewhat nonstandard notation.

876

12. MORE ON ALGEBRA

SA -module. For any choice of a ∈ I ⊂ m satisfying (1), (2), and (3) we denote MI,a the strict transform of M along R → R[ aI ]. It is a finite module over SI,a = R[ aI ][x1 , . . . , xn ]. By Algebra, Lemma 7.55.7 we have A = colimI,a R[ aI ]. This implies that SA = colim SI,a and MA = colimI,a MI,a . Thus we may choose a ∈ ⊕r I ⊂ R such that k1 , . . . , kt are elements of SI,a and map to zero in MI,a . For any such pair (I, a) we set X ⊕r 0 MI,a = SI,a / SI,a kj . P ⊕r 0 Since MA = SA / SA kj we see that also MA = colimI,a MI,a . At this point we 0 may apply Algebra, Lemma 7.151.1 (3) to conclude that MI,a is flat for some pair (I, a). (This lemma does not apply a priori to the system MI,a as the transition maps may not satisfy the assumptions of the lemma.) Since flatness implies torsion 0 free ( Lemma 12.19.3), we also conclude that MI,a = MI,a for such a pair and we win.  12.22. Completion and flatnes In this section we discuss when the completion of a “big” flat module is flat. Lemma 12.22.1. Let R be a ring. Let I ⊂ R be an ideal. Let A be a set. Assume R is Noetherian and complete with respect to I. There is a canonical map M ∧ Y R −→ R α∈A

α∈A

from the I-adic completion of the direct sum into the product which is universally injective. Proof. By is x = (xn ) where xn = Ldefinitionn an element x of the left hand side n (xn,α ) ∈ R/I such that x = x mod I . As R = R∧ we see that n,α n+1,α α∈A n for any α there exists a yα ∈ R such that xn,α = yα mod I . Note that for each n there are onlyQ finitely many α such that the elements xn,α are nonzero. Conversely, given (yα ) ∈ α R such that for each n there are only finitely many α such that yα mod I n is nonzero, then this defines an element of the left hand side. Hence P we can think of an element of the left hand side as infinite “convergent sums” α yα with yα ∈ R such that for each n there are only finitely many yα which are nonzero modulo I n . The displayed map maps this element to the element to (yα ) in the product. In particular the map is injective. Let Q be a finite R-module. We have to show that the map M ∧ Y  Q ⊗R R −→ Q ⊗R R α∈A

α∈A

is injective, see Algebra, Theorem 7.77.3. Choose a presentation R⊕k → R⊕m → Q → 0 and denote q1 , . . . , qm ∈ Q the corresponding generators for Q. By ArtinRees (Algebra, Lemma 7.48.4) there exists a constant c such that Im(R⊕k → R⊕m ) ∩ (I N )⊕m ⊂ Im((I N −c )⊕k → R⊕m ). Let us contemplate the diagram
∧
∧
∧ Lk L / Q ⊗R L /0 / Lm L j=1 α∈A R α∈A R l=1 α∈A R

Lk

l=1

 Q

α∈A R




/

Lm

j=1

 Q

α∈A R




/ Q ⊗R

Q

α∈A

R




/0

12.23. THE KOSZUL COMPLEX

877


∧ P P L L with exact rows. Pick an element j α yj,α of R . If this j=1,...,m α∈A
Q element maps to zero in the module Q ⊗R α∈A R , then we see in particular that P q ⊗ y = 0 in Q for each α. Thus we can find an element (z1,α , . . . , zk,α ) ∈ j,α Lj j L Nα l=1,...,k R which maps to (y1,α , . . . , ym,α ) ∈ j=1,...,m R. Moreover, if yj,α ∈ I Nα −c for j = 1, . . . , m, then we may assume that zl,α ∈ I for l = 1, . . . , k. Hence
∧ P P L L the sum l α zl,α is “convergent” and defines an element of l=1,...,k α∈A R P P which maps to the element j α yj,α we started out with. Thus the right vertical arrow is injective and we win.  Lemma 12.22.2. Let R be a ring.L Let I ⊂ R be an ideal. Let A be a set. Assume R is Noetherian. The completion ( α∈A R)∧ is a flat R-module. ∧ Proof. Denote R∧ the completion of R with respect L to I. ∧As R → R ∧is flat by Algebra, Lemma 7.91.3 it suffices to prove that ( α∈A R) is a flat R -module (use Algebra, Lemma 7.36.3). Since M M ( R)∧ = ( R∧ )∧ α∈A

α∈A

we may replace R by R∧ and assume that R is complete with respectLto I (see Algebra, Lemma 7.91.8). In this case Lemma 12.22.1 tells us the map ( α∈A R)∧ → Q α∈A QR is universally injective. Thus, by Algebra, Lemma 7.77.7 it suffices to show that α∈A RQis flat. By Algebra, Proposition 7.85.5 (and Algebra, Lemma 7.85.4) we see that α∈A R is flat.  12.23. The Koszul complex We define the Koszul complex as follows. Definition 12.23.1. Let R be a ring. Let ϕ : E → R be an R-module map. The Koszul complex K• (ϕ) associated to ϕ is the commutative differential graded algebra defined as follows: (1) the underlying graded algebra is the exterior algebra K• (ϕ) = ∧(E), (2) the differential d : K• (ϕ) → K• (ϕ) is the unique derivation such that d(e) = ϕ(e) for all e ∈ E = K1 (ϕ). Explicitly, if e1 ∧ . . . ∧ en is one of the generators of degree n in K• (ϕ), then X d(e1 ∧ . . . ∧ en ) = (−1)i+1 ϕ(ei )e1 ∧ . . . ∧ ebi ∧ . . . ∧ en . i=1,...,n

It is straightforward to see that this gives a well defined derivation on the tensor algebra, which annihilates e ⊗ e and hence factors through the exterior algebra. We often assume that E is a finite free module, say E = R⊕n . In this case the map ϕ is given by a sequence of elements f1 , . . . , fn ∈ R. Definition 12.23.2. Let R be a ring and let f1 , . . . , fn ∈ R. The Koszul complex on f1 , . . . , fr is the Koszul complex associated to the map (f1 , . . . , fn ) : R⊕n → R. Notation K• (f• ), K• (f1 , . . . , fn ), K• (R, f1 , . . . , fn ), or K• (R, f• ). Of course, if E is finite locally free, then K• (ϕ) is locally on Spec(R) isomorphic to a Koszul complex K• (f1 , . . . , fn ). This complex has many interesting formal properties.

878

12. MORE ON ALGEBRA

Lemma 12.23.3. Let ϕ : E → R and ϕ : E 0 → R be an R-module maps. Let ψ : E → E 0 be an R-module map such that ϕ0 ◦ψ = ϕ. Then ψ induces a homomorphism of differential graded algebras K• (ϕ) → K• (ϕ0 ). Proof. This is immediate from the definitions.



Lemma 12.23.4. Let f1 , . . . , fc ∈ R be a sequence. Let (xij ) be an invertible c × c-matrix with coefficients in R. Then the complexes K• (f• ) and X X X K• ( x1j fj , x2j fj , . . . , xcj fj ) are isomorphic. P Proof. Set gi = xij fj . The matrix (xij ) gives an isomorphism x : R⊕c → R⊕c such that (g1 , . . . , gc ) ◦ x = (f1 , . . . , fc ). Hence this follows from the functoriality of the Koszul complex described in Lemma 12.23.3.  Lemma 12.23.5. Let R be a ring. Let ϕ : E → R be an R-module map. Let e ∈ E with image f = ϕ(e) in R. Then f = de + ed as endomorphisms of K• (ϕ). Proof. This is true because d(ea) = d(e)a − ed(a) = f a − ed(a).



Lemma 12.23.6. Let R be a ring. Let f1 , . . . , fc ∈ R be a sequence. Multiplication by fi on K• (f• ) is homotopic to zero, and in particular the cohomology modules Hi (K• (f• )) are annihilated by the ideal (f1 , . . . , fr ). Proof. Special case of Lemma 12.23.5.



In Derived Categories, Section 11.8 we defined the cone of a morphism of cochain complexes. The cone C(f )• of a morphism of chain complexes f : A• → B• is the complex C(f )• given by C(f )n = Bn ⊕ An−1 and differential   dB,n fn−1 (12.23.6.1) dC(f ),n = 0 −dA,n−1 It comes equipped with canonical morphisms of complexes i : B• → C(f )• and p : C(f )• → A• [−1] induced by the obvious maps Bn → C(f )n → An−1 . Lemma 12.23.7. Let R be a ring. Let ϕ : E → R be an R-module map. Let f ∈ R. Set E 0 = E ⊕ R and define ϕ0 : E 0 → R by ϕ on E and multiplication by f on R. The complex K• (ϕ0 ) is isomorphic to the cone of the map of complexes f : K• (ϕ) −→ K• (ϕ). Proof. Denote e0 ∈ E 0 the element 1 ∈ R ⊂ R ⊕ E. By our definition of the cone above we see that C(f )n = Kn (ϕ) ⊕ Kn−1 (ϕ) = ∧n (E) ⊕ ∧n−1 (E) = ∧n (E 0 ) where in the last = we map (0, e1 ∧ . . . ∧ en−1 ) to e0 ∧ e1 ∧ . . . ∧ en−1 in ∧n (E 0 ). A computation shows that this isomorphism is compatible with differentials. Namely, this is clear for elements of the first summand as ϕ0 |E = ϕ and dC(f ) restricted to the first summand is just dK• (ϕ) . On the other hand, if e1 ∧ . . . ∧ en−1 is in the first summand, then dC(f ) (0, e1 ∧ . . . ∧ en−1 ) = f e1 ∧ . . . ∧ en−1 − dK• (ϕ) (e1 ∧ . . . ∧ en−1 )

12.23. THE KOSZUL COMPLEX

879

and on the other hand 0 (e0 ∧ e1 ∧ . . . ∧ en−1 ) dKP • (ϕ ) = i=0,...,n−1 (−1)i ϕ0 (ei )e0 ∧ . . . ∧ ebi ∧ . . . ∧ en−1 P = f e1 ∧ . . . ∧ en−1 + i=1,...,n−1 (−1)i ϕ(ei )e0 ∧ . . . ∧ ebi ∧ . . . ∧ en−1 P  i+1 = f e1 ∧ . . . ∧ en−1 − e0 (−1) ϕ(e )e ∧ . . . ∧ e b ∧ . . . ∧ e i 1 i n−1 i=1,...,n−1

which is the image of the result of the previous computation.



Lemma 12.23.8. Let R be a ring. Let f1 , . . . , fn be a sequence of elements of R. The complex K• (f1 , . . . , fn ) is isomorphic to the cone of the map of complexes fn : K• (f1 , . . . , fn−1 ) −→ K• (f1 , . . . , fn−1 ). Proof. Special case of Lemma 12.23.7.



Lemma 12.23.9. Let R be a ring. Let A• be a complex of R-modules. Let f, g ∈ R. Let C(f )• be the cone of f : A• → A• . Define similarly C(g)• and C(f g)• . Then C(f g)• is homotopy equivalent to the cone of a map C(f )• [1] −→ C(g)• Proof. We first prove this if A• is the complex consisting of R placed in degree 0. In this case the map we use is 0

/0

 0

 /R

/R 1

g

 /R

f

/R

/0

 /0

 /0

The cone of this is the chain complex consisting of R ⊕ R placed in degrees 1 and 0 and differential (12.23.6.1)   g 1 : R⊕2 −→ R⊕2 0 −f We leave it to the reader to show this this chain complex is homotopic to the complex f g : R → R. In general we write C(f )• and C(g)• as the total complex of the double complexes f

(R − → R) ⊗R A•

and

g

(R − → R) ⊗R A•

and in this way we deduce the result from the special case discussed above. Some details omitted.  Lemma 12.23.10. Let R be a ring. Let ϕ : E → R be an R-module map. Let f, g ∈ R. Set E 0 = E ⊕ R and define ϕ0f , ϕ0g , ϕ0f g : E 0 → R by ϕ on E and multiplication by f, g, f g on R. The complex K• (ϕ0f g ) is isomorphic to the cone of a map of complexes K• (ϕ0f )[1] −→ K• (ϕ0g ). Proof. By Lemma 12.23.7 the complex K• (ϕ0f ) is isomorphic to the cone of multiplication by f on K• (ϕ) and similarly for the other two cases. Hence the the lemma follows from Lemma 12.23.9. 

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Lemma 12.23.11. Let R be a ring. Let f1 , . . . , fn−1 be a sequence of elements of R. Let f, g ∈ R. The complex K• (f1 , . . . , fn−1 , f g) is homotopy equivalent to the cone of a map of complexes K• (f1 , . . . , fn−1 , f )[1] −→ K• (f1 , . . . , fn−1 , g) Proof. Special case of Lemma 12.23.10.



Lemma 12.23.12. Let A be a ring. Let f1 , . . . , fn , g1 , . . . , gm be elements of A. Then there is an isomorphism of Koszul complexes K• (A, f1 , . . . , fn , g1 , . . . , gm ) = Tot(K• (A, f1 , . . . , fn ) ⊗A K• (A, g1 , . . . , gm )). Proof. We first show that K• (A, f1 , . . . , fn , g1 , . . . , gm ) is isomorphic to the tensor product K• (A, f1 , . . . , fn ) ⊗A K• (A, g1 , . . . , gm ) as a differential graded A-algebra. This is clear as the multiplication map ∧(A⊕n ) ⊗A ∧(A⊕m ) −→ ∧(A⊕n ⊕ A⊕m ) is an isomorphism and the fact that the d of generators agree. Thus the lemma follows from Homology, Lemma 10.25.5.  12.24. Koszul regular sequences Please take a look at Algebra, Sections 7.66 and 7.67 before looking at this one. Definition 12.24.1. Let R be a ring. A sequence of elements f1 , . . . , fc of R is called Koszul-regular if Hi (K• (f1 , . . . , fr )) = 0 for all i 6= 0. A sequence of elements f1 , . . . , fc of R is called H1 -regular if H1 (K• (f1 , . . . , fr )) = 0. Clear a Koszul-regular sequence is H1 -regular. If f = f1 ∈ R is a length 1 sequence then it is clear that the following are all equivalent (1) f is a regular sequence of length one, (2) f is a Koszul-regular sequence of length one, and (3) f is a H1 -regular sequence of length one. It is also clear that these imply that f is a quasi-regular sequence of length one. But there do exist quasi-regular sequences of length 1 which are not regular sequences. Namely, let R = k[x, y0 , y1 , . . .]/(xy0 , xy1 − y0 , xy2 − y1 , . . .) L and let f be the image of x in R. Then f is a zerodivisor, but (f n )/(f n+1 ) ∼ = n≥0

k[x] is a polynomial ring. Lemma 12.24.2. A regular sequence is Koszul-regular. Proof. Let f1 , . . . , fc be a regular sequence. Then f1 is a nonzero divisor in R. Hence f1

0 → K• (R, f2 , . . . , fc ) −→ K• (R, f2 , . . . , fc ) → K• (R/(f1 ), f 2 , . . . , f c ) → 0 is a short exact sequence of complexes. By Lemma 12.23.8 the complex K• (R, f1 , . . . , fc ) is isomorphic to the cone of the first map. Hence K• (R/(f1 ), f 2 , . . . , f c ) is quasiisomorphic to K• (R, f1 , . . . , fc ). As f 2 , . . . , f c is a regular sequence in R/(f1 ) the result follows from the case c = 1 discussed above and induction.  Lemma 12.24.3. Let f1 , . . . , fc−1 ∈ R be a sequence and f, g ∈ R.

12.24. KOSZUL REGULAR SEQUENCES

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(1) If f1 , . . . , fc−1 , f and f1 , . . . , fc−1 , g are H1 -regular then f1 , . . . , fc−1 , f g is an H1 -regular sequence too. (2) If f1 , . . . , fc−1 , f and f1 , . . . , fc−1 , f are Koszul-regular then f1 , . . . , fc−1 , f g is a Koszul-regular sequence too. Proof. By Lemma 12.23.11 we have exact sequences Hi (K• (f1 , . . . , fc−1 , f )) → Hi (K• (f1 , . . . , fc−1 , f g)) → Hi (K• (f1 , . . . , fc−1 , g)) for all i.



Lemma 12.24.4. Let ϕ : R → S be a flat ring map. (1) If f1 , . . . , fr is a H1 -regular sequence in R, then ϕ(f1 ), . . . , ϕ(fr ) is a H1 regular sequence in S. (2) If f1 , . . . , fr is a Koszul-regular sequence in R, then ϕ(f1 ), . . . , ϕ(fr ) is a Koszul-regular sequence in S. Proof. This is true because K• (f1 , . . . , fr ) ⊗R S = K• (ϕ(f1 ), . . . , ϕ(fr )).



Lemma 12.24.5. An H1 -regular sequence is quasi-regular. Proof. Let f1 , . . . , fc be an H1 -regular sequence. Denote J = (f1 , . . . , fc ). The assumption means that we have an exact sequence ∧2 (Rc ) → R⊕c → J → 0 where the first arrow is given by ei ∧ ej 7→ fi ej − fj ei . In particular this implies that J/J 2 = J ⊗R R/J = (R/J)c is a finite free module. To finish the proof we have to prove for every n ≥ 2 the following: if X ξ= aI f1i1 . . . fcic ∈ J n+1 |I|=n,I=(i1 ,...,ic )

then aI ∈ J for all I. Note that f1 , . . . , fc−1 , fcn is a H1 -regular sequence by Lemma 12.24.3. Hence we see that the required result holds for the multi-index I = (0, . . . , 0, n). It turns out that we can reduce the general case to this case as follows. Let S = R[x1 , x2 , . . . , xc , 1/xc ]. The ring map R → S is faithfully flat, hence f1 , . . . , fc is an H1 -regular sequence in S, see Lemma 12.24.4. By Lemma 12.23.4 we see that g1 = f1 − x1 /xc fc , . . . gc−1 = fc−1 − xc−1 /xc fc , gc = (1/xc )fc is an H1 -regular sequence in S. Finally, note that our element ξ can be rewritten X ξ= aI (g1 + xc gc )i1 . . . (gc−1 + xc gc )ic−1 (xc gc )ic |I|=n,I=(i1 ,...,ic )

and the coefficient of gcn in this expression is X aI xi11 . . . xicc ∈ JS. Since the monomials xi11 . . . xicc form part of an R-basis of S over R we conclude that aI ∈ J for all I as desired.  Lemma 12.24.6. Let A be a ring. Let I ⊂ A be an ideal. Let g1 , . . . , gm be a sequence in A whose image in A/I is H1 -regular. Then I ∩ (g1 , . . . , gm ) = I(g1 , . . . , gm ).

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Proof. Consider the exact sequence of complexes 0 → I ⊗A K• (A, g1 , . . . , gm ) → K• (A, g1 , . . . , gm ) → K• (A/I, g1 , . . . , gm ) → 0 Since the complex on the right has H1 = 0 by assumption we see that Coker(I ⊕m → I) −→ Coker(A⊕m → A) is injective. This is equivalent to the assertion of the lemma.



Lemma 12.24.7. Let A be a ring. Let I ⊂ J ⊂ A be ideals. Assume that J/I ⊂ A/I is generated by an H1 -regular sequence. Then I ∩ J 2 = IJ. Proof. To prove this choose g1 , . . . , gm ∈ J whose images in A/I form a H1 -regular sequence which generates J/I. In particular J = I + (g1 , . . . , gm ). Suppose that x ∈ I ∩ J 2 . Because x ∈ J 2 can write X X x= aij gi gj + aj gj + a P with aij ∈ A, aj ∈ I and P a ∈ I 2 . Then aij gi gj ∈ I ∩ (g1 , . . . , gm ) hence by Lemma 12.24.6 we see that aij gi gj ∈ I(g1 , . . . , gm ). Thus x ∈ IJ as desired.  Lemma 12.24.8. Let A be a ring. Let I be an ideal generated by a quasi-regular sequence f1 , . . . , fn in A. Let g1 , . . . , gm ∈ A be elements whose images g 1 , . . . , g m form an H1 -regular sequence in A/I. Then f1 , . . . , fn , g1 , . . . , gm is a quasi-regular sequence in A. Proof. We claim that g1 , . . . , gm forms an H1 -regular sequence in A/I d for every d. By induction assume that this holds in A/I d−1 . We have a short exact sequence of complexes 0 → K• (A, g• ) ⊗A I d−1 /I d → K• (A/I d , g• ) → K• (A/I d−1 , g• ) → 0 Since f1 , . . . , fn is quasi-regular we see that the first complex is a direct sum of copies of K• (A/I, g1 , . . . , gm ) hence acyclic in degree 1. By induction hypothesis the last complex is acyclic in degree 1. Hence also the middle complex is. In particular, the sequence g1 , . . . , gm forms a quasi-regular sequence in A/I d for every d ≥ 1, see Lemma 12.24.5. Now we are ready to prove that f1 , . . . , fn , g1 , . . . , gm is a quasiregular sequence in A. Namely, set J = (f1 , . . . , fn , g1 , . . . , gm ) and suppose that (with multinomial notation) X aN,M f N g M ∈ J d+1 |N |+|M |=d

for some aN,M ∈ A. We have to show that aN,M ∈ J for all N, M . Let e ∈ {0, 1, . . . , d}. Then X aN,M f N g M ∈ (g1 , . . . , gm )e+1 + I d−e+1 |N |=d−e, |M |=e

Because g1 , . . . , gm is a quasi-regular sequence in A/I d−e+1 we deduce X aN,M f N ∈ (g1 , . . . , gm ) + I d−e+1 |N |=d−e

for each M with |M | =P e. By Lemma 12.24.6 applied to I d−e /I d−e+1 in the ring d−e+1 A/I this implies |N |=d−e aN,M f N ∈ I d−e (g1 , . . . , gm ). Since f1 , . . . , fn is quasi-regular in A this implies that aN,M ∈ J for each N, M with |N | = d − e and |M | = e. This proves the lemma. 

12.24. KOSZUL REGULAR SEQUENCES

883

Lemma 12.24.9. Let A be a ring. Let I be an ideal generated by an H1 -regular sequence f1 , . . . , fn in A. Let g1 , . . . , gm ∈ A be elements whose images g 1 , . . . , g m form an H1 -regular sequence in A/I. Then f1 , . . . , fn , g1 , . . . , gm is an H1 -regular sequence in A. Proof. We have to show that H1 (A, f1 , . . . , fn , g1 , . . . , gm ) = 0. To do this consider the commutative diagram ∧2 (A⊕n+m )

/ A⊕n+m

/A

/0

 ∧2 (A/I ⊕m )

 / A/I ⊕m

 / A/I

/0

Consider an element (a1 , . . . , an+m ) ∈ A⊕n+m which maps to zero in A. Because g 1 , . . . , g m form an H1 -regular sequence in A/I we see that (an+1 , . . . , an+m ) is the image of some element α of ∧2 (A/I ⊕m ). We can lift α to an element α ∈ ∧2 (A⊕n+m ) and substract the image of it in A⊕n+m from our element (a1 , . . . , an+m ). Thus we may assume that an+1 , . . . , an+m ∈ I. Since I = (f1 , . . . , fn ) we can modify our element (a1 , . . . , an+m ) by linear combinations of the elements (0, . . . , gj , 0, . . . , 0, fi , 0, . . . , 0) in the image of the top left horizontal arrow to reduce to the case that an+1 , . . . , an+m are zero. In this case (a1 , . . . , an , 0, . . . , 0) defines an element of H1 (A, f1 , . . . , fn ) which we assumed to be zero.  Lemma 12.24.10. Let A be a ring. Let f1 , . . . , fn , g1 , . . . , gm ∈ A be an H1 regular sequence. Then the images g 1 , . . . , g m in A/(f1 , . . . , fn ) form an H1 -regular sequence. P Proof. Set I = (f1 , . . . , fn ). We have to show that any relation j=1,...,m aj g j in A/I is a linear combination of trivial relations. Because I = (f1 , . . . , fn ) we can lift this relation to a relation X X aj gj + bi fi = 0 j=1,...,m

i=1,...,n

in A. By assumption this relation in A is a linear combination of trivial relations. Taking the image in A/I we obtain what we want.  Lemma 12.24.11. Let A be a ring. Let I be an ideal generated by a Koszulregular sequence f1 , . . . , fn in A. Let g1 , . . . , gm ∈ A be elements whose images g 1 , . . . , g m form a Koszul-regular sequence in A/I. Then f1 , . . . , fn , g1 , . . . , gm is a Koszul-regular sequence in A. Proof. Our assumptions say that K• (A, f1 , . . . , fn ) is a finite free resolution of A/I and K• (A/I, g 1 , . . . , g m ) is a finite free resolution of A/(fi , gj ) over A/I. Then K• (A, f1 , . . . , fn , g1 , . . . , gm ) = Tot(K• (A, f1 , . . . , fn ) ⊗A K• (A, g1 , . . . , gm )) ∼ Tot(A/I ⊗A K• (A, g1 , . . . , gm )) = = K• (A/I, g 1 , . . . , g m ) ∼ = A/(fi , gj ) The first equality by Lemma 12.23.12. The first quasi-isomorphism by Lemma 12.5.8. The second equality is clear. The last quasi-isomorphism by assumption. Hence we win. 

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To conclude in the following lemma it is necessary to assume that both f1 , . . . , fn and f1 , . . . , fn , g1 , . . . , gm are Koszul-regular. A counter example to dropping the assumption that f1 , . . . , fn is Koszul-regular is Examples, Lemma 66.6.1. Lemma 12.24.12. Let A be a ring. Let f1 , . . . , fn , g1 , . . . , gm ∈ A. If both f1 , . . . , fn and f1 , . . . , fn , g1 , . . . , gm are Koszul-regular sequences in A, then g 1 , . . . , g m in A/(f1 , . . . , fn ) form a Koszul-regular sequence. Proof. Set I = (f1 , . . . , fn ). Our assumptions say that K• (A, f1 , . . . , fn ) is a finite free resolution of A/I and K• (A, f1 , . . . , fn , g1 , . . . , gm ) is a finite free resolution of A/(fi , gj ) over A. Then A/(fi , gj ) ∼ = K• (A, f1 , . . . , fn , g1 , . . . , gm ) = Tot(K• (A, f1 , . . . , fn ) ⊗A K• (A, g1 , . . . , gm )) ∼ = Tot(A/I ⊗A K• (A, g1 , . . . , gm )) = K• (A/I, g 1 , . . . , g m ) The first quasi-isomorphism by assumption. The first equality by Lemma 12.23.12. The second quasi-isomorphism by Lemma 12.5.8. The second equality is clear. Hence we win.  Lemma 12.24.13. Let R be a ring. Let I be an ideal generated by f1 , . . . , fr ∈ R. (1) If I can be generated by a quasi-regular sequence of length r, then f1 , . . . , fr is a quasi-regular sequence. (2) If I can be generated by an H1 -regular sequence of length r, then f1 , . . . , fr is an H1 -regular sequence. (3) If I can be generated by a Koszul-regular sequence of length r, then f1 , . . . , fr is a Koszul-regular sequence. In other words, a minimal set of generators of an ideal generated by a quasi-regular (resp. H1 -regular, Koszul-regular) sequence is a quasi-regular (resp. H1 -regular, Koszul-regular) sequence. Proof. If I can be generated by a quasi-regular sequence of length r, then I/I 2 is free of rank r over R/I. Since f1 , . . . , fr generate by assumption we see that the images f i form a basis of I/I 2 over R/I. It follows that f1 , . . . , fr is a quasiregular sequence as all this means, besides the freeness of I/I 2 , is that the maps SymnR/I (I/I 2 ) → I n /I n+1 are isomorphisms. We continue to assumePthat I can be generated by a quasi-regular sequence, say g1 , . . . , gr . Write gj = aij fi . As f1 , . . . , fr is quasi-regular according to the previous paragraph, we see that det(aij ) is invertible mod I. The matrix aij gives a map R⊕r → R⊕r which induces a map of Koszul complexes α : K• (R, f1 , . . . , fr ) → K• (R, g1 , . . . , gr ), see Lemma 12.23.3. This map becomes an isomorphism on inverting det(aij ). Since the cohomology modules of both K• (R, f1 , . . . , fr ) and K• (R, g1 , . . . , gr ) are annihilated by I, see Lemma 12.23.6, we see that α is a quasiisomorphism. Hence if g1 , . . . , gr is H1 -regular, then so is f1 , . . . , fr . Similarly for Koszul-regular.  Lemma 12.24.14. Let A → B be a ring map. Let f1 , . . . , fr be a sequence in B such that B/(f1 , . . . , fr ) is A-flat. Let A → A0 be a ring map. Then the canonical map H1 (K• (B, f1 , . . . , fr )) ⊗A A0 −→ H1 (K• (B 0 , f10 , . . . , fr0 ))

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is surjective, where B 0 = B ⊗A A0 and fi0 ∈ B 0 is the image of fi . Proof. The sequence ∧2 (B ⊕r ) → B ⊕r → B → B/J → 0 is a complex of A-modules with B/J flat over A and cohomology group H1 = H1 (K• (B, f1 , . . . , fr )) in the spot B ⊕r . If we tensor this with A0 we obtain a complex ∧2 ((B 0 )⊕r ) → (B 0 )⊕r → B 0 → B 0 /J 0 → 0 which is exact at B 0 and B 0 /J 0 . In order to compute its cohomology group H10 = H1 (K• (B 0 , f10 , . . . , fr0 )) at (B 0 )⊕r we split the first sequence above into short exact sequences 0 → J → B → B/J → 0 and 0 → K → B ⊕r → J → 0 and ∧2 (B ⊕r ) → K → H1 → 0. Tensoring with A0 over A we obtain the exact sequences 0 → J ⊗A A0 → B ⊗A A0 → (B/J) ⊗A A0 → 0 K ⊗A A0 → B ⊕r ⊗A A0 → J ⊗A A0 → 0 ∧2 (B ⊕r ) ⊗A A0 → K ⊗A A0 → H1 ⊗A A0 → 0 where the first one is exact as B/J is flat over A, see Algebra, Lemma 7.36.11. Hence we conclude what we want.  Lemma 12.24.15. Let R be a ring. Let a1 , . . . , an ∈ R beP elements such that R → R⊕n , x 7→ (xa1 , . . . , xan ) is injective. Then the element ai ti of the polynomial ring R[t1 , . . . , tn ] is a nonzerodivisor. Proof. If one of the ai is a unit this is just the statement that any element of the form t1 + a2 t2 + . . . + an tn is a nonzero divisor in the polynomial ring over R. Case I: R is Noetherian. Let qj , j = 1, . . . , m be the associated primes of R. We have to show that each of the maps X ai ti : Symd (R⊕n ) −→ Symd+1 (R⊕n ) is injective. As Symd (R⊕n ) is a free R-module its associated primes are qj , j = 1, . . . , m. For each j there exists an i = i(j) such that ai 6∈ qj because there exists an x ∈ R with qj x = 0 but ai x 6= 0 for some i by assumption. Hence ai is a unit in Rqj and the map is injective after localizing at qj . Thus the map is injective, see Algebra, Lemma 7.61.18. Case II: R general. We can write R as the union of Noetherian rings Rλ with a1 , . . . , an ∈ Rλ . For each Rλ the result holds, hence the result holds for R.  Lemma 12.24.16. Let R be a ring. Let f1 , . . . , fn be a Koszul-regular sequence in R. Consider the faithfully flat, smooth ring map −1 −1 R −→ S = R[{tij }i≤j , t−1 11 , t22 , . . . , tnn ]

For 1 ≤ i ≤ n set gi =

X i≤j

tij fj ∈ S.

Then g1 , . . . , gn is a regular sequence in S and (f1 , . . . , fn )S = (g1 , . . . , gn ).

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Proof. The equality of ideals is obvious  t11 t12  0 t22  0 0 ... ...

as the matrix  t13 . . . t23 . . .  t33 . . . ... ...

is invertible in S. Because f1 , . . . , fn is a Koszul-regular sequence we see that the kernel of R → R⊕n , x 7→ (xf1 , . . . , xfn ) is zero (as it computes the nthe Koszul homology of R w.r.t. f1 , . . . , fn ). Hence by Lemma 12.24.15 we see that g1 = f1 t11 + . . . + fn t1n is a nonzerodivisor in S 0 = R[t11 , t12 , . . . , t1n , t−1 11 ]. We see that g1 , f2 , . . . , fn is a Koszul-sequence in S 0 by Lemma 12.24.4 and 12.24.13. We conclude that f 2 , . . . , f n is a Koszul-regular sequence in S 0 /(g2 ) by Lemma 12.24.12. Hence by induction on n we see that the images g 2 , . . . , g n of g2 , . . . , gn in −1 S 0 /(g2 )[{tij }2≤i≤j , t−1 22 , . . . , tnn ] form a regular sequence. This in turn means that g1 , . . . , gn forms a regular sequence in S.  12.25. Regular ideals We will discuss the notion of a regular ideal sheaf in great generality in Divisors, Section 26.12. Here we define the corresponding notion in the affine case, i.e., in the case of an ideal in a ring. Definition 12.25.1. Let R be a ring and let I ⊂ R be an ideal. (1) We say I is a regular ideal if for every p ∈ V (I) there exists a g ∈ R, g 6∈ p and a regular sequence f1 , . . . , fr ∈ Rg such that Ig is generated by f1 , . . . , f r . (2) We say I is a Koszul-regular ideal if for every p ∈ V (I) there exists a g ∈ R, g 6∈ p and a Koszul-regular sequence f1 , . . . , fr ∈ Rg such that Ig is generated by f1 , . . . , fr . (3) We say I is a H1 -regular ideal if for every p ∈ V (I) there exists a g ∈ R, g 6∈ p and an H1 -regular sequence f1 , . . . , fr ∈ Rg such that Ig is generated by f1 , . . . , fr . (4) We say I is a quasi-regular ideal if for every p ∈ V (I) there exists a g ∈ R, g 6∈ p and a quasi-regular sequence f1 , . . . , fr ∈ Rg such that Ig is generated by f1 , . . . , fr . It is clear that given I ⊂ R we have the implications I is a regular ideal ⇒ I is a Koszul-regular ideal ⇒ I is a H1 -regular ideal ⇒ I is a quasi-regular ideal see Lemmas 12.24.2 and 12.24.5. Such an ideal is always finitely generated. Lemma 12.25.2. A quasi-regular ideal is finitely generated. Proof. Let I ⊂ R be a quasi-regular ideal. Since V (I) is quasi-compact, there exist g1 , . . . , gm ∈ R such that V (I) ⊂ D(g1 ) ∪ . . . ∪ D(gm ) and such that Igj is e 0 generated by a quasi-regular sequence gj1 , . . . , gjrj ∈ Rgj . Write gji = gji /gj ij P 0 for some gij ∈ I. Write 1 + x = gj hj for some x ∈ I which is possible S as V (I) ⊂ D(g1 ) ∪ . . . ∪ D(gm ). Note that Spec(R) = D(g1 ) ∪ . . . ∪ D(gm ) D(x) 0 Then I is generated by the elements gij and x as these generate on each of the pieces of the cover, see Algebra, Lemma 7.22.2. 

12.26. LOCAL COMPLETE INTERSECTION MAPS

887

We prove flat descent for Koszul-regular, H1 -regular, quasi-regular ideals. Lemma 12.25.3. Let A → B be a faithfully flat ring map. Let I ⊂ A be an ideal. If IB is a Koszul-regular (resp. H1 -regular, resp. quasi-regular) ideal in B, then I is a Koszul-regular (resp. H1 -regular, resp. quasi-regular) ideal in A. Proof. We fix the prime p ⊃ I throughout the proof. Assume IB is quasi-regular. By Lemma 12.25.2 IB is a finite module, hence I is a finite A-module by Algebra, Lemma 7.78.2. As A → B is flat we see that I/I 2 ⊗A/I B/IB = I/I 2 ⊗A B = IB/(IB)2 . As IB is quasi-regular, the B/IB-module IB/(IB)2 is finite locally free. Hence I/I 2 is finite projective, see Algebra, Proposition 7.78.3. In particular, after replacing A by Af for some f ∈ A, f 6∈ p we may assume that I/I 2 is free of rank r. Pick f1 , . . . , fr ∈ I which give a basis of I/I 2 . By Nakayama’s lemma (see Algebra, Lemma 7.18.1) we see that, after another replacement A Af as above, I is generated by f1 , . . . , fr . Proof of the “quasi-regular” case. Above we have seen that I/I 2 is free on the r-generators f1 , . . . , fr . To finish the proof in this case we have to show that the maps Symd (I/I 2 ) → I d /I d+1 are isomorphisms for each d ≥ 2. This is clear as the faithfully flat base changes Symd (IB/(IB)2 ) → (IB)d /(IB)d+1 are isomorphisms locally on B by assumption. Details omitted. Proof of the “H1 -regular” and “Koszul-regular” case. Consider the sequence of elements f1 , . . . , fr generating I we constructed above. By Lemma 12.24.13 we see that f1 , . . . , fr map to a H1 -regular or Koszul-regular sequence in Bg for any g ∈ B such that IB is generated by an H1 -regular or Koszul-regular sequence. Hence K• (A, f1 , . . . , fr ) ⊗A Bg has vanishing H1 or Hi , i > 0. Since the homology of K• (B, f1 , . . . , fr ) = K• (A, S f1 , . . . , fr ) ⊗A B is annihilated by IB (see Lemma 12.23.6) and since V (IB) ⊂ g as above D(g) we conclude that K• (A, f1 , . . . , fr ) ⊗A B has vanishing homology in degree 1 or all positive degrees. Using that A → B is faithfully flat we conclude that the same is true for K• (A, f1 , . . . , fr ).  Lemma 12.25.4. Let A be a ring. Let I ⊂ J ⊂ A be ideals. Assume that J/I ⊂ A/I is a H1 -regular ideal. Then I ∩ J 2 = IJ. Proof. Follows immediately from Lemma 12.24.7 by localizing.



12.26. Local complete intersection maps We can use the material above to define a local complete intersection map between rings using presentations by (finite) polynomial algebras. Lemma 12.26.1. Let A → B be a finite type ring map. If for some presentation α : A[x1 , . . . , xn ] → B the kernel I is a Koszul-regular ideal then for any presentation β : A[y1 , . . . , ym ] → B the kernel J is a Koszul-regular ideal. Proof. Choose fj ∈ A[x1 , . . . , xn ] with α(fj ) = β(yj ) and gi ∈ A[y1 , . . . , ym ] with β(gi ) = α(xi ). Then we get a commutative diagram A[x1 , . . . , xn , y1 , . . . , ym ]

yj 7→fj

/ A[x1 , . . . , xn ]

xi 7→gi

 A[y1 , . . . , ym ]

 /B

888

12. MORE ON ALGEBRA

Note that the kernel K of A[xi , yj ] → B is equal to K = (I, yj −fj ) = (J, xi −fi ). In particular, as I is finitely generated by Lemma 12.25.2 we see that J = K/(xi − fi ) is finitely generated too. Pick a prime q ⊂ B. Since I/I 2 ⊕ B ⊕m = J/J 2 ⊕ B ⊕n (Algebra, Lemma 7.124.13) we see that dim J/J 2 ⊗B κ(q) + n = dim I/I 2 ⊗B κ(q) + m. Pick p1 , . . . , pt ∈ I which map to a basis of I/I 2 ⊗ κ(q) = I ⊗A[xi ] κ(q). Pick q1 , . . . , qs ∈ J which map to a basis of J/J 2 ⊗κ(q) = J ⊗A[yj ] κ(q). So s+n = t+m. By Nakayama’s lemma there exist h ∈ A[xi ] and h0 ∈ A[yj ] both mapping to a nonzero element of κ(q) such that Ih = (p1 , . . . , pt ) in A[xi , 1/h] and Jh0 = (q1 , . . . , qs ) in A[yj , 1/h0 ]. As I is Koszul-regular we may also assume that Ih is generated by a Koszul regular sequence. This sequence must necessarily have length t = dim I/I 2 ⊗B κ(q), hence we see that p1 , . . . , pt is a Koszul-regular sequence by Lemma 12.24.13. As also y1 − f1 , . . . , ym − fm is a regular sequence we conclude y1 − f1 , . . . , ym − fm , p1 , . . . , pt is a Koszul-regular sequence in A[xi , yj , 1/h] (see Lemma 12.24.11). This sequence generates the ideal Kh . Hence the ideal Khh0 is generated by a Koszul-regular sequence of length m + t = n + s. But it is also generated by the sequence x1 − g1 , . . . , xn − gn , q1 , . . . , qs of the same length which is thus a Koszul-regular sequence by Lemma 12.24.13. Finally, by Lemma 12.24.12 we conclude that the images of q1 , . . . , qs in A[xi , yj , 1/hh0 ]/(x1 − g1 , . . . , xn − gn ) ∼ = A[yj , 1/h00 ] form a Koszul-regular sequence generating Jh00 . Since h00 is the image of hh0 it doesn’t map to zero in κ(q) and we win.  This lemma allows us to make the following definition. Definition 12.26.2. A ring map A → B is called a local complete intersection if it is of finite type and for some (equivalently any) presentation B = A[x1 , . . . , xn ]/I the ideal I is Koszul-regular. This notion is local. Lemma 12.26.3. Let R → S be a ring map. Let g1 , . . . , gm ∈ S generate the unit ideal. If each R → Sgj is a local complete intersection so is R → S. Proof. Let S = R[x1 , . . . , xn ]/I be a presentation. Pick hj ∈ R[x1 , . . . , xn ] mapping to gj in S. Then R[x1 , . . . , xn , xn+1 ]/(I, xn+1 hj − 1) is a presentation of Sgj . Hence Ij = (I, xn+1 hj − 1) is a Koszul-regular ideal in R[x1 , . . . , xn , xn+1 ]. Pick a prime I ⊂ q ⊂ R[x1 , . . . , xn ]. Then hj 6∈ q for some j and qj = (q, xn+1 hj − 1) is a prime ideal of V (Ij ) lying over q. Pick f1 , . . . , fr ∈ I which map to a basis of I/I 2 ⊗κ(q). Then xn+1 hj −1, f1 , . . . , fr is a sequence of elements of Ij which map to a basis of Ij ⊗ κ(qj ). By Nakayma’s lemma there exists an h ∈ R[x1 , . . . , xn , xn+1 ] such that (Ij )h is generated by xn+1 hj − 1, f1 , . . . , fr . We may also assume that (Ij )h is generated by a Koszul regular sequence of some length e. Looking at the dimension of Ij ⊗ κ(qj ) we see that e = r + 1. Hence by Lemma 12.24.13 we see that xn+1 hj − 1, f1 , . . . , fr is a Koszul-regular sequence generating (Ij )h for some h ∈ R[x1 , . . . , xn , xn+1 ], h 6∈ qj . By Lemma 12.24.12 we see that Ih0 is generated by a Koszul-regular sequence for some h0 ∈ R[x1 , . . . , xn ], h0 6∈ q as desired. 

12.27. CARTIER’S EQUALITY AND GEOMETRIC REGULARITY

889

Lemma 12.26.4. Let R be a ring. Let R[x1 , . . . , xn ]. If R[x1 , . . . , xn ]/(f1 , . . . , fc ) be a relative global complete intersection. Then f1 , . . . , fc is a Koszul regular sequence. Proof. Recall that the homology groups Hi (K• (f• )) are annihilated by the ideal (f1 , . . . , fc ). Hence it suffices to show that Hi (K• (f• ))q is zero for all primes q ⊂ R[x1 , . . . , xn ] containing (f1 , . . . , fc ). This follows from Algebra, Lemma 7.126.13 and the fact that a regular sequence is Koszul regular (Lemma 12.24.2).  Lemma 12.26.5. A syntomic ring map is a local complete intersection. Proof. Combine Lemmas 12.26.4 and 12.26.3 and Algebra, Lemma 7.126.16.



For a local complete intersection R → S we have Hn (LS/R ) = 0 for n ≥ 2. Since we haven’t (yet) defined the full cotangent complex we can’t state and prove this, but we can deduce one of the consequences. Lemma 12.26.6. Let A → B → C be ring maps. Assume B → C is a local complete intersection homomorphism. Choose a presentation α : A[xs , s ∈ S] → B with kernel I. Choose a presentation β : B[y1 , . . . , ym ] → C with kernel J. Let γ : A[xs , yt ] → C be the induced presentation of C with kernel K. Then we get a canonical commutative diagram 0

/ ΩA[x ]/A ⊗ C s O

/ ΩA[x ,y ]/A ⊗ C s t O

/ ΩB[y ]/B ⊗ C t O

/0

0

/ I/I 2 ⊗ C

/ K/K 2

/ J/J 2

/0

with exact rows. In particular, the six term exact sequence of Algebra, Lemma 7.124.3 can be completed with a zero on the left, i.e., the sequence 0 → H1 (N LB/A ⊗B C) → H1 (LC/A ) → H1 (LC/B ) → ΩB/A ⊗B C → ΩC/A → ΩC/B → 0 is exact. Proof. The only thing to prove is the injectivity of the map I/I 2 ⊗ C → K/K 2 . By assumption the ideal J is Koszul-regular. Hence we have IA[xs , yj ] ∩ K 2 = IK by Lemma 12.25.4. This means that the kernel of K/K 2 → J/J 2 is isomorphic to IA[xs , yj ]/IK. Since I/I 2 ⊗A C = IA[xs , yj ]/IK this provides us with the desired injectivity of I/I 2 ⊗A C → K/K 2 so that the result follows from the snake lemma, see Homology, Lemma 10.3.23.  Lemma 12.26.7. Let A → B → C be ring maps. If B → C is a filtered colimit of local complete intersection homomorphisms then the conclusion of Lemma 12.26.6 remains valid. Proof. Follows from Lemma 12.26.6 and Algebra, Lemma 7.124.8.



12.27. Cartier’s equality and geometric regularity A reference for this section and the next is [Mat70, Section 39]. In order to comfortably read this section the reader should be familiar with the naive cotangent complex and its properties, see Algebra, Section 7.124.

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12. MORE ON ALGEBRA

Lemma 12.27.1 (Cartier equality). Let K/k be a finitely generated field extension. Then ΩK/k and H1 (LK/k ) are finite dimensional and trdegk (K) = dimK ΩK/k − dimk H1 (LK/k ). Proof. We can find a global complete intersection A = k[x1 , . . . , xn ]/(f1 , . . . , fc ) over k such that K is isomorphic to the fraction field of A, see Algebra, Lemma 7.142.10 and its proof. In this case we see that N LK/k is homotopy equivalent to the complex M M K −→ Kdxi j=1,...,c

i=1,...,n

by Algebra, Lemmas 7.124.2 and 7.124.11. The transcendence degree of K over k is the dimension of A (by Algebra, Lemma 7.108.1) which is n − c and we win.  Lemma 12.27.2. Let K ⊂ L ⊂ M be field extensions. Then the Jacobi-Zariski sequence 0 → H1 (LL/K )⊗L M → H1 (LM/K ) → H1 (LM/L ) → ΩL/K ⊗L M → ΩM/K → ΩM/L → 0 is exact. Proof. Combine Lemma 12.26.7 with Algebra, Lemma 7.142.10.



Lemma 12.27.3. Given a commutative diagram of fields KO

/ K0 O

k

/ k0

with k ⊂ k 0 and K ⊂ K 0 finitely generated field extensions the kernel and cokernel of the maps α : ΩK/k ⊗K K 0 → ΩK 0 /k0

and

β : H1 (LK/k ) ⊗K K 0 → H1 (LK 0 /k0 )

are finite dimensional and dim Ker(α) − dim Coker(α) − dim Ker(β) + dim Coker(β) = trdegk (k 0 ) − trdegK (K 0 ) Proof. The Jacobi-Zariski sequences for k ⊂ k 0 ⊂ K 0 and k ⊂ K ⊂ K 0 are 0 → H1 (Lk0 /k )⊗K 0 → H1 (LK 0 /k ) → H1 (LK 0 /k0 ) → Ωk0 /k ⊗K 0 → ΩK 0 /k → ΩK 0 /k → 0 and 0 → H1 (LK/k )⊗K 0 → H1 (LK 0 /k ) → H1 (LK 0 /K ) → ΩK/k ⊗K 0 → ΩK 0 /k → ΩK 0 /K → 0 By Lemma 12.27.1 the vector spaces Ωk0 /k , ΩK 0 /K , H1 (LK 0 /K ), and H1 (Lk0 /k ) are finite dimensional and the alternating sum of their dimensions is trdegk (k 0 ) − trdegK (K 0 ). The lemma follows.  12.28. Geometric regularity Let k be a field. Let (A, m, K) be a Noetherian local k-algebra. The Jacobi-Zariski sequence (Algebra, Lemma 7.124.3) is a canonical exact sequence H1 (LK/k ) → m/m2 → ΩA/k ⊗A K → ΩK/k → 0 because H1 (LK/A ) = m/m2 by Algebra, Lemma 7.124.5. We will show that exactness on the left of this sequence characterizes whether or not a regular local ring A is geometrically regular over k. We will link this to the notion of formal smoothness in Section 12.32.

12.28. GEOMETRIC REGULARITY

891

Proposition 12.28.1. Let k be a field of characteristic p > 0. Let (A, m, K) be a Noetherian local k-algebra. The following are equivalent (1) A is geometrically regular over k, (2) for all k ⊂ k 0 ⊂ k 1/p finite over k the ring A ⊗k k 0 is regular, (3) A is regular and the canonical map H1 (LK/k ) → m/m2 is injective, and (4) A is regular and the map Ωk/Fp ⊗k K → ΩA/Fp ⊗A K is injective. Proof. Proof of (3) ⇒ (1). Assume (3). Let k ⊂ k 0 be a finite purely inseparable extension. Set A0 = A ⊗k k 0 . This is a local ring with maximal ideal m0 . Set K 0 = A0 /m0 . We get a commutative diagram / H1 (LK/k ) ⊗ K 0

0

/ m/m2 ⊗ K 0

β

 / m0 /(m0 )2

 H1 (LK 0 /k0 )

/ ΩA/k ⊗A K 0 ∼ =

/ ΩA0 /k0

/ ΩK/k ⊗ K 0 α

 ⊗A0 K 0

 / ΩK 0 /k0

with exact rows. The third vertical arrow is an isomorphism by base change for modules of differentials (Algebra, Lemma 7.123.12). Thus α is surjective. By Lemma 12.27.3 we have dim Ker(α) − dim Ker(β) + dim Coker(β) = 0 (and these dimensions are all finite). A diagram chase shows that dim m0 /(m0 )2 ≤ dim m/m2 . However, since A → A0 is finite flat we see that dim(A) = dim(A0 ), see Algebra, Lemma 7.104.6. Hence A0 is regular by definition. Equivalence of (3) and (4). Consider the Jacobi-Zariski sequences for rows of the commutative diagram /A /K Fp O O O /k

Fp

/K

to get a commutative diagram 0

/ m/m2 O

/ ΩA/F ⊗A K p O

/ ΩK/F O p

/0 O

0

/ H1 (LK/k )

/ Ωk/F ⊗k K p

/ ΩK/F p

/ ΩK/k

/0

with exact rows. We have used that H1 (LK/A ) = m/m2 and that H1 (LK/Fp ) = 0 as K/Fp is separable, see Algebra, Proposition 7.142.8. Thus it is clear that the kernels of H1 (LK/k ) → m/m2 and Ωk/Fp ⊗k K → ΩA/Fp ⊗A K have the same dimension. Proof of (2) ⇒ (4) following Faltings, see [Fal78]. Let a1 , . . . , an ∈ k be elements such that da1 , . . . , dan are linearly independent in Ωk/Fp . Consider the 1/p

1/p

/0

field extension k 0 = k(a1 , . . . , an ). By Algebra, Lemma 7.142.2 we see that k 0 = k[x1 , . . . , xn ]/(xp1 − a1 , . . . , xpn − an ). In particularL we see that the naive L cotan-0 0 gent complex of k 0 /k is homotopic to the complex j=1,...,n k → i=1,...,n k with the zero differential as d(xpj − aj ) = 0 in Ωk[x1 ,...,xn ]/k . Set A0 = A ⊗k k 0 and K 0 = A0 /m0 as above. By Algebra, Lemma 7.124.7 we see that N LA0 /A is homotopy

/0

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L L 0 0 equivalent to the complex j=1,...,n A → i=1,...,n A with the zero differential, i.e., H1 (LA0 /A ) and ΩA0 /A are free of rank n. The Jacobi-Zariski sequence for Fp → A → A0 is H1 (LA0 /A ) → ΩA/Fp ⊗A A0 → ΩA0 /Fp → ΩA0 /A → 0 Using the presentation A[x1 , . . . , xn ] → A0 with kernel (xpj − aj ) we see, unwinding the maps in Algebra, Lemma 7.124.3, that the jth basis vector of H1 (LA0 /A ) maps to daj ⊗ 1 in ΩA/Fp ⊗ A0 . As ΩA0 /A is free (hence flat) we get on tensoring with K 0 an exact sequence β

K 0⊕n → ΩA/Fp ⊗A K 0 − → ΩA0 /Fp ⊗A0 K 0 → K 0⊕n → 0 We conclude that the elements daj ⊗ 1 generate Ker(β) and we have to show that are linearly independent, i.e., we have to show dim(ker(β)) = n. Consider the following big diagram 0

α

0

/ ΩK 0 /F O p

/ ΩA0 /F ⊗ K 0 p O

/ m0 /(m0 )2 O

γ

β

/ m/m2 ⊗ K 0

/0

/ ΩK/F ⊗ K 0 p

/ ΩA/F ⊗ K 0 p

/0

By Lemma 12.27.1 and the Jacobi-Zariski sequence for Fp → K → K 0 we see that the kernel and cokernel of γ have the same finite dimension. By assumption A0 is regular (and of the same dimension as A, see above) hence the kernel and cokernel of α have the same dimension. It follows that the kernel and cokernel of β have the same dimension which is what we wanted to show. The implication (1) ⇒ (2) is trivial. This finishes the proof of the proposition.



Lemma 12.28.2. Let k be a field of characteristic p > 0. Let (A, m, K) be a Noetherian local k-algebra. Assume A is geometrically regular over k. Let k ⊂ F ⊂ K be a finitely generated subextension. Let ϕ : k[y1 , . . . , ym ] → A be a k-algebra map such that yi maps to an element of F in K and such that dy1 , . . . , dym map to a basis of ΩF/k . Set p = ϕ−1 (m). Then k[y1 , . . . , ym ]p → A is flat and A/pA is regular. Proof. Set A0 = k[y1 , . . . , ym ]p with maximal ideal m0 and residue field K0 . Note that ΩA0 /k is free of rank m and ΩA0 /k ⊗ K0 → ΩK0 /k is an isomorphism. It is clear that A0 is geometrically regular over k. Hence H1 (LK0 /k ) → m0 /m20 is an isomorphism, see Proposition 12.28.1. Now consider H1 (LK0 /k ) ⊗ K

/ m0 /m20 ⊗ K

 H1 (LK/k )

 / m/m2

Since the left vertical arrow is injective by Lemma 12.27.2 and the lower horizontal by Proposition 12.28.1 we conclude that the right vertical one is too. Hence a regular system of parameters in A0 maps to part of a regular system of parameters in A. We win by Algebra, Lemmas 7.120.2 and 7.99.3. 

12.29. TOPOLOGICAL RINGS AND MODULES

893

12.29. Topological rings and modules Let’s quickly discuss some properties of topological abelian groups. An abelian group M is a topological abelian group if M is endowed with a topology such that addition M ×M → M is continuous. A homomorphism of topological abelian groups is just a homomorphism of abelian groups which is continuous. The category of commutative topological groups is additive and has kernels and cokernels, but is not abelian (as the axiom Im = Coim doesn’t hold). If N ⊂ M is a subgroup, then we think of N and M/N as topological groups also, namely using the induced topology on N and the quotient topology on M/N (i.e., such that M → M/N is submersive). Note that if N ⊂ M is an open subgroup, then the topology on M/N is discrete. We say the topology on M is linear if there exists a fundamental system of neighbourhoods of 0 consisting of subgroups. If so then these subgroups are also open. An example is the following. Let I be a directed partially ordered set and let Gi be an inverse system of (discrete) abelian groups over I. Then G = limi∈I Gi with the inverse limit topology is linearly topologized with a fundamental system of neighbourhoods of 0 given by Ker(G → Gi ). Conversely, let M be a linearly topologized abelian group. Choose any fundamental system of open subgroups Ui ⊂ M , i ∈ I (i.e., the Ui form a fundamental system of open neighbourhoods and each Ui is a subgroup of M ). Setting i ≥ i0 ⇔ Ui ⊂ Ui0 we see that I is a directed partially ordered set. We obtain a homomorphism of linearly topologized abelian groups c : M −→ limi∈I M/Ui . It is clear that M is separated (as a topological space) if and only if c is injective. We say that M is complete if c is an isomorphism2. We leave it to the reader to check that this condition is independent of the choice of fundamental system of open subgroups {Ui }i∈I chosen above. In fact the topological abelian group M ∧ = limi∈I M/Ui is independent of this choice and is sometimes called the completion of M . Any G = lim Gi as above is complete, in particular, the completion M ∧ is always complete. Definition 12.29.1 (Topological rings). Let R be a ring and let M be an Rmodule. (1) We say R is a topological ring if R is endowed with a topology such that both addition and multiplication are continuous as maps R×R → R where R × R has the product topology. In this case we say M is a topological module if M is endowed with a topology such that addition M × M → M and scalar multiplication R × M → M are continuous. (2) A homomorphism of topological modules is just a continuous R-module map. A homomorphism of topological rings is a ring homomorphism which is continuous for the given topologies. 2We include being separated as part of being complete as we’d like to have a unique limits in complete groups. There is a definition of completeness for any topological group, agreeing, modulo the separation issue, with this one in our special case.

894

12. MORE ON ALGEBRA

(3) We say M is linearly topologized if 0 has a fundamental system of neighbourhoods consisting of submodules. We say R is linearly topologized if 0 has a fundamental system of neighbourhoods consisting of ideals. (4) If R is linearly topologized, we say that I ⊂ R is an ideal of definition if I is open and if every neighbourhood of 0 contains I n for some n. (5) If R is linearly topologized, we say that R is pre-admissible if R has an ideal of definition. (6) If R is linearly topologized, we say that R is admissible if it is preadmissible and complete3. (7) If R is linearly topologized, we say that R is pre-adic if there exists an ideal of definition I such that {I n }n≥0 forms a fundamental system of neighbourhoods of 0. (8) If R is linearly topologized, we say that R is adic if R is pre-adic and complete. Note that a (pre)adic ring is the same thing as a (pre)admissible ring which has an ideal of definition I such that I n is open for all n ≥ 1. Let R be a ring and let M be an R-module. Let I ⊂ R be an ideal. Then we can consider the linear topology on R which has {I n }n≥0 as a fundamental system of neighbourhoods of 0. This topology is called the I-adic topology; R is a pre-adic toplogical ring in the I-adic topology4. Moreover, the linear topology on M which has {I n M }n≥0 as a fundamental system of open neighbourhoods of 0 turns M into a topological R-module. This is called the I-adic topology on M . We see that M is I-adically complete (as defined in Algebra, Definition 7.91.5) if and only M is complete in the I-adic topology5. In particular, we see that R is I-adically complete if and only if R is an adic topological ring in the I-adic topology. As a special case, note that the discrete topology is the 0-adic topology and that any ring in the discrete topology is adic. Lemma 12.29.2. Let ϕ : R → S be a ring map. Let I ⊂ R and J ⊂ S be ideals and endow R with the I-adic topology and S with the J-adic topology. Then ϕ is a homomorphism of topological rings if and only if ϕ(I n ) ⊂ J for some n ≥ 1. Proof. Omitted.



12.30. Formally smooth maps of topological rings There is a version of formal smoothness which applies to homomorphisms of topological rings. Definition 12.30.1. Let R → S be a homomorphism of topological rings with R and S linearly topologized. We say S is formally smooth over R if for every 3By our conventions this includes separated. 4Thus the I-adic topology is sometimes called the I-pre-adic topology. 5 It may happen that the I-adic completion M ∧ is not I-adically complete, even though M ∧

is always complete with respect to the limit topology. If I is finitely generated then the I-adic topology and the limit topology on M ∧ agree, see Algebra, Lemma 7.91.7 and its proof.

12.30. FORMALLY SMOOTH MAPS OF TOPOLOGICAL RINGS

commutative solid diagram SO

895

/ A/J O

! /A R of homomorphisms of topological rings where A is a discrete ring and J ⊂ A is an ideal of square zero, a dotted arrow exists which makes the diagram commute. We will mostly use this notion when given ideals m ⊂ R and n ⊂ S and we endow R with the m-adic topology and S with the n-adic topology. Continuity of ϕ : R → S holds if and only if ϕ(mm ) ⊂ n for some m ≥ 1, see Lemma 12.29.2. It turns out that in this case only the topology on S is relevant. Lemma 12.30.2. Let ϕ : R → S be a ring map. (1) If R → S is formally smooth in the sense of Algebra, Definition 7.128.1, then R → S is formally smooth for any linear topology on R and any pre-adic topology on S such that R → S is continuous. (2) Let n ⊂ S and m ⊂ R ideals such that ϕ is continuous for the m-adic topology on R and the n-adic topology on S. Then the following are equivalent (a) ϕ is formally smooth for the m-adic topology on R and the n-adic topology on S, and (b) ϕ is formally smooth for the discrete topology on R and the n-adic topology on S. Proof. Assume R → S is formally smooth in the sense of Algebra, Definition 7.128.1. If S has a pre-adic topology, then there exists an ideal n ⊂ S such that S has the n-adic topology. Suppose given a solid commutative diagram as in Definition 12.30.1. Continuity of S → A/J means that nk maps to zero in A/J for some k ≥ 1, see Lemma 12.29.2. We obtain a ring map ψ : S → A from the assumed formal smoothness of S over R. Then ψ(nk ) ⊂ J hence ψ(n2k ) = 0 as J 2 = 0. Hence ψ is continuous by Lemma 12.29.2. This proves (1). The proof of (2)(b) ⇒ (2)(a) is the same as the proof of (1). Assume (2)(a). Suppose given a solid commutative diagram as in Definition 12.30.1 where we use the discrete topology on R. Since ϕ is continuos we see that ϕ(mn ) ⊂ n for some m ≥ 1. As S → A/J is continuous we see that nk maps to zero in A/J for some k ≥ 1. Hence mnk maps in J under the map R → A. Thus m2nk maps to zero in A and we see that R → A is continuous in the m-adic topology. Thus (2)(a) gives a dotted arrow as desired.  Definition 12.30.3. Let R → S be a ring map. Let n ⊂ S be an ideal. If the equivalent conditions (2)(a) and (2)(b) of Lemma 12.30.2 hold, then we say R → S is formally smooth for the n-adic topology. This property is inherited by the completions. Lemma 12.30.4. Let (R, m) and (S, n) be rings endowed with finitely generated ideals. Endow R and S with the m-adic and n-adic topologies. Let R → S be a homomorphism of topological rings. The following are equivalent (1) R → S is formally smooth for the n-adic topology, (2) R → S ∧ is formally smooth for the n∧ -adic topology,

896

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(3) R∧ → S ∧ is formally smooth for the n∧ -adic topology. Here R∧ and S ∧ are the m-adic and n-adic completions of R and S. Proof. The assumption that m is finitely generated implies that R∧ is mR∧ -adically complete, that mR∧ = m∧ and that R∧ /mn R∧ = R/mn , see Algebra, Lemma 7.91.7 and its proof. Similarly for (S, n). Thus it is clear that diagrams as in Definition 12.30.1 for the cases (1), (2), and (3) are in 1-to-1 correspondence.  The advantage of working with adic rings is that one gets a stronger lifting property. Lemma 12.30.5. Let R → S be a ring map. Let n be an ideal of S. Assume that R → S is formally smooth in the n-adic topology. Consider a solid commutative diagram / A/J SO ψ O ! /A R of homomorphisms of topological rings where A is adic and A/J is the quotient (as topological ring) of A by a closed ideal J ⊂ A such that J t is contained in an ideal of definition of A for some t ≥ 1. Then there exists a dotted arrow in the category of topological rings which makes the diagram commute. Proof. Let I ⊂ A be an ideal of definition so that I ⊃ J t for some n. Then A = lim A/I n and A/J = lim A/J + I n because J is assumed closed. Consider the following diagram of discrete R algebras An,m = A/J n + I m : A/J 3 + I 3

/ A/J 2 + I 3

/ A/J + I 3

 A/J 3 + I 2

 / A/J 2 + I 2

 / A/J + I 2

 A/J 3 + I

 / A/J 2 + I

 / A/J + I

Note that each of the commutative squares defines a surjection An+1,m+1 −→ An+1,m ×An,m An,m+1 of R-algebras whose kernel has square zero. We will inductively construct R-algebra maps ϕn,m : S → An,m . Namely, we have the maps ϕ1,m = ψ mod J + I m . Note that each of these maps is continuous as ψ is. We can inductively choose the maps ϕn,1 by starting with our choice of ϕ1,1 and lifting up, using the formal smoothness of S over R, along the right column of the diagram above. We construct the remaining maps ϕn,m by induction on n + m. Namely, we choose ϕn+1,m+1 by lifting the pair (ϕn+1,m , ϕn,m+1 ) along the displayed surjection above (again using the formal smoothness of S over R). In this way all of the maps ϕn,m are compatible with the transition maps of the system. As J t ⊂ I we see that for example ϕn = ϕnt,n mod I n induces a map S → A/I n . Taking the limit ϕ = lim ϕn we obtain a map S → A = lim A/I n . The composition into A/J agrees with ψ as we have seen that A/J = lim A/J + I n . Finally we show that ϕ is continuous. Namely, we know that ψ(nr ) ⊂ J + I r /J for some r by our assumption that ψ is

12.30. FORMALLY SMOOTH MAPS OF TOPOLOGICAL RINGS

897

a morphism of topological rings, see Lemma 12.29.2. Hence ϕ(nr ) ⊂ J + I hence ϕ(nrt ) ⊂ I as desired.  Lemma 12.30.6. Let R → S be a ring map. Let n ⊂ n0 ⊂ S be ideals. If R → S is formally smooth for the n-adic topology, then R → S is formally smooth for the n0 -adic topology. Proof. Omitted.



Lemma 12.30.7. A composition of formally smooth continuous homomorphisms of linearly topologized rings is formally smooth. Proof. Omitted. (Hint: This is completely formal, and follows from considering a suitable diagram.)  Lemma 12.30.8. Let R, S be rings. Let n ⊂ S be an ideal. Let R → S be formally smooth for the n-adic topology. Let R → R0 be any ring map. Then R0 → S 0 = S ⊗R R0 is formally smooth in the n0 = nS 0 -adic topology. Proof. Let a solid diagram SO

/ S0 O

R

/ R0

/ A/J O !(

/A

as in Definition 12.30.1 be given. Then the composition S → S 0 → A/J is continuous. By assumption the longer dotted arrow exists. By the universal property of tensor product we obtain the shorter dotted arrow.  We have seen descent for formal smoothness along faithfully flat ring maps in Algebra, Lemma 7.128.15. Something similar holds in the current setting of topological rings. However, here we just prove the following very simple and easy to prove version which is already quite useful. Lemma 12.30.9. Let R, S be rings. Let n ⊂ S be an ideal. Let R → R0 be a ring map. Set S 0 = S ⊗R R0 and n0 = nS. If (1) the map R → R0 embeds R as a direct summand of R0 as an R-module, and (2) R0 → S 0 is formally smooth for the n0 -adic topology, then R → S is formally smooth in the n-adic topology. Proof. Let a solid diagram

/ A/J O

SO

/A R 0 as in Definition 12.30.1 be given. Set A = A ⊗R R0 and J 0 = Im(J ⊗R R0 → A0 ). The base change of the diagram above is the diagram SO 0 ψ

R0

/ A0 /J 0 O

0

"

/ A0

898

12. MORE ON ALGEBRA

with continuous arrows. By condition (2) we obtain the dotted arrow ψ 0 : S 0 → A0 . Using condition (1) choose a direct summand decomposition R0 = R ⊕ C as Rmodules. (Warning: C isn’t an ideal in R0 .) Then A0 = A ⊕ A ⊗R C. Set J 00 = Im(J ⊗R C → A ⊗R C) ⊂ J 0 ⊂ A0 . Then J 0 = J ⊕ J 00 as A-modules. The image of the composition ψ : S → A0 of ψ 0 with S → S 0 is contained in A + J 0 = A ⊕ J 00 . However, in the ring A + J 0 = A ⊕ J 00 the A-submodule J 00 is an ideal! (Use that J 2 = 0.) Hence the composition S → A + J 0 → (A + J 0 )/J 00 = A is the arrow we were looking for.  The following lemma will be improved on in Section 12.32. Lemma 12.30.10. Let k be a field and let (A, m, K) be a Noetherian local k-algebra. If k → A is formally smooth for the m-adic topology, then A is a regular local ring. Proof. Let k0 ⊂ k be the prime field. Then k0 is perfect, hence k/k0 is separable, hence formally smooth by Algebra, Lemma 7.142.6. By Lemmas 12.30.2 and 12.30.7 we see that k0 → A is formally smooth for the m-adic topology on A. Hence we may assume k = Q or k = Fp . By Algebra, Lemmas 7.91.4 and 7.103.8 it suffices to prove the completion A∧ is regular. By Lemma 12.30.4 we may replace A by A∧ . Thus we may assume that A is a Noetherian complete local ring. By the Cohen structure theorem (Algebra, Theorem 7.144.8) there exist a map K → A. As k is the prime field we see that K → A is a k-algebra map. Let x1 , . . . , xn ∈ m be elements whose images form a basis of m/m2 . Set T = K[[X1 , . . . , Xn ]]. Note that A/m2 ∼ = K[x1 , . . . , xn ]/(xi xj ) and T /m2T ∼ = K[X1 , . . . , Xn ]/(Xi Xj ). Let A/m2 → T /m2T be the local K-algebra isomorphism given by mapping the class of xi to the class of Xi . Denote f1 : A → T /m2T the composition of this isomorphism with the quotient map A → A/m2 . The assumption that k → A is formally smooth in the m-adic topology means we can lift f1 to a map f2 : A → T /m3T , then to a map f3 : A → T /m4T , and so on, for all n ≥ 1. Warning: the maps fn are continuous k-algebra maps and may not be K-algebra maps. We get an induced map f : A → T = lim T /mnT of local k-algebras. By our choice of f1 , the map f induces an isomorphism m/m2 → mT /m2T hence each fn is surjective and we conclude f is surjective as A is complete. This implies dim(A) ≥ dim(T ) = n. Hence A is regular by definition. (It also follows that f is an isomorphism.)  The following result will be improved on in Section 12.32 Lemma 12.30.11. Let k be a field. Let (A, m, K) be a regular local k-algebra such that K/k is separable. Then k → A is formally smooth in the m-adic topology. Proof. It suffices to prove that the completion of A is formally smooth over k, see Lemma 12.30.4. Hence we may assume that A is a complete local regular kalgebra with residue field K separable over k. Since K is formally smooth over k

12.31. SOME RESULTS ON POWER SERIES RINGS

899

by Algebra, Proposition 7.142.8 we can successively find maps K ... s

/ A/m4

u

/ A/m3

z

 / A/m2

/K

of k-algebras. Since A is complete this defines a k-algebra map K → A. Pick a1 , . . . , an ∈ m which map to a K-basis of m/m2 . Consider the K-algebra map c : K[[x1 , . . . , xn ]] −→ A which maps xi to ai (existence of c follows from the universal property of the powerseries ring). By construction the maps K[[x1 , . . . , xn ]] → A/me are surjective for all e ≥ 1. Since K[[x1 , . . . , xn ]] is complete we see that c is surjective. Since dim(A) = n as A is regular and since K[[x1 , . . . , xn ]] is a domain of dimension n we see that the kernel of c is zero. Hence c is an isomorphism. We win because the power series ring K[[x1 , . . . , xn ]] is formally smooth over k. Namely, K is formally smooth over k and K[x1 , . . . , xn ] is formally smooth over K as a polynomial algebra. Hence K[x1 , . . . , xn ] is formally smooth over k by Algebra, Lemma 7.128.3. It follows that k → K[x1 , . . . , xn ] is formally smooth for the (x1 , . . . , xn )-adic topology by Lemma 12.30.2. Finally, it follows that k → K[[x1 , . . . , xn ]] is formally smooth for the (x1 , . . . , xn )-adic topology by Lemma 12.30.4.  Lemma 12.30.12. Let A → B be a finite type ring map with A Noetherian. Let q ⊂ B be a prime ideal lying over p ⊂ A. The following are equivalent (1) A → B is smooth at q, and (2) Ap → Bq is formally smooth in the q-adic topology. Proof. The implication (2) ⇒ (1) follows from Algebra, Lemma 7.131.2. Conversely, if A → B is smooth at q, then A → Bg is smooth for some g ∈ B, g 6∈ q. Then A → Bg is formally smooth by Algebra, Proposition 7.128.13. Hence Ap → Bq is formally smooth as localization preserves formal smoothness (for example by the criterion of Algebra, Proposition 7.128.8 and the fact that the cotangent complex behaves well with respect to localization, see Algebra, Lemmas 7.124.10 and 7.124.11). Finally, Lemma 12.30.2 implies that Ap → Bq is formally smooth in the q-adic topology.  12.31. Some results on power series rings Questions on formally smooth maps between Noetherian local rings can often be reduced to questions on maps between power series rings. In this section we prove some helper lemmas to facilitate this kind of argument. Lemma 12.31.1. Let K be a field of characteristic 0 and A = K[[x1 , . . . , xn ]]. Let L be a field of characteristic p > 0 and B = L[[x1 , . . . , xn ]]. Let Λ be a Cohen ring. Let C = Λ[[x1 , . . . , xn ]]. (1) Q → A is formally smooth in the m-adic topology. (2) Fp → B is formally smooth in the m-adic topology. (3) Z → C is formally smooth in the m-adic topology. Proof. By the universal property of power series rings it suffices to prove:

900

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(1) Q → K is formally smooth. (2) Fp → L is formally smooth. (3) Z → Λ is formally smooth in the m-adic topology. The first two are Algebra, Proposition 7.142.8. The third follows from Algebra, Lemma 7.144.7 since for any test diagram as in Definition 12.30.1 some power of p will be zero in A/J and hence some power of p will be zero in A.  Lemma 12.31.2. Let K be a field and A = K[[x1 , . . . , xn ]]. Let Λ be a Cohen ring and let B = Λ[[x1 , . . . , xn ]]. (1) If y1 , . . . , yn ∈ A is a regular system of parameters then K[[y1 , . . . , yn ]] → A is an isomorphism. (2) If z1 , . . . , zr ∈ A form part of a regular system of parameters for A, then r ≤ n and A/(z1 , . . . , zr ) ∼ = K[[y1 , . . . , yn−r ]]. (3) If p, y1 , . . . , yn ∈ B is a regular system of parameters then Λ[[y1 , . . . , yn ]] → B is an isomorphism. (4) If p, z1 , . . . , zr ∈ B form part of a regular system of parameters for B, then r ≤ n and B/(z1 , . . . , zr ) ∼ = Λ[[y1 , . . . , yn−r ]]. Proof. Proof of (1). Set A0 = K[[y1 , . . . , yn ]]. It is clear that the map A0 → A induces an isomorphism A0 /mnA0 → A/mnA for all n ≥ 1. Since A and A0 are both complete we deduce that A0 → A is an isomorphism. Proof of (2). Extend z1 , . . . , zr to a regular system of parameters z1 , . . . , zr , y1 , . . . , yn−r of A. Consider the map A0 = K[[z1 , . . . , zr , y1 , . . . , yn−r ]] → A. This is an isomorphism by (1). Hence (2) follows as it is clear that A0 /(z1 , . . . , zr ) ∼ = K[[y1 , . . . , yn−r ]]. The proofs of (3) and (4) are exactly the same as the proofs of (1) and (2).  Lemma 12.31.3. Let A → B be a local homomorphism of Noetherian complete local rings. Then there exists a commutative diagram SO

/B O

R

/A

with the following properties: (1) the horizontal arrows are surjective, (2) if the characteristic of A/mA is zero, then S and R are power series rings over fields, (3) if the characteristic of A/mA is p > 0, then S and R are power series rings over Cohen rings, and (4) R → S maps a regular system of parameters of R to part of a regular system of parameters of S. In particular R → S is flat (see Algebra, Lemma 7.120.2) with regular fibre S/mR S (see Algebra, Lemma 7.99.3). Proof. Use the Cohen structure theorem (Algebra, Theorem 7.144.8) to choose a surjection S → B as in the statement of the lemma where we choose S to be a power series over a Cohen ring if the residue characteristic is p > 0 and a power series over a field else. Let J ⊂ S be the kernel of S → B. Next, choose a surjection R = Λ[[x1 , . . . , xn ]] → A where we choose Λ to be a Cohen ring if the residue characteristic of A is p > 0 and Λ equal to the residue field of A otherwise.

12.32. GEOMETRIC REGULARITY AND FORMAL SMOOTHNESS

901

We lift the composition Λ[[x1 , . . . , xn ]] → A → B to a map ϕ : R → S. This is possible because Λ[[x1 , . . . , xn ]] is formally smooth over Z in the m-adic topology (see Lemma 12.31.1) by an application of Lemma 12.30.5. Finally, we replace ϕ by the map ϕ0 : R = Λ[[x1 , . . . , xn ]] → S 0 = S[[y1 , . . . , yn ]] with ϕ0 |Λ = ϕ|Λ and ϕ0 (xi ) = ϕ(xi ) + yi . We also replace S → B by the map S 0 → B which maps yi to zero. After this replacement it is clear that a regular system of parameters of R maps to part of a regular sequence in S 0 and we win.  12.32. Geometric regularity and formal smoothness In this section we combine the results of the previous sections to prove the following characterization of geometrically regular local rings over fields. We then recycle some of our arguments to prove a characterization of formally smooth maps in the m-adic topology between Noetherian local rings. Theorem 12.32.1. Let k be a field. Let (A, m, K) be a Noetherian local k-algebra. If the characteristic of k is zero then the following are equivalent (1) A is a regular local ring, and (2) k → A is formally smooth in the m-adic topology. If the characteristic of k is p > 0 then the following are equivalent (1) A is geometrically regular over k, (2) k → A is formally smooth in the m-adic topology. (3) for all k ⊂ k 0 ⊂ k 1/p finite over k the ring A ⊗k k 0 is regular, (4) A is regular and the canonical map H1 (LK/k ) → m/m2 is injective, and (5) A is regular and the map Ωk/Fp ⊗k K → ΩA/Fp ⊗A K is injective. Proof. If the characteristic of k is zero, then the equivalence of (1) and (2) follows from Lemmas 12.30.10 and 12.30.11. If the characteristic of k is p > 0, then it follows from Proposition 12.28.1 that (1), (3), (4), and (5) are equivalent. Assume (2) holds. By Lemma 12.30.8 we see that k 0 → A0 = A ⊗k k 0 is formally smooth for the m0 = mA-adic topology. Hence if k ⊂ k 0 is finite purely inseparable, then A0 is a regular local ring by Lemma 12.30.10. Thus we see that (1) holds. Finally, we will prove that (5) implies (2). Choose a solid diagram AO

¯ ψ

/ B/J O π

i

k

ϕ

!

/B

as in Definition 12.30.1. As J 2 = 0 we see that J has a canonical B/J module structure and via ψ¯ an A-module structure. As ψ¯ is continuous for the m-adic topology we see that mn J = 0 for some n. Hence we can filter J by B/J-submodules 0 ⊂ J1 ⊂ J2 ⊂ . . . ⊂ Jn = J such that each quotient Jt+1 /Jt is annihilated by m. Considering the sequence of ring maps B → B/J1 → B/J2 → . . . → B/J we see that it suffices to prove the existence of the dotted arrow when J is annihilated by m, i.e., when J is a K-vector space. Assume given a diagram as above such that J is annihilated by m. By Lemma 12.30.11 we see that Fp → A is formally smooth in the m-adic topology. Hence

902

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¯ Then ψ ◦ i, ϕ : k → B we can find a ring map ψ : A → B such that π ◦ ψ = ψ. are two maps whose compositions with π are equal. Hence D = ψ ◦ i − ϕ : k → J is a derivation. By Algebra, Lemma 7.123.3 we can write D = ξ ◦ d for some klinear map ξ : Ωk/Fp → J. Using the K-vector space structure on J we extend ξ to a K-linear map ξ 0 : Ωk/Fp ⊗k K → J. Using (5) we can find a K-linear map ξ 00 : ΩA/Fp ⊗A K whose restriction to Ωk/Fp ⊗k K is ξ 0 . Write ξ 00

d

D0 : A − → ΩA/Fp → ΩA/Fp ⊗A K −→ J. Finally, set ψ 0 = ψ − D0 : A → B. The reader verifies that ψ 0 is a ring map such that π ◦ ψ 0 = ψ¯ and such that ψ 0 ◦ i = ϕ as desired.  Example 12.32.2. Let k be a field of characteristic p > 0. Suppose that a ∈ k is an element which is not a pth power. A standard example of a geometrically regular local k-algebra whose residue field is purely inseparable over k is the ring A = k[x, y](x,yp −a) /(y p − a − x) Namely, A is a localization of a smooth algebra over k hence k → A is formally smooth, hence k → A is formally smooth for the m-adic topology. A closely related example is the following. Let k = Fp (s) and K = Fp (t)perf . We claim the ring map k −→ A = K[[x]], s 7−→ t + x is formally smooth for the (x)-adic topology on A. Namely, Ωk/Fp is 1-dimensional with basis ds. It maps to the element dx + dt = dx in ΩA/Fp . We leave it to the reader to show that ΩA/Fp is free on dx as an A-module. Hence we see that condition (5) of Theorem 12.32.1 holds and we conclude that k → A is formally smooth in the (x)-adic topology. Lemma 12.32.3. Let A → B be a local homomorphism of Noetherian local rings. Assume A → B is formally smooth in the mB -adic topology. Then A → B is flat. Proof. We may assume that A and B a Noetherian complete local rings by Lemma 12.30.4 and Algebra, Lemma 7.91.10 (this also uses Algebra, Lemma 7.36.8 and 7.91.4 to see that flatness of the map on completions implies flatness of A → B). Choose a commutative diagram /B SO O R

/A

as in Lemma 12.31.3 with R → S flat. Let I ⊂ R be the kernel of R → A. Because B is formally smooth over A we see that the A-algebra map S/IS −→ B has a section, see Lemma 12.30.5. Hence B is a direct summand of the flat Amodule S/IS (by base change of flatness, see Algebra, Lemma 7.36.6), whence flat.  Proposition 12.32.4. Let A → B be a local homomorphism of Noetherian local rings. Let k be the residue field of A and B = B ⊗A k the special fibre. The following are equivalent (1) A → B is flat and B is geometrically regular over k,

12.32. GEOMETRIC REGULARITY AND FORMAL SMOOTHNESS

903

(2) A → B is flat and k → B is formally smooth in the mB -adic topology, and (3) A → B is formally smooth in the mB -adic topology. Proof. The equivalence of (1) and (2) follows from Theorem 12.32.1. Assume (3). By Lemma 12.32.3 we see that A → B is flat. By Lemma 12.30.8 we see that k → B is formally smooth in the mB -adic topology. Thus (2) holds. Assume (2). Lemma 12.30.4 tells us formal smoothness is preserved under completion. The same is true for flatness by Algebra, Lemma 7.91.4. Hence we may replace A and B by their respective completions and assume that A and B are Noetherian complete local rings. In this case choose a diagram /B SO O R

/A

as in Lemma 12.31.3. We will use all of the properties of this diagram without further mention. Fix a regular system of parameters t1 , . . . , td of R with t1 = p in case the characteristic of k is p > 0. Set S = S ⊗R k. Consider the short exact sequence 0→J →S→B→0 Since B is flat over A we see that J ⊗R k is the kernel of S → B. As B and S are regular we see that J ⊗R k is generated by elements x1 , . . . , xr which form part of a regular system of parameters of S, see Algebra, Lemma 7.99.4. Lift these elements to x1 , . . . , xr ∈ J. Then t1 , . . . , td , x1 , . . . , xr is part of a regular system of parameters for S. Hence S/(x1 , . . . , xr ) is a power series ring over a field (if the characteristic of k is zero) or a power series ring over a Cohen ring (if the characteristic of k is p > 0), see Lemma 12.31.2. Moreover, it is still the case that R → S/(x1 , . . . , xr ) maps t1 , . . . , td to a part of a regular system of parameters of S/(x1 , . . . , xr ). In other words, we may replace S by S/(x1 , . . . , xr ) and assume we have a diagram /B SO O R

/A

as in Lemma 12.31.3 with moreover S = B. In this case the map S ⊗R A −→ B is an isomorphism as it is surjective and an isomorphism on special fibres, see Algebra, Lemma 7.92.1. Thus by Lemma 12.30.8 it suffices to show that R → S is formally smooth in the mS -adic topology. Of course, since S = B, we have that S is formally smooth over k = R/mR . Choose elements y1 , . . . , ym ∈ S such that t1 , . . . , td , y1 , . . . , ym is a regular system of parameters for S. If the characteristic of k is zero, choose a coefficient field K ⊂ S and if the characteristic of k is p > 0 choose a Cohen ring Λ ⊂ S with residue field K. At this point the map K[[t1 , . . . , td , y1 , . . . , ym ]] → S (characteristic zero case) or Λ[[t2 , . . . , td , y1 , . . . , ym ]] → S (characteristic p > 0 case) is an isomorphism, see Lemma 12.31.2. From now on we think of S as the above power series ring.

904

12. MORE ON ALGEBRA

The rest of the proof is analogous to the argument in the proof of Theorem 12.32.1. Choose a solid diagram / N/J SO ¯ O ψ π

i

!

/N R as in Definition 12.30.1. As J 2 = 0 we see that J has a canonical N/J module structure and via ψ¯ a S-module structure. As ψ¯ is continuous for the mS -adic topology we see that mnS J = 0 for some n. Hence we can filter J by N/J-submodules 0 ⊂ J1 ⊂ J2 ⊂ . . . ⊂ Jn = J such that each quotient Jt+1 /Jt is annihilated by mS . Considering the sequence of ring maps N → N/J1 → N/J2 → . . . → N/J we see that it suffices to prove the existence of the dotted arrow when J is annihilated by mS , i.e., when J is a K-vector space. ϕ

Assume given a diagram as above such that J is annihilated by mS . As Q → S (characteristic zero case) or Z → S (characteristic p > 0 case) is formally smooth in the mS -adic topology (see Lemma 12.31.1), we can find a ring map ψ : S → N ¯ Since S is a power series ring in t1 , . . . , td (characteristic such that π ◦ ψ = ψ. zero) or t2 , . . . , td (characteristic p > 0) over a subring, it follows from the universal property of power series rings that we can change our choice of ψ so that ψ(ti ) equals ϕ(ti ) (automatic for t1 = p in the characteristic p case). Then ψ ◦ i and ϕ : R → N are two maps whose compositions with π are equal and which agree on t1 , . . . , td . Hence D = ψ ◦ i − ϕ : R → J is a derivation which annihilates t1 , . . . , td . By Algebra, Lemma 7.123.3 we can write D = ξ ◦ d for some R-linear map ξ : ΩR/Z → J which annihilates dt1 , . . . , dtd (by construction) and mR ΩR/Z (as J is annihilated by mR ). Hence ξ factors as a composition ξ0

ΩR/Z → Ωk/Z −→ J where ξ 0 is k-linear. Using the K-vector space structure on J we extend ξ 0 to a K-linear map ξ 00 : Ωk/Z ⊗k K −→ J. Using that S/k is formally smooth we see that Ωk/Z ⊗k K → ΩS/Z ⊗S K is injective by Theorem 12.32.1 (this is true also in the characteristic zero case as it is even true that Ωk/Z → ΩK/Z is injective in characteristic zero, see Algebra, Proposition 7.142.8). Hence we can find a K-linear map ξ 000 : ΩS/Z ⊗S K → J whose restriction to Ωk/Z ⊗k K is ξ 00 . Write d

ξ 000

D0 : S − → ΩS/Z → ΩS/Z → ΩS/Z ⊗S K −−→ J. Finally, set ψ 0 = ψ − D0 : S → N . The reader verifies that ψ 0 is a ring map such that π ◦ ψ 0 = ψ¯ and such that ψ 0 ◦ i = ϕ as desired.  As an application of the result above we prove that deformations of formally smooth algebras are unobstructed. Lemma 12.32.5. Let A be a Noetherian complete local ring with residue field k. Let B be a Noetherian complete local k-algebra. Assume k → B is formally smooth in the mB -adic topology. Then there exists a Noetherian complete local ring C and

12.32. GEOMETRIC REGULARITY AND FORMAL SMOOTHNESS

905

a local homomorphism A → C which is formally smooth in the mC -adic topology such that C ⊗A k ∼ = B. Proof. Choose a diagram SO

/B O

/A R as in Lemma 12.31.3. Let t1 , . . . , td be a regular system of parameters for R with t1 = p in case the characteristic of k is p > 0. As B and S = S⊗A k are regular we see that Ker(S → B) is generated by elements x1 , . . . , xr which form part of a regular system of parameters of S, see Algebra, Lemma 7.99.4. Lift these elements to x1 , . . . , xr ∈ S. Then t1 , . . . , td , x1 , . . . , xr is part of a regular system of parameters for S. Hence S/(x1 , . . . , xr ) is a power series ring over a field (if the characteristic of k is zero) or a power series ring over a Cohen ring (if the characteristic of k is p > 0), see Lemma 12.31.2. Moreover, it is still the case that R → S/(x1 , . . . , xr ) maps t1 , . . . , td to a part of a regular system of parameters of S/(x1 , . . . , xr ). In other words, we may replace S by S/(x1 , . . . , xr ) and assume we have a diagram SO

/B O

R

/A

as in Lemma 12.31.3 with moreover S = B. In this case R → S is formally smooth in the mS -adic topology by Proposition 12.32.4. Hence the base change C = S ⊗R A is formally smooth over A in the mC -adic topology by Lemma 12.30.8.  Remark 12.32.6. The assertion of Lemma 12.32.5 is quite strong. Namely, suppose that we have a diagram BO / A0 A of local homomorphisms of Noetherian complete local rings where A → A0 induces an isomorphism of residue fields k = A/mA = A0 /mA0 and with B ⊗A0 k formally smooth over k. Then we can extend this to a commutative diagram /B CO O A

/ A0

of local homomorphisms of Noetherian complete local rings where A → C is formally smooth in the mC -adic topology and where C ⊗A k ∼ = B ⊗A0 k. Namely, pick A → C as in Lemma 12.32.5 lifting B ⊗A0 k over k. By formal smoothness we can 0 find the arrow C → B, see Lemma 12.30.5. Denote C ⊗∧ A A the completion of 0 C ⊗A A0 with respect to the ideal C ⊗A mA0 . Note that C ⊗∧ A A is a Noetherian complete local ring (see Algebra, Lemma 7.91.9) which is flat over A0 (see Algebra, Lemma 7.92.10). We have moreover 0 (1) C ⊗∧ A A → B is surjective,

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(2) if A → A0 is surjective, then C → B is surjective, (3) if A → A0 is finite, then C → B is finite, and 0 ∼ (4) if A0 → B is flat, then C ⊗∧ A A = B. Namely, by Nakayama’s lemma for nilpotent ideals (see Algebra, Lemma 7.18.1) we see that C ⊗A k ∼ = B ⊗A0 k implies that C ⊗A A0 /mnA0 → B/mnA0 B is surjective for all n. This proves (1). Parts (2) and (3) follow from part (1). Part (4) follows from Algebra, Lemma 7.92.1. 12.33. Regular ring maps Let k be a field. Recall that a Noetherian k-algebra A is said to be geometrically regular over k if and only if A ⊗k k 0 is regular for all finite purely inseparable extensions k 0 of k, see Algebra, Definition 7.149.2. Moreover, if this is the case then A ⊗k k 0 is regular for every finitely generated field extension k ⊂ k 0 , see Algebra, Lemma 7.149.1. We use this notion in the following definition. Definition 12.33.1. A ring map R → Λ is regular if it is flat and for every prime p ⊂ R the fibre ring Λ ⊗R κ(p) = Λp /pΛp is Noetherian and geometrically regular over κ(p). If R → Λ is a ring map with Λ Noetherian, then the fibre rings are always Noetherian. Lemma 12.33.2 (Regular is a local property). Let R → Λ be a ring map with Λ Noetherian. Then R → Λ is regular if and only if the local ring maps Rp → Λq are regular for all q ⊂ Λ lying over p ⊂ R. Proof. This is true because a Noetherian ring is regular if and only if all the local rings are regular local rings, see Algebra, Definition 12.33.1 and a ring map is flat if and only if all the induced maps of local rings are flat, see Algebra, Lemma 7.36.19.  Lemma 12.33.3 (Regular maps and base change). Let R → Λ be a regular ring map. For any finite type ring map R → R0 the base change R0 → Λ ⊗R R0 is regular too. Proof. Flatness is preserved under any base change, see Algebra, Lemma 7.36.6. Consider a prime p0 ⊂ R0 lying over p ⊂ R. The residue field extension κ(p) ⊂ κ(p0 ) is finitely generated as R0 is of finite type over R. Hence the fibre ring (Λ ⊗R R0 ) ⊗R0 κ(p0 ) = Λ ⊗R κ(p) ⊗κ(p) κ(p0 ) is Noetherian by Algebra, Lemma 7.29.7 and the assumption on the fibre rings of R → Λ. Geometric regularity of the fibres is preserved by Algebra, Lemma 7.149.1.  Lemma 12.33.4 (Composition of regular maps). Let A → B → C be regular ring maps. If the fibre rings of A → C are Noetherian, then A → C is regular. Proof. Let p ⊂ A be a prime. Let κ(p) ⊂ k be a finite purely inseparable extension. We have to show that C ⊗A k is regular. By Lemma 12.33.3 we may assume that A = k and we reduce to proving that C is regular. The assumption is that B is regular and that B → C is flat with regular fibres. Then C is regular by Algebra, Lemma 7.104.8. Some details omitted. 

12.35. PERMANENCE OF PROPERTIES UNDER COMPLETION

907

Lemma 12.33.5. Let R be a ring. Let (Ai , ϕii0 ) be a directed system of smooth R-algebras. Set Λ = colim Ai . If the fibre rings Λ ⊗R κ(p) are Noetherian for all p ⊂ R, then R → Λ is regular. Proof. Note that Λ is flat over R by Algebra, Lemmas 7.36.2 and 7.127.10. Let κ(p) ⊂ k be a finite purely inseparable extension. Note that Λ ⊗R κ(p) ⊗κ(p) k = Λ ⊗R k = colim Ai ⊗R k is a colimit of smooth k-algebras, see Algebra, Lemma 7.127.4. Since each local ring of a smooth k-algebra is regular by Algebra, Lemma 7.130.3 we conclude that all local rings of Λ ⊗R k are regular by Algebra, Lemma 7.99.8. This proves the lemma.  Let’s see when a field extension defines a regular ring map. Lemma 12.33.6. Let k ⊂ K be a field extension. Then k → K is a regular ring map if and only if K is a separable field extension of k. Proof. If k → K is regular, then K is geometrically reduced over k, hence K is separable over k by Algebra, Proposition 7.142.8. Conversely, if K/k is separable, then K is a colimit of smooth k-algebras, see Algebra, Lemma 7.142.10 hence is regular by Lemma 12.33.5.  Lemma 12.33.7. Let A → B → C be ring maps. If A → C is regular and B → C is flat and surjective on spectra, then A → B is regular. Proof. By Algebra, Lemma 7.36.9 we see that A → B is flat. Let p ⊂ A be a prime. The ring map B ⊗A κ(p) → C ⊗A κ(p) is flat and surjective on spectra. Hence B ⊗A κ(p) is geometrically regular by Algebra, Lemma 7.149.3.  12.34. Ascending properties along regular ring maps This section is the analogue of Algebra, Section 7.146 but where the ring map R → S is regular. Lemma 12.34.1. Let ϕ : R → S be a ring map. Assume (1) ϕ is regular, (2) S is Noetherian, and (3) R is Noetherian and reduced. Then S is reduced. Proof. For Noetherian rings being reduced is the same as having properties (S1 ) and (R0 ), see Algebra, Lemma 7.141.3. Hence we may apply Algebra, Lemmas 7.146.4 and 7.146.5.  12.35. Permanence of properties under completion Given a Noetherian local ring A we denote A∧ the completion of A with respect to its maximal ideal. We will use without further mention that A∧ is a Noetherian complete local ring (Algebra, Lemmas 7.91.10 and 7.91.7) and that A → A∧ is flat (Algebra, Lemma 7.91.3). Lemma 12.35.1. Let A be a Noetherian local ring. Then dim(A) = dim(A∧ ). Proof. See for example Algebra, Lemma 7.104.7.



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Lemma 12.35.2. Let A be a Noetherian local ring. Then depth(A) = depth(A∧ ). Proof. See Algebra, Lemma 7.146.1.



Lemma 12.35.3. Let A be a Noetherian local ring. Then A is Cohen-Macaulay if and only if A∧ is so. Proof. A local ring A is Cohen-Macaulay if and only dim(A) = depth(A). As both of these invariants are preserved under completion (Lemmas 12.35.1 and 12.35.2) the claim follows.  Lemma 12.35.4. Let A be a Noetherian local ring. Then A is regular if and only if A∧ is so. Proof. If A∧ is regular, then A is regular by Algebra, Lemma 7.103.8. Assume A is regular. Let m be the maximal ideal of A. Then dimκ(m) m/m2 = dim(A) = dim(A∧ ) (Lemma 12.35.1). On the other hand, mA∧ is the maximal ideal of A∧ and hence mA∧ is generated by at most dim(A∧ ) elements. Thus A∧ is regular. (You can also use Algebra, Lemma 7.104.8.)  Lemma (1) (2) (3)

12.35.5. Let A be a Noetherian local ring. If A∧ is reduced, then so is A. In general A reduced does not imply A∧ is reduced. If A is Nagata, then A is reduced if and only if A∧ is reduced.

Proof. As A → A∧ is faithfully flat we have (1) by Algebra, Lemma 7.147.2. For (2) see Algebra, Example 7.111.4 (there are also examples in characteristic zero, see Algebra, Remark 7.111.5). For (3) see Algebra, Lemmas 7.145.27 and 7.145.24.  12.36. Permanence of properties under henselization Given a local ring R we denote Rh , resp. Rsh the henselization, resp. strict henselization of R, see Algebra, Definition 7.140.14. Many of the properties of R are reflected in Rh and Rsh as we will show in this section. Lemma (1) (2) (3) (4)

12.36.1. Let (R, m, κ) be a local ring. Then we have the following R → Rh → Rsh are faithfully flat ring maps, mRh = mh and mRsh = mh Rsh = msh , R/mn = Rh /mn Rh for all n, there exist elements xi ∈ Rsh such that Rsh /mn Rsh is a free R/mn -module on xi mod mn Rsh .

Proof. By construction Rh is a colimit of ´etale R-algebras, see Algebra, Lemma 7.140.12. Since ´etale ring maps are flat (Algebra, Lemma 7.133.3) we see that Rh is flat over R by Algebra, Lemma 7.36.2. As a flat local ring homomorphism is faithfully flat (Algebra, Lemma 7.36.16) we see that R → Rh is faithfully flat. The ring map Rh → Rsh is a colimit of finite ´etale ring maps, see proof of Algebra, Lemma 7.140.13. Hence the same arguments as above show that Rh → Rsh is faithfully flat. Part (2) follows from Algebra, Lemmas 7.140.12 and 7.140.13. Part (3) follows from Algebra, Lemma 7.94.1 because R/m → Rh /mRh is an isomorphism and R/mn → Rh /mn Rh is flat as a base change of the flat ring map R → Rh (Algebra, Lemma 7.36.6). Let κsep be the residue field of Rsh (it is a separable algebraic

12.36. PERMANENCE OF PROPERTIES UNDER HENSELIZATION

909

closure of κ). Choose xi ∈ Rsh mapping to a basis of κsep as a κ-vector space. Then (4) follows from Algebra, Lemma 7.94.1 in exactly the same way as above.  Lemma 12.36.2. Let (R, m, κ) be a local ring. Then (1) R → Rh , Rh → Rsh , and R → Rsh are formally ´etale, (2) R → Rh , Rh → Rsh , resp. R → Rsh are formally smooth in the mh , msh , resp. msh -topology. Proof. Part (1) follows from the fact that Rh and Rsh are directed colimits of ´etale algebras (by construction), that ´etale algebras are formally ´etale (Algebra, Lemma 7.138.2), and that colimits of formally ´etale algebras are formally etale (Algebra, Lemma 7.138.3). Part (2) follows from the fact that a formally ´etale ring map is formally smooth and Lemma 12.30.2.  Lemma 12.36.3. Let R be a local ring. The following are equivalent (1) R is Noetherian, (2) Rh is Noetherian, and (3) Rsh is Noetherian. In this case we have (a) (Rh )∧ and (Rsh )∧ are Noetherian complete local rings, (b) R∧ → (Rh )∧ is an isomorphism, (c) Rh → (Rh )∧ and Rsh → (Rsh )∧ are flat, (d) R∧ → (Rsh )∧ is formally smooth in the m(Rsh )∧ -adic topology. Proof. Since R → Rh → Rsh are faithfully flat (Lemma 12.36.1), we see that Rh or Rsh being Noetherian implies that R is Noetherian, see Algebra, Lemma 7.147.1. In the rest of the proof we assume R is Noetherian. As m ⊂ R is finitely generated it follows that mh = mRh and msh = mRsh are finitely generated, see Lemma 12.36.1. Hence (Rh )∧ and (Rsh )∧ are Noetherian by Algebra, Lemma 7.144.3. This proves (a). Note that (b) is immediate from Lemma 12.36.1. In particular we see that (Rh )∧ is flat over R, see Algebra, Lemma 7.91.4. Next, we show that Rh → (Rh )∧ is flat. Write Rh = colimi Ri as a directed colimit of localizations of ´etale R-algebras. By Algebra, Lemma 7.36.5 if (Rh )∧ is flat over each Ri , then Rh → (Rh )∧ is flat. Note that Rh = Rih (by construction). Hence Ri∧ = (Rh )∧ by part (b) is flat over Ri as desired. To finish the proof of (c) we show that Rsh → (Rsh )∧ is flat. To do this, by a limit argument as above, it suffices to show that Rsh is flat over R. Note that it follows from Lemma 12.36.1 that (Rsh )∧ is the completion of a free R-module. By Lemma 12.22.2 we see this is flat over R as desired. This finishes the proof of (c). Part (d) follows from Lemma 12.36.2 and Lemma 12.30.4.



Lemma 12.36.4. Let R be a local ring. The following are equivalent: R is reduced, the henselization Rh of R is reduced, and the strict henselization Rsh of R is reduced. Proof. The ring maps R → Rh → Rsh are faithfully flat. Hence one direction of the implications follows from Algebra, Lemma 7.147.2. Conversely, assume R is reduced. Since Rh and Rsh are filtered colimits of ´etale, hence smooth R-algebras, the result follows from Algebra, Lemma 7.146.6. 

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Lemma 12.36.5. Let R be a local ring. The following are equivalent: R is a normal domain, the henselization Rh of R is a normal domain, and the strict henselization Rsh of R is a normal domain. Proof. A preliminary remark is that a local ring is normal if and only if it is a normal domain (see Algebra, Definition 7.34.10). The ring maps R → Rh → Rsh are faithfully flat. Hence one direction of the implications follows from Algebra, Lemma 7.147.3. Conversely, assume R is normal. Since Rh and Rsh are filtered colimits of ´etale, hence smooth R-algebras, the result follows from Algebra, Lemma 7.146.7.  Lemma 12.36.6. If A → B is an ´etale ring map and q is a prime of B lying over p ⊂ A, then dim(Ap ) = dim(Bq ). Proof. Namely, because Ap → Bq is flat we have going down, and hence the inequality dim(Ap ) ≤ dim(Bq ), see Algebra, Lemma 7.104.1. On the other hand, suppose that q0 ⊂ q1 ⊂ . . . ⊂ qn is a chain of primes in Bq . Then the corresponding sequence of primes p0 ⊂ p1 ⊂ . . . ⊂ pn (with pi = qi ∩ Ap ) is chain also (i.e., no equalities in the sequence) as an ´etale ring map is quasi-finite (see Algebra, Lemma 7.133.6) and a quasi-finite ring map induces a map of spectra with discrete fibres (by definition). This means that dim(Ap ) ≥ dim(Bq ) as desired.  Lemma 12.36.7. Given any local ring R we have dim(R) = dim(Rh ) = dim(Rsh ). Proof. Since R → Rsh is faithfully flat (Lemma 12.36.1) we see that dim(Rsh ) ≥ dim(R) by going down, see Algebra, Lemma 7.104.1. For the converse, we write Rsh = colim Ri as a directed colimit of local rings Ri each of which is a localization of an ´etale R-algebra. Now if q0 ⊂ q1 ⊂ . . . ⊂ qn is a chain of prime ideals in Rsh , then for some sufficiently large i the sequence Ri ∩ q0 ⊂ Ri ∩ q1 ⊂ . . . ⊂ Ri ∩ qn is a chain of primes in Ri . Thus we see that dim(Rsh ) ≤ supi dim(Ri ). But by the result of Lemma 12.36.6 we have dim(Ri ) = dim(R) for each i and we win.  Lemma 12.36.8. Given a Noetherian local ring R we have depth(R) = depth(Rh ) = depth(Rsh ). Proof. By Lemma 12.36.3 we know that Rh and Rsh are Noetherian. Hence the lemma follows from Algebra, Lemma 7.146.1.  Lemma 12.36.9. Let R be a Noetherian local ring. The following are equivalent: R is Cohen-Macaulay, the henselization Rh of R is Cohen-Macaulay, and the strict henselization Rsh of R is Cohen-Macaulay. Proof. By Lemma 12.36.3 we know that Rh and Rsh are Noetherian, hence the lemma makes sense. Since we have depth(R) = depth(Rh ) = depth(Rsh ) and dim(R) = dim(Rh ) = dim(Rsh ) by Lemmas 12.36.8 and 12.36.7 we conclude.  Lemma 12.36.10. Let R be a Noetherian local ring. The following are equivalent: R is a regular local ring, the henselization Rh of R is a regular local ring, and the strict henselization Rsh of R is a regular local ring.

12.37. FIELD EXTENSIONS, REVISITED

911

Proof. By Lemma 12.36.3 we know that Rh and Rsh are Noetherian, hence the lemma makes sense. Let m be the maximal ideal of R. Let x1 , . . . , xt ∈ m be a minimal system of generators of m, i.e., such that the images in m/m2 form a basis over κ = R/m. Because R → Rh and R → Rsh are faithfully flat, it follows sh sh that the images xh1 , . . . , xht in Rh , resp. xsh are a minimal system 1 , . . . , xt in R h h sh sh of generators for m = mR , resp. m = mR . Regularity of R by definition means t = dim(R) and similarly for Rh and Rsh . Hence the lemma follows from the equality of dimensions dim(R) = dim(Rh ) = dim(Rsh ) of Lemma 12.36.7  Lemma 12.36.11. Let R be a Noetherian local ring. Let p ⊂ R be a prime. Then Y Y Rh ⊗R κ(p) = κ(qi ) resp. Rsh ⊗R κ(p) = κ(ri ) i=1,...,t

i=1,...,s

h

sh

where q1 , . . . , qt , resp. r1 , . . . , rs are the prime of R , resp. R lying over p. Moreover, the field extensions κ(p) ⊂ κ(qi ) resp. κ(p) ⊂ κ(qi ) are separable algebraic. Proof. We will use without further mention the results of Lemmas 12.36.1 and 12.36.3. Note that Rh /pRh , resp. Rsh /pRsh is the henselization, resp. strict henselization of R/p, see Algebra, Lemma 7.140.18 resp. Algebra, Lemma 7.140.22. Hence we may replace R by R/p and assume that R is a Noetherian local domain and that p = (0). Since Rh , resp. Rsh is Noetherian, it has finitely many minimal primes q1 , . . . , qt , resp. r1 , . . . , rs . Since R → Rh , resp. R → Rsh is flat these are exactly the primes lying over p = (0) (by going down). Finally, as R is a domain, we see that Rh , resp. Rsh is reduced, see Lemma 12.36.4. Thus we see that Rh ⊗R f.f.(R) = Rh ⊗R κ(p) resp. Rsh ⊗R f.f.(R) = Rsh ⊗R κ(p) is a reduced Noetherian ring with finitely many primes, all of which are minimal (and hence maximal). Thus these rings are Artinian and are products of their localizations at maximal ideals, each necessarily a field (see Algebra, Proposition 7.58.6 and Algebra, Lemma 7.24.3). The final statement follows from the fact that R → Rh , resp. R → Rsh is a colimit of ´etale ring maps and hence the induced residue field extensions are colimits of finite separable extensions, see Algebra, Lemma 7.133.5.  12.37. Field extensions, revisited In this section we study some peculiarities of field extensions in characteristic p > 0. Definition 12.37.1. Let p be a prime number. Let k → K be an extension of fields of characteristic p. Denote kK p the compositum of k and K p in K. (1) A {xi } ⊂ K is called p-independent over k if the elements xE = Q subset ei xi where 0 ≤ ei < p are linearly independent over kK p . (2) A subset {xi } of K is called a p-basis of K over k if the elements xE form a basis of K over kK p . This is related to the notion of a p-basis of a Fp -algebra which we will discuss later (insert future reference here). Lemma 12.37.2. Let k ⊂ K be a field extension. Assume k has characteristic p > 0. Let {xi } be a subset of K. The following are equivalent (1) the elements {xi } are p-independent over k, and (2) the elements dxi are K-linearly independent in ΩK/k .

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Any p-independent collection can be extended to a p-basis of K over k. In particular, the field K has a p-basis over k. Moreover, the following are equivalent: (a) {xi } is a p-basis of K over k, and (b) dxi is a basis of the K-vector space ΩK/k . P Proof. Assume (2) and suppose that aE xE = 0 is a linear relation with aE ∈ p kK . Let θi : K → K be a k-derivation such that θi (xj ) = δij (Kronecker delta). Note that any k-derivation of K annihilates kK p . Applying θi to the given relation we obtain new relations X ei aE xe11 . . . xiei −1 . . . xenn = 0 E,ei >0 P P Hence if we pick aE xE as the relation with minimal total degree |E| = ei for some aE 6= 0, then we get a contradiction. Hence (2) holds. p p If {xi } is a p-basis for K over k, then K ∼ = kK p [Xi ]/(Xi − xi ). Hence we see that dxi forms a basis for ΩK/k over K. Thus (a) implies (b). Let {xi } be a p-independent subset of K over k. An application of Zorn’s lemma shows that we can enlarge this to a maximal p-independent subset of K over k. We claim that any maximal p-independent subset {xi } of K is a p-basis of K over k. The claim will imply that (1) implies (2) and establish the existence of p-bases. To prove the claim let L be the subfield of K generated by kK p and the xi . We have to show that L = K. If x ∈ K but x 6∈ L, then xp ∈ L and L(x) ∼ = L[z]/(z p − x). Hence {xi } ∪ {x} is p-independent over k, a contradiction. Finally, we have to show that (b) implies (a). By the equivalence of (1) and (2) we see that {xi } is a maximal p-independent subset of K over k. Hence by the claim above it is a p-basis.  Lemma 12.37.3. Let k ⊂ K be a field extension. Let {Kα }α∈A be a collection of subfields of K with the following properties (1) k ⊂ T Kα for all α ∈ A, (2) k = α∈A Kα , (3) for α, α0 ∈ A there exists an α00 ∈ A such that Kα00 ⊂ Kα ∩ Kα0 . Then for n ≥ 1 and V ⊂ K ⊕n a K-vector space we have V ∩ k ⊕n 6= 0 if and only if V ∩ Kα⊕n 6= 0 for all α ∈ A. Proof. By induction on n. The case n = 1 follows from the assumptions. Assume the result proven for subspaces of K ⊕n−1 . Assume that V ⊂ K ⊕n has nonzero intersection with Kα⊕n for all α ∈ A. If V ∩ 0 ⊕ k ⊕n−1 is nonzero then we win. Hence we may assume this is not the case. By induction hypothesis we can find an α such that V ∩ 0 ⊕ Kα⊕n−1 is zero. Let v = (x1 , . . . , xn ) ∈ V ∩ Kα be a nonzero element. By our choice of α we see that x1 is not zero. Replace v by x−1 1 v so that v = (1, x2 , . . . , xn ). Note that if v 0 = (x01 , . . . , x0n ) ∈ V ∩ Kα , then v 0 − x01 v = 0 by our choice of α. Hence we see that V ∩ Kα⊕n = Kα v. If we choose some α0 such that Kα0 ⊂ Kα , then we see that necessarily v ∈ V ∩ Kα⊕n (by the same arguments 0 applied to α0 ). Hence \ x2 , . . . , x n ∈ Kα0 0 α ∈A,Kα0 ⊂Kα

which equals k by (2) and (3).



12.37. FIELD EXTENSIONS, REVISITED

913

Lemma 12.37.4. Let K be a field of characteristic p. Let {Kα }α∈A be a collection of subfields of K with the following properties (1) K p ⊂ K Tα for all α ∈ A, (2) K p = α∈A Kα , (3) for α, α0 ∈ A there exists an α00 ∈ A such that Kα00 ⊂ Kα ∩ Kα0 . Then (1) the intersection of the kernels of the maps ΩK/F T p → ΩK/Kα is zero, (2) for any finite extension K ⊂ L we have Lp = α∈A Lp Kα . Proof. Proof of (1). Choose a p-basis {xi } for K over Fp . Suppose that η = P 0 y dx 0 i i maps to zero in ΩK/Kα for every α ∈ A. Here the index set I is finite. i∈I By Lemma 12.37.2 this means that for every α there exists a relation X aE,α xE , aE,α ∈ Kα E

where E runs over multi-indices E = (ei )i∈I 0 with 0 ≤ On the other hand, Pei < p. Lemma 12.37.2 guarantees there is no such relation aE xE = 0 with aE ∈ K p . This is a contradiction by Lemma 12.37.3. Proof of (2). Suppose that we have a tower K ⊂ M ⊂ L of finite extensions of p p p fields. Set T Mα = M Kα and Lα = L Kα = LpMα .TThen we can first prove that p M = α∈A Mα , and after that prove that L = α∈A Lα . Hence it suffices to prove (2) for primitive field extensions having no nontrivial subfields. First, assume that L = K(θ) is separable over K. Then L is generated by θp over K, hence we may assume that θ ∈ Lp . In this case we see that Lp = K p ⊕ K p θ ⊕ . . . K p θd−1

and Lp Kα = Kα ⊕ Kα θ ⊕ . . . Kα θd−1

where d = [L : K]. Thus the conclusion is clear in this case. The other case is where L = K(θ) with θp = t ∈ K, t 6∈ K p . In this case we have Lp = K p ⊕ K p t ⊕ . . . K p tp−1

and Lp Kα = Kα ⊕ Kα t ⊕ . . . Kα tp−1

Again the result is clear.



Lemma 12.37.5. Let k be a field of characteristic p > 0. Let n, m ≥ 0. As k 0 ranges through all subfields k p ⊂ k 0 ⊂ k with [k : k 0 ] < ∞ the subfields p f.f.(k 0 [[xp1 , . . . , xpn ]][y1p , . . . , ym ]) ⊂ f.f.(k[[x1 , . . . , xd ]][y1 , . . . , ym ])

form a family of subfields as in Lemma 12.37.4. Moreover, each of the ring extenp sions k 0 [[xp1 , . . . , xpn ]][y1p , . . . , ym ] ⊂ k[[x1 , . . . , xn ]][y1 , . . . , ym ] is finite. p Proof. Write A = k[[x1 , . . . , xn ]][y1 , . . . , ym ] and A0 = k 0 [[xp1 , . . . , xpn ]][y1p , . . . , ym ]. p p 0 0 0 We also set K = f.f.(A) and K = f.f.(A ). The ring extension k [[x1 , . . . , xd ]] ⊂ k[[x1 , . . . , xd ]] is finite by Algebra, Lemma 7.91.16 which implies that A → A0 is finite. For f ∈ A we see that f p ∈ A0 . Hence K p ⊂ K 0 . Any element of K 0 can be written as a/bp with a ∈ A0 and b ∈ A nonzero. Suppose that f /g p ∈ K, f, g ∈ A, g 6= 0 is contained in K 0 for every choice of k 0 . Fix a choice of k 0 for the moment. By the above we see f /g p = a/bp for some a ∈ A0 and some nonzero b ∈ A. Hence bp f ∈ A0 . For any A0 -derivation D : A → A we see that 0 = D(bp f ) = bp D(f ) hence D(f ) = 0 as A is a domain. Taking D = ∂xi and D = ∂yj we conclude that that f ∈ k[[xp1 , . . . , xpn ]][y1p , . . . , ydp ]. Applying a k 0 -derivation θ : k → k we similarly conclude that all coefficients of f are in k 0 , i.e., f ∈ A0 . Since it is clear T 0 that A = k0 A where k 0 ranges over all subfields as in the lemma we win. 

914

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12.38. The singular locus Let R be a Noetherian ring. The regular locus Reg(X) of X = Spec(R) is the set of primes p such that Rp is a regular local ring. The singular locus Sing(X) of X = Spec(R) is the complement X \Reg(X), i.e., the set of primes p such that Rp is not a regular local ring. By the discussion preceding Algebra, Definition 7.103.6 we see that Reg(X) is stable under generalization In the section we study conditions that guarantee that Reg(X) is open. Definition 12.38.1. (1) We say R is (2) We say R is (3) We say R is

Let R be a Noetherian ring. Let X = Spec(R). J-0 if Reg(X) contains a nonempty open. J-1 if Reg(X) is open. J-2 if any finite type R-algebra is J-1.

The ring Q[x]/(x2 ) does not satisfy J-0. On the other hand J-1 implies J-0 for domains and even reduced rings as such a ring is regular at the minimal primes. Here is a characterization of the J-1 property. Lemma 12.38.2. Let R be a Noetherian ring. Let X = Spec(R). The ring R is J-1 if and only if V (p) ∩ Reg(X) contains a nonempty open subset of V (p) for all p ∈ Reg(X). Proof. This follows immediately from Topology, Lemma 5.11.5.



Lemma 12.38.3. Let R be a Noetherian ring. Let X = Spec(R). Assume that for all p ⊂ R the ring R/p is J-0. Then R is J-1. Proof. We will show that the criterion of Lemma 12.38.2 applies. Let p ∈ Reg(X) be a prime of height r. Pick f1 , . . . , fr ∈ p which map to generators of pRp . Since p ∈ Reg(X) we see that f1 , . . . , fr maps to a regular sequence in Rp , see Algebra, Lemma 7.99.3. Thus by Algebra, Lemma 7.66.8 we see that after replacing R by Rg for some g ∈ R, g 6∈ p the sequence f1 , . . . , fr is a regular sequence in R. Next, let p ⊂ q be a prime ideal such that (R/p)q is a regular local ring. By the assumption of the lemma there exists a non-empty open subset of V (p) consisting of such primes, hence it suffices to prove Rq is regular. Note that f1 , . . . , fr is a regular sequence in Rq such that Rq /(f1 , . . . , fr )Rq is regular. Hence Rq is regular by Algebra, Lemma 7.99.7.  Lemma 12.38.4. Let R → S be a ring map. Assume that (1) R is a Noetherian domain, (2) R → S is injective and of finite type, and (3) S is a domain and J-0. Then R is J-0. Proof. After replacing S by Sg for some nonzero g ∈ S we may assume that S is a regular ring. By generic flatness we may assume that also R → S is faithfully flat, see Algebra, Lemma 7.110.1. Then R is regular by Algebra, Lemma 7.147.4.  Lemma 12.38.5. Let R → S be a ring map. Assume that (1) R is a Noetherian domain and J-0, (2) R → S is injective and of finite type, and (3) S is a domain and f.f.(R) → f.f.(S) is separable. Then S is J-0.

12.39. REGULARITY AND DERIVATIONS

915

Proof. We may replace R by a principal localization and assume R is a regular ring. By Algebra, Lemma 7.130.9 the ring map R → S is smooth at (0). Hence after replacing S by a principal localization we may assume that S is smooth over R. Then S is regular too, see Algebra, Lemma 7.146.8.  Lemma (1) (2) (3) (4)

12.38.6. Let R be a Noetherian ring. The following are equivalent R is J-2, every finite type R-algebra which is a domain is J-0, every finite R-algebra is J-1, for every prime p and every finite purely inseparable extension κ(p) ⊂ L there exists a finite R-algebra R0 which is a domain, which is J-0, and whose field of fractions is L.

Proof. It is clear that we have the implications (1) ⇒ (2) and (2) ⇒ (4). Recall that a domain which is J-1 is J-0. Hence we also have the implications (1) ⇒ (3) and (3) ⇒ (4). Let R → S be a finite type ring map and let’s try to show S is J-1. By Lemma 12.38.3 it suffices to prove that S/q is J-0 for every prime q of S. In this way we see (2) ⇒ (1). Assume (4). We will show that (2) holds which will finish the proof. Let R → S be a finite type ring map with S a domain. Let p = Ker(R → S). Set K = f.f.(S). There exists a diagram of fields KO

/ K0 O

κ(p)

/L

where the horizontal arrows are finite purely inseparable field extensions and where K 0 /L is separable, see Algebra, Lemma 7.40.4. Choose R0 ⊂ L as in (4) and let S 0 be the image of the map S ⊗R R0 → K 0 . Then S 0 is a domain whose fraction field is K 0 , hence S 0 is J-0 by Lemma 12.38.5 and our choice of R0 . Then we apply Lemma 12.38.4 to see that S is J-0 as desired.  12.39. Regularity and derivations Let R → S be a ring map. Let D : R → R be a derivation. We say that D extends to S if there exists a derivation D0 : S → S such that /S SO O D0 R

D

/R

is commutative. Lemma 12.39.1. Let R be a ring. Let D : R → R be a derivation. (1) For any ideal I ⊂ R the derivation D extends canonically to a derication D∧ : R∧ → R∧ on the I-adic completion. (2) For any mulitplicative subset S ⊂ R the derivation D extends uniquely to the localization S −1 R of R.

916

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If R ⊂ R0 is an finite type extension of rings such that Rg ∼ = Rg0 for some nonzeroN 0 divisor g ∈ R, then g D extends to R for some N ≥ 0. Proof. Proof of (1). For n ≥ 2 we have D(I n ) ⊂ I n−1 by the Leibniz rule. Hence D induces maps Dn : R/I n → R/I n−1 . Taking the limit we obtain D∧ . We omit the verification that D∧ is a derivation. Proof of (2). To extend D to S −1 R just set D(r/s) = D(r)/s − rD(s)/s2 and check the axioms. Proof of the final statement. Let x1 , . . . , xn ∈ R0 be generators of R0 over R. Choose an N such that g N xi ∈ R. Consider g N +1 D. By (2) this extends to Rg . Moreover, by the Leibniz rule and our construction of the extension above we have g N +1 D(xi ) = g N +1 D(g −N g N xi ) = −N g N xi D(g) + gD(g N xi ) and both terms are in R. This implies that X g N +1 D(xe11 . . . xenn ) = ei xe11 . . . xiei −1 . . . xenn g N +1 D(xi ) is an element of R0 . Hence every element of R0 (which can be written as a sum of monomials in the xi with coefficients in R) is mapped to an element of R0 by g N +1 D and we win.  Lemma 12.39.2. Let R be a regular ring. Let f ∈ R. Assume there exists a derivation D : R → R such that D(f ) is a unit of R/(f ). Then R/(f ) is regular. Proof. It suffices to prove this when R is a local ring with maximal ideal m and residue field κ. In this case it suffices to prove that f 6∈ m2 , see Algebra, Lemma 7.99.3. However, if f ∈ m2 then D(f ) ∈ m by the Leibniz rule, a contradiction.  Lemma 12.39.3. Let R be a regular Fp -algebra. Let f ∈ R. Assume there exists a derivation D : R → R such that D(f ) is a unit of R. Then R[z]/(z p − f ) is regular. Proof. Apply Lemma 12.39.2 to the extension of D to R[z] which maps z to zero.  Lemma 12.39.4. Let p be a prime number. Let B be a domain with p = 0 in B. Let f ∈ B be an element which is not a pth power in the fraction field of B. If B is of finite type over a Noetherian complete local ring, then there exists a derivation D : B → B such that D(f ) is not zero. Proof. Let R be a Noetherian complete local ring such that there exists a finite type ring map R → B. Of course we may replace R by its image in B, hence we may assume R is a domain of characteristic p > 0 (as well as Noetherian complete local). By Algebra, Lemma 7.144.10 we can write R as a finite extension of k[[x1 , . . . , xn ]] for some field k and integer n. Hence we may replace R by k[[x1 , . . . , xn ]]. Next, we use Algebra, Lemma 7.107.7 to factor R → B as R ⊂ R[y1 , . . . , yd ] ⊂ B 0 ⊂ B with B 0 finite over R[y1 , . . . , yd ] and Bg0 ∼ = Bg for some nonzero g ∈ R. Note that f 0 = g pN f ∈ B 0 for some large integer N . It is clear that f 0 is not a pth power in f.f.(B 0 ) = f.f.(B). If we can find a derivation D0 : B 0 → B 0 with D0 (f 0 ) 6= 0, then Lemma 12.39.1 guarantees that D = g M D0 extends to S for some M > 0. Then D(f ) = g N D0 (f ) = g M D0 (g −pN f 0 ) = g M −pN D0 (f 0 ) is nonzero. Thus it suffices to prove the lemma in case B is a finite exteion of A = k[[x1 , . . . , xn ]][y1 , . . . , ym ].

12.40. FORMAL SMOOTHNESS AND REGULARITY

917

Note that df is not zero in Ωf.f.(B)/Fp , see Algebra, Lemma 7.142.1. We apply Lemma 12.37.5 to find a subfield k 0 ⊂ k of finite index such that with A0 = p ] the element df does not map to zero in Ωf.f.(B)/f.f.(A0 ) . k 0 [[xp1 , . . . , xpn ]][y1p , . . . , ym Thus we can choose a f.f.(A0 )-derivation D0 : f.f.(B) → f.f.(B) with D0 (f ) 6= 0. Since A0 ⊂ A and A ⊂ B are finite by construction we see that A0 ⊂ B is finite. Choose b1 , . . . , bt ∈ B which generate B as an A0 -module. Then D0 (bi ) = fi /gi for some fi , gi ∈ B with gi 6= 0. Setting D = g1 . . . gt D0 we win.  Lemma 12.39.5. Let A be a Noetherian complete local domain. Then A is J-0. Proof. By Algebra, Lemma 7.144.10 we can find a regular subring A0 ⊂ A with A finite over A0 . If f.f.(A0 ) ⊂ f.f.(A) is separable, then we are done by Lemma 12.38.5. If not, then A0 and A have characteristic p > 0. For any subextension f.f.(A0 ) ⊂ M ⊂ f.f.(A) there exists a finite subextension A0 ⊂ B ⊂ A such that f.f.(B) = M . Hence, arguing by induction on [f.f.(A) : f.f.(A0 )] we may assume there exists A0 ⊂ B ⊂ A such that B is J-0 and f.f.(B) ⊂ f.f.(A) has no nontrivial subextensions. In this case, if f.f.(B) ⊂ f.f.(A) is separable, then we see that A is J-0 by Lemma 12.38.5. If not, then f.f.(A) = f.f.(B)[z]/(z p − b) for some b ∈ B which is not a pth power in f.f.(B). By Lemma 12.39.4 we can find a derivation D : B → B with D(f ) 6= 0. Applying Lemma 12.39.3 we see that Ap is regular for any prime p of A lying over a regular prime of B and not containing D(f ). As B is J-0 we conclude A is too.  Proposition 12.39.6. The following types of rings are J-2: (1) fields, (2) Noetherian complete local rings, (3) Z, (4) Dedekind domains with fraction field of characteristic zero, (5) finite type ring extensions of any of the above. Proof. For fields, Z and Dedekind domains of characteristic zero you just check condition (4) of Lemma 12.38.6. In the case of Noetherian complete local rings, note that if R → R0 is finite and R is a Noetherian complete local ring, then R0 is a product of Noetherian complete local rings, see Algebra, Lemma 7.144.2. Hence it suffices to prove that a Noetherian complete local ring which is a domain is J-0, which is Lemma 12.39.5.  12.40. Formal smoothness and regularity The title of this section refers to Proposition 12.40.2. Lemma 12.40.1. Let A → B be a local homomorphism of Noetherian local rings. Let D : A → A be a derivation. Assume that B is complete and A → B is formally smooth in the mB -adic topology. Then there exists an extension D0 : B → B of D. Proof. Denote B[] = B[x]/(x2 ) the ring of dual numbers over B. Consider the ring map ψ : A → B[], a 7→ a + D(a). Consider the commutative diagram BO A

1

ψ

/B O / B[]

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By Lemma 12.30.5 and the assumption of formal smoothness of B/A we find a map ϕ : B → B[] fitting into the diagram. Write ϕ(b) = b + D0 (b). Then D0 : B → B is the desired extension.  Proposition 12.40.2. Let A → B be a local homomorphism of Noetherian complete local rings. The following are equivalent (1) A → B is regular, (2) A → B is flat and B is geometrically regular over k, (3) A → B is flat and k → B is formally smooth in the mB -adic topology, and (4) A → B is formally smooth in the mB -adic topology. Proof. We have seen the equivalence of (2), (3), and (4) in Proposition 12.32.4. It is clear that (1) implies (2). Thus we assume the equivalent conditions (2), (3), and (4) hold and we prove (1). Let p be a prime of A. We will show that B ⊗A κ(p) is geometrically regular over κ(p). By Lemma 12.30.8 we may replace A by A/p and B by B/pB. Thus we may assume that A is a domain and that p = (0). Choose A0 ⊂ A as in Algebra, Lemma 7.144.10. We will use all the properties stated in that lemma without further mention. As A0 → A induces an isomorphism on residue fields, and as B/mA B is geometrically regular over A/mA we can find a diagram /B CO O /A A0 with A0 → C formally smooth in the mC -adic topology such that B = C ⊗A0 A, see Remark 12.32.6. (Completion in the tensor product is not needed as A0 → A is finite, see Algebra, Lemma 7.91.2.) Hence it suffices to show that C ⊗A0 f.f.(A0 ) is a geometrically regular algebra over f.f.(A0 ). The upshot of the preceding paragraph is that we may assume that A = k[[x1 , . . . , xn ]] where k is a field or A = Λ[[x1 , . . . , xn ]] where Λ is a Cohen ring. In this case B is a regular ring, see Algebra, Lemma 7.104.8. Hence B ⊗A f.f.(A) is a regular ring too and we win if the characteristic of f.f.(A) is zero. Thus we are left with the case where A = k[[x1 , . . . , xn ]] and k is a field of characteristic p > 0. Set K = f.f.(A). Let L ⊃ K be a finite purely inseparable field extension. We will show by induction on [L : K] that B ⊗A L is regular. The base case is L = K which we’ve seen above. Let K ⊂ M ⊂ L be a subfield such that L is a degree p extension of M obtained by adjoining a pth root of an element f ∈ M . Let A0 be a finite A-subalgebra of M with fraction field M . Clearing denominators, we may and do assume f ∈ A0 . Set A00 = A0 [z]/(z p − f ) and note that A0 ⊂ A00 is finite and that the fraction field of A00 is L. By induction we know that B ⊗A M ring is regular. We have B ⊗A L = B ⊗A M [z]/(z p − f ) By Lemma 12.39.4 we know there exists a derivation D : A0 → A0 such that D(f ) 6= 0. As A0 → B ⊗A A0 is formally smooth in the m-adic topology by Lemma 12.30.9 we can use Lemma 12.40.1 to extend D to a derivation D0 : B ⊗A A0 → B ⊗A A0 .

12.41. G-RINGS

919

Note that D0 (f ) = D(f ) is a unit in B ⊗A M as D(f ) is not zero in A0 ⊂ M . Hence B ⊗A L is regular by Lemma 12.39.3 and we win.  12.41. G-rings Let A be a Noetherian local ring A. In Section 12.35 we have seen that some but not all properties of A are reflected in the completion A∧ of A. To study this further we introduce some terminology. For a prime q of A the fibre ring (A∧ ) ⊗A κ(q) = (A∧ )q /q(A∧ )q is called a formal fibre of A. We think of the formal fibre as an algebra over κ(q). Thus A → A∧ is a regular ring homomorphism if and only if all the formal fibres are geometrically regular algebras. Definition 12.41.1. A ring R is called a G-ring if R is Noetherian and for every prime p of R the ring map Rp → (Rp )∧ is regular. By the discussion above we see that R is a G-ring if and only if every local ring Rp has geometrically regular formal fibres. Note that if Q ⊂ R, then it suffices to check the formal fibres are regular. Another way to express the G-ring condition is described in the following lemma. Lemma 12.41.2. Let R be a Noetherian ring. Then R is a G-ring if and only if for every pair of primes q ⊂ p ⊂ R the algebra (R/q)∧ p ⊗R/q κ(q) is geometrically regular over κ(q). Proof. This follows from the fact that Rp∧ ⊗R κ(q) = (R/q)∧ p ⊗R/q κ(q) as algebras over κ(q).



Lemma 12.41.3. Let R → R0 be a finite type map of Noetherian rings and let q0

/ p0

/ R0 O

q

/p

/R

be primes. Assume R → R0 is quasi-finite at p0 . (1) If the formal fibre Rp∧ ⊗R κ(q) is geometrically regular over κ(q), then the formal fibre Rp0 0 ⊗R0 κ(q0 ) is geometrically regular over κ(q0 ). (2) If the formal fibres of Rp are geometrically regular, then the formal fibres of Rp0 0 are geometrically regular. (3) If R → R0 is quasi-finite and R is a G-ring, then R0 is a G-ring. Proof. It is clear that (1) ⇒ (2) ⇒ (3). Assume Rp∧ ⊗R κ(q) is geometrically regular over κ(q). By Algebra, Lemma 7.116.3 we see that Rp∧ ⊗R R0 = (Rp0 0 )∧ × B for some Rp∧ -algebra B. Hence Rp0 0 → (Rp0 0 )∧ is a factor of a base change of the map Rp → Rp∧ . It follows that (Rp0 0 )∧ ⊗R0 κ(q0 ) is a factor of Rp∧ ⊗R R0 ⊗R0 κ(q0 ) = Rp∧ ⊗R κ(q) ⊗κ(q) κ(q0 ).

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Thus the result follows as extension of base field preserves geometric regularity, see Algebra, Lemma 7.149.1.  Lemma 12.41.4. Let R be a Noetherian ring. Then R is a G-ring if and only if for every finite free ring map R → S the formal fibres of S are regular rings. Proof. Assume that for any finite free ring map R → S the ring S has regular formal fibres. Let q ⊂ p ⊂ R be primes and let κ(q) ⊂ L be a finite purely inseparable extension. To show that R is a G-ring it suffices to show that Rp∧ ⊗R κ(q) ⊗κ(q) L is a regular ring. Choose a finite free extension R → R0 such that q0 = qR0 is a prime and such that κ(q0 ) is isomorphic to L over κ(q), see Algebra, Lemma 7.143.2. By Algebra, Lemma 7.91.17 we have Y Rp∧ ⊗R R0 = (Rp0 0i )∧ where p0i are the primes of R0 lying over p. Thus we have Y Rp∧ ⊗R κ(q) ⊗κ(q) L = Rp∧ ⊗R R0 ⊗R0 κ(q0 ) = (Rp0 0i )∧ ⊗R0 0 κ(q0 ) p i

Our assumption is that the rings on the right are regular, hence the ring on the left is regular too. Thus R is a G-ring. The converse follows from Lemma 12.41.3.  Lemma 12.41.5. Let k be a field of characteristic p. Let A = k[[x1 , . . . , xn ]][y1 , . . . , yn ] and denote K = f.f.(A). Let p ⊂ A be a prime. Then A∧ p ⊗A K is geometrically regular over K. Proof. Let L ⊃ K be a finite purely inseparable field extension. We will show by induction on [L : K] that A∧ p ⊗ L is regular. The base case is L = K: as A is regular, A∧ is regular (Lemma 12.35.4), hence the localization A∧ p p ⊗ K is regular. Let K ⊂ M ⊂ L be a subfield such that L is a degree p extension of M obtained by adjoining a pth root of an element f ∈ M . Let B be a finite A-subalgebra of M with fraction field M . Clearing denominators, we may and do assume f ∈ B. Set C = B[z]/(z p − f ) and note that B ⊂ C is finite and that the fraction field of C is L. Since A ⊂ B ⊂ C are finite and L/M/K are purely inseparable we see that for every element of B or C some power of it lies in A. Hence there is a unique prime r ⊂ B, resp. q ⊂ C lying over p. Note that ∧ A∧ p ⊗A M = B r ⊗B M

see Algebra, Lemma 7.91.17. By induction we know that this ring is regular. In the same manner we have ∧ ∧ p A∧ p ⊗A L = Cr ⊗C L = Br ⊗B M [z]/(z − f )

the last equality because the completion of C = B[z]/(z p −f ) equals Br∧ [z]/(z p −f ). By Lemma 12.39.4 we know there exists a derivation D : B → B such that D(f ) 6= 0. In other words, g = D(f ) is a unit in M ! By Lemma 12.39.1 D extends to a derivation of Br , Br∧ and Br∧ ⊗B M (successively extending through a localization, a completion, and a localization). Since it is an extension we end up with a derivation of Br∧ ⊗B M which maps f to g and g is a unit of the ring Br∧ ⊗B M . Hence A∧ p ⊗A L is regular by Lemma 12.39.3 and we win.  Proposition 12.41.6. A Noetherian complete local ring is a G-ring.

12.41. G-RINGS

921

Proof. Let A be a Noetherian complete local ring. By Lemma 12.41.2 it suffices to check that B = A/q has geometrically regular formal fibres over the minimal prime (0) of B. Thus we may assume that A is a domain and it suffices to check the condition for the formal fibres over the minimal prime (0) of A. Set K = f.f (A). We can choose a subring A0 ⊂ A which is a regular complete local ring such that A is finite over A0 , see Algebra, Lemma 7.144.10. Moreover, we may assume that A0 is a power series ring over a field or a Cohen ring. By Lemma 12.41.3 we see that it suffices to prove the result for A0 . Assume that A is a power series ring over a field or a Cohen ring. Since A is regular the localizations Ap are regular (see Algebra, Definition 7.103.6 and the discussion preceding it). Hence the completions A∧ p are regular, see Lemma 12.35.4. ∧ Hence the fibre A∧ p ⊗A K is, as a localization of Ap , also regular. Thus we are done if the characteristic of K is 0. The positive characteristic case is the case A = k[[x1 , . . . , xd ]] which is a special case of Lemma 12.41.5.  Lemma 12.41.7. Let R be a Noetherian ring. Then R is a G-ring if and only if Rm has geometrically regular formal fibres for every maximal ideal m of R. ∧ is regular for every maximal ideal m of R. Let p be a Proof. Assume Rm → Rm ∧ is faithfully flat we prime of R and choose a maximal ideal p ⊂ m. Since Rm → Rm 0 ∧ can choose a prime p if Rm lying over pRm . Consider the commutative diagram ∧ Rm O

∧ 0 / (Rm ) O p

∧ ∧ / (Rm ) 0 O p

Rm

/ Rp

/ Rp∧

∧ ∧ 0 By assumption the ring map Rm → Rm is regular. By Proposition 12.41.6 (Rm )p → ∧ ∧ ∧ ∧ (Rm )p0 is regular. Hence Rm → (Rm )p0 is regular and since it factors through the ∧ ∧ )p0 is regular. Thus we may apply localization Rp , also the ring map Rp → (Rm ∧  Lemma 12.33.7 to see that Rp → Rp is regular.

Lemma 12.41.8. Let R be a Noetherian local ring ring which is a G-ring. Then the henselization Rh and the strict henselization Rsh are G-rings. Proof. We will use the criterion of Lemma 12.41.7. Let q ⊂ Rh be a prime and h over p, set p = R ∩ q. Set q1 = Qq and let q2 , . . . , qt be the other primes of R lying h so that R ⊗R κ(p) = i=1,...,t κ(qi ), see Lemma 12.36.11. Using that (Rh )∧ = R∧ (Lemma 12.36.3) we see Y (Rh )∧ ⊗Rh κ(qi ) = (Rh )∧ ⊗Rh (Rh ⊗R κ(p)) = R∧ ⊗R κ(p) i=1,...,t

Hence (Rh )∧ ⊗Rh κ(qi ) is geometrically regular over κ(p) by assumption. Since κ(qi ) is separable algebraic over κ(p) it follows from Algebra, Lemma 7.149.6 that (Rh )∧ ⊗Rh κ(qi ) is geometrically regular over κ(qi ). Let r ⊂ Rsh be a prime and set p = R ∩ r. Set r1 = Q r and let r2 , . . . , rs be the other primes of Rsh lying over p, so that Rsh ⊗R κ(p) = i=1,...,t κ(qi ), see Lemma 12.36.11. Then we see that Y (Rsh )∧ ⊗Rsh κ(ri ) = (Rsh )∧ ⊗Rsh (Rsh ⊗R κ(p)) = (Rsh )∧ ⊗R κ(p) i=1,...,t

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Note that R∧ → (Rsh )∧ is formally smooth in the m(Rsh )∧ -adic topology, see Lemma 12.36.3. Hence R∧ → (Rsh )∧ is regular by Proposition 12.40.2. We conclude that (Rsh )∧ ⊗Rh κ(qi ) is regular over κ(p) by Lemma 12.33.4 as R∧ ⊗R κ(p) is regular over κ(p) by assumption. Since κ(ri ) is separable algebraic over κ(p) it follows from Algebra, Lemma 7.149.6 that (Rsh )∧ ⊗Rsh κ(ri ) is geometrically regular over κ(ri ).  Lemma 12.41.9. Let p be a prime number. Let A be a Noetherian complete local domain with fraction field K of characteristic p. Let q ⊂ A[x] be a maximal ideal lying over the maximal ideal of A and let r ⊂ q be a prime lying over (0) ⊂ A. Then A[x]∧ q ⊗A[x] κ(r) is geometrically regular over κ(r). Proof. Note that K ⊂ κ(r) is finite. Hence, given a finite purely inseparable extension κ(r) ⊂ L there exists a finite extension of Noetherian complete local domains A ⊂ B such that κ(r) ⊗A B surjects onto L. Namely, you take B ⊂ L a finite A-subalgebra whose field of fractions is L. Denote r0 ⊂ B[x] the kernel of the map B[x] = A[x] ⊗A B → κ(r) ⊗A B → L so that κ(r0 ) = L. Then Y ∧ 0 0 A[x]∧ B[x]∧ q ⊗A[x] L = A[x]q ⊗A[x] B[x] ⊗B[x] κ(r ) = qi ⊗B[x] κ(r ) where q1 , . . . , qt are the primes of B[x] lying over q, see Algebra, Lemma 7.91.17. 0 Thus we see that it suffices to prove the rings B[x]∧ qi ⊗B[x] κ(r ) are regular. This ∧ reduces us to showing that A[x]q ⊗A[x] κ(r) is regular in the special case that K = κ(r). Assume K = κ(r). In this case we see that rK[x] is generated by x − f for some f ∈ K and ∧ A[x]∧ q ⊗A[x] κ(r) = (A[x]q ⊗A K)/(x − f )

The derivation D = d/dx of A[x] extends to K[x] and maps x − f to a unit of ∧ K[x]. Moreover D extends to A[x]∧ q ⊗A K by Lemma 12.39.1. As A → A[x]q is ∧ formally smooth (see Lemmas 12.30.2 and 12.30.4) the ring A[x]q ⊗A K is regular by Proposition 12.40.2 (the arguments of the proof of that proposition simplify significantly in this particular case). We conclude by Lemma 12.39.2.  Proposition 12.41.10. Let R be a G-ring. If R → S is essentially of finite type then S is a G-ring. Proof. Since being a G-ring is a property of the local rings it is clear that a localization of a G-ring is a G-ring. Conversely, if every localization at a prime is a G-ring, then the ring is a G-ring. Thus it suffices to show that Sq is a G-ring for every finite type R-algebra S and every prime q of S. Writing S as a quotient of R[x1 , . . . , xn ] we see from Lemma 12.41.3 that it suffices to prove that R[x1 , . . . , xn ] is a G-ring. By induction on n it suffices to prove that R[x] is a G-ring. Let q ⊂ R[x] be a maximal ideal. By Lemma 12.41.7 it suffices to show that R[x]q −→ R[x]∧ q is regular. If q lies over p ⊂ R, then we may replace R by Rp . Hence we may assume that R is a Noetherian local G-ring with maximal ideal m and that q ⊂ R[x] lies over m. Note that there is a unique prime q0 ⊂ R∧ [x] lying over q. Consider the

12.41. G-RINGS

923

diagram R[x]∧ O q

/ (R∧ [x]q0 )∧ O

R[x]q

/ R∧ [x]q0

Since R is a G-ring the lower horizontal arrow is regular (as a localization of a base change of the regular ring map R → R∧ ). Suppose we can prove the right vertical arrow is regular. Then it follows that the composition R[x]q → (R∧ [x]q0 )∧ is regular, and hence the left vertical arrow is regular by Lemma 12.33.7. Hence we see that we may assume R is a Noetherian complete local ring and q a prime lying over the maximal ideal of R. Let R be a Noetherian complete local ring and let q ⊂ R[x] be a maximal ideal lying over the maximal ideal of R. Let r ⊂ q be a prime ideal. We want to show that R[x]∧ q ⊗R[x] κ(r) is a geometrically regular algebra over κ(r). Set p = R ∩ r. Then we can replace R by R/p and q and r by their images in R/p[x], see Lemma 12.41.2. Hence we may assume that R is a domain and that r ∩ R = (0). By Algebra, Lemma 7.144.10 we can find R0 ⊂ R which is regular and such that R is finite over R0 . Applying Lemma 12.41.3 we see that it suffices to prove R[x]∧ q ⊗R[x] κ(r) is geometrically regular over κ(r) when, in addition to the above, R is a regular complete local ring. Now R is a regular complete local ring, we have q ⊂ r ⊂ R[x], we have (0) = R ∩ r and q is a maximal ideal lying over the maximal ideal of R. Since R is regular the ring R[x] is regular (Algebra, Lemma 7.146.8). Hence the localization R[x]q is regular. Hence the completions R[x]∧ q are regular, see Lemma 12.35.4. Hence the ⊗ κ(r) is, as a localization of R[x]∧ fibre R[x]∧ R[x] q , also regular. Thus we are done q if the characteristic of f.f.(R) is 0. If the characteristic of R is positive, then R = k[[x1 , . . . , xn ]]. In this case we split the argument in two subcases: (1) The case r = (0). The result is a direct consequence of Lemma 12.41.5. (2) The case r 6= (0). This is Lemma 12.41.9.  Remark 12.41.11. Let R be a G-ring and let I ⊂ R be an ideal. In general it is not the case that the I-adic completion R∧ is a G-ring. An example was given by Nishimura in [Nis81]. A generalization and, in some sense, clarification of this example can be found in the last section of [Dum00]. Proposition 12.41.12. The following types of rings are G-rings: (1) fields, (2) Noetherian complete local rings, (3) Z, (4) Dedekind domains with fraction field of characteristic zero, (5) finite type ring extensions of any of the above. Proof. For fields, Z and Dedekind domains of characteristic zero this follows immediately from the definition and the fact that the completion of a discrete valuation ring is a discrete valuation ring. A Noetherian complete local ring is a

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G-ring by Proposition 12.41.6. The statement on finite type overrings is Proposition 12.41.10.  12.42. Excellent rings In this section we discuss Grothendieck’s notion of excellent rings. For the definitions of G-rings, J-2 rings, and universally catenary rings we refer to Definition 12.41.1, Definition 12.38.1, and Algebra, Definition 7.98.5. Definition 12.42.1. Let R be a ring. (1) We say R is quasi-excellent if R is Noetherian, a G-ring, and J-2. (2) We say R is excellent if R is quasi-excellent and universally catenary. Thus a Noetherian ring is quasi-excellent if it has geometrically regular formal fibres and if any finite type algebra over it has closed singular set. For such a ring to be excellent we require in addition that there exists (locally) a good dimension function. Lemma 12.42.2. Any localization of a finite type ring over a (quasi-)excellent ring is (quasi-)excellent. Proof. For finite type algebras this follows from the definitions for the properties J-2 and universally catenary. For G-rings, see Proposition 12.41.10. We omit the proof that localization preserves (quasi-)excellency.  Lemma 12.42.3. A quasi-excellent ring is Nagata. Proof. Let R be quasi-excellent. Using that a finite type algebra over R is quasiexcellent (Lemma 12.42.2) we see that it suffices to show that any quasi-excellent domain is N-1, see Algebra, Lemma 7.145.17. Applying Algebra, Lemma 7.145.29 (and using that a quasi-excellent ring is J-2) we reduce to showing that a quasiexcellent local domain R is N-1. As R → R∧ is regular we see that R∧ is reduced by Lemma 12.34.1. In other words, R is analytically unramified. Hence R is N-1 by Algebra, Lemma 7.145.24.  Proposition 12.42.4. The following types of rings are excellent: (1) (2) (3) (4) (5)

fields, Noetherian complete local rings, Z, Dedekind domains with fraction field of characteristic zero, finite type ring extensions of any of the above.

Proof. See Propositions 12.41.12 and 12.39.6 to see that these rings are G-rings and have J-2. Any Cohen-Macaulay ring is universally catenary (in particular fields, Dedekind rings, and more generally regular rings are universally catenary). Via the Cohen structure theorem we see that complete local rings are universally catenary, see Algebra, Remark 7.144.9.  12.43. Pseudo-coherent modules Suppose that R is a ring. Recall that an R-module M is of finite type if there exists a surjection R⊕a → M and of finite presentation if there exists a presentation

12.43. PSEUDO-COHERENT MODULES

925

R⊕a1 → R⊕a0 → M → 0. Similarly, we can consider those R-modules for which there exists a length n resolution (12.43.0.1)

R⊕an → R⊕an−1 → . . . → R⊕a0 → M → 0

by finite free R-modules. A module is called pseudo-coherent of we can find such a resolution for every n. Here is the formal definition. Definition 12.43.1. Let R be a ring. Denote D(R) its derived category. Let m ∈ Z. (1) An object K • of D(R) is m-pseudo-coherent if there exists a bounded complex E • of finite free R-modules and a morphism α : E • → K • such that H i (α) is an isomorphism for i > m and H m (α) is surjective. (2) An object K • of D(R) is pseudo-coherent if it is quasi-isomorphic to a bounded above complex of finite free R-modules. (3) An R-module M is called m-pseudo-coherent if if M [0] is an m-pseudocoherent object of D(R). (4) An R-module M is called pseudo-coherent6 if M [0] is a pseudo-coherent object of D(R). As usual we apply this terminology also to complexes of R-modules. Since any morphism E • → K • in D(R) is represented by an actual map of complexes, see Derived Categories, Lemma 11.18.8, there is no ambiguity. It turns out that K • is pseudo-coherent if and only if K • is m-pseudo-coherent for all m ∈ Z, see Lemma 12.43.5. Also, if the ring is Noetherian the condition can be understood as a finite generation condition on the cohomology, see Lemma 12.43.16. Let us first relate this to the informal discussion above. Lemma 12.43.2. Let R be a ring and m ∈ Z. Let (K • , L• , M • , f, g, h) be a distinguished triangle in D(R). (1) If is (2) If (3) If is

K • is (m + 1)-pseudo-coherent and L• is m-pseudo-coherent then M • m-pseudo-coherent. K • , M • are m-pseudo-coherent, then L• is m-pseudo-coherent. L• is (m + 1)-pseudo-coherent and M • is m-pseudo-coherent, then K • (m + 1)-pseudo-coherent.

Proof. Proof of (1). Choose α : P • → K • with P • a bounded complex of finite free modules such that H i (α) is an isomorphism for i > m + 1 and surjective for i = m + 1. We may replace P • by σ≥m+1 P • and hence we may assume that P i = 0 for i < m + 1. Choose β : E • → L• with E • a bounded complex of finite free modules such that H i (β) is an isomorphism for i > m and surjective for i = m. By Derived Categories, Lemma 11.18.11 we can find a map α : P • → E • such that the diagram / L• KO • O P•

α

/ E•

6This clashes with what is meant by a pseudo-coherent module in [Bou61].

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is commutative in D(R). The cone C(α)• is a bounded complex of finite free R-modules, and the commutativity of the diagram implies that there exists a morphism of distinguished triangles (P • , E • , C(α)• ) −→ (K • , L• , M • ). It follows from the induced map on long exact cohomology sequences and Homology, Lemmas 10.3.24 and 10.3.25 that C(α)• → M • induces an isomorpism on cohomology in degrees > m and a surjection in degree m. Hence M • is m-pseudo-coherent. Assertions (2) and (3) follow from (1) by rotating the distinguished triangle.



•

Lemma 12.43.3. Let R be a ring. Let K be a complex of R-modules. Let m ∈ Z. (1) If K • is m-pseudo-coherent and H i (K • ) = 0 for i > m, then H m (K • ) is a finite type R-module. (2) If K • is m-pseudo-coherent and H i (K • ) = 0 for i > m+1, then H m+1 (K • ) is a finitely presented R-module. Proof. Proof of (1). Choose a bounded complex E • of finite projective R-modules and a map α : E • → K • which induces an isomorphism on cohomology in degrees > m and a surjection in degree m. It is clear that it suffices to prove the result for E • . Let n be the largest integer such that E n 6= 0. If n = m, then the result is clear. If n > m, then E n−1 → E n is surjective as H n (E • ) = 0. As E n is finite projective we see that E n−1 = E 0 ⊕ E n . Hence it suffices to prove the result for the complex (E 0 )• which is the same as E • except has E 0 in degree n − 1 and 0 in degree n. We win by induction on n. Proof of (2). Choose a bounded complex E • of finite projective R-modules and a map α : E • → K • which induces an isomorphism on cohomology in degrees > m and a surjection in degree m. As in the proof of (1) we can reduce to the case that E i = 0 for i > m + 1. Then we see that H m+1 (K • ) ∼ = H m+1 (E • ) = Coker(E m → E m+1 ) which is of finite presentation.  Lemma 12.43.4. Let R be a ring. Let M be an R-module. Then (1) M is 0-pseudo-coherent if and only if M is a finite type R-module, (2) M is (−1)-pseudo-coherent if and only if M is a finitely presented Rmodule, (3) M is (−d)-pseudo-coherent if and only if there exists a resolution R⊕ad → R⊕ad−1 → . . . → R⊕a0 → M → 0 of length d, and (4) M is pseudo-coherent if and only if there exists an infinite resolution . . . → R⊕a1 → R⊕a0 → M → 0 by finite free R-modules. Proof. If M is of finite type (resp. of finite presentation), then M is 0-pseudocoherent (resp. (−1)-pseudo-coherent) as follows from the discussion preceding Definition 12.43.1. Conversely, if M is 0-pseudo-coherent, then M = H 0 (M [0]) is of finite type by Lemma 12.43.3. If M is (−1)-pseudo-coherent, then it is 0pseudo-coherent hence of finite type. Choose a surjection R⊕a → M and denote K = Ker(R⊕a → M ). By Lemma 12.43.2 we see that K is 0-pseudo-coherent, hence of finite type, whence M is of finite presentation.

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To prove the third and fourth statement use induction and an argument similar to the above (details omitted).  Lemma 12.43.5. Let R be a ring. Let K • be a complex of R-modules. The following are equivalent (1) K • is pseudo-coherent, (2) K • is m-pseudo-coherent for every m ∈ Z, and (3) K • is quasi-isomorphic to a bounded above complex of finite projective R-modules. Proof. We see that (1) ⇒ (3) as a finite free module is a finite projective R-module. Conversely, suppose P • is a bounded above complex of finite projective R-modules. Say P i = 0 for i > n0 . We choose a direct sum decompositions F n0 = P n0 ⊕ C n0 with F n0 a finite free R-module, and inductively F n−1 = P n−1 ⊕ C n ⊕ C n−1 for n ≤ n0 with F n0 a finite free R-module. As a complex F • has maps F n−1 → F n which agree with P n−1 → P n , induce the identity C n → C n , and are zero on C n−1 . The map F • → P • is a quasi-isomorphism (even a homotopy equivalence) and hence (3) implies (1). Assume (1). Let E • be a bounded above complex of finite free R-modules and let E • → K • be a quasi-isomorphism. Then the induced maps σ≥m E • → K • from the stupid truncation of E • to K • show that K • is m-pseudo-coherent. Hence (1) implies (2). Assume (2). We first apply (2) for n = 0 to obtain a map of complexes α : F • → K • where F • is bounded above, consists of finite free R-modules and such that H i (α) is an isomorphism for i > 0 and surjective for i = 0. Note that these conditions remain satisfied after replacing F • by σ≥0 F • . Picture F0 α

K −1

 / K0

/ F1

/ ...

α

 / K1

/ ...

By induction on n < 0 we are going to extend F • to a complex F n → F n+1 → . . . → F −1 → F 0 → . . . of finite free R-modules and extend α such that H i (α) is an isomorphism for i > n and surjective for i = n. By shifting it suffices to prove the induction step for n = −1. By Lemma 12.43.3 the kernel of H 0 (F • ) = Ker(d0F ) → H 0 (K • ) is a finitely generated R-module. Hence we can choose a finite −1 free R-module F −1 and a map d−1 → F 0 whose image is this kernel. Then F : F −1 −1 −1 α(Im(dF )) ⊂ Im(dK ) and as F is projective we can a lift α : F −1 → K −1 fitting into the diagram F −1

/ F0



 / K0

K −1

α

/ F1

/ ...

α

 / K1

/ ...

By Lemma 12.43.3 the cokernel of H −1 (F • ) → H −1 (K • ) is a finitely generated R-module. Hence we can add a finite free summand to F −1 which is annihilated by d−1  F but via α maps to generators of this cokernel. This proves the lemma.

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Lemma 12.43.6. Let R be a ring. Let (K • , L• , M • , f, g, h) be a distinguished triangle in D(R). If two out of three of K • , L• , M • are pseudo-coherent then the third is also pseudo-coherent. Proof. Combine Lemmas 12.43.2 and 12.43.5.



Lemma 12.43.7. Let R be a ring. Let K • be a complex of R-modules. Let m ∈ Z. (1) If H i (K • ) = 0 for all i ≥ m, then K • is m-pseudo-coherent. (2) If H i (K • ) = 0 for i > m and H m (K • ) is a finite R-module, then K • is m-pseudo-coherent. (3) If H i (K • ) = 0 for i > m + 1, the module H m+1 (K • ) is of finite presentation, and H m (K • ) is of finite type, then K • is m-pseudo-coherent. Proof. It suffices to prove (3). Set M = H m+1 (K • ). Note that τ≥m+1 K • is quasi-isomorphic to M [−m − 1]. By Lemma 12.43.4 we see that M [−m − 1] is m-pseudo-coherent. Since we have the distinguished triangle (τ≤m K • , K • , τ≥m+1 K • ) by Lemma 12.43.2 it suffices to prove that τ≤m K • is pseudo-coherent. By assumption H m (τ≤m K • ) is a finite type R-module. Hence we can find a finite free m R-module E and a map E → Ker(dm K ) such that the composition E → Ker(dK ) → m • • H (τ≤m K ) is surjective. Then E[−m] → τ≤m K witnesses the fact that τ≤m K • is m-pseudo-coherent.  Lemma 12.43.8. Let R be a ring. Let m ∈ Z. If K • ⊕ L• is m-pseudo-coherent (resp. pseudo-coherent) so are K • and L• . Proof. In this proof we drop the superscript • . Assume that K ⊕ L is m-pseudocoherent. It is clear that K, L ∈ D− (R). Note that there is a distinguished triangle (K ⊕ L, K ⊕ L, K ⊕ L[1]) = (K, K, 0) ⊕ (L, L, L ⊕ L[1]) see Derived Categories, Lemma 11.4.8. By Lemma 12.43.2 we see that L ⊕ L[1] is m-pseudo-coherent. Hence also L[1] ⊕ L[2] is m-pseudo-coherent. By induction L[n] ⊕ L[n + 1] is m-pseudo-coherent. By Lemma 12.43.7 we see that L[n] is mpseudo-coherent for large n. Hence working backwards, using the distinguished triangles (L[n], L[n] ⊕ L[n − 1], L[n − 1]) we conclude that L[n], L[n − 1], . . . , L are m-pseudo-coherent as desired. pseudo-coherent case follows from this and Lemma 12.43.5.

The 

Lemma 12.43.9. Let R be a ring. Let m ∈ Z. Let K • be a bounded above complex of R-modules such that K i is (m − i)-pseudo-coherent for all i. Then K • is mpseudo-coherent. In particular, if K • is a bounded above complex of pseudo-coherent R-modules, then K • is pseudo-coherent. Proof. We may replace K • by σ≥m−1 K • (for example) and hence assume that K • is bounded. Then the complex K • is m-pseudo-coherent as each K i [−i] is mpseudo-coherent by induction on the length of the complex: use Lemma 12.43.6 and the stupid truncations. For the final statement, it suffices to prove that K • is m-pseudo-coherent for all m ∈ Z, see Lemma 12.43.5. This follows from the first part. 

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Lemma 12.43.10. Let R be a ring. Let m ∈ Z. Let K • ∈ D− (R) such that H i (K • ) is (m − i)-pseudo-coherent (resp. pseudo-coherent) for all i. Then K • is m-pseudo-coherent (resp. pseudo-coherent). Proof. Assume K • is an object of D− (R) such that each H i (K • ) is (m−i)-pseudocoherent. Let n be the largest integer such that H n (K • ) is nonzero. We will prove the lemma by induction on n. If n < m, then K • is m-pseudo-coherent by Lemma 12.43.7. If n ≥ m, then we have the distinguished triangle (τ≤n−1 K • , K • , H n (K • )[−n]) Since H n (K • )[−n] is m-pseudo-coherent by assumption, we can use Lemma 12.43.2 to see that it suffices to prove that τ≤n−1 K • is m-pseudo-coherent. By induction on n we win. (The pseudo-coherent case follows from this and Lemma 12.43.5.)  Lemma 12.43.11. Let A → B be a ring map. Assume that B is pseudo-coherent as an A-module. Let K • be a complex of B-modules. The following are equivalent (1) K • is m-pseudo-coherent as a complex of B-modules, and (2) K • is m-pseudo-coherent as a complex of A-modules. The same equivalence holds for pseudo-coherence. Proof. Assume (1). Choose a bounded complex of finite free B-modules E • and a map α : E • → K • which is an isomorphism on cohomology in degrees > m and a surjection in degree m. Consider the distinguished triangle (E • , K • , C(α)• ). By Lemma 12.43.7 C(α)• is m-pseudo-coherent as a complex of A-modules. Hence it suffices to prove that E • is pseudo-coherent as a complex of A-modules, which follows from Lemma 12.43.9. The pseudo-coherent case of (1) ⇒ (2) follows from this and Lemma 12.43.5. Assume (2). Let n be the largest integer such that H n (K • ) 6= 0. We will prove that K • is m-pseudo-coherent as a complex of B-modules by induction on n − m. The case n < m follows from Lemma 12.43.7. Choose a bounded complex of finite free A-modules E • and a map α : E • → K • which is an isomorphism on cohomology in degrees > m and a surjection in degree m. Consider the induced map of complexes α ⊗ 1 : E • ⊗A B → K • . Note that C(α⊗1)• is acyclic in degrees ≥ n as H n (E) → H n (E • ⊗A B) → H n (K • ) is surjective by construction and since H i (E • ⊗A B) = 0 for i > n by the spectral sequence of Example 12.8.4. On the other hand, C(α ⊗ 1)• is m-pseudo-coherent as a complex of A-modules because both K • and E • ⊗A B (see Lemma 12.43.9) are so, see Lemma 12.43.2. Hence by induction we see that C(α⊗1)• is m-pseudo-coherent as a complex of B-modules. Finally another application of Lemma 12.43.2 shows that K • is m-pseudo-coherent as a complex of B-modules (as clearly E • ⊗A B is pseudo-coherent as a complex of B-modules). The pseudo-coherent case of (2) ⇒ (1) follows from this and Lemma 12.43.5.  Lemma 12.43.12. Let A → B be a ring map. Let K • be an m-pseudo-coherent (resp. pseudo-coherent) complex of A-modules. Then K • ⊗L A B is an m-pseudocoherent (resp. pseudo-coherent) complex of B-modules. Proof. First we note that the statement of the lemma makes sense as K • is bounded above and hence K • ⊗L A B is defined by Equation (12.4.0.2). Having said this, choose a bounded complex E • of finite free A-modules and α : E • → K • with

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H i (α) an isomorphism for i > m and surjective for i = m. Then the cone C(α)• is acyclic in degrees ≥ m. Since − ⊗L A B is an exact functor we get a distinguished triangle • L • L (E • ⊗L A B, K ⊗A B, C(α) ⊗A B) of complexes of B-modules. By the dual to Derived Categories, Lemma 11.16.1 • we see that H i (C(α)• ⊗L A B) = 0 for i ≥ m. Since E is a complex of projective • L • R-modules we see that E ⊗A B = E ⊗A B and hence E • ⊗A B −→ K • ⊗L AB is a morphism of complexes of B-modules that witnesses the fact that K • ⊗L A B is m-pseudo-coherent. The case of pseudo-coherent complexes follows from the case of m-pseudo-coherent complexes via Lemma 12.43.5.  Lemma 12.43.13. Let A → B be a flat ring map. Let M be an m-pseudo-coherent (resp. pseudo-coherent) A-module. Then M ⊗A B is an m-pseudo-coherent (resp. pseudo-coherent) B-module. Proof. Immediate consequence of Lemma 12.43.12 and the fact that M ⊗L A B = M ⊗A B because B is flat over A.  The following lemma also follows from the stronger Lemma 12.43.14. Lemma 12.43.14. Let R be a ring. Let f1 , . . . , fr ∈ R be elements which generate the unit ideal. Let m ∈ Z. Let K • be a complex of R-modules. If for each i the complex K • ⊗R Rfi is m-pseudo-coherent (resp. pseudo-coherent), then K • is m-pseudo-coherent (resp. pseudo-coherent). Proof. We will use without further mention that − ⊗R Rfi is an exact functor and that therefore H i (K • )fi = H i (K • ) ⊗R Rfi = H i (K • ⊗R Rfi ). Assume K • ⊗R Rfi is m-pseudo-coherent for i = 1, . . . , r. Let n ∈ Z be the largest integer such that H n (K • ⊗R Rfi ) is nonzero for some i. This implies in particular that H i (K • ) = 0 for i > n (and that H n (K • ) 6= 0) see Algebra, Lemma 7.22.2. We will prove the lemma by induction on n − m. If n < m, then the lemma is true by Lemma 12.43.7. If n ≥ m, then H n (K • )fi is a finite Rfi -module for each i, see Lemma 12.43.3. Hence H n (K • ) is a finite R-module, see Algebra, Lemma 7.22.2. Choose a finite free R-module E and a surjection E → H n (K • ). As E is projective we can lift this to a map of complexes α : E[−n] → K • . Then the cone C(α)• has vanishing cohomology in degrees ≥ n. On the other hand, the complexes C(α)• ⊗R Rfi are m-pseudo-coherent for each i, see Lemma 12.43.2. Hence by induction we see that C(α)• is m-pseudo-coherent as a complex of Rmodules. Applying Lemma 12.43.2 once more we conclude.  Lemma 12.43.15. Let R be a ring. Let m ∈ Z. Let K • be a complex of Rmodules. Let R → R0 be a faithfully flat ring map. If the complex K • ⊗R R0 is m-pseudo-coherent (resp. pseudo-coherent), then K • is m-pseudo-coherent (resp. pseudo-coherent). Proof. We will use without further mention that − ⊗R R0 is an exact functor and that therefore H i (K • ) ⊗R R0 = H i (K • ⊗R R0 ).

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Assume K • ⊗R R0 is m-pseudo-coherent. Let n ∈ Z be the largest integer such that H n (K • ) is nonzero; then n is also the largest integer such that H n (K • ⊗R R0 ) is nonzero. We will prove the lemma by induction on n−m. If n < m, then the lemma is true by Lemma 12.43.7. If n ≥ m, then H n (K • ) ⊗R R0 is a finite R0 -module, see Lemma 12.43.3. Hence H n (K • ) is a finite R-module, see Algebra, Lemma 7.78.2. Choose a finite free R-module E and a surjection E → H n (K • ). As E is projective we can lift this to a map of complexes α : E[−n] → K • . Then the cone C(α)• has vanishing cohomology in degrees ≥ n. On the other hand, the complex C(α)• ⊗R R0 is m-pseudo-coherent, see Lemma 12.43.2. Hence by induction we see that C(α)• is m-pseudo-coherent as a complex of R-modules. Applying Lemma 12.43.2 once more we conclude.  Lemma 12.43.16. Let R be a Noetherian ring. Then (1) A complex of R-modules K • is m-pseudo-coherent if and only if K • ∈ D− (R) and H i (K • ) is a finite R-module for i ≥ m. (2) A complex of R-modules K • is pseudo-coherent if and only if K • ∈ D− (R) and H i (K • ) is a finite R-module for all i. (3) An R-module is pseudo-coherent if and only if it is finite. Proof. In Algebra, Lemma 7.68.1 we have seen that any finite R-module is pseudocoherent. On the other hand, a pseudo-coherent module is finite, see Lemma 12.43.4. Hence (3) holds. Suppose that K • is an m-pseudo-coherent complex. Then there exists a bounded complex of finite free R-modules E • such that H i (K • ) is isomorphic to H i (E • ) for i > m and such that H m (K • ) is a quotient of H m (E • ). Thus it is clear that each H i (K • ), i ≥ m is a finite module. The converse implication in (1) follows from Lemma 12.43.10 and part (3). Part (2) follows from (1) and Lemma 12.43.5.  12.44. Tor dimension Instead of resolving by projective modules we can look at resolutions by flat modules. This leads to the following concept. Definition 12.44.1. Let R be a ring. Denote D(R) its derived category. Let a, b ∈ Z. (1) An object K • of D(R) has tor-amplitude in [a, b] if H i (K • ⊗L R M ) = 0 for all R-modules M and all i 6∈ [a, b]. (2) An object K • of D(R) has finite tor dimension if it has tor-amplitude in [a, b] for some a, b. (3) An R-module M has tor dimension ≤ d if if M [0] as an object of D(R) has tor-amplitude in [−d, 0]. (4) An R-module M has finite tor dimension if M [0] as an object of D(R) has finite tor dimension. We observe that if K • has finite tor dimension, then K • ∈ Db (R). Lemma 12.44.2. Let R be a ring. Let K • be a bounded above complex of flat R-modules with tor-amplitude in [a, b]. Then Coker(da−1 K ) is a flat R-module. Proof. As K • is a bounded above complex of flat modules we see that K • ⊗R M = K • ⊗L R M . Hence for every R-module M the sequence K a−2 ⊗R M → K a−1 ⊗R M → K a ⊗R M

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is exact in the middle. Since K a−2 → K a−1 → K a → Coker(da−1 K ) → 0 is a flat a−1 resolution this implies that TorR (Coker(d ), M ) = 0 for all R-modules M . This 1 K means that Coker(da−1 ) is flat, see Algebra, Lemma 7.70.7.  K Lemma 12.44.3. Let R be a ring. Let K • be an object of D(R). Let a, b ∈ Z. The following are equivalent (1) K • has tor-amplitude in [a, b]. (2) K • is quasi-isomorphic to a complex E • of flat R-modules with E i = 0 for i 6∈ [a, b]. • Proof. If (2) holds, then we may compute K • ⊗L R M = E ⊗R M and it is clear that • (1) holds. Assume that (1) holds. We may replace K by a projective resolution. Let n be the largest integer such that K n 6= 0. If n > b, then K n−1 → K n is surjective as H n (K • ) = 0. As K n is projective we see that K n−1 = K 0 ⊕ K n . Hence it suffices to prove the result for the complex (K 0 )• which is the same as K • except has K 0 in degree n − 1 and 0 in degree n. Thus, by induction on n, we reduce to the case that K • is a complex of projective R-modules with K i = 0 for i > b.

Set E • = τ≥a K • . Everything is clear except that E a is flat which follows immediately from Lemma 12.44.2 and the definitions.  Lemma 12.44.4. Let R be a ring and m ∈ Z. Let (K • , L• , M • , f, g, h) be a distinguished triangle in D(R). Let a, b ∈ Z. (1) If K • has tor-amplitude in [a + 1, b + 1] and L• has tor-amplitude in [a, b] then M • has tor-amplitude in [a, b]. (2) If K • , M • have tor-amplitude in [a, b], then L• has tor-amplitude in [a, b]. (3) If L• has tor-amplitude in [a + 1, b + 1] and M • has tor-amplitude in [a, b], then K • has tor-amplitude in [a + 1, b + 1]. Proof. Omitted. Hint: This just follows from the long exact cohomology sequence associated to a distinguished triangle and the fact that − ⊗L R M preserves distinguished triangles. The easiest one to prove is (2) and the others follow from it by translation.  Lemma 12.44.5. Let R be a ring. Let M be an R-module. Let d ≥ 0. The following are equivalent (1) M has tor dimension ≤ d, and (2) there exists a resolution 0 → Fd → . . . → F1 → F0 → M → 0 with Fi a flat R-module. In particular an R-module has tor dimension 0 if and only if it is a flat R-module. Proof. Assume (2). Then the complex E • with E −i = Fi is quasi-isomorphic to M . Hence the Tor dimension of M is at most d by Lemma 12.44.3. Conversely, assume (1). Let P • → M be a projective resolution of M . By Lemma 12.44.2 we see that τ≥−d P • is a flat resolution of M of length d, i.e., (2) holds.  Lemma 12.44.6. Let R be a ring. Let a, b ∈ Z. If K • ⊕ L• has tor amplitude in [a, b] so do K • and L• . Proof. Clear from the fact that the Tor functors are additive.



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Lemma 12.44.7. Let R be a ring. Let K • be a bounded complex of R-modules such that K i has tor amplitude in [a − i, b − i] for all i. Then K • has tor amplitude in [a, b]. In particular if K • is a finite complex of R-modules of finite tor dimension, then K • has finite tor dimension. Proof. Follows by induction on the length of the finite complex: use Lemma 12.44.4 and the stupid truncations.  Lemma 12.44.8. Let R be a ring. Let a, b ∈ Z. Let K • ∈ Db (R) such that H i (K • ) has tor amplitude in [a − i, b − i] for all i. Then K • has tor amplitude in [a, b]. In particular if K • ∈ D− (R) and all its cohomology groups have finite tor dimension then K • has finite tor dimension. Proof. Follows by induction on the length of the finite complex: use Lemma 12.44.4 and the canonical truncations.  Lemma 12.44.9. Let A → B be a ring map. Assume that B is flat as an Amodule. Let K • be a complex of B-modules. Let a, b ∈ Z. If K • as a complex of B-modules has tor amplitude in [a, b], then K • as a complex of A-modules has tor amplitude in [a, b]. • L Proof. This is true because K • ⊗L A M = K ⊗B (M ⊗A B) since any projective • resolution of K as a complex of B-modules is a flat resolution of K • as a complex of A-modules and can be used to compute K • ⊗L  A M.

Lemma 12.44.10. Let A → B be a ring map. Assume that B has tor dimension ≤ d as an A-module. Let K • be a complex of B-modules. Let a, b ∈ Z. If K • as a complex of B-modules has tor amplitude in [a, b], then K • as a complex of A-modules has tor amplitude in [a − d, b]. Proof. Let M be an A-module. Choose a free resolution F • → M . Then • • • • • L L K • ⊗L A M = Tot(K ⊗A F ) = Tot(K ⊗B (F ⊗A B)) = K ⊗B (M ⊗A B).

By our assumption on B as an A-module we see that M ⊗L A B has cohomology only in degrees −d, −d + 1, . . . , 0. Because K • has tor amplitude in [a, b] we see from the L spectral sequence in Example 12.8.4 that K • ⊗L B (M ⊗A B) has cohomology only in degrees [−d + a, b] as desired.  Lemma 12.44.11. Let A → B be a ring map. Let a, b ∈ Z. Let K • be a complex of A-modules with tor amplitude in [a, b]. Then K • ⊗L A B as a complex of B-modules has tor amplitude in [a, b]. Proof. By Lemma 12.44.3 we can find a quasi-isomorphism E • → K • where E • is a complex of flat A-modules with E i = 0 for i 6∈ [a, b]. Then E • ⊗A B computes i K • ⊗L A B by construction and each E ⊗A B is a flat B-module by Algebra, Lemma 7.36.6. Hence we conclude by Lemma 12.44.3.  Lemma 12.44.12. Let A → B be a flat ring map. Let d ≥ 0. Let M be an A-module of tor dimension ≤ d. Then M ⊗A B is a B-module of tor dimension ≤ d. Proof. Immediate consequence of Lemma 12.44.11 and the fact that M ⊗L A B = M ⊗A B because B is flat over A. 

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Lemma 12.44.13. Let R be a ring. Let f1 , . . . , fr ∈ R be elements which generate the unit ideal. Let a, b ∈ Z. Let K • be a complex of R-modules. If for each i the complex K • ⊗R Rfi has tor amplitude in [a, b], then K • has tor amplitude in [a, b]. Proof. Note that − ⊗R Rfi is an exact functor and that therefore H i (K • )fi = H i (K • ) ⊗R Rfi = H i (K • ⊗R Rfi ). and similarly for every R-module M we have i • L i • L H i (K • ⊗L R M )fi = H (K ⊗R M ) ⊗R Rfi = H (K ⊗R Rfi ⊗Rf Mfi ). i

Hence the result follows from the fact that an R-module N is zero if and only if Nfi is zero for each i, see Algebra, Lemma 7.22.2.  Lemma 12.44.14. Let R be a ring. Let a, b ∈ Z. Let K • be a complex of Rmodules. Let R → R0 be a faithfully flat ring map. If the complex K • ⊗R R0 has tor amplitude in [a, b], then K • has tor amplitude in [a, b]. Proof. Let M be an R-module. Since R → R0 is flat we see that • 0 0 L • 0 (M ⊗L R K ) ⊗R R = ((M ⊗R R ) ⊗R0 (K ⊗R R ) • and taking cohomology commutes with tensoring with R0 . Hence TorR i (M, K ) = R0 0 • 0 0 Tori (M ⊗R R , K ⊗R R ). Since R → R is faithfully flat, the vanishing of 0 R • 0 • 0 TorR i (M ⊗R R , K ⊗R R ) for i 6∈ [a, b] implies the same thing for Tori (M, K ). 

Lemma 12.44.15. Let R be a ring of finite global dimension d. Then (1) every module has finite tor dimension ≤ d, (2) a complex of R-modules K • with H i (K • ) 6= 0 only if i ∈ [a, b] has tor amplitude in [a − d, b], and (3) a complex of R-modules K • has finite tor dimension if and only if K • ∈ Db (R). Proof. The assumption on R means that every module has a finite projective resolution of length at most d, in particular every module has finite tor dimension. The second statement follows from Lemma 12.44.8 and the definitions. The third statement is a rephrasing of the second.  12.45. Perfect complexes A perfect complex is a pseudo-coherent complex of finite tor dimension. But we can also define the directly as follows. Definition 12.45.1. Let R be a ring. Denote D(R) the derived category of the abelian category of R-modules. (1) An object K of D(R) is perfect if it is quasi-isomorphic to a bounded complex of finite projective R-modules. (2) An R-module M is perfect if M [0] is a perfect object in D(R). Lemma 12.45.2. Let K • be an object of D(R). The following are equivalent (1) K • is perfect, and (2) K • is pseudo-coherent and has finite tor dimension.

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Proof. It is clear that (1) implies (2), see Lemmas 12.43.5 and 12.44.3. Assume (2). Choose a bounded above complex F • of finite free R-modules and a quasiisomorphism F • → K • . Assume that K • has tor-amplitude in [a, b]. Set E • = τ≥a F • . Note that E i is finite free except E a which is a finitely presented R-module. By Lemma 12.44.2 E a is flat. Hence by Algebra, Lemma 7.73.2 we see that E a is finite projective.  Lemma 12.45.3. Let M be a module over a ring R. The following are equivalent (1) M is a perfect module, and (2) there exists a resolution 0 → Fd → . . . → F1 → F0 → M → 0 with each Fi a finite projective R-module. Proof. Assume (2). Then the complex E • with E −i = Fi is quasi-isomorphic to M [0]. Hence M is perfect. Conversely, assume (1). By Lemmas 12.45.2 and 12.43.4 we can find resolution E • → M with E −i a finite free R-module. By Lemma 12.44.2 we see that Fd = Coker(E d−1 → E d ) is flat for some d sufficiently large. By Algebra, Lemma 7.73.2 we see that Fd is finite projective. Hence 0 → Fd → E −d+1 → . . . → E 0 → M → 0 is the desired resolution.



Lemma 12.45.4. Let R be a ring. Let (K • , L• , M • , f, g, h) be a distinguished triangle in D(R). If two out of three of K • , L• , M • are perfect then the third is also perfect. Proof. Combine Lemmas 12.45.2, 12.43.6, and 12.44.4.



Lemma 12.45.5. Let R be a ring. If K • ⊕ L• is perfect, then so are K • and L• . Proof. Follows from Lemmas 12.45.2, 12.43.8, and 12.44.6.



Lemma 12.45.6. Let R be a ring. Let K • be a bounded complex of perfect Rmodules. Then K • is a perfect complex. Proof. Follows by induction on the length of the finite complex: use Lemma 12.45.4 and the stupid truncations.  Lemma 12.45.7. Let R be a ring. If K • ∈ D− (R) and all its cohomology modules are perfect, then K • is perfect. Proof. Follows by induction on the length of the finite complex: use Lemma 12.45.4 and the canonical truncations.  Lemma 12.45.8. Let A → B be a ring map. Assume that B is perfect as an A-module. Let K • be a perfect complex of B-modules. Then K • is perfect as a complex of A-modules. Proof. Using Lemma 12.45.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 12.44.10 and Lemma 12.43.11 for those results.  Lemma 12.45.9. Let A → B be a ring map. Let K • be a perfect complex of A-modules. Then K • ⊗L A B is a perfect complex of B-modules.

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Proof. Using Lemma 12.45.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 12.44.11 and Lemma 12.43.12 for those results.  Lemma 12.45.10. Let A → B be a flat ring map. Let M be a perfect A-module. Then M ⊗A B is a perfect B-module. Proof. By Lemma 12.45.3 the assumption implies that M has a finite resolution F• by finite projective R-modules. As A → B is flat the complex F• ⊗A B is a finite length resolution of M ⊗A B by finite projective modules over B. Hence M ⊗A B is perfect.  Lemma 12.45.11. Let R be a ring. Let f1 , . . . , fr ∈ R be elements which generate the unit ideal. Let K • be a complex of R-modules. If for each i the complex K • ⊗R Rfi is perfect, then K • is perfect. Proof. Using Lemma 12.45.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 12.44.13 and Lemma 12.43.14 for those results.  Lemma 12.45.12. Let R be a ring. Let a, b ∈ Z. Let K • be a complex of Rmodules. Let R → R0 be a faithfully flat ring map. If the complex K • ⊗R R0 has tor amplitude in [a, b], then K • has tor amplitude in [a, b]. Proof. Using Lemma 12.45.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 12.44.14 and Lemma 12.43.15 for those results.  Lemma 12.45.13. Let R be a regular ring of finite dimension. Then (1) an R-module is perfect if and only if it is a finite R-module, and (2) a complex of R-modules K • is perfect if and only if K • ∈ Db (R) and each H i (K • ) is a finite R-module. Proof. By Algebra, Lemma 7.103.7 the assumption on R means that R has finite global dimension. Hence every module has finite tor dimension, see Lemma 12.44.15. On the other hand, as R is Noetherian, a module is pseudo-coherent if and only if it is finite, see Lemma 12.43.16. This proves part (1). Let K • be a complex of R-modules. If K • is perfect, then it is in Db (R) and it is quasi-isomorphic to a finite complex of finite projective R-modules so certainly each H i (K • ) is a finite R-module (as R is Noetherian). Conversely, suppose that K • is in Db (R) and each H i (K • ) is a finite R-module. Then by (1) each H i (K • )  is a perfect R-module, whence K • is perfect by Lemma 12.45.7 Lemma 12.45.14. Let R be a ring. Let p ⊂ R be a prime ideal. Let i ∈ Z. Let K • be a pseudo-coherent complex of R-modules such that H i (K • ⊗L R κ(p)) = 0. Then there exists an f ∈ R, f 6∈ p such that K • ⊗R Rf = τ≥i+1 K • ⊗R Rf ⊕ τ≤i−1 K • ⊗R Rf in D(Rf ) with τ≥i+1 K • ⊗R Rf a perfect complex with tor amplitude in [i + 1, j] for some j ∈ Z.

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Proof. We may assume that K • is a bounded above complex of finite free Rmodules. Let us inspect what is happening in degree i: . . . → K i−2 → R⊕l → R⊕m → R⊕n → K i+2 → . . . Let A be the m × l matrix corresponding to K i−1 → K i and let B be the n × m matrix corresponding to K i → K i+1 . The assumption is that A mod p has rank r and that B mod p has rank m − r. In other words, there is some r × r minor a of A which is not in p and there is some (m − r) × (m − r)-minor b of B which is not in p. Set f = ab. Then after inverting f we can find direct sum decompositions K i−1 = R⊕l−r ⊕ R⊕r , K i = R⊕r ⊕ R⊕m−r , K i+1 = R⊕m−r ⊕ R⊕n−m+r such that the module map K i−1 → K i kills of R⊕l−r and induces an isomorphism of R⊕r onto the corresponding summand of K i and such that the module map K i → K i+1 kills of R⊕r and induces an isomorphism of R⊕m−r onto the corresponding summand of K i+1 . Thus K • becomes quasi-isomorphic to . . . → K i−2 → R⊕l−r → 0 → R⊕n−m+r → K i+2 → . . . and everything is clear.



Lemma 12.45.15. Let R be a ring. Let a, b ∈ Z. Let K • be a pseudo-coherent complex of R-modules. The following are equivalent (1) K • is perfect with tor amplitude in [a, b], (2) for every prime p we have H i (K • ⊗L R κ(p)) = 0 for all i 6∈ [a, b], and (3) for every maximal ideal m we have H i (K • ⊗L R κ(m)) = 0 for all i 6∈ [a, b]. Proof. We omit the proof of the implications (1) ⇒ (2) ⇒ (3). Assume (3). Let i ∈ Z with i 6∈ [a, b]. By Lemma 12.45.14 we see that the assumption implies that H i (K • )m = 0 for all maximal ideals of R. Hence H i (K • ) = 0, see Algebra, Lemma 7.22.1. Moreover, Lemma 12.45.14 now also implies that for every maximal ideal m there exists an element f ∈ R, f 6∈ m such that K • ⊗R Rf is perfect with tor amplitude in [a, b]. Hence we conclude by applealing to Lemmas 12.45.11 and 12.44.13.  Lemma modules. (1) (2) (3) (4) (5)

12.45.16. Let R be a ring. Let K • be a pseudo-coherent complex of RThe following are equivalent K • is perfect, for every prime ideal p the complex K • ⊗R Rp is perfect, for every prime p we have H i (K • ⊗L R κ(p)) = 0 for all i  0, for every maximal ideal m the complex K • ⊗R Rm is perfect, for every maximal ideal m we have H i (K • ⊗L R κ(m)) = 0 for all i  0.

Proof. Assume (5). Pick a maximal ideal m of R. By Lemma 12.45.14 we see that the assumption implies that K • ⊗R Rf is a perfect complex for some f ∈ R, f 6∈ m. Since Spec(R) is quasi-compact we conclude that K • is perfect by Lemmas 12.45.11. The proof of the other implications is omitted.  The following lemma useful in order to find perfect complexes over a polynomial ring B = A[x1 , . . . , xd ]. Lemma 12.45.17. Let A → B be a ring map. Let a, b ∈ Z. Let d ≥ 0. Let K • be a complex of B-modules. Assume (1) the ring map A → B is flat, (2) for every prime p ⊂ A the ring B ⊗A κ(p) has finite global dimension ≤ d,

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(3) K • is pseudo-coherent as a complex of B-modules, and (4) K • has tor amplitude in [a, b] as a complex of A-modules. Then K • is perfect as a complex of B-modules with tor amplitude in [a − d, b]. Proof. We may assume that K • is a bounded above complex of finite free Bmodules. In particular, K • is flat as a complex of A-modules and K • ⊗A M = K • ⊗L A for any A-module M . For every prime p of A the complex K • ⊗A κ(p) is a bounded above complex of finite free modules over B ⊗A κ(p) with vanishing H i except for i ∈ [a, b]. As B ⊗A κ(p) has global dimension d we see from Lemma 12.44.15 that K • ⊗A κ(p) has tor amplitude in [a − d, b]. Let q be a prime of B lying over p. Since K • ⊗A κ(p) is a bounded above complex of free B ⊗A κ(q)-modules we see that • K • ⊗L B κ(q) = K ⊗B κ(q)

= (K • ⊗A κ(p)) ⊗B⊗A κ(q) κ(q) = (K • ⊗A κ(p)) ⊗L B⊗A κ(q) κ(q) Hence the arguments above imply that H i (K • ⊗L B κ(q)) = 0 for i 6∈ [a − d, b]. We conclude by Lemma 12.45.15.  Lemma 12.45.18. Let K • be a perfect complex over a ring A. There exists a perfect complex E • such that we have functorial isomorphisms 0 • • • H 0 (K • ⊗L A L ) = ExtA (E , L )

for L• ∈ D(A). Proof. We may assume that K • is a finite complex of finite projective A-modules. The cohomology group on the left is simply H 0 (Tot(K • ⊗A L• )). Set E • = HomA (K • , A), i.e., E n = HomA (K −n , A) with differentials the transpose of the differentials of K • . Observe that E • is a finite complex of finite projective Amodules. The group on the right is MorK(ModA ) (E • , L• ) by Derived Categories, Lemma 11.18.8 and the definition of Ext groups, see Derived Categories, Section 11.26. By definition this is the cohomology of Y Y Y HomA (E n , Ln−1 ) → HomA (E n , Ln ) → HomA (E n , Ln+1 ) n

n

n

Using HomA (E n , L) = K −n ⊗A L as K −n is finite projective, we see that the cohomology groups are the same.  12.46. Characterizing perfect complexes Let R be a ring. Recall that D(R) has direct sums which are given simply by taking direct sums of complexes, see Injectives, Lemma 17.17.4. We will use this in the lemmas of this section without further mention. Lemma 12.46.1. Let R be a ring. Let K ∈ D(R) be an object such that for every countable set of objects En ∈ D(R) the canonical map M M HomD(R) (K, En ) −→ HomD(R) (K, En ) is a bijection. Then, given any system L•n of complexes over N we have that colim HomD(R) (K, L•n ) −→ HomD(R) (K, L• )

12.46. CHARACTERIZING PERFECT COMPLEXES

939

is a bijection, where L• is the termwise colimit, i.e., Lm = colim Lm n for all m ∈ Z. Proof. Consider the short exact sequence of complexes M M 0→ L•n → L•n → L• → 0 where the first map is given by 1 − tn in degree n where tn : L•n → L•n+1 is the transition map. By Derived Categories, Lemma 11.11.1 this is a distinguished triangle in D(R). Apply the homological functor HomD(R) (K, −), see Derived Categories, Lemma 11.4.2. Thus a long exact cohomology sequence

HomD(R) (K,

HomD(R) (K,

L

L

L•n )

r

L•n [1])

r

...

/ HomD(R) (K, colim L•n [−1])

/ HomD(R) (K, L L•n )

/ HomD(R) (K, colim L•n )

/ ...

L L HomD(R) (K, L•n ) Since we have assumed that HomD(R) (K, L•n ) is equal to we see that the first map on every row of the diagram is injective (by the explicit description of this map as the sum of the maps induced by 1 − tn ). Hence we conclude that HomD(R) (K, colim L•n ) is the cokernel of the first map of the middle row in the diagram above which is what we had to show.  Definition 12.46.2. Let D be an additive category with arbitrary direct sums. A compact object of D is an object K such that the map M M HomD (K, Ei ) −→ HomD (K, Ei ) i∈I

i∈I

is bijective for any set I and objects Ei ∈ Ob(D) parametrized by i ∈ I. The following proposition shows up in various places. See for example [Ric89, proof of Proposition 6.3] (this treats the bounded case), [TT90, Theorem 2.4.3] (the statement doesn’t match exactly), and [BN93, Proposition 6.4] (watch out for horrendous notational conventions). Proposition 12.46.3. Let R be a ring. For an object K of D(R) the following are equivalent (1) K is perfect, and (2) K is a compact object of D(R). Proof. Assume K is perfect, i.e., K is quasi-isomorphic to a bounded complex P • of finite projectiveLmodules, see Definition 12.45.1. If Ei is represented by the complex Ei• , then Ei is represented by the complex whose degree n term is L Ein . On the other hand, as P n is projective for all n we have HomD(R) (P • , K • ) = HomK(R) (P • , K • ) for every complex of R-modules K • , see Derived Categories, Lemma 11.18.8. Thus HomD(R) (P • , E • ) is the cohomology of the complex Y Y Y HomR (P n , E n−1 ) → HomR (P n , E n ) → HomR (P n , E n+1 ). Q L Since P • is bounded we see that we may replace the signs by signs in Lthe complex above. Since each P n is a finite R-module we see that HomR (P n , i Eim ) =

940

12. MORE ON ALGEBRA

HomR (P n , Eim ) for all n, m. Combining these remarks we see that the map of Definition 12.46.2 is a bijection.

L

i

Conversely, assume K is compact. Represent K by a complex K • and consider the map M K • −→ τ≥n K • n≥0

where we have used the canonical truncations, see Homology, Section 10.11. This makes sense as in each degree the direct sum on the right is finite. By assumption this map factors through a finite direct sum. We conclude that K → τ≥n K is zero for at least one n, i.e., K is in D− (R). Since K ∈ D− (R) and since every R-module is a quotient of a free module, we may represent K by a bounded above complex K • of free R-modules, see Derived Categories, Lemma 11.15.5. Note that we have [ K• = σ≥n K • n≤0

where we have used the stupid truncations, see Homology, Section 10.11. Hence by Lemma 12.46.1 we see that 1 : K • → K • factors through σ≥n K • → K • in D(R). Thus we see that 1 : K • → K • factors as ϕ

ψ

K• − → L• − → K• in D(R) for some complex L• which is bounded and whose terms are free R-modules. Say Li = 0 for i 6∈ [a, b]. Fix a, b from now on. Let c be the largest integer ≤ b + 1 such that we can find a factorization of 1K • as above with Li is finite Lfree for i < c. We will show by induction that c = b + 1. Namely, write Lc = λ∈Λ R. Since 0 c−1 Lc−1 is finite free we can find a finite subset Λ ⊂ Λ such that L → Lc factors L c through λ∈Λ0 R ⊂ L . Consider the map of complexes M π : L• −→ ( R)[−i] 0 λ∈Λ\Λ

given by the projection onto the factors corresponding to Λ \ Λ0 in degree i. By our assumption on K we see that, after possibly replacing Λ0 by a larger finite subset, we may assume that π ◦ ϕ = 0 in D(R). Let (L0 )• ⊂ L• be the kernel of π. Since π is surjective we get a short exact sequence of complexes, which gives a distinguished triangle in D(R) (see Derived Categories, Lemma 11.11.1). Since HomD(R) (K, −) is homological (see Derived Categories, Lemma 11.4.2) and π ◦ ϕ = 0, we can find a morphism ϕ0 : K • → (L0 )• in D(R) whose composition with (L0 )• → L• gives ϕ. Setting ψ 0 equal to the composition of ψ with (L0 )• → L• 0 • • we obtain a new L factorization. Since (L ) agrees with L except in degree c and 0 c since (L ) = λ∈Λ0 R the induction step is proved. The conclusion of the discussion of the preceding paragraph is that 1K : K → K factors as ϕ ψ K− →L− →K in D(R) where L can be represented by a finite complex of free R-modules. In particular we see that L is perfect. Note that e = ϕ ◦ ψ ∈ EndD(R) (L) is an idempotent. By Derived Categories, Lemma 11.4.12 we see that L = Ker(e) ⊕ Ker(1 − e) (see also discussion preceding Derived Categories, Lemma 11.4.11). The map ϕ : K → L induces an isomorphism with Ker(1 − e) in D(R). Hence we finally conclude that K is perfect by Lemma 12.45.5. 

12.46. CHARACTERIZING PERFECT COMPLEXES

941

Lemma 12.46.4. Let R be a ring. Let I ⊂ R be an ideal. Let K be an object of D(R). Assume that (1) K ⊗L R R/I is perfect in D(R/I), and (2) I is a nilpotent ideal. Then K is perfect in D(R). Proof. Assumption (2) means that I n = 0 for some n. The result holds if n = 1. Below we will prove the result holds for n = 2. This will imply that the complex 2 2 2 dn/2e K ⊗L = 0 we see that we win by R R/I is perfect in D(R/I ). Since (I ) induction on n. We prove the lemma in case I 2 = 0. First, we may represent K by a K-flat complex K • with all K n flat, see Lemma 12.5.10. Then we see that we have a short exact sequence of complexes 0 → K • ⊗R I → K • → K • ⊗R R/I → 0 Note that K • ⊗R R/I represents K ⊗L R R/I by constuction of the derived tensor product. Also K • ⊗R I ∼ = K • ⊗R R/I ⊗R/I I L • represents K ⊗L R R/I ⊗R/I I because K ⊗R R/I is a K-flat complex over R/I, see Lemma 12.5.5. By assumption (1) we see that both K • ⊗R R/I and K • ⊗R I have finitely many nonzero cohomology groups (since a perfect complex has finite Toramplitude, see Lemma 12.45.2). We conclude that K ∈ Db (R) by the long exact cohomology sequence associated to short exact sequence of complexes displayed above. In particular we can represent K by a bounded above complex K • of free R-modules (see Derived Categories, Lemma 11.15.5). Then for any complex E • of R-modules we have

HomD(R) (K, E • ) = HomK(R) (K • , E • ) see Derived Categories, Lemma 11.18.8. We will now show that K is perfect using the criterion of Proposition 12.46.3. Thus we let Ej ∈ D(R) be a family of objects parametrized by a set J. We choose complexes Ej• with flat terms representing Ej , see for example Lemma 12.5.10. It is clear that 0 → Ej• ⊗R I → Ej• → Ej• ⊗R R/I → 0 is a short exact sequence of complexes. Taking direct sums we obtain a similar short exact sequence M M M 0→ Ej• ⊗R I → Ej• → Ej• ⊗R R/I → 0 (Note that − ⊗R I and − ⊗R R/I commute with direct sums.) This short exact sequence determines a distinguished triangle in D(R), see Derived Categories, Lemma 11.11.1. Apply the homological functor HomD(R) (K, −) (see Derived Categories,

942

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Lemma 11.4.2) to get a commutative diagram L / HomD(R) (K • , L E • ⊗R R/I)[−1] HomD(R) (K • , Ej• ⊗R R/I)[−1] j

L

 HomD(R) (K • , Ej• ⊗R I)

 / HomD(R) (K • , L E • ⊗R I) j

 HomD(R) (K • , Ej• )

 / HomD(R) (K • , L E • )

L

j

L

 HomD(R) (K • , Ej• ⊗R R/I)

/ HomD(R) (K • ,

 L

L

 HomD(R) (K • , Ej• ⊗R I)[1]

/ HomD(R) (K • ,

 L

Ej• ⊗R R/I)

Ej• ⊗R I)[1]

with exact columns. Note that the complexes Ej• ⊗R R/I and Ej• ⊗R I have terms annihilated by I. Hence the 5 lemma (see Homology, Lemma 10.3.25) shows it suffices to show that given any collection of complexes Mj• with IMjn = 0 for all n, j the map M M HomD(R) (K • , Mj• ) −→ HomD(R) (K • , Mj• ) is a bijection. By our choice of K • we can rewrite this as M M HomK(R) (K • , Mj• ) −→ HomK(R) (K • , Mj• ) Since I annihilates Mjn we see this is equal to M M HomK(R/I) (K • ⊗R R/I, Mj• ) −→ HomK(R) (K • ⊗R R/I, Mj• ). Using that K • ⊗R R/I is a bounded above complex of free R/I-modules we see this is equal to the map M M HomD(R/I) (K • ⊗R R/I, Mj• ) −→ HomD(R) (K • ⊗R R/I, Mj• ) by Derived Categories, Lemma 11.18.8 as before. The complex K • ⊗R R/I repre• • sents K ⊗L R R/I since K is K-flat (Lemma 12.5.8). We conclude that K ⊗R R/I is compact (by Proposition 12.46.3). Hence the last displayed map is a bijection and we win.  12.47. Relatively finitely presented modules Let R be a ring. Let A → B be a finite map of finite type R-algebras. Let M be a finite B-module. In this case it is not true that M of finite presentation over B ⇔ M of finite presentation over A A counter example is R = k[x1 , x2 , x3 , . . .], A = R, B = R/(xi ), and M = B. To “fix” this we introduce a relative notion of finite presentation. Lemma 12.47.1. Let R → A be a ring map of finite type. Let M be an A-module. The following are equivalent

12.47. RELATIVELY FINITELY PRESENTED MODULES

943

(1) for some presentation α : R[x1 , . . . , xn ] → A the module M is a finitely presented R[x1 , . . . , xn ]-module, (2) for all presentations α : R[x1 , . . . , xn ] → A the module M is a finitely presented R[x1 , . . . , xn ]-module, and (3) for any surjection A0 → A where A0 is a finitely presented R-algebra, the module M is finitely presented as A0 -module. In this case M is a finitely presented A-module. Proof. If α : R[x1 , . . . , xn ] → A and β : R[y1 , . . . , ym ] → A are presentations. Choose fj ∈ R[x1 , . . . , xn ] with α(fj ) = β(yj ) and gi ∈ R[y1 , . . . , ym ] with β(gi ) = α(xi ). Then we get a commutative diagram R[x1 , . . . , xn , y1 , . . . , ym ] 

/ R[x1 , . . . , xn ]

yj 7→fj

xi 7→gi

 /A

R[y1 , . . . , ym ]

Hence the equivalence of (1) and (2) follows by applying Algebra, Lemmas 7.6.4 and 7.7.4. The equivalence of (2) and (3) follows by choosing a presentation A0 = R[x1 , . . . , xn ]/(f1 , . . . , fm ) and using Algebra, Lemma 7.7.4 to show that M is finitely presented as A0 -module if and only if M is finitely presented as a R[x1 , . . . , xn ]-module.  Definition 12.47.2. Let R → A be a finite type ring map. Let M be an Amodule. We say M is an A-module finitely presented relative to R if the equivalent conditions of Lemma 12.47.1 hold. Note that if R → A is of finite presentation, then M is an A-module finitely presented relative to R if and only if M is a finitely presented A-module. It is equally clear that A as an A-module is finitely presented relative to R if and only if A is of finite presentation over R. If R is Noetherian the notion is uninteresting. Now we can formulate the result we were looking for. Lemma 12.47.3. Let R be a ring. Let A → B be a finite map of finite type Ralgebras. Let M be a B-module. Then M is an A-module finitely presented relative to R if and only if M is a B-module finitely presented relative to R. Proof. Choose a surjection R[x1 , . . . , xn ] → A. Choose y1 , . . . , ym ∈ B which generate B over A. As A → B is finite each yi satisfies a monic equation with coefficients in A. Hence we can find monic polynomials Pj (T ) ∈ R[x1 , . . . , xn ][T ] such that Pj (yj ) = 0 in B. Then we get a commutative diagram R[x1 , . . . , xn ]

/ R[x1 , . . . , xn , y1 , . . . , ym ]/(Pj (yj ))

 A

 /B

Since the top arrow is a finite and finitely presented ring map we conclude by Algebra, Lemma 7.7.4 and the definition.  With this result in hand we see that the relative notion makes sense and behaves well with regards to finite maps of rings of finite type over R. It is also stable under localization, stable under base change, and ”glues” well.

944

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Lemma 12.47.4. Let R be a ring, f ∈ R an element, Rf → A is a finite type ring map, g ∈ A, and M an A-module. If M of finite presentation relative to Rf , then Mg is an Ag -module of finite presentation relative to R. Proof. Choose a presentation Rf [x1 , . . . , xn ] → A. We write Rf = R[x0 ]/(f x0 − 1). Consider the presentation R[x0 , x1 , . . . , xn , xn+1 ] → Ag which extends the given map, maps x0 to the image of 1/f , and maps xn+1 to 1/g. Choose g 0 ∈ R[x0 , x1 , . . . , xn ] which maps to g (this is possible). Suppose that Rf [x1 , . . . , xn ]⊕s → Rf [x1 , . . . , xn ]⊕t → M → 0 is a presentation of M given by a matrix (hij ). Pick h0ij ∈ R[x0 , x1 , . . . , xn ] which map to hij . Then R[x0 , x1 , . . . , xn , xn+1 ]⊕s+2t → R[x0 , x1 , . . . , xn , xn+1 ]⊕t → Mg → 0 is a presentation of Mf . Here the t × (s + 2t) matrix defining the map has a first t × s block consisting of the matrix h0ij , a second t × t block which is (x0 f −)It , and a third block which is (xn+1 g 0 − 1)It .  Lemma 12.47.5. Let R → A be a finite type ring map. Let M be an A-module finitely presented relative to R. For any ring map R → R0 the A ⊗R R0 -module M ⊗ A A0 = M ⊗ R R 0 is finitely presented relative to R0 . Proof. Choose a surjection R[x1 , . . . , xn ] → A. Choose a presentation R[x1 , . . . , xn ]⊕s → R[x1 , . . . , xn ]⊕t → M → 0 Then R0 [x1 , . . . , xn ]⊕s → R0 [x1 , . . . , xn ]⊕t → M ⊗R R0 → 0 is a presentation of the base change and we win.



Lemma 12.47.6. Let R → A be a finite type ring map. Let M be an A-module finitely presented relative to R. Let A → A0 be a ring map of finite presentation. A0 -module M ⊗A A0 is finitely presented relative to R. Proof. Choose a surjection R[x1 , . . . , xn ] → A. Choose a presentation A0 = A[y1 , . . . , ym ]/(g1 , . . . , gl ). Pick gi0 ∈ R[x1 , . . . , xn , y1 , . . . , ym ] mapping to gi . Say R[x1 , . . . , xn ]⊕s → R[x1 , . . . , xn ]⊕t → M → 0 is a presentation of M given by a matrix (hij ). Then R[x1 , . . . , xn , y1 , . . . , ym ]⊕s+tl → R[x0 , x1 , . . . , xn , y1 , . . . , ym ]⊕t → M ⊗A A0 → 0 is a presentation of M ⊗A A0 . Here the t × (s + lt) matrix defining the map has a first t × s block consisting of the matrix hij , followed by l blocks of size t × t which are gi0 It .  Lemma 12.47.7. Let R → A → B be finite type ring maps. Let M be a B-module. If M is finitely presented relative to A and A is of finite presentation over R, then M is finitely presented relative to R.

12.48. RELATIVELY PSEUDO-COHERENT MODULES

945

Proof. Choose a surjection A[x1 , . . . , xn ] → B. Choose a presentation A[x1 , . . . , xn ]⊕s → A[x1 , . . . , xn ]⊕t → M → 0 given by a matrix (hij ). Choose a presentation A = R[y1 , . . . , ym ]/(g1 , . . . , gu ). Choose h0ij ∈ R[y1 , . . . , ym , x1 , . . . , xn ] mapping to hij . Then we obtain the presentation R[y1 , . . . , ym , x1 , . . . , xn ]⊕s+tu → R[y1 , . . . , ym , x1 , . . . , xn ]⊕t → M → 0 where the t × (s + tu)-matrix is given by a first t × s block consisting of h0ij followed by u blocks of size t × t given by gi It , i = 1, . . . , u.  Lemma 12.47.8. Let R → A be a finite type ring map. Let M be an A-module. Let f1 , . . . , fr ∈ A generate the unit ideal. The following are equivalent (1) each Mfi is finitely presented relative to R, and (2) M is finitely presented relative to R. Proof. The implication (2) ⇒ (1) is in Lemma 12.47.4. Assume (1). Write 1 = P fi gi in A. Choose a surjection R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr ] → A. such that yi maps to fi and zi maps to gi . Then we see that there exists a surjection X P = R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr ]/( yi zi − 1) −→ A. By Lemma 12.47.1 we see that Mfi is a finitely presented Afi -module, hence by Algebra, Lemma 7.22.2 we see that M is a finitely presented A-module. Hence M is a finite P -module (with P as above). Choose a surjection P ⊕t → M . We have to show that the kernel K of this map is a finite P -module. Since Pyi surjects onto Afi we see by Lemma 12.47.1 and Algebra, Lemma 7.5.3 that the localization Kyi is a finitely generated Pyi -module. Choose elements ki,j ∈ K, i = 1, . . . , r, j = 1, . . . , si such that the images of ki,j in Kyi generate. Set K 0 ⊂ K equal to the P -module generated by the elements ki,j . Then K/K 0 is a module whose localization at yi is zero for all i. Since (y1 , . . . , yr ) = P we see that K/K 0 = 0 as desired.  Lemma 12.47.9. Let R → A be a finite type ring map. Let 0 → M 0 → M → M 00 → 0 be a short exact sequence of A-modules. (1) If M 0 , M 00 are finitely presented relative to R, then so is M . (2) If M 0 is finite a type A-module and M is finitely presented relative to R, then M 00 is finitely presented relative to R. Proof. Follows immediately from Algebra, Lemma 7.5.4.



Lemma 12.47.10. Let R → A be a finite type ring map. Let M, M 0 be A-modules. If M ⊕ M 0 is finitely presented relative to R, then so are M and M 0 . Proof. Omitted. 12.48. Relatively pseudo-coherent modules This section is the analogue of Section 12.47 for pseudo-coherence.



946

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Lemma 12.48.1. Let R be a ring. Let K • be an object of D− (R). Consider the R-algebra map R[x] → R which maps x to zero. Then ∼ K • ⊕ K • [1] K • ⊗L R = R[x]

in D(R). Proof. Choose a projective resolution P • → K • over R. Then x

P • ⊗R R[x] − → P • ⊗R R[x] is a double complex of projective R[x]-modules whose associated total complex is quasi-isomorphic to P • . Hence x 0 • ∼ K • ⊗L → P • ⊗R R[x]) ⊗R[x] R = Tot(P • − → P •) R[x] R = Tot(P ⊗R R[x] − = P • ⊕ P • [1] ∼ = K • ⊕ K • [1]

as desired.



Lemma 12.48.2. Let R be a ring and K • a complex of R-modules. Let m ∈ Z. Consider the R-algebra map R[x] → R which maps x to zero. Then K • is mpseudo-coherent as a complex of R-modules if and only if K • is m-pseudo-coherent as a complex of R[x]-modules. Proof. This is a special case of Lemma 12.43.11. We also prove it in another way as follows. Note that 0 → R[x] → R[x] → R → 0 is exact. Hence R is pseudo-coherent as an R[x]-module. Thus one implication of the lemma follows from Lemma 12.43.11. To prove the other implication, assume that K • is m-pseudo-coherent as a complex of R[x]-modules. By Lemma 12.43.12 we see that K • ⊗L R[x] R is m-pseudo-coherent as a comples of R-modules. By Lemma 12.48.1 we see that K • ⊕ K • [1] is m-pseudocoherent as a complex of R-modules. Finally, we conclude that K • is m-pseudocoherent as a complex of R-modules from Lemma 12.43.8.  Lemma 12.48.3. Let R → A be a ring map of finite type. Let K • be a complex of A-modules. Let m ∈ Z. The following are equivalent (1) for some presentation α : R[x1 , . . . , xn ] → A the complex K • is an mpseudo-coherent complex of R[x1 , . . . , xn ]-modules, (2) for all presentations α : R[x1 , . . . , xn ] → A the complex K • is an mpseudo-coherent complex of R[x1 , . . . , xn ]-modules. In particular the same equivalence holds for pseudo-coherence. Proof. If α : R[x1 , . . . , xn ] → A and β : R[y1 , . . . , ym ] → A are presentations. Choose fj ∈ R[x1 , . . . , xn ] with α(fj ) = β(yj ) and gi ∈ R[y1 , . . . , ym ] with β(gi ) = α(xi ). Then we get a commutative diagram R[x1 , . . . , xn , y1 , . . . , ym ] 

yj 7→fj

/ R[x1 , . . . , xn ]

xi 7→gi

R[y1 , . . . , ym ]

 /A

After a change of coordinates the ring homomorphism R[x1 , . . . , xn , y1 , . . . , ym ] → R[x1 , . . . , xn ] is isomorphic to the ring homomorphism which maps each yi to zero. Similarly for the left vertical map in the diagram. Hence, by induction on the

12.48. RELATIVELY PSEUDO-COHERENT MODULES

947

number of variables this lemma follows from Lemma 12.48.2. The pseudo-coherent case follows from this and Lemma 12.43.5.  Definition 12.48.4. Let R → A be a finite type ring map. Let K • be a complex of A-modules. Let M be an A-module. Let m ∈ Z. (1) We say K • is m-pseudo-coherent relative to R if the equivalent conditions of Lemma 12.48.3 hold. (2) We say K • is pseudo-coherent relative to R if K • is m-pseudo-coherent relative to R for all m ∈ Z. (3) We say M is m-pseudo-coherent relative to R if M [0] is m-pseudo-coherent. (4) We say M is pseudo-coherent relative to R if M [0] is pseudo-coherent relative to R. Part (2) means that K • is pseudo-coherent as a complex of R[x1 , . . . , xn ]-modules for any surjection R[y1 , . . . , ym ] → A, see Lemma 12.43.5. This definition has the following pleasing property. Lemma 12.48.5. Let R be a ring. Let A → B be a finite map of finite type R-algebras. Let m ∈ Z. Let K • be a complex of B-modules. Then K • is m-pseudocoherent (resp. pseudo-coherent) relative to R if and only if K • seen as a complex of A-modules is m-pseudo-coherent (pseudo-coherent) relative to R. Proof. Choose a surjection R[x1 , . . . , xn ] → A. Choose y1 , . . . , ym ∈ B which generate B over A. As A → B is finite each yi satisfies a monic equation with coefficients in A. Hence we can find monic polynomials Pj (T ) ∈ R[x1 , . . . , xn ][T ] such that Pj (yj ) = 0 in B. Then we get a commutative diagram R[x1 , . . . , xn , y1 , . . . , ym ]

R[x1 , . . . , xn ]

 / R[x1 , . . . , xn , y1 , . . . , ym ]/(Pj (yj ))

 A

 /B

The top horizontal arrow and the top right vertial arrow satisfy the assumptions of Lemma 12.43.11. Hence K • is m-pseudo-coherent (resp. pseudo-coherent) as a complex of R[x1 , . . . , xn ]-modules if and only if K • is m-pseudo-coherent (resp. pseudo-coherent) as a complex of R[x1 , . . . , xn , y1 , . . . , ym ]-modules.  Lemma 12.48.6. Let R be a ring. Let R → A be a finite type ring map. Let m ∈ Z. Let (K • , L• , M • , f, g, h) be a distinguished triangle in D(A). (1) If K • is (m+1)-pseudo-coherent relative to R and L• is m-pseudo-coherent relative to R then M • is m-pseudo-coherent relative to R. (2) If K • , M • are m-pseudo-coherent relative to R, then L• is m-pseudocoherent relative to R. (3) If L• is (m + 1)-pseudo-coherent relative to R and M • is m-pseudocoherent relative to R, then K • is (m + 1)-pseudo-coherent relative to R. Moreover, if two out of three of K • , L• , M • are pseudo-coherent relative to R, the so is the third.

948

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Proof. Follows immediately from Lemma 12.43.2 and the definitions.



Lemma 12.48.7. Let R → A be a finite type ring map. Let M be an A-module. Then (1) M is 0-pseudo-coherent relative to R if and only if M is a finite type A-module, (2) M is (−1)-pseudo-coherent relative to R if and only if M is a finitely presented relative to R, (3) M is (−d)-pseudo-coherent relative to R if and only if for every surjection R[x1 , . . . , xn ] → A there exists a resolution R[x1 , . . . , xn ]⊕ad → R[x1 , . . . , xn ]⊕ad−1 → . . . → R[x1 , . . . , xn ]⊕a0 → M → 0 of length d, and (4) M is pseudo-coherent relative to R if and only if for every presentation R[x1 , . . . , xn ] → A there exists an infinite resolution . . . → R[x1 , . . . , xn ]⊕a1 → R[x1 , . . . , xn ]⊕a0 → M → 0 by finite free R[x1 , . . . , xn ]-modules. Proof. Follows immediately from Lemma 12.43.4 and the definitions.



Lemma 12.48.8. Let R → A be a finite type ring map. Let m ∈ Z. Let K • , L• ∈ D(A). If K • ⊕ L• is m-pseudo-coherent (resp. pseudo-coherent) relative to R so are K • and L• . Proof. Immediate from Lemma 12.43.8 and the definitions.



Lemma 12.48.9. Let R → A be a finite type ring map. Let m ∈ Z. Let K • be a bounded above complex of A-modules such that K i is (m − i)-pseudo-coherent relative to R for all i. Then K • is m-pseudo-coherent relative to R. In particular, if K • is a bounded above complex of A-modules pseudo-coherent relative to R, then K • is pseudo-coherent relative to R. Proof. Immediate from Lemma 12.43.9 and the definitions.



Lemma 12.48.10. Let R → A be a finite type ring map. Let m ∈ Z. Let K • ∈ D− (A) such that H i (K • ) is (m−i)-pseudo-coherent (resp. pseudo-coherent) relative to R for all i. Then K • is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Proof. Immediate from Lemma 12.43.10 and the definitions.



Lemma 12.48.11. Let R be a ring, f ∈ R an element, Rf → A is a finite type ring map, g ∈ A, and K • a complex of A-modules. If K • is m-pseudo-coherent (resp. pseudo-coherent) relative to Rf , then K • ⊗A Ag is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Proof. First we show that K • is m-pseudo-coherent relative to R. Namely, suppose Rf [x1 , . . . , xn ] → A is surjective. Write Rf = R[x0 ]/(f x0 − 1). Then R[x0 , x1 , . . . , xn ] → A is surjective, and Rf [x1 , . . . , xn ] is pseudo-coherent as an R[x0 , . . . , xn ]-module. Hence by Lemma 12.43.11 we see that K • is m-pseudocoherent as a complex of R[x0 , x1 , . . . , xn ]-modules.

12.48. RELATIVELY PSEUDO-COHERENT MODULES

949

Choose an element g 0 ∈ R[x0 , x1 , . . . , xn ] which maps to g ∈ A. By Lemma 12.43.12 we see that 1 1 • K • ⊗L R[x0 ,x1 ,...,xn ] R[x0 , x1 , . . . , xn , 0 ] = K ⊗R[x0 ,x1 ,...,xn ] R[x0 , x1 , . . . , xn , 0 ] g g = K • ⊗A Af is m-pseudo-coherent as a complex of R[x0 , x1 , . . . , xn , g10 ]-modules. write R[x0 , x1 , . . . , xn ,

1 ] = R[x0 , . . . , xn , xn+1 ]/(xn+1 g 0 − 1). g0

As R[x0 , x1 , . . . , xn , g10 ] is pseudo-coherent as a R[x0 , . . . , xn , xn+1 ]-module we conclude (see Lemma 12.43.11) that K • ⊗A Ag is m-pseudo-coherent as a complex of R[x0 , . . . , xn , xn+1 ]-modules as desired.  Lemma 12.48.12. Let R → A be a finite type ring map. Let m ∈ Z. Let K • be a complex of A-modules which is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Let R → R0 be a ring map such that A and R0 are Tor independent over R. 0 Set A0 = A ⊗R R0 . Then K • ⊗L A A is is m-pseudo-coherent (resp. pseudo-coherent) 0 relative to R . Proof. Choose a surjection R[x1 , . . . , xn ] → A. Note that 0 • L 0 • L 0 K • ⊗L A A = K ⊗R R = K ⊗R[x1 ,...,xn ] R [x1 , . . . , xn ]

by Lemma 12.7.2 applied twice. Hence we win by Lemma 12.43.12.



Lemma 12.48.13. Let R → A → B be finite type ring maps. Let m ∈ Z. Let K • be a complex of A-modules. Assume B as a B-module is pseudo-coherent relative to A. If K • is m-pseudo-coherent (resp. pseudo-coherent) relative to R, then K • ⊗L AB is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Proof. Choose a surjection A[y1 , . . . , ym ] → B. Choose a surjection R[x1 , . . . , xn ] → A. Combined we get a surjection R[x1 , . . . , xn , y1 , . . . ym ] → B. Choose a resolution E • → B of B by a complex of finite free A[y1 , . . . , yn ]-modules (which is possible by our assumption on the ring map A → B). We may assume that K • is a bounded above complex of flat A-modules. Then • K • ⊗L A B = Tot(K ⊗A B[0])

= Tor(K • ⊗A A[y1 , . . . , ym ] ⊗A[y1 ,...,ym ] B[0])
∼ = Tot (K • ⊗A A[y1 , . . . , ym ]) ⊗A[y ,...,y ] E • 1

m

= Tot(K • ⊗A E • ) in D(A[y1 , . . . , ym ]). The quasi-isomorphism ∼ = comes from an application of Lemma 12.5.8. Thus we have to show that Tot(K • ⊗A E • ) is m-pseudo-coherent as a complex of R[x1 , . . . , xn , y1 , . . . ym ]-modules. Note that Tot(K • ⊗A E • ) has a filtration by subcomplexes with successive quotients the complexes K • ⊗A E i [−i]. Note that for i  0 the complexes K • ⊗A E i [−i] have zero cohomology in degrees ≤ m and hence are m-pseudo-coherent (over any ring). Hence, applying Lemma 12.48.6 and induction, it suffices to show that K • ⊗A E i [−i] is pseudo-coherent relative to R for all i. Note that E i = 0 for i > 0. Since also E i is finite free this reduces to proving that K • ⊗A A[y1 , . . . , ym ] is m-pseudo-coherent relative to R which follows from Lemma 12.48.12 for instance. 

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Lemma 12.48.14. Let R → A → B be finite type ring maps. Let m ∈ Z. Let M be an A-module. Assume B as a B-module is flat and pseudo-coherent relative to A. If M is m-pseudo-coherent (resp. pseudo-coherent) relative to R, then M ⊗A B is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Proof. Immediate from Lemma 12.48.13.



Lemma 12.48.15. Let R be a ring. Let A → B be a map of finite type R-algebras. Let m ∈ Z. Let K • be a complex of B-modules. Assume A is pseudo-coherent relative to R. Then the following are equivalent (1) K • is m-pseudo-coherent (resp. pseudo-coherent) relative to A, and (2) K • is m-pseudo-coherent (resp. pseudo-coherent) relative to R. Proof. Choose a surjection R[x1 , . . . , xn ] → A. Choose a surjection A[y1 , . . . , ym ] → B. Then we get a surjection R[x1 , . . . , xn , y1 , . . . , ym ] → A[y1 , . . . , ym ] which is a flat base change of R[x1 , . . . , xn ] → A. By assumption A is a pseudocoherent module over R[x1 , . . . , xn ] hence by Lemma 12.43.13 we see that A[y1 , . . . , ym ] is pseudo-coherent over R[x1 , . . . , xn , y1 , . . . , ym ]. Thus the lemma follows from Lemma 12.43.11 and the definitions.  Lemma 12.48.16. Let R → A be a finite type ring map. Let K • be a complex of A-modules. Let m ∈ Z. Let f1 , . . . , fr ∈ A generate the unit ideal. The following are equivalent (1) each K • ⊗A Afi is m-pseudo-coherent relative to R, and (2) K • is m-pseudo-coherent relative to R. The same equivalence holds for pseudo-coherence. Proof. P The implication (2) ⇒ (1) is in Lemma 12.48.11. Assume (1). Write 1 = fi gi in A. Choose a surjection R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr ] → A. such that yi maps to fi and zi maps to gi . Then we see that there exists a surjection X P = R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr ]/( yi zi − 1) −→ A. Note that P is pseudo-coherent as an R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr ]-module and that P [1/yi ] is pseudo-coherent as an R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr , 1/yi ]module. Hence by Lemma 12.43.11 we see that K • ⊗A Afi is an m-pseudo-coherent complex of P [1/yi ]-modules for each i. Thus by Lemma 12.43.14 we see that K • is pseudo-coherent as a complex of P -modules, and Lemma 12.43.11 shows that K • is pseudo-coherent as a complex of R[x1 , . . . , xn , y1 , . . . , yr , z1 , . . . , zr ]-modules.  Lemma 12.48.17. Let R be a Noetherian ring. Let R → A be a finite type ring map. Then (1) A complex of A-modules K • is m-pseudo-coherent relative to R if and only if K • ∈ D− (A) and H i (K • ) is a finite A-module for i ≥ m. (2) A complex of A-modules K • is pseudo-coherent relative to R if and only if K • ∈ D− (A) and H i (K • ) is a finite A-module for all i. (3) An A-module is pseudo-coherent relative to R if and only if it is finite. Proof. Immediate consequence of Lemma 12.43.16 and the definitions.



12.49. PSEUDO-COHERENT AND PERFECT RING MAPS

951

12.49. Pseudo-coherent and perfect ring maps We can define these types of ring maps as follows. Definition 12.49.1. Let A → B be a ring map. (1) We say A → B is a pseudo-coherent ring map if it is of finite type and B, as a B-module, is pseudo-coherent relative to A. (2) We say A → B is a perfect ring map if it is a pseudo-coherent ring map such that B as an A-module has finite tor dimension. This terminology may be nonstandard. Using Lemma 12.48.7 we see that A → B is pseudo-coherent if and only if B = A[x1 , . . . , xn ]/I and B as an A[x1 , . . . , xn ]module has a resolution by finite free A[x1 , . . . , xn ]-modules. The motivation for the definition of a perfect ring map is Lemma 12.45.2. The following lemmas gives a more useful and intuitive characterization of a perfect ring map. Lemma 12.49.2. A ring map A → B is perfect if and only if B = A[x1 , . . . , xn ]/I and B as an A[x1 , . . . , xn ]-module has a finite resolution by finite projective A[x1 , . . . , xn ]modules. Proof. If A → B is perfect, then B = A[x1 , . . . , xn ]/I and B is pseudo-coherent as an A[x1 , . . . , xn ]-module and has finite tor dimension as an A-module. Hence Lemma 12.45.17 implies that B is perfect as a A[x1 , . . . , xn ]-module, i.e., it has a finite resolution by finite projective A[x1 , . . . , xn ]-modules (Lemma 12.45.3). Conversely, if B = A[x1 , . . . , xn ]/I and B as an A[x1 , . . . , xn ]-module has a finite resolution by finite projective A[x1 , . . . , xn ]-modules then B is pseudo-coherent as an A[x1 , . . . , xn ]-module, hence A → B is pseudo-coherent. Moreover, the given resolution over A[x1 , . . . , xn ] is a finite resolution by flat A-modules and hence B has finite tor dimension as an A-module.  Lots of the results of the preceding sections can be reformulated in terms of this terminology. We also refer to More on Morphisms, Sections 33.37 and 33.38 for the corresponding discussion concerning morphisms of schemes. Lemma 12.49.3. A finite type ring map of Noetherian rings is pseudo-coherent. Proof. See Lemma 12.48.17.



Lemma 12.49.4. A ring map which is flat and of finite presentation is perfect. Proof. Let A → B be a ring map which is flat and of finite presentation. It is clear that B has finite tor dimension. By Algebra, Lemma 7.151.1 there exists a finite type Z-algebra A0 ⊂ A and a flat finite type ring map A0 → B0 such that B = B0 ⊗A0 A. By Lemma 12.48.17 we see that A0 → B0 is pseudo-coherent. As A0 → B0 is flat we see that B0 and A are tor independent over A0 , hence we may use Lemma 12.48.12 to conclude that A → B is pseudo-coherent.  Lemma 12.49.5. Let A → B be a finite type ring map with A a regular ring of finite dimension. Then A → B is perfect. Proof. By Algebra, Lemma 7.103.7 the assumption on A means that A has finite global dimension. Hence every module has finite tor dimension, see Lemma 12.44.15, in particular B does. By Lemma 12.49.3 the map is pseudo-coherent.  Lemma 12.49.6. A local complete intersection homomorphism is perfect.

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Proof. Let A → B he a local complete intersection homomorphism. By Definition 12.26.2 this means that B = A[x1 , . . . , xn ]/I where I is a Koszul ideal in A[x1 , . . . , xn ]. By Lemmas 12.49.2 and 12.45.3 it suffices to show that I is a perfect module over A[x1 , . . . , xn ]. By Lemma 12.45.11 this is a local question. Hence we may assume that I is generated by a Koszul-regular sequence (by Definition 12.25.1). Of course this means that I has a finite free resolution and we win.  12.50. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology

(39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 13

Smoothing Ring Maps 13.1. Introduction The main result of this chapter is the following: A regular map of Noetherian rings is a filtered colimit of smooth ones. This theorem is due to Popescu, see [Pop90]. A readable exposition of Popescu’s proof was given by Richard Swan, see [Swa98] who used notes by Andr´e and a paper of Ogoma, see [Ogo94]. Our exposition follows Swan’s, but we first prove an intermediate result which let’s us work in a slightly simpler situation. Here is an overview. We first solve the following “lifting problem”: A flat infinitesimal deformation of a filtered colimit of smooth algebras is a filtered colimit of smooth algebras. This result essentially says that it suffices to prove the main theorem for maps between reduced Noetherian rings. Next we prove two very clever lemmas called the “lifting lemma” and the “desingularization lemma”. We show that these lemmas combined reduce the main theorem to proving a Noetherian, geometrically regular k-algebra Λ is a filtered limit of smooth k-algebras. Next, we discuss the necessary local tricks that go into the Popescu-Ogoma-Swan-Andr´e proof. Finally, in the last three sections we give the proof. We end this introduction with some pointers to references. Let A be a henselian Noetherian local ring. We say A has the approximation property if for any f1 , . . . , fm ∈ A[x1 , . . . , xn ] the system of equations f1 = 0, . . . , fm = 0 has a solution in the completion of A if and only if it has a solution in A. This definition is due to Artin. Artin first proved the approximation property for analytic systems of equations, see [Art68]. In [Art69a] Artin proved the approximation property for local rings essentially of finite type over an excellent discrete valuation ring. Artin conjectured (page 26 of [Art69a]) that every excellent henselian local ring should have the approximation property. At some point in time it became a conjecture that that every regular homomorphism of Noetherian rings is a filtered colimit of smooth algebras (see for example [Ray72], [Pop81], [Art82], [AD83]). We’re not sure who this conjecture1 is due to. The relationship with the approximation property is that if A → A∧ is a colimit of smooth algebras, then the approximation property holds (insert future reference here). Moreover, the main theorem applies to the map A → A∧ if A is an excellent local ring, as one of the conditions of an excellent local ring is that the formal 1The question/conjecture as formulated in [Art82], [AD83], and [Pop81] is stronger and was shown to be equivalent to the original version in [CP84]. 953

954

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fibres are geometrically regular. Note that excellent local rings were defined by Grothendieck and their definition appeared in print in 1965. In [Art82] it was shown that R → R∧ is a filtered colimit of smooth algebras for any local ring R essentially of finite type over a field. In [AR88] it was shown that R → R∧ is a filtered colimit of smooth algebras for any local ring R essentially of finite type over an excellent discrete valution ring. Finally, the main theorem was shown in [Pop85], [Pop86], [Pop90], [Ogo94], and [Swa98] as discussed above. Conversely, using some of the results above, in [Rot90] it was shown that any local ring with the approximation property is excellent. The paper [Spi99] provides an alternative approach to the main theorem, but it seems hard to read (for example [Spi99, Lemma 5.2] appears to be an incorrectly reformulated version of [Elk73, Lemma 3]). There is also a Bourbaki lecture about this material, see [Tei95]. 13.2. Colimits In Categories, Section 4.17 we discuss filtered colimits. In particular, note that Categories, Lemma 4.19.3 tells us that colimits over filtered index categories are the same thing as colimits over directed partially ordered sets. Lemma 13.2.1. Let R → Λ be a ring map. Let E be a set of R-algebras such that each A ∈ E is of finite presentation over R. Then the following two statements are equivalent (1) Λ is a filtered colimit of elements of E, and (2) for any R algebra map A → Λ with A of finite presentation over R we can find a factorization A → B → Λ with B ∈ E. Proof. Suppose that I → E, i 7→ Ai is a diagram such that Λ = colimi Ai . Let A → Λ with A of finite presentation over R. Pick a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fm ). Say A → Λ maps xs to λs ∈ Λ. We can find an i ∈ Ob(I) and elements as ∈ Ai whose image in Λ is λs . Increasing i if necessary we may also assume that ft (a1 , . . . , an ) = 0 in Ai . Hence we can factor A → Λ through Ai by mapping xs to as . Conversely, suppose that (2) holds. Consider the category I whose objects are Ralgebra maps A → Λ with A ∈ E and whose morphisms are commutative diagrams / A0

A 

Λ

~

of R-algebras. We claim that I is a filtered index category and that Λ = colimI A. To see that I is filtered, let A → Λ and A0 → Λ be two objects. Then we can factor A ⊗R A0 → Λ through an object of I by assumption (2) and the fact that the elements of E are of finite presentation over R. Suppose that ϕ, ψ : A → A0 are two morphisms of I. Let x1 , . . . , xn be generators of A as an R-algebra. By assumption (2) we can factor the R-algebra map A0 /(ϕ(xi ) − ψ(xi )) → Λ through an object of I. This proves that I is filtered. We omit the proof that Λ = colimI A. 

13.3. SINGULAR IDEALS

955

13.3. Singular ideals Let R → A be a ring map. The singular ideal of A over R is the radical ideal in A cutting out the singular locus of the morphism Spec(A) → Spec(R). Here is a formal definition. Definition 13.3.1. Let R → A be a ring map. The singular ideal of A over R, denoted HA/R is the unique radical ideal HA/R ⊂ A with V (HA/R ) = {q ∈ Spec(A) | R → A not smooth at q} This makes sense because the set of primes where R → A is smooth is open, see Algebra, Definition 7.127.11. In order to find an explicit set of generators for the singular ideal we first prove the following lemma. Lemma 13.3.2. Let R be a ring. Let A = R[x1 , . . . , xn ]/(f1 , . . . , fm ). Let q ⊂ A. Assume R → A is smooth at q. Then there exists an a ∈ A, a 6∈ q, an integer c, 0 ≤ c ≤ min(n, m), subsets U ⊂ {1, . . . , n}, V ⊂ {1, . . . , m} of cardinality c such that a = a0 det(∂fj /∂xi )j∈V,i∈U for some a0 ∈ A and af` ∈ (fj , j ∈ V ) + (f1 , . . . , fm )2 for all ` ∈ {1, . . . , m}. Proof. Set I = (f1 , . . . , fm ) so that L the the naive cotangent complex of A over R is homotopy equivalent to I/I 2 → Adxi , see Algebra, Lemma 7.124.2. We will use the formation of the naive cotangent complex commutes with localization, see Algebra, Section 7.124, especially Algebra, Lemma L 7.124.11. By Algebra, Definitions 7.127.1 and 7.127.11 we see that (I/I 2 )a → Aa dxi is a split injection for some a ∈ A, a 6∈ p. After renumbering x1 , . . . , xn and f1 , . . . , fm we may assume that f1 , . . . , fc form a basis for the vector space I/I 2 ⊗A κ(q) and that dxc+1 , . . . , dxn map to a basis of ΩA/R ⊗A κ(q). Hence after replacing a by aa0 for some a0 ∈ A, a0 6∈ q we may assume f1 , . . . , fc form a basis for (I/I 2 )a and that dxc+1 , . . . , dxn map to a basis of (ΩA/R )a . In this situation aN for some large integer N satisfies the conditions of the lemma (with U = V = {1, . . . , c}).  We will use the notion of a strictly standard element in a A over R. Our notion is slightly weaker than the one in Swan’s paper [Swa98]. We also define an elementary standard element to be one of the type we found in the lemma above. We compare the different types of elements in Lemma 13.4.7. Definition 13.3.3. Let R → A be a ring map of finite presentation. We say an element a ∈ A is elementary standard in A over R if there exists a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) and 0 ≤ c ≤ min(n, m) such that (13.3.3.1)

a = a0 det(∂fj /∂xi )i,j=1,...,c

for some a0 ∈ A and (13.3.3.2)

afc+j ∈ (f1 , . . . , fc ) + (f1 , . . . , fm )2

for j = 1, . . . , m − c. We say a ∈ A is strictly standard in A over R if there exists a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) and 0 ≤ c ≤ min(n, m) such that X (13.3.3.3) a= aI det(∂fj /∂xi )j=1,...,c, i∈I I⊂{1,...,n}, |I|=c

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for some aI ∈ A and (13.3.3.4)

afc+j ∈ (f1 , . . . , fc ) + (f1 , . . . , fm )2

for j = 1, . . . , m − c. The following lemma is useful to find implications of (13.3.3.3). Lemma 13.3.4. Let R be a ring. Let A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) and write I =L (f1 , . . . , fn ). Let a ∈ A. Then (13.3.3.3) implies there exists an A-linear map ψ : i=1,...,n Adxi → A⊕c such that the composition (f1 ,...,fc ) f 7→df M ψ A⊕c −−−−−−→ I/I 2 −−−−→ Adxi − → A⊕c i=1,...,n

is multiplication by a. Conversely, if such a ψ exists, then ac satisfies (13.3.3.3). Proof. This is a special case of Algebra, Lemma 7.14.5.



Lemma 13.3.5 (Elkik). Let R → A be a ring map of finite presentation. The singular ideal HA/R is the radical of the ideal generated by strictly standard elements in A over R and also the radical of the ideal generated by elementary standard elements in A over R. Proof. Assume a is strictly standard in A over R. We claim that Aa is smooth over R, which proves that a ∈ HA/R . Namely, let A = R[x1 , . . . , xn ]/(f1 , . . . , fm ), c, and a0 ∈ A be as in Definition 13.3.3. Write I =L(f1 , . . . , fm ) so that the naive cotangent complex of A over R is given by I/I 2 → Adxi . Assumption (13.3.3.4) implies that (I/I 2 )a is generated by the classes of f , . . . , fc . Assumption (13.3.3.3) 1 L implies that the differential (I/I 2 )a → Aa dxi has a left inverse, see Lemma 13.3.4. Hence R → Aa is smooth by definition and Algebra, Lemma 7.124.11. Let He , Hs ⊂ A be the radical of the ideal generated by elementary, resp. strictly standard elements of A over R. By definition and what we just proved we have He ⊂ Hs ⊂ HA/R . The inclusion HA/R ⊂ He follows from Lemma 13.3.2.  Example 13.3.6. The set of points where a finitely presented ring map is smooth needn’t be a quasi-compact open. For example, let RS= k[x, y1 , y2 , y3 , . . .]/(xyi ) and A = R/(x). Then the smooth locus of R → A is D(yi ) which is not quasicompact. Lemma 13.3.7. Let R → A be a ring map of finite presentation. Let R → R0 be a ring map. If a ∈ A is elementary, resp. strictly standard in A over R, then a ⊗ 1 is elementary, resp. strictly standard in A ⊗R R0 over R0 . Proof. If A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) is a presentation of A over R, then A ⊗R 0 R0 = R0 [x1 , . . . , xn ]/(f10 , . . . , fm ) is a presentation of A ⊗R R0 over R0 . Here fj0 is 0 the image of fj in R [x1 , . . . , xn ]. Hence the result follows from the definitions.  Lemma 13.3.8. Let R → A → Λ be ring maps with A of finite presentation over R. Assume that HA/R Λ = Λ. Then there exists a factorization A → B → Λ with B smooth over R. P Proof. Choose f1 , . . . , fr ∈ HA/R and λ1 , . . . , λr ∈ Λ such that fi λi = 1 in Λ. Set B = A[x1 , . . . , xr ]/(f1 x1 + . . . + fr xr − 1) and define B → Λ by mapping xi to λi . Details omitted. 

13.4. PRESENTATIONS OF ALGEBRAS

957

13.4. Presentations of algebras Some of the results in this section are due to Elkik. Note that the algebra C in the following lemma is a symmetric algebra over A. Moreover, if R is Noetherian, then C is of finite presentation over R. Lemma 13.4.1. Let R be a ring and let A be a finitely presented R-algebra. There exists finite type R-algebra map A → C which has a retraction with the following two properties (1) for each a ∈ A such that Aa is syntomic2 over R the ring Ca is smooth over Aa and has a presentation Ca = R[y1 , . . . , ym ]/J such that J/J 2 is free over Ca , and (2) for each a ∈ A such that Aa is smooth over R the module ΩCa /R is free over Ca . Proof. Choose a presentation A = R[x1 , . . . , xn ]/I and write I = (f1 , . . . , fm ). Define the A-module K by the short exact sequence 0 → K → A⊕m → I/I 2 → 0 where the jth basis vector ej in the middle is mapped to the class of fj on the right. Set C = Sym∗A (I/I 2 ). The retraction is just the projection onto the degree 0 part of C. We have a surjection R[x1 , . . . , xn , y1 , . . . , ym ] → C which maps yj to the class of fj in I/I 2 . The P kernel J of this map is generated by thePelements f1 , . . . , fm and by elements hj yj with hj ∈ R[x1 , . . . , xn ] such that hj ej defines an element of K. By Algebra, Lemma 7.124.3 applied to R → A → C and the presentations above and More on Algebra, Lemma 12.11.10 there is a short exact sequence (13.4.1.1)

I/I 2 ⊗A C → J/J 2 → K ⊗A C → 0

of C-modules. Let h ∈ R[x1 , . . . , xn ] be an element with image a ∈ A. We will use as presentations for the localized rings Aa = R[x0 , x1 , . . . , xn ]/I 0

and Ca = R[x0 , x1 , . . . , xn , y1 , . . . , ym ]/J 0

where I 0 = (hx0 − 1, I) and J 0 = (hx0 − 1, J). Hence I 0 /(I 0 )2 = Ca ⊕ I/I 2 ⊗A Ca and J 0 /(J 0 )2 = Ca ⊕ (J/J 2 )a as Ca -modules. Thus we obtain (13.4.1.2)

Ca ⊕ I/I 2 ⊗A Ca → Ca ⊕ (J/J 2 )a → K ⊗A Ca → 0

as the sequence of Algebra, Lemma 7.124.3 corresponding to R → Aa → Ca and the presentations above. Next, assume that a ∈ A is such that Aa is syntomic over R. Then (I/I 2 )a is finite projective over Aa , see Algebra, Lemma 7.126.17. Hence we see Ka ⊕ (I/I 2 )a ∼ = A⊕m is free. In particular K is finite projective too. By More on Algebra, Lemma a a 12.26.6 the sequence (13.4.1.2) is exact on the left. Hence J 0 /(J 0 )2 ∼ = Ca ⊕ I/I 2 ⊗A Ca ⊕ K ⊗A Ca ∼ = Ca⊕m+1 2Or just that R → A is a local complete intersection, see More on Algebra, Definition a 12.26.2.

958

13. SMOOTHING RING MAPS

This proves (1). Finally, suppose that in addition Aa is smooth over R. Then the same presentation shows that ΩCa /R is the cokernel of the map J 0 /(J 0 )2 −→

M i

Ca dxi ⊕

M j

Ca dyj

The summand Ca of J 0 /(J 0 )2 in the decomposition above corresponds to hx0 − 1 and hence maps isomorphically L to the summand Ca dx0 . The summand I/I 2 ⊗A Ca 0 0 2 of J /(J ) maps injectively to i=1,...,n Ca dxi with quotient ΩAa /R ⊗Aa Ca . The L summand K ⊗A Ca maps injectively to j≥1 Ca dyj with quotient isomorphic to I/I 2 ⊗A Ca . Thus the cokernel of the last displayed map is the module I/I 2 ⊗A Ca ⊕ ΩAa /R ⊗Aa Ca . Since (I/I 2 )a ⊕ ΩAa /R is free (from the definition of smooth ring maps) we see that (2) holds.  The following proposition was proved for henselian pairs by Elkik in [Elk73]. In the form stated below it can be found in [Ara01], where they also prove that ring maps between smooth algebras can be lifted. Proposition 13.4.2. Let R be a ring and let I ⊂ R be an ideal. Let R/I → A be a smooth ring map. Then there exists a smooth ring map R → A such that A/IA is isomorphic to A. 2

∗ (J/J ). Proof. Choose a presentation A = (R/I)[x1 , . . . , xn ]/J. Set C = SymA 2

Note that J/J is a finite projective A-module (follows from the definition of smoothness). By Lemma 13.4.1 and its proof the ring map A → C is smooth 2 and we can find a presentation C = R/I[y1 , . . . , ym ]/K with K/K free over C. By Algebra, Lemma 7.126.6 we can even assume that C = R/I[y1 , . . . , ym ]/(f 1 , . . . , f c ) 2 where f 1 , . . . , f c maps to a basis of K/K over C. Choose f1 , . . . , fc ∈ R[y1 , . . . , yc ] lifting f 1 , . . . , f c and set C = R[y1 , . . . , ym ]/(f1 , . . . , fc ) By construction C/IC = C. Consider the naive cotangent complex M (f1 , . . . , fc )/(f1 , . . . , fc )2 −→ Cdyi i=1,...,m

associated to the presentation of LC. For every prime q ⊃ IC of C the images dfj are linearly independent in κ(q)dyi because C is smooth over R/I. Hence we conclude that ((f , . . . , f )/(f , . . . , fc )2 )q is free of rank c and maps to a direct 1 c 1 L summand of Cq dyj . Hence R → C is smooth at q, see Algebra, Lemma 7.127.12. Thus we can find a g ∈ C mapping to an invertible element of C/IC such that R → Cg is smooth, see More on Algebra, Lemma 12.11.4. We conclude that there ∗ exists a finite projective A-module P such that C = SymA (P ) is isomorphic to C/IC for some smooth R-algebra C. ⊕n

Choose an integer n and a direct sum decomposition A = P ⊕ Q. By More on Algebra, Lemma 12.11.9 we can find an ´etale ring map C → C 0 which induces an isomorphism C/IC → C 0 /IC 0 and a finite projective C 0 -module Q such that Q/IQ is isomorphic to Q ⊗A C/IC. Then D = Sym∗C 0 (Q) is a smooth C 0 -algebra (see

13.4. PRESENTATIONS OF ALGEBRAS

959

More on Algebra, Lemma 12.11.11). Picture /C

R  R/I

/A

 / C/IC

∼ =

/ C0

/D

 / C 0 /IC 0

 / D/ID

Observe that our choice of Q gives D/ID = Sym∗C/IC (Q ⊗A C/IC) = Sym∗A (Q) ⊗A C/IC = Sym∗A (Q) ⊗A Sym∗A (P ) = Sym∗A (Q ⊕ P ) = Sym∗A (A

⊕n

)

= A[x1 , . . . , xn ] Choose f1 , . . . , fn ∈ D which map to x1 , . . . , xn in D/ID = A[x1 , . . . , xn ]. Set A = D/(f1 , . . . , fn ). Note that A = A/IA. By an argument similar to the argument in the first paragraph of the proof we see that R → A is smooth at all primes of IA. Hence, after replacing A by Af for a suitable f ∈ A (see More on Algebra, Lemma 12.11.4) we win.  We know that any syntomic ring map R → A is locally a relative global complete intersection, see Algebra, Lemma 7.126.16. The next lemma says that a vector bundle over Spec(A) is a relative global complete intersection. Lemma 13.4.3. Let R → A be a syntomic ring map. Then there exists a smooth R-algebra map A → C with a retraction such that C is a global relative complete intersection over R, i.e., C∼ = R[x1 , . . . , xn ]/(f1 , . . . , fc ) flat over R and all fibres of dimension n − c. Proof. Apply Lemma 13.4.1 to get A → C. By Algebra, Lemma 7.126.6 we can write C = R[x1 , . . . , xn ]/(f1 , . . . , fc ) with fi mapping to a basis of J/J 2 . The ring map R → C is syntomic (hence flat) as it is a composition of a syntomic and a smooth ring map. The dimension of the fibres is n − c by Algebra, Lemma 7.125.4 (the fibres are local complete intersections, so the lemma applies).  Lemma 13.4.4. Let R → A be a smooth ring map. Then there exists a smooth R-algebra map A → B with a retraction such that B is standard smooth over R, i.e., B∼ = R[x1 , . . . , xn ]/(f1 , . . . , fc ) and det(∂fj /∂xi )i,j=1,...,c is invertible in B. Proof. Apply Lemma 13.4.3 to get a smooth R-algebra map A → C with a retraction such that C = R[x1 , . . . , xn ]/(f1 , . . . , fc ) is a relative global complete intersection over R. As C is smooth over R we have a short exact sequence M M 0→ Cfj → Cdxi → ΩC/R → 0 j=1,...,c

i=1,...,n

960

13. SMOOTHING RING MAPS

Since ΩC/R is a projective C-module this sequence is split. Choose a left inverse t P P ∂f to the first map. Say t(dxi ) = cij fj so that i ∂xji ci` = δj` (Kronecker delta). Let B 0 = C[y1 , . . . , yc ] = R[x1 , . . . , xn , y1 , . . . , yc ]/(f1 , . . . , fc ) The R-algebra map C → B 0 has a retraction given by mapping yj to zero. We claim that the map X R[z1 , . . . , zn ] −→ B 0 , zi 7−→ xi − cij yj j

is ´etale at every point in the image of Spec(C) → Spec(B 0 ). In ΩB 0 /R[z1 ,...,zn ] we have X ∂fj X ∂fj dzi ≡ ci` dy` ≡ dyj mod (y1 , . . . , yc )ΩB 0 /R[z1 ,...,zn ] 0 = dfj − i,` ∂xi i ∂xi P 0 B dyj + (y1 , . . . , yc )ΩB 0 /R[z1 ,...,zn ] we conclude that Since 0 = dzi = dxi modulo ΩB 0 /R[z1 ,...,zn ] /(y1 , . . . , yc )ΩB 0 /R[z1 ,...,zn ] = 0. As ΩB 0 /R[z1 ,...,zn ] is a finite B 0 -module by Nakayama’s lemma there exists a g ∈ 1 + (y1 , . . . , yc ) that (ΩB 0 /R[z1 ,...,zn ] )g = 0. This proves that R[z1 , . . . , zn ] → Bg0 is unramified, see Algebra, Definition 7.139.1. For any ring map R → k where k is a field we obtain an unramified ring map k[z1 , . . . , zn ] → (Bg0 ) ⊗R k between smooth k-algebras of dimension n. It follows that k[z1 , . . . , zn ] → (Bg0 ) ⊗R k is flat by Algebra, Lemmas 7.120.1 and 7.130.2. By the crit`ere de platitude par fibre (Algebra, Lemma 7.120.8) we conclude that R[z1 , . . . , zn ] → Bg0 is flat. Finally, Algebra, Lemma 7.133.7 implies that R[z1 , . . . , zn ] → Bg0 is ´etale. Set B = Bg0 . Note that C → B is smooth and has a retraction, so also A → B is smooth and has a retraction. Moreover, R[z1 , . . . , zn ] → B is ´etale. By Algebra, Lemma 7.133.2 we can write B = R[z1 , . . . , zn , w1 , . . . , wc ]/(g1 , . . . , gc ) with det(∂gj /∂wi ) invertible in B. This proves the lemma.  Lemma 13.4.5. Let R → Λ be a ring map. If Λ is a filtered colimit of smooth R-algebras, then Λ is a filtered colimit of standard smooth R-algebras. Proof. Let A → Λ be an R-algebra map with A of finite presentation over R. According to Lemma 13.2.1 we have to factor this map through a standard smooth algebra, and we know we can factor it as A → B → Λ with B smooth over R. Choose an R-algebra map B → C with a retraction C → B such that C is standard smooth over R, see Lemma 13.4.4. Then the desired factorization is A → B → C → B → Λ.  Lemma 13.4.6. Let R → A be a standard smooth ring map. Let E ⊂ A be a finite subset of order |E| = n. Then there exists a presentation A = R[x1 , . . . , xn+m ]/(f1 , . . . , fc ) with c ≥ n, with det(∂fj /∂xi )i,j=1,...,c invertible in A, and such that E is the set of congruence classes of x1 , . . . , xn . Proof. Choose a presentation A = R[y1 , . . . , ym ]/(g1 , . . . , gd ) such that the image of det(∂gj /∂yi )i,j=1,...,d is invertible in A. Choose an enumerations E = {a1 , . . . , an } and choose hi ∈ R[y1 , . . . , ym ] whose image in A is ai . Consider the presentation A = R[x1 , . . . , xn , y1 , . . . , ym ]/(x1 − h1 , . . . , xn − hn , g1 , . . . , gd )

13.4. PRESENTATIONS OF ALGEBRAS

and set c = n + d.

961



Lemma 13.4.7. Let R → A be a ring map of finite presentation. Let a ∈ A. Consider the following conditions on a: (1) Aa is smooth over R, (2) Aa is smooth over R and ΩAa /R is stably free, (3) Aa is smooth over R and ΩAa /R is free, (4) Aa is standard smooth over R, (5) a is strictly standard in A over R, (6) a is elementary standard in A over R. Then we have (a) (4) ⇒ (3) ⇒ (2) ⇒ (1), (b) (6) ⇒ (5), (c) (6) ⇒ (4), (d) (5) ⇒ (2), (e) (2) ⇒ the elements ae , e ≥ e0 are strictly standard in A over R, (f) (4) ⇒ the elements ae , e ≥ e0 are elementary standard in A over R. Proof. Part (a) is clear from the definitions and Algebra, Lemma 7.127.7. Part (b) is clear from Definition 13.3.3. Proof of (c). Choose a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) such that (13.3.3.1) and (13.3.3.2) hold. Choose h ∈ R[x1 , . . . , xn ] mapping to a. Then Aa = R[x0 , x1 , . . . , xn ]/(x0 h − 1, f1 , . . . , fn ). Write J = (x0 h − 1, f1 , . . . , fn ). By (13.3.3.2) we see that the Aa -module J/J 2 is generated by x0 h − 1, f1 , . . . , fc over Aa . Hence, as in the proof of Algebra, Lemma 7.126.6, we can choose a g ∈ 1 + J such that Aa = R[x0 , . . . , xn , xn+1 ]/(x0 h − 1, f1 , . . . , fn , gxn+1 − 1). At this point (13.3.3.1) implies that R → Aa is standard smooth (use the coordinates x0 , x1 , . . . , xc , xn+1 to take derivatives). Proof of (d). Choose a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) such that (13.3.3.3) and (13.3.3.4) hold. We already know that Aa is smooth over R, see Lemma 13.3.5. As above we get a presentation Aa = R[x0 , x1 , . . . , xn ]/J with J/J 2 by the definition of smooth ring maps, hence free. Then ΩAa /R ⊕ J/J 2 ∼ = A⊕n+1 a we see that ΩAa /R is stably free. Proof of (e). Choose a presentation A = R[x1 , . . . , xn ]/I with I finitely generated. By assumption we have a short exact sequence M 0 → (I/I 2 )a → Aa dxi → ΩAa /R → 0 i=1,...,n

which is split exact. Hence we see that (I/I 2 )a ⊕ ΩAa /R is a free Aa -module. Since ΩAa /R is stably free we see that (I/I 2 )a is stably free as well. Thus replacing the presentation chosen above by A = R[x1 , . . . , xn , xn+1 , . . . , xn+r ]/J with J = (I, xn+1 , . . . , xn+r ) for some r we get that (J/J 2 )a is (finite) free. Choose f1 , . . . , fc ∈ J which map to a basis of (J/J 2 )a . Extend this to a list of generators f1 , . . . , fm ∈ J. Consider the presentation A = R[x1 , . . . , xn+r ]/(f1 , . . . , fm ). Then e (13.3.3.4) holds L for a for all sufficiently large e by construction. Moreover, since 2 (J/J )a → i=1,...,n Aa dxi is a split injection we can find an Aa -linear left inverse.

962

13. SMOOTHING RING MAPS

Writing this left inverse in terms of the basis f1 , . . . , fc and clearing denominators we find a linear map ψ0 : A⊕n → A⊕c such that (f1 ,...,fc ) f 7→df M ψ0 A⊕c −−−−−−→ J/J 2 −−−−→ Adxi −−→ A⊕c i=1,...,n

is multiplication by ae0 for some e0 ≥ 1. By Lemma 13.3.4 we see (13.3.3.3) holds for all ace0 and hence for ae for all e with e ≥ ce0 . Proof of (f). Choose a presentation Aa = R[x1 , . . . , xn ]/(f1 , . . . , fc ) such that det(∂fj /∂xi )i,j=1,...,c is invertible in Aa . We may assume that for some m < n the classes of the elements x1 , . . . , xm correspond ai /1 where a1 , . . . , am ∈ A are generators of A over R, see Lemma 13.4.6. After replacing xi by aN xi for m < i ≤ n we may assume the class of xi is ai /1 ∈ Aa for some ai ∈ A. Consider the ring map Ψ : R[x1 , . . . , xn ] −→ A,

xi 7−→ ai .

This is a surjective ring map. By replacing fj by aN fj we may assume that fj ∈ R[x1 , . . . , xn ] and that Ψ(fj ) = 0 (since after all fj (a1 /1, . . . , an /1) = 0 in Aa ). Let J = Ker(Ψ). Then A = R[x1 , . . . , xn ]/J is a presentation and f1 , . . . , fc ∈ J are elements such that (J/J 2 )a is freely generated by f1 , . . . , fc and such that det(∂fj /∂xi )i,j=1,...,c maps to an invertible element of Aa . It follows that (13.3.3.1) and (13.3.3.2) hold for ae and all large enough e as desired.  13.5. The lifting problem The goal in this section is to prove (Proposition 13.5.3) that the collection of algebras which are filtered colimits of smooth algebras is closed under infinitesimal flat deformations. The proof is elementary and only uses the results on presentations of smooth algebras from Section 13.4. Lemma 13.5.1. Let R → Λ be a ring map. Let I ⊂ R be an ideal. Assume that (1) I 2 = 0, and (2) Λ/IΛ is a filtered colimit of smooth R/I-algebras. Let ϕ : A → Λ be an R-algebra map with A of finite presentation over R. Then there exists a factorization A → B/J → Λ where B is a smooth R-algebra and J ⊂ IB is a finitely generated ideal. Proof. Choose a factorization ¯ → Λ/IΛ A/IA → B ¯ standard smooth over R/I; this is possible by assumption and Lemma with B 13.4.5. Write ¯ = A/IA[t1 , . . . , tr ]/(¯ B g1 , . . . , g¯s ) ¯ → Λ/IΛ maps ti to the class of λi modulo IΛ. Choose g1 , . . . , gs ∈ and say B P A[t1 , . . . , tr ] lifting g¯1 , . . . , g¯s . Write ϕ(gi )(λ1 , . . . , λr ) = ij µij for some ij ∈ I and µij ∈ Λ. Define X A0 = A[t1 , . . . , tr , δi,j ]/(gi − ij δij ) and consider the map A0 −→ Λ,

a 7−→ ϕ(a),

ti 7−→ λi ,

δij 7−→ µij

13.5. THE LIFTING PROBLEM

963

We have ∼ B[δ ¯ ij ] A0 /IA0 = A/IA[t1 , . . . , tr ]/(¯ g1 , . . . , g¯s )[δij ] = ¯ This is a standard smooth algebra over R/I as B is standard smooth. Choose a presentation A0 /IA0 = R/I[x1 , . . . , xn ]/(f¯1 , . . . , f¯c ) with det(∂ f¯j /∂xi )i,j=1,...,c invertible in A0 /IA0 . Choose lifts f1 , . . . , fc ∈ R[x1 , . . . , xn ] of f¯1 , . . . , f¯c . Then B = R[x1 , . . . , xn , xn+1 ]/(f1 , . . . , fc , xn+1 det(∂fj /∂xi )i,j=1,...,c − 1) is smooth over R. Since smooth ring maps are formally smooth (Algebra, Proposition 7.128.13) there exists an R-algebra map B → A0 which is an isomorphism modulo I. Then B → A0 is surjective by Nakayama’s lemma (Algebra, Lemma 7.18.1). Thus A0 = B/J with J ⊂ IB finitely generated (see Algebra, Lemma 7.6.3).  Lemma 13.5.2. Let R → Λ be a ring map. Let I ⊂ R be an ideal. Assume that (1) I 2 = 0, (2) Λ/IΛ is a filtered colimit of smooth R/I-algebras, and (3) R → Λ is flat. Let ϕ : B → Λ be an R-algebra map with B smooth over R. Let J ⊂ IB be a finitely generated ideal. Then there exists R-algebra maps α

β

B− → B0 − →Λ such that B 0 is smooth over R, such that α(J) = 0 and such that β ◦ α = ϕ mod IΛ. Proof. If we can prove the lemma in case J = (h), then we can prove the lemma by induction on the number of generators of J. Namely, suppose that J can be generated by n elements h1 , . . . , hn and the lemma holds for all cases where J is generated by n−1 elements. Then we apply the case n = 1 to produce B → B 0 → Λ where the first map kills of hn . Then we let J 0 be the ideal of B 0 generated by the images of h1 , . . . , hn−1 and we apply the case for n − 1 to produce B 0 → B 00 → Λ. It is easy to verify that B → B 00 → Λ does the job. P Assume J = P (h) and write h = i bi for some i ∈ I and bi ∈ B. Note that 0 = ϕ(h) = i ϕ(bi ). As Λ is flat over R, the equational criterion for flatness (Algebra, Lemma 7.36.10)Pimplies that P we can find λj ∈ Λ, j = 1, . . . , m and aij ∈ R such that ϕ(bi ) = j aij λj and i i aij = 0. Set X C = B[x1 , . . . , xm ]/(bi − aij xj ) with C → Λ given by ϕ and xj 7→ λj . Choose a factorization C → B 0 /J 0 → Λ as in Lemma 13.5.1. Since B is smooth over R we can lift the map B → C → B 0 /J 0 to a map ψ : B → B 0 . We claim that ψ(h) = 0. Namely, the fact that ψ agrees with B → C → B 0 /J 0 mod I implies that X ψ(bi ) = aij ξj + θi for some ξi ∈ B 0 and θi ∈ IB 0 . Hence we see that X X X ψ(h) = ψ(  i bi ) = i aij ξj + i θ i = 0 because of the relations above and the fact that I 2 = 0.



964

13. SMOOTHING RING MAPS

Proposition 13.5.3. Let R → Λ be a ring map. Let I ⊂ R be an ideal. Assume that (1) I is nilpotent, (2) Λ/IΛ is a filtered colimit of smooth R/I-algebras, and (3) R → Λ is flat. Then Λ is a colimit of smooth R-algebras. Proof. Since I n = 0 for some n, it follows by induction on n that it suffices to consider the case where I 2 = 0. Let ϕ : A → Λ be an R-algebra map with A of finite presentation over R. We have to find a factorization A → B → Λ with B smooth over R, see Lemma 13.2.1. By Lemma 13.5.1 we may assume that A = B/J with B smooth over R and J ⊂ IB a finitely generated ideal. By Lemma 13.5.2 we can find a (possibly noncommutative) diagram B

/ B0

α ϕ



~

Λ

β

of R-algebras which commutes modulo I and such that α(J) = 0. The map D : B −→ IΛ,

b 7−→ ϕ(b) − β(α(b))

is a derivation over R hence we can write it as D = ξ ◦ dB/R for some B-linear map P ξ : ΩB/R → IΛ. Since ΩB/R is a finite projective B-module we can write ξ = i=1,...,n i Ξi for some i ∈ I and B-linear maps Ξi : ΩB/R → Λ. (Details omitted. Hint: write ΩB/R as a direct sum of a finite free module to reduce to the finite free case.) We define M  B 00 = Sym∗B 0 ΩB/R ⊗B,α B 0 i=1,...,n

0

00

0

and we define β : B → Λ by β on B and by β 0 |ith summand ΩB/R ⊗B,α B 0 = Ξi ⊗ β and α0 : B → B 00 by α0 (b) = α(b) ⊕

X

i dB/R (b) ⊗ 1 ⊕ 0 ⊕ . . .

At this point the diagram B

/ B 00

α0 ϕ



Λ

~

β0

does commute. Moreover, it is direct from the definitions that α0 (J) = 0 as I 2 = 0. Hence the desired factorization.  13.6. The lifting lemma Here is a fiendishly clever lemma. Lemma 13.6.1. Let R be a Noetherian ring. Let Λ be an R-algebra. Let π ∈ R and assume that AnnR (π) = AnnR (π 2 ) and AnnΛ (π) = AnnΛ (π 2 ). Suppose we

13.6. THE LIFTING LEMMA

965

have R-algebra maps R/π 2 R → C¯ → Λ/π 2 Λ with C¯ of finite presentation. Then there exists an R-algebra homomorphism D → Λ and a commutative diagram R/π 2 R

/ C¯

/ Λ/π 2 Λ

 R/πR

 / D/πD

 / Λ/πΛ

with the following properties (a) D is of finite presentation, (b) R → D is smooth at any prime q with π 6∈ q, (c) R → D is smooth at any prime q with π ∈ q lying over a prime of C¯ where R/π 2 R → C¯ is smooth, and ¯ C¯ → D/πD is smooth at any prime lying over a prime of C¯ where (d) C/π R/π 2 R → C¯ is smooth. Proof. We choose a presentation C¯ = R[x1 , . . . , xn ]/(f1 , . . . , fm ) We also denote I = (f1 , . . . , fm ) and I¯ the image of I in R/π 2 R[x1 , . . . , xn ]. Since ¯ Hence the smooth locus of R/π 2 R → C¯ is quasi-compact, R is Noetherian, so is C. see Topology, Lemma 5.6.2. Applying Lemma 13.3.2 we may choose a finite list of elements a1 , . . . , ar ∈ R[x1 , . . . , xn ] such that ¯ cover the smooth (1) the union of the open subspaces Spec(C¯ak ) ⊂ Spec(C) 2 ¯ locus of R/π R → C, and (2) for each k = 1, . . . , r there exists a finite subset Ek ⊂ {1, . . . , m} such that ¯ I¯2 )a is freely generated by the classes of fj , j ∈ Ek . (I/ k Set Ik = (fj , j ∈ Ek ) ⊂ I and denote I¯k the image of Ik in R/π 2 R[x1 , . . . , xn ]. By ¯ I¯k )a is annihilated by 1 + b0 for some (2) and Nakayama’s lemma we see that (I/ k k 0 0 ¯ bk ∈ Iak . Suppose bk is the image of bk /(ak )N for some bk ∈ I and some integer N . After replacing ak by ak bk we get ¯a . (3) (I¯k )a = (I) k

k

Thus, after possibly replacing ak by a high power, we may write P (4) ak f` = j∈Ek hjk,` fj + π 2 gk,` for any ` ∈ {1, . . . , m} and some hji,` , gi,` ∈ R[x1 , . . . , xn ]. If ` ∈ Ek we choose hjk,` = ak δ`,j (Kronecker delta) and gk,` = 0. Set D = R[x1 , . . . , xn , z1 , . . . , zm ]/(fj − πzj , pk,` ). Here j ∈ {1, . . . , m}, k ∈ {1, . . . , r}, ` ∈ {1, . . . , m}, and X pk,` = ak z` − hjk,` zj − πgk,` . j∈Ek

Note that for ` ∈ Ek we have pk,` = 0 by our choices above. The map R → D is the given one. Say C¯ → Λ/π 2 Λ maps xi to the class of λi modulo π 2 . For an element f ∈ R[x1 , . . . , xn ] we denote f (λ) ∈ Λ the result of

966

13. SMOOTHING RING MAPS

subsituting λi for xi . Then we know that fj (λ) = π 2 µj for some µj ∈ Λ. Define D → Λ by the rules xi 7→ λi and zj 7→ πµj . This is well defined because X pk,` 7→ ak (λ)πµ` − hj (λ)πµj − πgk,` (λ) j∈Ek k,`   X = π ak (λ)µ` − hjk,` (λ)µj − gk,` (λ) j∈Ek

Substituting xi = λi in (4) above we see that the expression inside the brackets is annihilated by π 2 , hence it is annihilated by π as we have assumed AnnΛ (π) = AnnΛ (π 2 ). The map C¯ → D/πD is determined by xi 7→ xi (clearly well defined). Thus we are done if we can prove (b), (c), and (d). Using (4) we obtain the following key equality X πpk,` = πak z` − πhjk,` zj − π 2 gk,` j∈Ek X X = −ak (f` − πz` ) + ak f` + hj (fj − πzj ) − hj fj − π 2 gk,` j∈Ek k,` j∈Ek k,` X = −ak (f` − πz` ) + hjk,` (fj − πzj ) j∈Ek

The end result is an element of the ideal generated by fj − πzj . In particular, we see that D[1/π] is isomorphic to R[1/π][x1 , . . . , xn , z1 , . . . , zm ]/(fj − πzj ) which is isomorphic to R[1/π][x1 , . . . , xn ] hence smooth over R. This proves (b). For fixed k ∈ {1, . . . , r} consider the ring Dk = R[x1 , . . . , xn , z1 , . . . , zm ]/(fj − πzj , j ∈ Ek , pk,` ) The number of equations is m = |Ek | + (m − |Ek |) as pk,` is zero if ` ∈ Ek . Also, note that (Dk /πDk )ak = R/πR[x1 , . . . , xn , 1/ak , z1 , . . . , zm ]/(fj , j ∈ Ek , pk,` ) X ¯ C) ¯ a [z1 , . . . , zm ]/(ak z` − = (C/π hjk,` zj ) k j∈Ek

∼ ¯ C) ¯ a [zj , j ∈ Ek ] = (C/π k ¯ C) ¯ a . By our choice of ak we have In particular (Dk /πDk )ak is smooth over (C/π k ¯ ¯ that (C/π C)ak is smooth over R/πR of relative dimension n − |Ek |, see (2). Hence for a prime qk ⊂ Dk containing π and lying over Spec(C¯ak ) the fibre ring of R → Dk is smooth at qk of dimension n. Thus R → Dk is syntomic at qk by our count of the number of equations above, see Algebra, Lemma 7.126.11. Hence R → Dk is smooth at qk , see Algebra, Lemma 7.127.16. To finish the proof, let q ⊂ D be a prime containing π lying over a prime where R/π 2 R → C¯ is smooth. Then ak 6∈ q for some k by (1). We will show that the surjection Dk → D induces an isomorphism on local rings at q. Since we know that ¯ C¯ → Dk /πDk and R → Dk are smooth at the corresponding the ring maps C/π prime qk by the preceding paragraph this will prove (c) and (d) and thus finish the proof. P First, note that for any ` the equation πpk,` = −ak (f` − πz` ) + j∈Ek hjk,` (fj − πzj ) proved above shows that f` −πz` maps to zero in (Dk )ak and in particular in (Dk )qk .

13.7. THE DESINGULARIZATION LEMMA

The relations (4) imply that ak f` = on fj , j ∈ Ek we see that

P

ak0 hjk,` −

X

j∈Ek

967

hjk,` fj in I/I 2 . Since (I¯k /I¯k2 )ak is free 0

j 0 ∈Ek0

hjk0 ,` hjk,j 0

is zero in C¯ak for every k, k 0 , ` and j ∈ Ek . Hence we can find a large integer N such that   X j j0 j N ak ak0 hk,` − hk0 ,` hk,j 0 0 j ∈Ek0

2

is in Ik + π R[x1 , . . . , xn ]. Computing modulo π we have X j0 ak pk0 ,` − ak0 pk,` + hk0 ,` pk,j 0 X j0 X j X j0 X X j0 j = −ak hk0 ,` zj 0 + ak0 hk,` zj + hk0 ,` ak zj 0 − hk0 ,` hk,j 0 zj  X X 0 = ak0 hjk,` − hjk0 ,` hjk,j 0 zj +1 with Einstein summation convention. Combining with the above we see aN pk0 ,` k is contained in the ideal generated by Ik and π in R[x1 , . . . , xn , z1 , . . . , zm ]. Thus pk0 ,` maps into π(Dk )ak . On the other hand, the equation X 0 πpk0 ,` = −ak0 (f` − πz` ) + hjk0 ,` (fj 0 − πzj 0 ) 0 j ∈Ek0

shows that πpk0 ,` is zero in (Dk )ak . Since we have assumed that AnnR (π) = AnnR (π 2 ) and since (Dk )qk is smooth hence flat over R we see that Ann(Dk )qk (π) = Ann(Dk )qk (π 2 ). We conclude that pk0 ,` maps to zero as well, hence Dq = (Dk )qk and we win.  13.7. The desingularization lemma Here is another fiendishly clever lemma. Lemma 13.7.1. Let R be a Noetherian ring. Let Λ be an R-algebra. Let π ∈ R and assume that AnnΛ (π) = AnnΛ (π 2 ). Let A → Λ be an R-algebra map with A of finite presentation. Assume (1) the image of π is strictly standard in A over R, and (2) there exists a section ρ : A/π 4 A → R/π 4 R which is compatible with the map to Λ/π 4 Λ. Then we can find R-algebra maps A → B → Λ with B of finite presentation such that aB ⊂ HB/R where a = AnnR (AnnR (π 2 )/AnnR (π)). Proof. Choose a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fm ) and 0 ≤ c ≤ min(n, m) such that (13.3.3.3) holds for π and such that (13.7.1.1)

πfc+j ∈ (f1 , . . . , fc ) + (f1 , . . . , fm )2

for j = 1, . . . , m−c. Say ρ maps xi to the class of ri ∈ R. Then we can replace xi by xi − ri . Hence we may assume ρ(xi ) = 0 in R/π 4 R. This implies that fj (0) ∈ π 4 R and that A → Λ maps xi to π 4 λi for some λi ∈ Λ. Write X fj = fj (0) + rji xi + h.o.t. i=1,...,n

968

13. SMOOTHING RING MAPS

This implies that the constant term of ∂fj /∂xi is rji . Apply ρ to (13.3.3.3) for π and we see that X π= rI det(rji )j=1,...,c, i∈I mod π 4 R I⊂{1,...,n}, |I|=c

for some rI ∈ R. Thus we have X uπ =

I⊂{1,...,n}, |I|=c

rI det(rji )j=1,...,c,

i∈I

for some u ∈ 1 + π 3 R. By Algebra, Lemma 7.14.5 this implies there exists a n × c matrix (sik ) such that X uπδjk = rji cik for all j, k = 1, . . . , c i=1,...,n

(Kronecker delta). We introduce auxiliary variables v1 , . . . , vc , w1 , . . . , wn and we set X sij vj − π 3 wi hi = xi − π 2 j=1,...c

In the following we will use that R[x1 , . . . , xn , v1 , . . . , vc , w1 , . . . , wn ]/(h1 , . . . , hn ) = R[v1 , . . . , vc , w1 , . . . , wn ] without further mention. In R[x1 , . . . , xn , v1 , . . . , vc , w1 , . . . , wn ]/(h1 , . . . , hn ) we have fj = fj (x1 − h1 , . . . , xn − hn ) X X = π 2 rji sik vk + π 3 rji wi mod π 4 i i X = π 3 vj + π 3 rji wi mod π 4 for 1 ≤ j ≤ c. Hence P we can choose elements gj ∈ R[v31 , . . . , vc , w1 , . . . , wn ] such that gj = vj + rji wi mod π and such that fj = π gj in the R-algebra R[x1 , . . . , xn , v1 , . . . , vc , w1 , . . . , wn ]/(h1 , . . . , hn ). We set B = R[x1 , . . . , xn , v1 , . . . , vc , w1 , . . . , wn ]/(f1 , . . . , fn , h1 , . . . , hn , g1 , . . . , gc ). The map A → B is clear. We define B → Λ by mapping xi → π 4 λi , vi 7→ 0, and wi 7→ πλi . Then it is clear that the elements fj and hi are mapped to zero in Λ. Moreover, it is clear that gi is mapped to an element t of πΛ such that π 3 t = 0 (as fi = π 3 gi modulo the ideal generated by the h’s). Hence our assumption that AnnΛ (π) = AnnΛ (π 2 ) implies that t = 0. Thus we are done if we can prove the statement about smoothness. Note that Bπ ∼ = Aπ [v1 , . . . , vc ] because the equations gi = 0 are implied by fi = 0. Hence Bπ is smooth over R as Aπ is smooth over R by the assumption that π is strictly standard in A over R, see Lemma 13.3.5. P Set B 0 = R[v1 , . . . , vc , w1 , . . . , wn ]/(g1 , . . . , gc ). As gi = vi + rji wi mod π we see that B 0 /πB 0 = R/πR[w1 , . . . , wn ]. Hence R → B 0 is smooth of relative dimension n at every point of V (π) by Algebra, Lemmas 7.126.11 and 7.127.16 (the first lemma shows it is syntomic at those primes, in particular flat, whereupon the second lemma shows it is smooth). Let q ⊂ B be a prime with π ∈ q and for some r ∈ a, r 6∈ q. Denote q0 = B 0 ∩ q. We claim the surjection B 0 → B induces an isomorphism of local rings (B 0 )q0 → Bq . This will conclude the proof of the lemma. Note that Bq is the quotient of (B 0 )q0 by the ideal generated by fc+j , j = 1, . . . , m − c. We observe two things: first the

13.7. THE DESINGULARIZATION LEMMA

969

0 0 2 0 0 image of fc+j in (B P)q is divisible by π and second the image of πfc+j in (B )q can be written as bj1 j2 fc+j1 fc+j2 by (13.7.1.1). Thus we see that the image of each πfc+j is contained in the ideal generated by the elements π 2 fc+j 0 . Hence πfc+j = 0 in (B 0 )q0 as this is a Noetherian local ring, see Algebra, Lemma 7.48.6. As R → (B 0 )q0 is flat we see that
AnnR (π 2 )/AnnR (π) ⊗R (B 0 )q0 = Ann(B 0 )q0 (π 2 )/Ann(B 0 )q0 (π)

Because r ∈ a is invertible in (B 0 )q0 we see that this module is zero. Hence we see that the image of fc+j is zero in (B 0 )q0 as desired.  Lemma 13.7.2. Let R be a Noetherian ring. Let Λ be an R-algebra. Let π ∈ R and assume that AnnR (π) = AnnR (π 2 ) and AnnΛ (π) = AnnΛ (π 2 ). Let A → Λ and D → Λ be R-algebra maps with A and D of finite presentation. Assume (1) π is strictly standard in A over R, and (2) there exists an R-algebra map A/π 4 A → D/π 4 D compatible with the maps to Λ/π 4 Λ. Then we can find an R-algebra map B → Λ with B of finite presentation and R-algebra maps A → B and D → B compatible with the maps to Λ such that HD/R B ⊂ HB/D and HD/R B ⊂ HB/R . Proof. We apply Lemma 13.7.1 to D −→ A ⊗R D −→ Λ and the image of π in D. By Lemma 13.3.7 we see that π is strictly standard in A ⊗R D over D. As our section ρ : (A ⊗R D)/π 4 (A ⊗R D) → D/π 4 D we take the map induced by the map in (2). Thus Lemma 13.7.1 applies and we obtain a factorization A ⊗R D → B → Λ with B of finite presentation and aB ⊂ HB/D where a = AnnD (AnnD (π 2 )/AnnD (π)). For any prime q of D such that Dq is flat over R we have AnnDq (π 2 )/AnnDq (π) = 0 because annihilators of elements commutes with flat base change and we assumed AnnR (π) = AnnR (π 2 ). Because D is Noetherian we see that AnnD (π 2 )/AnnD (π) is a finite D-module, hence formation of its annihilator commutes with localization. Thus we see that a 6⊂ q. Hence we see that D → B is smooth at any prime of B lying over q. Since any prime of D where R → D is smooth is one where Dq is flat over R we conclude that HD/R B ⊂ HB/D . The final inclusion HD/R B ⊂ HB/R follows because compositions of smooth ring maps are smooth (Algebra, Lemma 7.127.14).  Lemma 13.7.3. Let R be a Noetherian ring. Let Λ be an R-algebra. Let π ∈ R and assume that AnnR (π) = AnnR (π 2 ) and AnnΛ (π) = AnnΛ (π 2 ). Let A → Λ be an R-algebra map with A of finite presentation and assume π is strictly standard in A over R. Let A/π 8 A → C¯ → Λ/π 8 Λ be a factorization with C¯ of finite presentation. Then we can find a factorization A → B → Λ with B of finite presentation such that Rπ → Bπ is smooth and such that q 8 HB/R Λ mod π 8 Λ. HC/(R/π 8 R) · Λ/π Λ ⊂ ¯

970

13. SMOOTHING RING MAPS

¯ 4 C¯ → Proof. Apply Lemma 13.6.1 to get R → D → Λ with a factorization C/π 4 4 D/π D → Λ/π Λ such that R → D is smooth at any prime not containing π and at ¯ 4 C¯ where R/π 8 R → C¯ is smooth. By Lemma any prime lying over a prime of C/π 13.7.2 we can find a finitely presented R-algebra B and factorizations A → B → Λ and D → B → Λ such that HD/R B ⊂ HB/R . We omit the verification that this is a solution to the problem posed by the lemma.  13.8. Warmup: reduction to a base field In this section we apply the lemmas in the previous sections to prove that it suffices to prove the main result when the base ring is a field, see Lemma 13.8.4. Situation 13.8.1. Here R → Λ is a regular ring map of Noetherian rings. Let R → Λ be as in Situation 13.8.1. We say PT holds for R → Λ if Λ is a filtered colimit of smooth R-algebras. Lemma 13.8.2. Let Ri → Λi , i = 1, 2 be as in Situation 13.8.1. If PT holds for Ri → Λi , i = 1, 2, then PT holds for R1 × R2 → Λ1 × Λ2 . Proof. Omitted. Hint: A product of colimits is a colimit.



Lemma 13.8.3. Let R → A → Λ be ring maps with A of finite presentation over R. Let S ⊂ R be a multiplicative set. Let S −1 A → B 0 → S −1 Λ be a factorization with B 0 smooth over S −1 R. Then we can find a factorization A → B → Λ such that some s ∈ S maps to an elementary standard element in B over R. Proof. We first apply Lemma 13.4.4 to S −1 R → B 0 . Thus we may assume B 0 is standard smooth over S −1 R. Write A = R[x1 , . . . , xn ]/(g1 , . . . , gt ) and say xi 7→ λi in Λ. We may write B 0 = S −1 R[x1 , . . . , xn+m ]/(f1 , . . . , fc ) for some c ≥ n where det(∂fj /∂xi )i,j=1,...,c is invertible in B 0 and such that A → B 0 is given by xi 7→ xi , see Lemma 13.4.6. After multiplying xi , i > n by an element of S and correspondingly modifying the equations fj we may assume B 0 → S −1 Λ maps xi to λi /1 for some λi ∈ Λ for i > n. Choose a relation X 1 = a0 det(∂fj /∂xi )i,j=1,...,c + a j fj j=1,...,c

for some aj ∈ S −1 R[x1 , . . . , xn+m ]. Since each element of S is invertible in B 0 we may (by clearing denominators) assume that fj , aj ∈ R[x1 , . . . , xn+m ] and that X s0 = a0 det(∂fj /∂xi )i,j=1,...,c + aj fj j=1,...,c

for some s0 ∈ S. Since gj maps to zero in S −1 R[x1 , . . . , xn+m ]/(f1 , . . . , xc ) we can find elements sj ∈ S such that sj gj = 0 in R[x1 , . . . , xn+m ]/(f1 , . . . , fc ). Since fj maps to zero in S −1 Λ we can find s0j ∈ S such that s0j fj (λ1 , . . . , λn+m ) = 0 in Λ. Consider the ring B = R[x1 , . . . , xn+m ]/(s01 f1 , . . . , s0c fc , g1 , . . . , gt ) and the factorization A → B → Λ with B → Λ given by xi 7→ λi . We claim that s = s0 s1 . . . st s01 . . . s0c is elementary standard in B over R which finishes the proof. Namely, sj gj ∈ (f1 , . . . , fc ) and hence sgj ∈ (s01 f1 , . . . , s0c fc ). Finally, we have X a0 det(∂s0j fj /∂xi )i,j=1,...,c + (s01 . . . sˆ0j . . . s0c )aj s0j fj = s0 s01 . . . s0c j=1,...,c

which divides s as desired.



13.9. LOCAL TRICKS

971

Lemma 13.8.4. If for every Situation 13.8.1 where R is a field PT holds, then PT holds in general. Proof. Assume PT holds for any Situation 13.8.1 where R is a field. Let R → Λ be as in Situation 13.8.1 arbitrary. Note that R/I → Λ/IΛ is another regular ring map of Noetherian rings, see More on Algebra, Lemma 12.33.3. Consider the set of ideals I = {I ⊂ R | R/I → Λ/IΛ does not have PT} We have to show that I is empty. If this set is nonempty, then it contains a maxmimal element because R is Noetherian. Replacing R by R/I and Λ by Λ/I we obtain a situation where PT holds for R/I → Λ/IΛ for any nonzero ideal of R. In particular, we see by applying Proposition 13.5.3 that R is a reduced ring. Let A → Λ be an R-algebra homomorphism with A of finite presentation. We have to find a factorization A → B → Λ with B smooth over R, see Lemma 13.2.1. Let S ⊂ R be the set of nonzerodivisors and consider the total ring of fractions Q = S −1 R of R. We know that Q = K1 ×. . .×Kn is a product of fields, see Algebra, Lemmas 7.23.2 and 7.29.6. By Lemma 13.8.2 and our assumption PT holds for the ring map S −1 R → S −1 Λ. Hence we can find a factorization S −1 A → B 0 → Λ with B 0 smooth over S −1 R. We apply Lemma 13.8.3 and find a factorization A → B → Λ such that some π ∈ S is elementary standard in B over R. After replacing A by B we may assume that π is elementary standard, hence strictly standard in A. We know that R/π 8 R → Λ/π 8 Λ satisfies PT. Hence we can find a factorization R/π 8 R → A/π 8 A → C¯ → Λ/π 8 Λ with R/π 8 R → C¯ smooth. By Lemma 13.6.1 we can find an R-algebra map D → Λ with D smooth over R and a factorization R/π 4 R → A/π 4 A → D/π 4 D → Λ/π 4 Λ. By Lemma 13.7.2 we can find A → B → Λ with B smooth over R which finishes the proof.  13.9. Local tricks Situation 13.9.1. We are given a Noetherian ring R and an R-algebra map A → Λ and a prime q ⊂ p Λ. We assume A is of finite presentation over R. In this situation we denote hA = HA/R Λ. Let R → A → Λ ⊃ q be as in Situation 13.9.1. We say R → A → Λ ⊃ q can be resolved if there exists a factorization A → B → Λ with B of finite presentation and hA ⊂ hB 6⊂ q. In this case we will call the factorization A → B → Λ a resolution of R → A → Λ ⊃ q. Lemma 13.9.2. Let R → A → Λ ⊃ q be as in Situation 13.9.1. Let r ≥ 1 and π1 , . . . , πr ∈ R map to elements of q. Assume (1) for i = 1, . . . , r we have 2 8 8 AnnR/(π18 ,...,πi−1 )R (πi ) = AnnR/(π18 ,...,πi−1 )R (πi )

and 2 8 8 AnnΛ/(π18 ,...,πi−1 )Λ (πi ) = AnnΛ/(π18 ,...,πi−1 )Λ (πi )

(2) for i = 1, . . . , r the element πi maps to a strictly standard element in A over R.

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Then, if R/(π18 , . . . , πr8 )R → A/(π18 , . . . , πr8 )A → Λ/(π18 , . . . , πr8 )Λ ⊃ q/(π18 , . . . , πr8 )Λ can be resolved, so can R → A → Λ ⊃ q. Proof. We are going to prove this by induction on r. The case r = 1. Here the assumption is that there exists a factorization A/π18 → C¯ → Λ/π18 which resolves the situation modulo π18 . Conditions (1) and (2) are the assumptions needed to apply Lemma 13.7.3. Thus we can “lift” the resolution C¯ to a resolution of R → A → Λ ⊃ q. The case r > 1. In this case we apply the induction hypothesis for r − 1 to the situation R/π18 → A/π18 → Λ/π18 ⊃ q/π18 Λ. Note that property (2) is preserved by Lemma 13.3.7.  Lemma 13.9.3. Let R → A → Λ ⊃ q be as in Situation 13.9.1. Let p = R ∩ q. Assume that q is minimal over hA and that Rp → Ap → Λq ⊃ qΛq can be resolved. Then there exists a factorization A → C → Λ with C of finite presentation such that HC/R Λ 6⊂ q. Proof. Let Ap → C → Λq be a resolution of Rp → Ap → Λq ⊃ qΛq . By our assumption that q is minimal over hA this means that HC/Rp Λq = Λq . By Lemma 13.3.8 we may assume that C is smooth over Λp . By Lemma 13.4.4 we may assume that C is standard smooth over Rp . Write A = R[x1 , . . . , xn ]/(g1 , . . . , gt ) and say A → Λ is given by xi 7→ λi . Write C = Rp [x1 , . . . , xn+m ]/(f1 , . . . , fc ) for some c ≥ n such that A → C maps xi to xi and such that det(∂fj /∂xi )i,j=1,...,c is invertible in C, see Lemma 13.4.6. After clearing denominators we may assume f1 , . . . , fc are elements of R[x1 , . . . , xn+m ]. Of course det(∂fj /∂xi )i,j=1,...,c is not invertible in R[x1 , . . . , xn+m ]/(f1 , . . . , fc ) but it becomes invertible after inverting some element s0 ∈ R, s0 6∈ p. As gj maps to zero under R[x1 , . . . , xn ] → A → C we can find sj ∈ R, sj 6∈ p such that sj gj is zero in R[x1 , . . . , xn+m ]/(f1 , . . . , fc ). Write fj = Fj (x1 , . . . , xn+m , 1) for some polynomial Fj ∈ R[x1 , . . . , xn , Xn+1 , . . . , Xn+m+1 ] homogeneous in Xn+1 , . . . , Xn+m+1 . Pick λn+i ∈ Λ, i = 1, . . . , m+1 with λn+m+1 6∈ q such that xn+i maps to λn+i /λn+m+1 in Λq . Then λn+m λn+1 ,..., , 1) λn+m+1 λn+m+1 λn+m λn+1 ,..., ) = (λn+m+1 )deg(Fj ) fj (λ1 , . . . , λn , λn+m+1 λn+m+1 =0

Fj (λ1 , . . . , λn+m+1 ) = (λn+m+1 )deg(Fj ) Fj (λ1 , . . . , λn ,

in Λq . Thus we can find λ0 ∈ Λ, λ0 6∈ q such that λ0 Fj (λ1 , . . . , λn+m+1 ) = 0 in Λ. Now we set B equal to R[x0 , . . . , xn+m+1 ]/(g1 , . . . , gt , x0 F1 (x1 , . . . , xn+m+1 ), . . . , x0 Fc (x1 , . . . , xn+m+1 )) which we map to Λ by mapping xi to λi . Let b be the image of x0 x1 s0 s1 . . . st in B. Then Bb is isomorphic to Rs0 s1 [x0 , x1 , . . . , xn+m+1 , 1/x0 xn+m+1 ]/(f1 , . . . , fc ) which is smooth over R by construction. Since b does not map to an element of q, we win. 

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Lemma 13.9.4. Let R → A → Λ ⊃ q be as in Situation 13.9.1. Let p = R ∩ q. Assume (1) q is minimal over hA , (2) Rp → Ap → Λq ⊃ qΛq can be resolved, and (3) dim(Λq ) = 0. Then R → A → Λ ⊃ q can be resolved. Proof. By (3) the ring Λq is Artinian local hence qΛq is nilpotent. Thus (hA )N Λq = 0 for some N > 0. Thus there exists a λ ∈ Λ, λ 6∈ q such that λ(hA )N = 0 in Λ. Say HA/R = (a1 , . . . , ar ) so that λaN i = 0 in Λ. By Lemma 13.9.3 we can find a factorization A → C → Λ with C of finite presentation such that hC 6⊂ q. Write C = A[x1 , . . . , xn ]/(f1 , . . . , fm ). Set X yi tij , zyi ) B = A[x1 , . . . , xn , y1 , . . . , yr , z, tij ]/(fj − where tij is a set of rm variables. Note that there is a map B → C[yi , z]/(yi z) given by setting tij equal to zero. The map B → Λ is the composition B → C[yi , z]/(yi z) → Λ where C[yi , z]/(yi z) → Λ is the given map C → Λ, maps z to λ, and maps yi to the image of aN i in Λ. We claim that B is a solution for R → A → Λ ⊃ q. First note that Bz is isomorphic to C[y1 , . . . , yr , z, z −1 ] and hence is smooth. On the other hand, By` ∼ = A[xi , yi , y`−1 , tij , i 6= `] which is smooth over A. Thus we see that z and a` y` (compositions of smooth maps are smooth) are all elements of HB/R . This proves the lemma.  13.10. Separable residue fields In this section we explain how to solve a local problem in the case of a separable residue field extension. Lemma 13.10.1 (Ogoma). Let A be a Noetherian ring and let M be a finite Amodule. Let S ⊂ A be a multiplicative set. If π ∈ A and Ker(π : S −1 M → S −1 M ) = Ker(π 2 : S −1 M → S −1 M ) then there exists an s ∈ S such that for any n > 0 we have Ker(sn π : M → M ) = Ker((sn π)2 : M → M ). Proof. Let K = Ker(π : M → M ) and K 0 = {m ∈ M | π 2 m = 0 in S −1 M } and Q = K 0 /K. Note that S −1 Q = 0 by assumption. Since A is Noetherian we see that Q is a finite A-module. Hence we can find an s ∈ S such that s annihilates Q. Then s works.  Lemma 13.10.2. Let Λ be a Noetherian ring. Let I ⊂ Λ be an ideal. Let I ⊂ q be a prime. Let n, e be positive integers Assume that qn Λq ⊂ IΛq and that Λq is a regular local ring of dimension d. Then there exists an n > 0 and π1 , . . . , πd ∈ Λ such that (1) (π1 , . . . , πd )Λq = qΛq , (2) π1n , . . . , πdn ∈ I, and (3) for i = 1, . . . , d we have 2 e e AnnΛ/(π1e ,...,πi−1 )Λ (πi ) = AnnΛ/(π1e ,...,πi−1 )Λ (πi ).

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Proof. Set S = Λ \ q so that Λq = S −1 Λ. First pick π1 , . . . , πd with (1) which is possible as Λq is regular. By assumption πin ∈ IΛq . Thus we can find s1 , . . . , sd ∈ S such that si πin ∈ I. Replacing πi by si πi we get (2). Note that (1) and (2) are preserved by further multiplying by elements of S. Suppose that (3) holds for i = 1, . . . , t for some t ∈ {0, . . . , d}. Note that π1 , . . . , πd is a regular sequence in S −1 Λ, see Algebra, Lemma 7.99.3. In particular π1e , . . . , πte , πt+1 is a regular sequence in S −1 Λ = Λq by Algebra, Lemma 7.66.10. Hence we see that 2 e e AnnS −1 Λ/(π1e ,...,πi−1 ) (πi ) = AnnS −1 Λ/(π1e ,...,πi−1 ) (πi ).

Thus we get (3) for i = t+1 after replacing πt+1 by sπt+1 for some s ∈ S by Lemma 13.10.1. By induction on t this produces a sequence satisfying (1), (2), and (3).  Lemma 13.10.3. Let k → A → Λ ⊃ q be as in Situation 13.9.1 where (1) k is a field, (2) Λ is Noetherian, (3) q is minimal over hA , (4) Λq is a regular local ring, and (5) the field extension k ⊂ κ(q) is separable. Then k → A → Λ ⊃ q can be resolved. Proof. Set d = dim Λq . Set R = k[x1 , . . . , xd ]. Choose n > 0 such that qn Λq ⊂ hA Λq which is possible as q is minimal over hA . Choose generators a1 , . . . , ar of HA/R . Set X B = A[x1 , . . . , xd , zij ]/(xni − zij aj ) Each Baj is smooth over R it is a polynomial algebra over Aaj [x1 , . . . , xd ] and Aaj is smooth over k. Hence Bxi is smooth over R. Let B → C be the R-algebra map constructed in Lemma 13.4.1 which comes with a R-algebra retraction C → B. In particular a map C → Λ fitting into the diagram above. By construction Cxi is a smooth R-algebra with ΩCxi /R free. Hence we can find c > 0 such that xci is strictly standard in C/R, see Lemma 13.4.7. Now choose π1 , . . . , πd ∈ P Λ as in Lemma 13.10.2 where n = n, e = 8c, q = q and I = hA . Write πin = λij aj for some πij ∈ Λ. There is a map B → Λ given by xi 7→ πi and zij 7→ λij . Set R = k[x1 , . . . , xd ]. Diagram RO

/B O

k

/A



/Λ

Now we apply Lemma 13.9.2 to R → C → Λ ⊃ q and the sequence of elements xc1 , . . . , xcd of R. Assumption (2) is clear. Assumption (1) holds for R by inspection and for Λ by our choice of π1 , . . . , πd . (Note that if AnnΛ (π) = AnnΛ (π 2 ), then we have AnnΛ (π) = AnnΛ (π c ) for all c > 0.) Thus it suffices to resolve R/(xe1 , . . . , xed ) → C/(xe1 , . . . , xed ) → Λ/(π1e , . . . , πde ) ⊃ q/(π1e , . . . , πde ) for e = 8c. By Lemma 13.9.4 it suffices to resolve this after localizing at q. But since x1 , . . . , xd map to a regular sequence in Λq we see that R → Λ is flat, see Algebra, Lemma 7.120.2. Hence R/(xe1 , . . . , xed ) → Λq /(π1e , . . . , πde )

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is a flat ring map of Artinian local rings. Moreover, this map induces a separable field extension on residue fields by assumption. Thus this map is a filtered colimit of smooth algebras by Algebra, Lemma 7.142.10 and Proposition 13.5.3. Existence of the desired solution follows from Lemma 13.2.1.  13.11. Inseparable residue fields In this section we explain how to solve a local problem in the case of an inseparable residue field extension. Lemma 13.11.1. Let k be a field of characteristic p > 0. Let (Λ, m, K) be an Artinian local k-algebra. Assume that dim H1 (LK/k ) < ∞. Then Λ is a filtered colimit of Artinian local k-algebras A with each map A → Λ flat, with mA Λ = m, and with A essentially of finite type over k. Proof. Note that the flatness of A → Λ implies that A → Λ is injective, so the lemma really tells us that Λ is a directed union of these types of subrings A ⊂ Λ. Let n be the minimal integer such that mn = 0. We will prove this lemma by induction on n. The case n = 1 is clear as a field extension is a union of finitely generated field extensions. Pick λ1 , . . . , λd ∈ m which generate m. As K is formally smooth over Fp (see Algebra, Lemma 7.142.6) we can find a ring map σ : K → Λ which is a section of the quotient map Λ → K. In general σ is not a k-algebra map. Given σ we define Ψσ : K[x1 , . . . , xd ] −→ Λ using σ on elements of K and mapping xi to λi . Claim: there exists a σ : K → Λ and a subfield k ⊂ F ⊂ K finitely generated over k such that the image of k in Λ is contained in Ψσ (F [x1 , . . . , xd ]). We will prove the claim by induction on the least integer n such that mn = 0. It is clear for n = 1. If n > 1 set I = mn−1 and Λ0 = Λ/I. By induction we may assume given σ 0 : K → Λ0 and k ⊂ F 0 ⊂ K finitely generated such that the image of k → Λ → Λ0 is contained in A0 = Ψσ0 (F 0 [x1 , . . . , xd ]). Denote τ 0 : k → A0 the induced map. Choose a lift σ : K → Λ of σ 0 (this is possible by the formal smoothness of K/Fp we mentioned above). For later reference we note that we can change σ to σ + D for some derivation D : K → I. Set A = F [x1 , . . . , xd ]/(x1 , . . . , xd )n . Then Ψσ induces a ring map Ψσ : A → Λ. The composition with the quotient map Λ → Λ0 induces a surjective map A → A0 with nilpotent kernel. Choose a lift τ : k → A of τ 0 (possible as k/Fp is formally smooth). Thus we obtain two maps k → Λ, namely Ψσ ◦ τ : k → Λ and the given map i : k → Λ. These maps agree modulo I, whence the difference is a derivation θ = i − Ψσ ◦ τ : k → I. Note that if we change σ into σ + D then we change θ into θ − D|k . Choose a set of elements {yj }j∈J of k whose differentials dyj form a basis of Ωk/Fp . The Jacobi-Zariski sequence for Fp ⊂ k ⊂ K is 0 → H1 (LK/k ) → Ωk/Fp ⊗ K → ΩK/Fp → ΩK/k → 0 As dim H1 (LK/k ) < ∞ weL can find a finite subset J0 ⊂ J such that the image of the first map is contained in j∈J0 Kdyj . Hence the elements dyj , j ∈ J \ J0 map to

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K-linearly independent elements of ΩK/Fp . Therefore we can choose a D : K → I such that θ − D|k = ξ ◦ d where ξ is a composition M M kdyj −→ I Ωk/Fp = kdyj −→ j∈J

j∈J0

Let fj = ξ(dy Pj ) ∈ I for j ∈ J0 . Change σ into P σ + D as above. Then we see that θ(a) = j∈J0 aj fj for a ∈ k where da = aj dyj in Ωk/Fp . Note that I is P ed e1 E = ei = n − 1 generated by the monomials P λ = Eλ1 . . . λd of total degree |E| in λ1 , . . . , λd . Write fj = E cj,E λ with cj,E ∈ K. Replace F 0 by F = F 0 (cj,E ). Then the claim holds. Choose σ and F as in the claim. The kernel of Ψσ is generated by finitely many polynomials g1 , . . . , gt ∈ K[x1 , . . . , xd ] and we may assume their coefficients are in F after enlarging F by adjoining finitely many elements. In this case it is clear that the map A = F [x1 , . . . , xd ]/(g1 , . . . , gt ) → K[x1 , . . . , xd ]/(g1 , . . . , gt ) = Λ is flat. By the claim A is a k-subalgebra of Λ. It is clear that Λ is the filtered colimit of these algebras, as K is the filtered union of the subfields F . Finally, these algebras are essentially of finite type over k by Algebra, Lemma 7.51.3.  Lemma 13.11.2. Let k be a field of characteristic p > 0. Let Λ be a Noetherian geometrically regular k-algebra. Let q ⊂ Λ be a prime ideal. Let n ≥ 1 be an integer and let E ⊂ Λq /qn Λq be a finite subset. Then we can find m ≥ 0 and ϕ : k[y1 , . . . , ym ] → Λ with the following properties (1) setting p = ϕ−1 (q) we have qΛq = pΛq and k[y1 , . . . , ym ]p → Λq is flat, (2) there is a factorization by homomorphisms of local Artinian rings k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p → D → Λq /qn Λq where the first arrow is essentially smooth and the second is flat, (3) E is contained in D modulo qn Λq . ¯ = Λq /qn Λq . Note that dim H1 (Lκ(q)/k ) < ∞ by More on Algebra, Proof. Set Λ ¯ containing E such that A is local Artinian, Proposition 12.28.1. Pick A ⊂ Λ ¯ is flat, and mA generates the essentially of finite type over k, the map A → Λ ¯ see Lemma 13.11.1. Denote F = A/mA the residue field so maximal ideal of Λ, ¯ such that that k ⊂ F ⊂ K. Pick λ1 , . . . , λt ∈ Λ which map to elements of A in Λ moreover the images of dλ1 , . . . , dλt form a basis of ΩF/k . Consider the map ϕ0 : k[y1 , . . . , yt ] → Λ sending yj to λj . Set p0 = (ϕ0 )−1 (q). By More on Algebra, Lemma 12.28.2 the ring map k[y1 , . . . , yt ]p0 → Λq is flat and Λq /p0 Λq is regular. Thus we ¯ and which can choose further elements λt+1 , . . . , λm ∈ Λ which map into A ⊂ Λ 0 map to a regular system of parameters of Λq /p Λq . We obtain ϕ : k[y1 , . . . , ym ] → Λ ¯ factors through having property (1) such that k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p → Λ n A. Thus k[y1 , . . . , ym ]p /p k[y1 , . . . , ym ]p → A is flat by Algebra, Lemma 7.36.8. By construction the residue field extension κ(p) ⊂ F is finitely generated and ΩF/κ(p) = 0. Hence it is finite separable by More on Algebra, Lemma 12.27.1. Thus k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p → A is finite by Algebra, Lemma 7.51.3. Finally, we conclude that it is ´etale by Algebra, Lemma 7.133.7. Since an ´etale ring map is certainly essentially smooth we win.  Lemma 13.11.3. Let ϕ : k[y1 , . . . , ym ] → Λ, n, q, p and k[y1 , . . . , ym ]p /pn → D → Λq /qn Λq

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be as in Lemma 13.11.2. Then for any λ ∈ Λ \ q there exists an integer q > 0 and a factorization k[y1 , . . . , ym ]p /pn → D → D0 → Λq /qn Λq such that D → D0 is an essentially smooth map of local Artinian rings, the last arrow is flat, and λq is in D0 . ¯ be the image of λ in Λ. ¯ = Λq /qn Λq . Let λ ¯ Let α ∈ κ(q) be the image Proof. Set Λ of λ in the residue field. Let k ⊂ F ⊂ κ(q) be the residue field of D. If α is in ¯ = 1 mod q. Hence (xλ) ¯ q = 1 mod (q)q F then we can find an x ∈ D such that xλ q ¯ if q is divisible by p. Hence λ is in D. If α is transcendental over F , then we ¯ m equal to the subring generated by D and λ ¯ localized at can take D0 = (D[λ]) ¯ ¯ ¯ m = D[λ] ∩ qΛ. This works because D[λ] is in fact a polynomial algebra over D in this case. Finally, if λ mod q is algebraic over F , then we can find a p-power q such that αq is separable algebraic over F , see Algebra, Section 7.39. Note that D and ¯ are henselian local rings, see Algebra, Lemma 7.140.11. Let D → D0 be a finite Λ ´etale extension whose residue field extension is F ⊂ F (αq ), see Algebra, Lemma ¯ is henselian and F (αq ) is contained in its residue field we can find 7.140.8. Since Λ ¯ qq0 ∈ D0 for ¯ By the first part of the argument we see that λ a factorization D0 → Λ. some q 0 > 0.  Lemma 13.11.4. Let k → A → Λ ⊃ q be as in Situation 13.9.1 where (1) k is a field of characteristic p > 0, (2) Λ is Noetherian and geometrically regular over k, (3) q is minimal over hA . Then k → A → Λ ⊃ q can be resolved. Proof. The lemma is proven by the following steps in the given order. We will justify each of these steps below. (1) Pick an integer N > 0 such that qN Λq ⊂ HA/k Λq . (2) Pick generators a1 , . . . , at ∈ A of the ideal HA/R . (3) Set d = dim(Λq ). P (4) Set B = A[x1 , . . . , xd , zij ]/(x2N − zij aj ). i (5) Consider B as a k[x1 , . . . , xd ]-algebra and let B → C be as in Lemma 13.4.1. We also obtain a section C → B. (6) Choose c > 0 such that each xci is strictly standard in C over k[x1 , . . . , xd ]. (7) Set n = N + dc and e = 8c. (8) Let E ⊂ Λq /qn Λq be the images of generators of A as a k-algebra. (9) Choose an integer m and a k-algebra map ϕ : k[y1 , . . . , ym ] → Λ and a factorization by local Artinian rings k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p → D → Λq /qn Λq such that the first arrow is essentially smooth, the second is flat, E is contained in D, with p = ϕ−1 (q) the map k[y1 , . . . , ym ]p → Λq is flat, and pΛq = qΛq . (10) Choose π1 , . . . , πd ∈ p which map to a regular system of parameters of k[y1 , . . . , ym ]p . (11) Let R = k[y1 , . . . , ym , t1 , . . . , tm ] and γi = πi ti . (12) If necessary modify the choice of πi such that for i = 1, . . . , d we have 2 e e AnnR/(γ1e ,...,γi−1 )R (γi ) = AnnR/(γ1e ,...,γi−1 )R (γi )

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(13) There exist δ1 , . . . , δd ∈ Λ, δi 6∈ q and a factorization D → D0 → Λq /qn Λq with D0 local Artinian, D → D0 essentially smooth, the map D0 → 0 Λq /qn Λq flat such for i = 1, . . . , d P that, with πi = δi πi , we have 0 2N (a) (πi ) = aj λij in Λ where λij mod qn Λq is an element of D0 , 2 (b) AnnΛ/(π0 e1 ,...,π0 ei−1 ) (π 0 i ) = AnnΛ/(π0 e1 ,...,π0ei−1 ) (π 0 i ), (c) δi mod qn Λq is an element of D0 . (14) Define B → Λ by sending xi to πi0 and zij to λij found above. Define C → Λ by composing the map B → Λ with the retraction C → B. (15) Map R → Λ by ϕ on k[y1 , . . . , ym ] and by sending ti to δi . Further introduce a map k[x1 , . . . , xd ] −→ R = k[y1 , . . . , ym , t1 , . . . , td ] by sending xi to γi = πi ti . (16) It suffices to resolve R → C ⊗k[x1 ,...,xd ] R → Λ ⊃ q (γ1e , . . . , γde )

(17) Set I = ⊂ R. (18) It suffices to resolve R/I → C ⊗k[x1 ,...,xd ] R/I → Λ/IΛ ⊃ q/IΛ (19) We denote r ⊂ R = k[y1 , . . . , ym , t1 , . . . , td ] the inverse image of q. (20) It suffices to resolve (R/I)r → C ⊗k[x1 ,...,xd ] (R/I)r → Λq /IΛq ⊃ qΛq /IΛq (21) Set J = (π1e , . . . , πde ) in k[y1 , . . . , ym ]. (22) It suffices to resolve (R/JR)p → C ⊗k[x1 ,...,xd ] (R/JR)p → Λq /JΛq ⊃ qΛq /JΛq (23) It suffices to resolve (R/pn R)p → C ⊗k[x1 ,...,xd ] (R/pn R)p → Λq /qn Λq ⊃ qΛq /qn Λq (24) It suffices to resolve (R/pn R)p → B ⊗k[x1 ,...,xd ] (R/pn R)p → Λq /qn Λq ⊃ qΛq /qn Λq (25) The ring D0 [t1 , . . . , td ] is given the structure of an Rp /pn Rp -algebra by the given map k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p → D0 and by sending ti to ti . It suffices to find a factorization B ⊗k[x1 ,...,xd ] (R/pn R)p → D0 [t1 , . . . , td ] → Λq /qn Λq where the second arrow sends ti to δi and induces the given homomorphism D0 → Λq /qn Λq . (26) Such a factorization exists by our choice of D0 above. We now give the justification for each of the steps, except that we skip justifying the steps which just introduce notation. p Ad (1). This is possible as q is minimal over hA = HA/k Λ. Ad (6). Note that Aai is smooth over k. Hence Baj , which is isomorphic to a polynomial algebra over Aaj [x1 , . . . , xd ], is smooth over k[x1 , . . . , xd ]. Thus Bxi is smooth over k[x1 , . . . , xd ]. By Lemma 13.4.1 we see that Cxi is smooth over k[x1 , . . . , xd ] with finite free module of differentials. Hence some power of xi is strictly standard in C over k[x1 , . . . , xn ] by Lemma 13.4.7.

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Ad (9). This follows by applying Lemma 13.11.2. Ad (10). Since k[y1 , . . . , ym ]p → Λq is flat and pΛq = qΛq by construction we see that dim(k[y1 , . . . , ym ]p ) = d by Algebra, Lemma 7.104.7. Thus we can find π1 , . . . , πd ∈ Λ which map to a regular system of parameters in Λq . Ad (12). By Algebra, Lemma 7.99.3 any permutation of the sequence π1 , . . . , πd is a regular sequence in k[y1 , . . . , ym ]p . Hence γ1 = π1 t1 , . . . , γd = πd td is a regular sequence in Rp = k[y1 , . . . , ym ]p [t1 , . . . , td ], see Algebra, Lemma 7.66.11. Let S = k[y1 , . . . , ym ] \ p so that Rp = S −1 R. Note that π1 , . . . , πd and γ1 , . . . , γd remain regular sequences if we multiply our πi by elements of S. Suppose that 2 e e AnnR/(γ1e ,...,γi−1 )R (γi ) = AnnR/(γ1e ,...,γi−1 )R (γi )

holds for i = 1, . . . , t for some t ∈ {0, . . . , d}. Note that γ1e , . . . , γte , γt+1 is a regular sequence in S −1 R by Algebra, Lemma 7.66.10. Hence we see that 2 e e AnnS −1 R/(γ1e ,...,γi−1 ) (γi ) = AnnS −1 R/(γ1e ,...,γi−1 ) (γi ).

Thus we get 2 AnnR/(γ1e ,...,γte )R (γt+1 ) = AnnR/(γ1e ,...,γte )R (γt+1 ) after replacing πt+1 by sπt+1 for some s ∈ S by Lemma 13.10.1. By induction on t this produces the desired sequence. ¯ = Λq /qn Λq . Suppose that Ad (13). Let S = Λ \ q so that Λq = S −1 Λ. Set Λ ¯ as we have a t ∈ {0, . . . , d} and δ1 , . . . , δt ∈ S and a factorization D → D0 → Λ N in (13) such that (a), (b), (c) hold for i = 1, . . . , t. We have πt+1 ∈ HA/k Λq as N ¯ Hence π N ∈ HA/k D0 as D0 → Λ ¯ qN Λq ⊂ HA/k Λq by (1). Hence πt+1 ∈ HA/k Λ. t+1 is faithfully flat, see Algebra, Lemma 7.77.11. Recall that HA/k = (a1 , . . . , at ). Say P P N N πt+1 = aj dj in D0 and choose cj ∈ Λq lifting dj ∈ D0 . Then πt+1 = cj aj +  P with  ∈ qn Λq ⊂ qn−N HA/k Λq . Write  = aj c0j for some c0j ∈ qn−N Λq . Hence P N 2N N N ¯ this trivial but πt+1 = (πt+1 cj + πt+1 c0j )aj . Note that πt+1 c0j maps to zero in Λ; key observation will ensure later that (a) holds. Now we choose s ∈ S such that N N there exist µt+1j ∈ Λ such that on the one hand πt+1 cj + πt+1 c0j = µt+1j /s2N in P −1 2N S Λ and on the other (sπt+1 ) = µt+1j aj in Λ (minor detail omitted). We may further replace s by a power and enlarge D0 such that s maps to an element of D0 . With these choices µt+1j maps to s2N dj which is an element of D0 . Note that π1 , . . . , πd are a regular sequence of parameters in S −1 Λ by our choice of ϕ. Hence π1 , . . . , πd forms a regular sequence in Λq by Algebra, Lemma 7.99.3. It follows e e that π 0 1 , . . . , π 0 t , sπt+1 is a regular sequence in S −1 Λ by Algebra, Lemma 7.66.10. Thus we get

AnnS −1 Λ/(π0 e1 ,...,π0et ) (sπt+1 ) = AnnS −1 Λ/(π0 e1 ,...,π0 et ) ((sπt+1 )2 ). Hence we may apply Lemma 13.10.1 to find an s0 ∈ S such that AnnΛ/(π0 e1 ,...,π0 et ) ((s0 )q sπt+1 ) = AnnΛ/(π0 e1 ,...,π0 et ) (((s0 )q sπt+1 )2 ). for any q > 0. By Lemma 13.11.3 we can choose q and enlarge D0 such that (s0 )q maps to an element of D0 . Setting δt+1 = (s0 )q s and we conclude that (a), (b), (c) hold for i = 1, . . . , t + 1. For (a) note that λt+1j = (s0 )2N q µt+1j works. By induction on t we win. Ad (16). By construction the radical of H(C⊗k[x1 ,...,xd ] R)/R Λ contains hA . Namely, the elements aj ∈ HA/k map to elements of HB/k[x1 ,...,xn ] , hence map to elements

980

13. SMOOTHING RING MAPS

of HC/k[x1 ,...,xn ] , hence aj ⊗ 1 map to elements of HC⊗k[x1 ,...,xd ] R/R . Moreover, if we have a solution C ⊗k[x1 ,...,xn ] R → T → Λ of R → C ⊗k[x1 ,...,xd ] R → Λ ⊃ q then HT /R ⊂ HT /k as R is smooth over k. Hence T will also be a solution for the original situation k → A → Λ ⊃ q. Ad (18). Follows on applying Lemma 13.9.2 to R → C ⊗k[x1 ,...,xd ] R → Λ ⊃ q and the sequence of elements γ1c , . . . , γdc . We note that since xci are strictly standard in C over k[x1 , . . . , xd ] the elements γic are strictly standard in C ⊗k[x1 ,...,xd ] R over R by Lemma 13.3.7. The other assumption of Lemma 13.9.2 holds by steps (12) and (13). Ad (20). Apply Lemma 13.9.4 to the situation in (18). In the rest of the arguments the target ring is local Artinian, hence we are looking for a factorization by a smooth algebra T over the source ring. Ad (22). Suppose that C ⊗k[x1 ,...,xd ] (R/JR)p → T → Λq /JΛq is a solution to (R/JR)p → C ⊗k[x1 ,...,xd ] (R/JR)p → Λq /JΛq ⊃ qΛq /JΛq Then C ⊗k[x1 ,...,xd ] (R/I)r → Tr → Λq /IΛq is a solution to the situation in (20). Ad (23). Our n = N + dc is large enough so that pn k[y1 , . . . , ym ]p ⊂ Jp and qn Λq ⊂ JΛq . Hence if we have a solution C ⊗k[x1 ,...,xd ] (R/pn R)p → T → Λq /qn Λq of (22 then we can take T /JT as the solution for (23). Ad (24). This is true because we have a section C → B in the category of Ralgebras. Ad (25). This is true because D0 is essentially smooth over the local Artinian ring k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p and Rp /pn Rp = k[y1 , . . . , ym ]p /pn k[y1 , . . . , ym ]p [t1 , . . . , td ]. Hence D0 [t1 , . . . , td ] is a filtered colimit of smooth Rp /pn Rp -algebras and B⊗k[x1 ,...,xd ] (Rp /pn Rp ) factors through one of these. Ad (26). The final twist of the proof is that we cannot just use the map B → D0 which maps xi to the image of πi0 in D0 and zij to the image of λij in D0 because we need the diagram BO

/ D0 [t1 , . . . , td ] O

k[x1 , . . . , xd ]

/ Rp /pn Rp

to commute and we need the compostion B → D0 [t1 , . . . , td ] → Λq /qn Λq to be the map of (14). This requires us to map xi to the image of πi ti in D0 [t1 , . . . , td ]. Hence 2N we map zij to the image of λij t2N in D0 [t1 , . . . , td ] and everything is clear.  i /δi

13.13. THE APPROXIMATION PROPERTY FOR G-RINGS

981

13.12. The main theorem In this section we wrap up the discussion. Theorem 13.12.1 (Popescu). Any regular homomorphism of Noetherian rings is a filtered colimit of smooth ring maps. Proof. By Lemma 13.8.4 it suffices to prove this for k → Λ where Λ is Noetherian and geometrically regular over k. Let k → A → Λ be a factorization with A a finite type k-algebra. It suffices to construct a factorization A → B → Λ with B of finite type such that hB = Λ, see Lemma 13.3.8. Hence we may perform Noetherian induction on the ideal hA . Pick a prime q ⊃ hA such that q is minimal over hA . It now suffices to resolve k → A → Λ ⊃ q (as defined in the text following Situation 13.9.1). If the characteristic of k is zero, this follows from Lemma 13.10.3. If the characteristic of k is p > 0, this follows from Lemma 13.11.4.  13.13. The approximation property for G-rings Let R be a Noetherian local ring. In this case R is a G-ring if and only if the ring map R → R∧ is regular, see More on Algebra, Lemma 12.41.7. In this case it is true that the henselization Rh and the strict henselization Rsh of R are G-rings, see More on Algebra, Lemma 12.41.8. Moreover, any algebra essentially of finite type over a field, over a complete local ring, over Z, or over a characteristic zero Dedekind ring is a G-ring, see More on Algebra, Proposition 12.41.12. This gives an ample supply of rings to which the result below applies. Let R be a ring. Let f1 , . . . , fm ∈ R[x1 , . . . , xn ]. Let S be an R-algebra. In this situation we say a vector (a1 , . . . , an ) ∈ S n is a solution in S if and only if fj (a1 , . . . , an ) = 0 in S, for j = 1, . . . , m Of course an important question in algebraic geometry is to see when systems of polynomial equations have solutions. The following theorem tells us that having solutions in the completion of a local Noetherian ring is often enough to show there exist solutions in the henselization of the ring. Theorem 13.13.1. Let R be a Noetherian local ring. Let f1 , . . . , fm ∈ R[x1 , . . . , xn ]. Suppose that (a1 , . . . , an ) ∈ (R∧ )n is a solution in R∧ . If R is a henselian G-ring, then for every integer N there exists a solution (b1 , . . . , bn ) ∈ Rn in R such that ai − bi ∈ mN R∧ . N Proof. Let ci ∈ R be an element such P that ai − ci ∈ m . Choose generators N m = (d1 , . . . , dM ). Write ai = ci + ai,l dl . Consider the polynomial ring R[xi,l ] and the elements X X gj = fj (c1 + x1,l dl , . . . , cn + xn,l dn,l ) ∈ R[xi,l ]

The system of equations gj = 0 has the solution (ai,l ). Suppose P that we can show that gj as a solution (bi,l ) in R. Then it follows that bi = ci + bi,l dl is a solution of fj = 0 which is congruent to ai modulo mN . Thus it suffices to show that solvability over R∧ implies solvability over R. Let A ⊂ R∧ be the R-subalgebra generated by a1 , . . . , an . Since we’ve assumed R is a G-ring, i.e., that R → R∧ is regular, we see that there exists a factorization A → B → R∧

982

13. SMOOTHING RING MAPS

with B smooth over R, see Theorem 13.12.1. Denote κ = R/m the residue field. It is also the residue field of R∧ , so we get a commutative diagram BO

/ R0

R

/ κ

Since the vertical arrow is smooth, More on Algebra, Lemma 12.11.12 implies that there exists an ´etale ring map R → R0 which induces an isomorphism R/m → R0 /mR0 and an R-algebra map B → R0 making the diagram above commute. Since R is henselian we see that R → R0 has a section, see Algebra, Lemma 7.140.3. Let bi ∈ R be the image of ai under the ring maps A → B → R0 → R. Since all of these maps are R-algebra maps, we see that (b1 , . . . , bn ) is a solution in R.  Given a Noetherian local ring (R, m), an ´etale ring map R → R0 , and a maximal ideal m0 ⊂ R0 lying over m with κ(m) = κ(m0 ), then we have inclusions R ⊂ Rm0 ⊂ Rh ⊂ R∧ , by Algebra, Lemma 7.140.15 and More on Algebra, Lemma 12.36.3. Theorem 13.13.2. Let R be a Noetherian local ring. Let f1 , . . . , fm ∈ R[x1 , . . . , xn ]. Suppose that (a1 , . . . , an ) ∈ (R∧ )n is a solution. If R is a G-ring, then for every integer N there exist (1) an ´etale ring map R → R0 , (2) a maximal ideal m0 ⊂ R0 lying over m (3) a solution (b1 , . . . , bn ) ∈ (R0 )n in R0 such that κ(m) = κ(m0 ) and ai − bi ∈ (m0 )N R∧ . Proof. We could deduce this theorem from Theorem 13.13.1 using that the henselization Rh is a G-ring by More on Algebra, Lemma 12.41.8 and writing Rh as a directed colimit of ´etale extension R0 . Instead we prove this by redoing the proof of the previous theorem in this case. N N Let ci ∈ R be an element such P that ai − ci ∈ m . Choose generators m = (d1 , . . . , dM ). Write ai = ci + ai,l dl . Consider the polynomial ring R[xi,l ] and the elements X X gj = fj (c1 + x1,l dl , . . . , cn + xn,l dn,l ) ∈ R[xi,l ]

The system of equations gj = 0 has the solution (ai,l ). Suppose that we can show that gj as a solution (bi,l ) in R0 for some ´etale ring map R → R0 endowedP with a maximal ideal m0 such that κ(m) = κ(m0 ). Then it follows that bi = ci + bi,l dl is a solution of fj = 0 which is congruent to ai modulo (m0 )N . Thus it suffices to show that solvability over R∧ implies solvability over some ´etale ring extension which induces a trivial residue field extension at some prime over m. Let A ⊂ R∧ be the R-subalgebra generated by a1 , . . . , an . Since we’ve assumed R is a G-ring, i.e., that R → R∧ is regular, we see that there exists a factorization A → B → R∧

13.14. OTHER CHAPTERS

983

with B smooth over R, see Theorem 13.12.1. Denote κ = R/m the residue field. It is also the residue field of R∧ , so we get a commutative diagram BO

/ R0

R

/ κ

Since the vertical arrow is smooth, More on Algebra, Lemma 12.11.12 implies that there exists an ´etale ring map R → R0 which induces an isomorphism R/m → R0 /mR0 and an R-algebra map B → R0 making the diagram above commute. Let bi ∈ R0 be the image of ai under the ring maps A → B → R0 . Since all of these maps are R-algebra maps, we see that (b1 , . . . , bn ) is a solution in R0 .  13.14. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness

(35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66)

Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples

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(67) (68) (69) (70)

Exercises Guide to Literature Desirables Coding Style

(71) Obsolete (72) GNU Free Documentation License (73) Auto Generated Index

CHAPTER 14

Simplicial Methods 14.1. Introduction This is a minimal introduction to simplicial methods. We just add here whenever something is needed later on. A general reference to this material is perhaps [GJ99]. An example of the things you can do is the paper by Quillen on Homotopical Algebra, see [Qui67] or the paper on Etale Homotopy by Artin and Mazur, see [AM69].

14.2. The category of finite ordered sets The category ∆ is the category with (1) objects [0], [1], [2], . . . with [n] = {0, 1, 2, . . . , n} and (2) a morphism [n] → [m] is the set of nondecreasing maps of the corresponding sets {0, 1, 2, . . . , n} → {0, 1, 2, . . . , m}. Here nondecreasing for a map ϕ : [n] → [m] means by definition that ϕ(i) ≥ ϕ(j) if i ≥ j. In other words, ∆ is a category equivalent to the “big” category of finite totally ordered sets and nondecreasing maps. There are exactly n + 1 morphisms [0] → [n] and there is exactly 1 morphism [n] → [0]. There are exactly (n + 1)(n + 2)/2 morphisms [1] → [n] and there are exactly n + 2 morphisms [n] → [1]. And so on and so forth. Definition 14.2.1. For any integer n ≥ 1, and any 0 ≤ j ≤ n we let δjn : [n − 1] → [n] denote the injective order preserving map skipping j. For any integer n ≥ 0, and any 0 ≤ j ≤ n we denote σjn : [n + 1] → [n] the surjective order preserving map with (σjn )−1 ({j}) = {j, j + 1}. Lemma 14.2.2. Any morphism in ∆ can be written as a composition of an identity morphism, and the morphisms δjn and σjn . Proof. Let ϕ : [n] → [m] be a morphism of ∆. If j 6∈ Im(ϕ), then we can write ϕ as δjm ◦ ψ for some morphism ψ : [n] → [m − 1]. If ϕ(j) = ϕ(j + 1) then we can write ϕ as ψ ◦ σjn−1 for some morphism ψ : [n − 1] → [m]. The result follows because each replacement as above lowers n + m and hence at some point ϕ is both injective and surjective, hence an identity morphism.  Lemma 14.2.3. The morphisms δjn and σjn satisfy the following relations. 985

986

14. SIMPLICIAL METHODS n . In other words the (1) If 0 ≤ i < j ≤ n + 1, then δjn+1 ◦ δin = δin+1 ◦ δj−1 diagram

< [n]

δin

δjn+1

" [n + 1]
2 we have a1 f1 + . . . + an fn = 1 for some a1 , . . . , an ∈ O(U ). By the case n = 2 we see that we have some covering {Uj → U }j∈J such that for each j either fn |Uj is invertible or a1 f1 + . . . + an−1 fn−1 |Uj is invertible. Say the first case happens for j ∈ Jn . Set J 0 = J \ Jn . By induction hypothesis, for each j ∈ J 0 we can find a covering {Ujk → Uj }k∈Kj such that for each k ∈ Kj there exists an i ∈ {1, . . . , n − 1} such that fi |Ujk is invertible. By the axioms of a site the family of morphisms {Uj → U }j∈Jn ∪ {Ujk → U }j∈J 0 ,k∈Kj is a covering which has the desired property. Assume (1). To see that the map in (3) is surjective, let (f, c) be a section of O × O over U . By assumption there exists a covering {Uj → U } such that for each j either f or 1 − f restricts to an invertible section. In the first case we can take a = c|Uj (f |Uj )−1 , and in the second case we can take b = c|Uj (1 − f |Uj )−1 . Hence 4This assumption is not necessary, see introduction to this section.

16.34. LOCALLY RINGED TOPOI

1113

(f, c) is in the image of the map on each of the members. Conversely, assume (3) holds. For any U and f ∈ O(U ) there exists a covering {Uj → U } of U such that the section (f, 1)|Uj is in the image of the map in (3) on sections over Uj . This means precisely that either f or 1 − f restricts to an invertible section over Uj , and we see that (1) holds.  Lemma 16.34.2. Let (C, O) be a ringed site. Consider the following conditions (1) For every object U of C and f ∈ O(U ) there exists a covering {Uj → U } such that for each j either f |Uj is invertible or (1 − f )|Uj is invertible. (2) For every point p of C the stalk Op is either the zero ring or a local ring. We always have (1) ⇒ (2). If C has enough points then (1) and (2) are equivalent. Proof. Assume (1). Let p be a point of C given by a functor u : C → Sets. Let fp ∈ Op . Since Op is computed by Sites, Equation (9.28.1.1) we may represent fp by a triple (U, x, f ) where x ∈ U (U ) and f ∈ O(U ). By assumption there exists a covering {Ui → U } such that for each i either f or 1−f is invertible on Ui . Because u defines a point of the site we see that for some i there exists an xi ∈ u(Ui ) which maps to x ∈ u(U ). By the discussion surrounding Sites, Equation (9.28.1.1) we see that (U, x, f ) and (Ui , xi , f |Ui ) define the same element of Op . Hence we conclude that either fp or 1 − fp is invertible. Thus Op is a ring such that for every element a either a or 1 − a is invertible. This means that Op is either zero or a local ring, see Algebra, Lemma 7.17.2. Assume (2) and assume that C has enough points. Consider the map of sheaves of sets O × O q O × O −→ O × O of Lemma 16.34.1 part (3). For any local ring R the corresponding map (R × R) q (R × R) → R × R is surjective, see for example Algebra, Lemma 7.17.2. Since each Op is a local ring or zero the map is surjective on stalks. Hence, by our assumption that C has enough points it is surjective and we win.  In Modules, Section 15.2 we pointed out how in a ringed space (X, OX ) there can be an open subspace over which the structure sheaf is zero. To prevent this we can require the sections 1 and 0 to have different values in every stalk of the space X. In the setting of ringed topoi and ringed sites the condition is that (16.34.2.1)

∅# −→ Equalizer(0, 1 : ∗ −→ O)

is an isomorphism of sheaves. Here ∗ is the singleton sheaf, resp. ∅# is the “empty sheaf”, i.e., the final, resp. initial object in the category of sheaves, see Sites, Example 9.10.2, resp. Section 9.37. In other words, the condition is that whenever U ∈ Ob(C) is not sheaf theoretically empty, then 1, 0 ∈ O(U ) are not equal. Let us state the obligatory lemma. Lemma 16.34.3. Let (C, O) be a ringed site. Consider the statements (1) (16.34.2.1) is an isomorphism, and (2) for every point p of C the stalk Op is not the zero ring. We always have (1) ⇒ (2) and if C has enough points then (1) ⇔ (2). Proof. Omitted. Lemmas 16.34.1, 16.34.2, and 16.34.3 motivate the following definition.



1114

16. MODULES ON SITES

Definition 16.34.4. A ringed site (C, O) is said to be locally ringed site if (16.34.2.1) is an isomorphism, and the equivalent properties of Lemma 16.34.1 are satisfied. In [AGV71, Expos´e IV, Exercice 13.9] the condition that (16.34.2.1) be an isomorphism is missing leading to a slightly different notion of a locally ringed site and locally ringed topos. As we are motivated by the notion of a locally ringed space we decided to add this condition (see explanation above). Lemma 16.34.5. Being a locally ringed site is an intrinsic property. More precisely, (1) if f : Sh(C 0 ) → Sh(C) is a morphism of topoi and (C, O) is a locally ringed site, then (C 0 , f −1 O) is a locally ringed site, and (2) if (f, f ] ) : (Sh(C 0 ), O0 ) → (Sh(C), O) is an equivalence of ringed topoi, then (C, O) is locally ringed if and only if (C 0 , O0 ) is locally ringed. Proof. It is clear that (2) follows from (1). To prove (1) note that as f −1 is exact we have f −1 ∗ = ∗, f −1 ∅# = ∅# , and f −1 commutes with products, equalizers and transforms isomorphisms and surjections into isomorphisms and surjections. Thus f −1 transforms the isomorphism (16.34.2.1) into its analogue for f −1 O and transforms the surjection of Lemma 16.34.1 part (3) into the corresponding surjection for f −1 O.  In fact Lemma 16.34.5 part (2) is the analogue of Schemes, Lemma 21.2.2. It assures us that the following definition makes sense. Definition 16.34.6. A ringed topos (Sh(C), O) is said to be locally ringed if the underlying ringed site (C, O) is locally ringed. Next, we want to work out what it means to have a morphism of locally ringed spaces. In order to do this we have the following lemma. Lemma 16.34.7. Let (f, f ] ) : (Sh(C), OC ) → (Sh(D), OD ) be a morphism of ringed topoi. Consider the following conditions (1) The diagram of sheaves ∗ f −1 (OD )

 f −1 (OD )

f]

f]

/ O∗ C  / OC

is cartesian. (2) For any point p of C, setting q = f ◦ p, the diagram ∗ OD,q

/ O∗

 OD,q

 / OC,p

C,p

of sets is cartesian. We always have (1) ⇒ (2). If C has enough points then (1) and (2) are equivalent. If (Sh(C), OC ) and (Sh(D), OD ) are locally ringed topoi then (2) is equivalent to (3) For any point p of C, setting q = f ◦ p, the ring map OD,q → OC,p is a local ring map.

16.34. LOCALLY RINGED TOPOI

1115

In fact, properties (2), or (3) for a conservative family of points implies (1). Proof. This lemma proves itself, in other words, it follows by unwinding the definitions.  Definition 16.34.8. Let (f, f ] ) : (Sh(C), OC ) → (Sh(D), OD ) be a morphism of ringed topoi. Assume (Sh(C), OC ) and (Sh(D), OD ) are locally ringed topoi. We say that (f, f ] ) is a morphism of locally ringed topoi if and only if the diagram of sheaves ∗ / O∗ f −1 (OD ) C ] f

 f −1 (OD )

f]

 / OC

(see Lemma 16.34.7) is cartesian. If (f, f ] ) is a morphism of ringed sites, then we say that it is a morphism of locally ringed sites if the associated morphism of ringed topoi is a morphism of locally ringed topoi. It is clear that an isomorphism of ringed topoi between locally ringed topoi is automatically an isomorphism of locally ringed topoi. Lemma 16.34.9. Let (f, f ] ) : (Sh(C1 ), O1 ) → (Sh(C2 ), O2 ) and (g, g ] ) : (Sh(C2 ), O2 ) → (Sh(C3 ), O3 ) be morphisms of locally ringed topoi. Then the composition (g, g ] ) ◦ (f, f ] ) (see Definition 16.7.1) is also a morphism of locally ringed topoi. Proof. Omitted.



Lemma 16.34.10. If f : Sh(C 0 ) → Sh(C) is a morphism of topoi. If O is a sheaf of rings on C, then f −1 (O∗ ) = (f −1 O)∗ . In particular, if O turns C into a locally ringed site, then setting f ] = id the morphism of ringed topoi (f, f ] ) : (Sh(C 0 ), f −1 O) → (Sh(C, O) is a morphism of locally ringed topoi. Proof. Note that the diagram /∗

O∗ u7→(u,u−1 )

 O×O

1

(a,b)7→ab

 /O

is cartesian. Since f −1 is exact we conclude that /∗

f −1 (O∗ ) u7→(u,u−1 )

 f −1 O × f −1 O

(a,b)7→ab



1

/ f −1 O

is cartesian which implies the first assertion. For the second, note that (C 0 , f −1 O) is a locally ringed site by Lemma 16.34.5 so that the assertion makes sense. Now the first part implies that the morphism is a morphism of locally ringed topoi.  Lemma 16.34.11. Localization of locally ringed sites and topoi.

1116

16. MODULES ON SITES

(1) Let (C, O) be a locally ringed site. Let U be an object of C. Then the localization (C/U, OU ) is a locally ringed site, and the localization morphism (jU , jU] ) : (Sh(C/U ), OU ) → (Sh(C), O) is a morphism of locally ringed topoi. (2) Let (C, O) be a locally ringed site. Let f : V → U be a morphism of C. Then the morphism (j, j ] ) : (Sh(C/V ), OV ) → (Sh(C/U ), OU ) of Lemma 16.19.4 is a morphism of locally ringed topoi. (3) Let (f, f ] ) : (C, O) −→ (D, O0 ) be a morphism of locally ringed sites where f is given by the continuous functor u : D → C. Let V be an object of D and let U = u(V ). Then the morphism (f 0 , (f 0 )] ) : (Sh(C/U ), OU ) → (Sh(D/V ), OV0 ) of Lemma 16.20.1 is a morphism of locally ringed sites. (4) Let (f, f ] ) : (C, O) −→ (D, O0 ) be a morphism of locally ringed sites where f is given by the continuous functor u : D → C. Let V ∈ Ob(D), U ∈ Ob(C), and c : U → u(V ). Then the morphism (fc , (fc )] ) : (Sh(C/U ), OU ) → (Sh(D/V ), OV0 ) of Lemma 16.20.2 is a morphism of locally ringed topoi. (5) Let (Sh(C), O) be a locally ringed topos. Let F be a sheaf on C. Then the localization (Sh(C)/F, OF ) is a locally ringed topos and the localization morphism ] (jF , jF ) : (Sh(C)/F, OF ) → (Sh(C), O)

is a morphism of locally ringed topoi. (6) Let (Sh(C), O) be a locally ringed topos. Let s : G → F be a map of sheaves on C. Then the morphism (j, j ] ) : (Sh(C)/G, OG ) −→ (Sh(C)/F, OF ) of Lemma 16.21.4 is a morphism of locally ringed topoi. (7) Let f : (Sh(C), O) −→ (Sh(D), O0 ) be a morphism of locally ringed topoi. Let G be a sheaf on D. Set F = f −1 G. Then the morphism (f 0 , (f 0 )] ) : (Sh(C)/F, OF ) −→ (Sh(D)/G, OG0 ) of Lemma 16.22.1 is a morphism of locally ringed topoi. (8) Let f : (Sh(C), O) −→ (Sh(D), O0 ) be a morphism of locally ringed topoi. Let G be a sheaf on D, let F be a sheaf on C, and let s : F → f −1 G be a morphism of sheaves. Then the morphism (fs , (fs )] ) : (Sh(C)/F, OF ) −→ (Sh(D)/G, OG0 ) of Lemma 16.22.3 is a morphism of locally ringed topoi. Proof. Part (1) is clear since OU is just the restriction of O, so Lemmas 16.34.5 and 16.34.10 apply. Part (2) is clear as the morphism (j, j ] ) is actually a localization of a locally ringed site so (1) applies. Part (3) is clear also since (f 0 )] is just the restriction of f ] to the topos Sh(C)/F, see proof of Lemma 16.22.1 (hence the diagram of Definition 16.34.8 for the morphism f 0 is just the restriction of the corresponding diagram for f , and restriction is an exact functor). Part (4)

16.35. LOWER SHRIEK FOR MODULES

1117

follows formally on combining (2) and (3). Parts (5), (6), (7), and (8) follow from their counterparts (1), (2), (3), and (4) by enlarging the sites as in Lemma 16.7.2 and translating everything in terms of sites and morphisms of sites using the comparisons of Lemmas 16.21.3, 16.21.5, 16.22.2, and 16.22.4. (Alternatively one could use the same arguments as in the proofs of (1), (2), (3), and (4) to prove (5), (6), (7), and (8) directly.)  16.35. Lower shriek for modules In this section we extend the construction of g! discussed in Section 16.16 to the case of sheaves of modules. Lemma 16.35.1. Let u : C → D be a continuous and concontinuous functor between sites. Denote g : Sh(C) → Sh(OD ) the associated morphism of topoi. Let OD be a sheaf of rings on D. Set OC = g −1 OD . Hence g becomes a morphism of ringed topoi with g ∗ = g −1 . In this case there exists a functor g! : Mod(OC ) −→ Mod(OD ) ∗

which is left adjoint to g . Proof. Let U be an object of C. For any OD -module G we have HomOC (jU ! OU , g −1 G) = g −1 G(U ) = G(u(U )) = HomOC (ju(U )! Ou(U ) , G) because g −1 is described by restriction, see Sites, Lemma 9.19.5. Of course a similar formula holds a direct sum of modules of the form jU ! OU . By Homology, Lemma 10.22.6 and Lemma 16.26.6 we see that g! exists.  Remark 16.35.2. Warning! Let u : C → D, g, OD , and OC be as in Lemma 16.35.1. In general it is not the case that the diagram Mod(OC )

/ Mod(OD )

g!

f orget

f orget

 Ab(C)

g!Ab

 / Ab(D)

commutes (here g!Ab is the one from Lemma 16.16.2). There is a transformation of functors g!Ab ◦ f orget −→ f orget ◦ g! From the proof of Lemma 16.35.1 we see that this is an isomorphism if and only if g! jU ! OU = g!Ab jU ! OU for all objects U of C, in other words, if and only if g!Ab jU ! OU = ju(U )! Ou(U ) for all objects U of C. Note that for such a U we obtain a commutative diagram C/U jU

 C

u0

/ D/u(U ) ju(U )

u

 /D

1118

16. MODULES ON SITES

of cocontinuous functors of sites, see Sites, Lemma 9.24.4. Hence we see that g! = g!Ab if the canonical map (16.35.2.1)

(g 0 )Ab ! OU −→ Ou(U )

is an isomorphism for all objects U of C. Here g 0 : Sh(C/U ) → Sh(D/u(U )) is the morphism of topoi induced by the cocontinuous functor u0 . 16.36. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology

(39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 17

Injectives 17.1. Introduction We will use the existence of sufficiently many injectives to do cohomology of abelian sheaves on a site. So we briefly explain why there are enough injectives. At the end we explain the more general story.

17.2. Abelian groups In this section we show the category of abelian groups has enough injectives. Recall that an abelian group M is divisible if and only if for every x ∈ M and every n ∈ N there exists a y ∈ M such that ny = x. Lemma 17.2.1. An abelian group J is an injective object in the category of abelian groups if and only if J is divisible. Proof. Suppose that J is not divisible. Then there exists an x ∈ J and n ∈ N such that there is no y ∈ J with ny = x. Then the morphism Z → J, m 7→ mx does not extend to n1 Z ⊃ Z. Hence J is not injective. Let A ⊂ B be abelian groups. Assume that J is a divisible abelian group. Let ϕ : A → J be a morphism. Consider the set of homomorphisms ϕ0 : A0 → J with A ⊂ A0 ⊂ B and ϕ0 |A = ϕ. Define (A0 , ϕ0 ) ≥ (A00 , ϕ00 ) if and only if A0 ⊃ A00 00 and ϕ0 |A00 = ϕS . If (Ai , ϕi )i∈I is a totally ordered collection of such pairs, then we obtain a map i∈I Ai → J defined by a ∈ Ai maps to ϕi (a). Thus Zorn’s lemma applies. To conclude we have to show that if the pair (A0 , ϕ0 ) is maximal then A0 = B. In other words, it suffices to show, given any subgroup A ⊂ B, A 6= B and any ϕ : A → J, then we can find ϕ0 : A0 → J with A ⊂ A0 ⊂ B such that (a) the inclusion A ⊂ A0 is strict, and (b) the morphism ϕ0 extends ϕ. To prove this, pick x ∈ B, x 6∈ A. If there exists no n ∈ N such that nx ∈ A, then A⊕Z ∼ = A + Zx. Hence we can extend ϕ to A0 = A + Zx by using ϕ on A and mapping x to zero for example. If there does exist an n ∈ N such that nx ∈ A, then let n be the minimal such integer. Let z ∈ J be an element such that nz = ϕ(nx). Define a morphism ϕ˜ : A ⊕ Z → J by (a, m) 7→ ϕ(a) + mz. By our choice of z the kernel of ϕ˜ contains the kernel of the map A ⊕ Z → B, (a, m) 7→ a + mx. Hence ϕ˜  factors through the image A0 = A + Zx, and this extends the morphism ϕ. We can use this lemma to show that every abelian group can be embbeded in a injective abelian group. But this is a special case of the result of the following section. 1119

1120

17. INJECTIVES

17.3. Modules As an example theorem let us try to prove that there are enough injective modules over a ring R. We start with the fact that Q/Z is an injective abelian group. This follows from Lemma 17.2.1 above. Definition 17.3.1. Let R be a ring. (1) For any R-module M over R we denote M ∨ = Hom(M, Q/Z) with its natural R-module structure. We think of M 7→ M ∨ as a contravariant functor from the category of R-modules to itself. (2) For any R-module M we denote M F (M ) = R[m] m∈M

the free module given by the elements [m] with m ∈ M . We let Pwith basisP F (M ) → M , fi [mi ] 7→ fi mi be the natural surjection of R-modules. We think of M 7→ (F (M ) → M ) as a functor from the category of Rmodules to the category of arrows in R-modules. Lemma 17.3.2. Let R be a ring. The functor M 7→ M ∨ is exact. Proof. This because Q/Z is an injective abelian group.



∨ ∨

There is a canonical map ev : M → (M ) given by evaluation: given x ∈ M we let ev(x) ∈ (M ∨ )∨ = Hom(M ∨ , Q/Z) be the map ϕ 7→ ϕ(x). Lemma 17.3.3. For any R-module M the evaluation map ev : M → (M ∨ )∨ is injective. Proof. You can check this using that Q/Z is an injective abelian group. Namely, if x ∈ M is not zero, then let M 0 ⊂ M be the cyclic group it generates. There exists a nonzero map M 0 → Q/Z which necessarily does not annihilate x. This extends to a map ϕ : M → Q/Z And then ev(x)(ϕ) = ϕ(x) 6= 0.  The canonical surjection F (M ) → M of R-modules turns into a a canonical injection, see above, of R-modules (M ∨ )∨ −→ (F (M ∨ ))∨ . Set J(M ) = (F (M ∨ ))∨ . The composition of ev with this the displayed map gives M → J(M ) functorially in M . Lemma 17.3.4. Let R be a ring. For every R-module M the R-module J(M ) is injective. Q Proof. Note that J(M ) ∼ = m∈M R∨ as an R-module. As the product of injective modules is injective, it suffices to show that R∨ is injective. For this we use that HomR (N, R∨ ) = HomR (N, HomZ (R, Q/Z)) = N ∨ and the fact that (−)∨ is an exact functor by Lemma 17.3.2.



Lemma 17.3.5. Let R be a ring. The construction above defines a covariant functor M 7→ (M → J(M )) from the category of R-modules to the category of arrows of R-modules such that for every module M the output M → J(M ) is an injective map of M into an injective R-module J(M ). Proof. Follows from the above.



17.6. BAER’S ARGUMENT FOR MODULES

1121

In particular, for any map of R-modules M → N there is an associated morphism J(M ) → J(N ) making the following diagram commute: M

/N

 J(M )

 / J(N )

This the kind of construction we would like to have in general. In Homology, Section 10.20 we introduced terminology to express this. Namely, we say this means that the category of R-modules has functorial injective embeddings. 17.4. Projective resolutions Totally unimportant. Skip this section. For any set S we let F (S) denote the free R-module on S. Then any left R-module has the following two step resolution F (M × M ) ⊕ F (R × M ) → F (M ) → M → 0. The first map is given by the rule [m1 , m2 ] ⊕ [r, m] 7→ [m1 + m2 ] − [m1 ] − [m2 ] + [rm] − r[m]. 17.5. Modules over noncommutative rings In the stacks project a ring is always commutative with 1. The material of Section 17.3 continues to work when R is only a noncommutative ring, except that if M is a right R-module, then M ∧ is a left R-module and vice-versa. The conclusion is that the category of right R-modules and the category of left R-modules have functorial injective embeddings. Precise statements and proofs omitted. 17.6. Baer’s argument for modules There is another, more set-theoretic approach to showing that any R-module M can be imbedded in an injective module. This approach constructs the injective module by a transfinite colimit of push-outs. While this method is somewhat abstract and more complicated than the one of Section 17.3, it is also more general. Apparently this method originates with Baer, and was revisited by Cartan and Eilenberg in [CE56] and by Grothendieck in [Gro57]. There Grothendieck uses it to show that many other abelian categories have enough injectives. We will get back to the general case later (insert future reference here). We begin with a few set theoretic remarks. Let {Bβ }β∈α be an inductive system of objects in some category C, indexed by an ordinal α. Assume that colimβ∈α Bβ exists in C. If A is an object of C, then there is a natural map (17.6.0.1)

colimβ∈α MorC (A, Bβ ) −→ MorC (A, colimβ∈α Bβ ).

because if one is given a map A → Bβ for some β, one naturally gets a map from A into the colimit by composing with Bβ → colimβ∈α Bα . Note that the left colimit is one of sets! In general, (17.6.0.1) is neither injective or surjective.

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17. INJECTIVES

Example 17.6.1. Consider the category of sets. Let A = N and Bn = {1, . . . , n} be the inductive system indexed by the natural numbers where Bn → Bm for n ≤ m is the obvious map. Then colim Bn = N, so there is a map A → lim Bn , −→ which does not factor as A → Bm for any m. Consequently, colim Mor(A, Bn ) → Mor(A, colim Bn ) is not surjective. Example 17.6.2. Next we give an example where the map fails to be injective. Let Bn = N/{1, 2, . . . , n}, that is, the quotient set of N with the first n elements collapsed to one element. There are natural maps Bn → Bm for n ≤ m, so the {Bn } form a system of sets over N. It is easy to see that colim Bn = {∗}: it is the one-point set. So it follows that Mor(A, colim Bn ) is a one-element set for every set A. However, colim Mor(A, Bn ) is not a one-element set. Consider the family of maps A → Bn which are just the natural projections N → N/{1, 2, . . . , n} and the family of maps A → Bn which map the whole of A to the class of 1. These two families of maps are distinct at each step and thus are distinct in colim Mor(A, Bn ), but they induce the same map A → colim Bn . Nonetheless, if we map out of a finite set then (17.6.0.1) is an isomorphism always. Lemma 17.6.3. Suppose that, in (17.6.0.1), C is the category of sets and A is a finite set, then the map is a bijection. Proof. Let f : A → colim Bβ . The range of f is finite, containing say elements c1 , . . . , cr ∈ colim Bβ . These all come from some elements in Bβ for β ∈ α large by definition of the colimit. Thus we can define fe : A → Bβ lifting f at a finite stage. This proves that (17.6.0.1) is surjective. Next, suppose two maps f : A → Bγ , f 0 : A → Bγ 0 define the same map A → colim Bβ . Then each of the finitely many elements of A gets sent to the same point in the colimit. By definition of the colimit for sets, there is β ≥ γ, γ 0 such that the finitely many elements of A get sent to the same points in Bβ under f and f 0 . This proves that (17.6.0.1) is injective.  The most interesting case of the lemma is when α = ω, i.e., when the system {Bβ } is a system {Bn }n∈N over the natural numbers as in Examples 17.6.1 and 17.6.2. The essential idea is that A is “small” relative to the long chain of compositions B1 → B2 → . . . , so that it has to factor through a finite step. A more general version of this lemma can be found in Sets, Lemma 3.7.1. Next, we generalize this to the category of modules. Definition 17.6.4. Let C be a category, let I ⊂ Arrow(C), and let α be an ordinal. An object A of C is said to be α-small with respect to I if whenever {Bβ } is a system over α with transition maps in I, then the map (17.6.0.1) is an isomorphism. In the rest of this section we shall restrict ourselves to the category of R-modules for a fixed commutative ring R. We shall also take I to be the collection of injective maps, i.e., the monomorphisms in the category of modules over R. In this case, for any system {Bβ } as in the definition each of the maps Bβ → colimβ∈α Bβ is an injection. It follows that the map (17.6.0.1) is an injection. We can in fact interpret the S Bβ ’s as submodules of the module B = colimβ∈α Bβ , and then we have B = β∈α Bβ . This is not an abuse of notation if we identify Bα with the

17.6. BAER’S ARGUMENT FOR MODULES

1123

image in the colimit. We now want to show that modules are always small for “large” ordinals α. Proposition 17.6.5. Let R be a ring. Let M be an R-module. Let κ the cardinality of the set of submodules of M . If α is an ordinal whose cofinality is bigger than κ, then M is α-small with respect to injections. Proof. The proof is straightforward, but let us first think about a special case. If M is finite, then the claim is that for any inductive system {Bβ } with injections between them, parametrized by a limit ordinal, any map M → colim Bβ factors through one of the Bβ . And this we proved in Lemma 17.6.3. Now we start the proof in the general case. We need only show that the map (17.6.0.1) is a surjection. Let f : M → colim Bβ be a map. Consider the subobjects S {f −1 (Bβ )} of M , where Bβ is considered as a subobject of the colimit B = β Bβ . If one of these, say f −1 (Bβ ), fills M , then the map factors through Bβ . So suppose to the contrary that all of the f −1 (Bβ ) were proper subobjects of M . However, we know that [  [ f −1 (Bβ ) = f −1 Bβ = M. Now there are at most κ different subobjects of M that occur among the f −1 (Bα ), by hypothesis. Thus we can find a subset S ⊂ α of cardinality at most κ such that as β 0 ranges over S, the f −1 (Bβ 0 ) range over all the f −1 (Bα ). However, S has an upper bound α e < α as α has cofinality bigger than κ. In particular, all the f −1 (Bβ 0 ), β 0 ∈ S are contained in f −1 (Bαe ). It follows that f −1 (Bαe ) = M . In particular, the map f factors through Bαe .  From this lemma we will be able to deduce the existence of lots of injectives. Let us recall the criterion of Baer. Lemma 17.6.6. Let R be a ring. An R-module Q is injective if and only if in every commutative diagram /Q a ?  R for a ⊂ R an ideal, the dotted arrow exists. Proof. Assume Q satisfies the assumption of the lemma. Let M ⊂ N be Rmodules, and let ϕ : M → Q be an R-module map. Arguing as in the proof of Lemma 17.2.1 we see that it suffices to prove that if M 6= N , then we can find an R-module M 0 , M ⊂ M 0 ⊂ N such that (a) the inclusion M ⊂ M 0 is strict, and (b) ϕ can be extended to M 0 . To find M 0 , let x ∈ N , x 6∈ M . Let ψ : R → N , r 7→ rx. Set a = ψ −1 (M ). By assumption the morphism ψ

ϕ

a− →M − →Q can be extended to a morphism ϕ0 : R → Q. Note that ϕ0 annihilates the kernel of ψ (as this is true for ϕ). Thus ϕ0 gives rise to a morphism ϕ00 : Im(ψ) → Q which agrees with ϕ on the intersection M ∩ Im(ψ) by construction. Thus ϕ and ϕ00 glue to give an extension of ϕ to the strictly bigger module M 0 = F + Im(ψ). 

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17. INJECTIVES

If M is an R-module, then in general we may have a semi-complete diagram as in Lemma 17.6.6. In it, we can form the push-out a

/Q

 R

 / R ⊕a Q.

Here the vertical map is injective, and the diagram commutes. The point is that we can extend a → Q to R if we extend Q to the larger module R ⊕a Q. The key point of Baer’s argument is to repeat this procedure transfinitely many times. To do this we first define, given an R-module M the following (huge) pushout L L /M a ϕ∈HomR (a,M ) a (17.6.6.1) L L a

 ϕ∈HomR (a,M )

 / M(M ).

R

Here the top horizontal arrow maps the element a ∈ a in the summand corresponding to ϕ to the element ϕ(a) ∈ M . The left vertical arrow maps a ∈ a in the summand corresponding to ϕ simply to the element a ∈ R in the summand corresponding to ϕ. The fundamental properties of this construction are formulated in the following lemma. Lemma 17.6.7. Let R be a ring. (1) The construction M 7→ (M → M(M )) is functorial in M . (2) The map M → M(M ) is injective. (3) For any ideal a and any R-module map ϕ : a → M there is an R-module map ϕ0 : R → M(M ) such that a  R

ϕ

ϕ0

/M  / M(M )

commutes. Proof. Parts (2) and (3) are immediate from the construction. To see (1), let χ : M → N be an R-module map. We claim there exists a canonical commutative diagram L L /M a ϕ∈HomR (a,M ) a χ

L L a

 ϕ∈HomR (a,M )

R

L+ L

ψ∈HomR (a,N )

a

L +L a

a



ψ∈HomR (a,N )

R

+/ N

17.7. G-MODULES

1125

which induces the desired map M(M ) → M(N ). The middle east-south-east arrow maps the summand a corresponding to ϕ via ida to the summand a corresponding to ψ = χ ◦ ϕ. Similarly for the lower east-south-east arrow. Details omitted.  The idea will now be to apply the functor M a transfinite number of times. We define for each ordinal α a functor Mα on the category of R-modules, together with a natural injection N → Mα (N ). We do this by transfinite induction. First, M1 = M is the functor defined above. Now, suppose given an ordinal α, and suppose Mα0 is defined for α0 < α. If α has an immediate predecessor α e, we let Mα = M ◦ Mαe . If not, i.e., if α is a limit ordinal, we let Mα (N ) = colimα0 0 and any open covering U : U = i∈I Ui of X. Then H p (U, F) = 0 for all p > 0 and any open U ⊂ X. Proof. Let F be a sheaf satisfying the assumption of the lemma. We will indicate this by saying “F has vanishing higher Cech cohomology for any open covering”. Choose an embedding F → I into an injective OX -module. By Lemma 18.11.1 I has vanishing higher Cech cohomology for any open covering. Let Q = I/F so that we have a short exact sequence 0 → F → I → Q → 0. By Lemma 18.11.6 and our assumptions this sequence is actually exact as a sequence ˇ of presheaves! In particular we have a long exact sequence of Cech cohomology groups for any open covering U, see Lemma 18.10.2 for example. This implies ˇ that Q is also an OX -module with vanishing higher Cech cohomology for all open coverings. Next, we look at the long exact cohomology sequence 0

/ H 0 (U, F)

H 1 (U, F) ... s

t

/ H 0 (U, I)

/ H 0 (U, Q)

/ H 1 (U, I)

/ H 1 (U, Q)

...

...

for any open U ⊂ X. Since I is injective we have H n (U, I) = 0 for n > 0 (see Derived Categories, Lemma 11.19.4). By the above we see that H 0 (U, I) → H 0 (U, Q) is surjective and hence H 1 (U, F) = 0. Since F was an arbitrary OX ˇ module with vanishing higher Cech cohomology we conclude that also H 1 (U, Q) = 0 since Q is another of these sheaves (see above). By the long exact sequence this in turn implies that H 2 (U, F) = 0. And so on and so forth.  Lemma 18.11.8. (Variant of Lemma 18.11.7.) Let X be a ringed space. Let B be a basis for the topology on X. Let F be an OX -module. Assume there exists a set of open coverings Cov with the following properties:

18.11. CECH COHOMOLOGY AND COHOMOLOGY

1157

S (1) For every U ∈ Cov with U : U = i∈I Ui we have U, Ui ∈ B and every Ui0 ...ip ∈ B. (2) For every U ∈ B the open coverings of U occuring in Cov is a cofinal system of open coverings of U . ˇ p (U, F) = 0 for all p > 0. (3) For every U ∈ Cov we have H Then H p (U, F) = 0 for all p > 0 and any U ∈ B. Proof. Let F and Cov be as in the lemma. We will indicate this by saying “F has vanishing higher Cech cohomology for any U ∈ Cov”. Choose an embedding F → I into an injective OX -module. By Lemma 18.11.1 I has vanishing higher ˇ Cech cohomology for any U ∈ Cov. Let Q = I/F so that we have a short exact sequence 0 → F → I → Q → 0. By Lemma 18.11.6 and our assumption (2) this sequence gives rise to an exact sequence 0 → F(U ) → I(U ) → Q(U ) → 0. ˇ for every U ∈ B. Hence for any U ∈ Cov we get a short exact sequence of Cech complexes 0 → Cˇ• (U, F) → Cˇ• (U, I) → Cˇ• (U, Q) → 0 ˇ since each term in the Cech complex is made up out of a product of values over elements of B by assumption (1). In particular we have a long exact sequence of ˇ Cech cohomology groups for any open covering U ∈ Cov. This implies that Q is ˇ also an OX -module with vanishing higher Cech cohomology for all U ∈ Cov. Next, we look at the long exact cohomology sequence 0

/ H 0 (U, F)

H 1 (U, F) ... s

t

/ H 0 (U, I)

/ H 0 (U, Q)

/ H 1 (U, I)

/ H 1 (U, Q)

...

...

for any U ∈ B. Since I is injective we have H n (U, I) = 0 for n > 0 (see Derived Categories, Lemma 11.19.4). By the above we see that H 0 (U, I) → H 0 (U, Q) is surjective and hence H 1 (U, F) = 0. Since F was an arbitrary OX -module with vanˇ ishing higher Cech cohomology for all U ∈ Cov we conclude that also H 1 (U, Q) = 0 since Q is another of these sheaves (see above). By the long exact sequence this in turn implies that H 2 (U, F) = 0. And so on and so forth.  Lemma 18.11.9. Let f : X → Y be a morphism of ringed spaces. Let I be an injective OX -module. Then ˇ p (V, f∗ I) = 0 for all p > 0 and any open covering V : V = S (1) H j∈J Vj of Y. (2) H p (V, f∗ I) = 0 for all p > 0 and every open V ⊂ Y . In other words, f∗ I is right acyclic for Γ(U, −) (see Derived Categories, Definition 11.15.3) for any U ⊂ X open.

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Proof. Set U : f −1 (V ) =

S

j∈J

f −1 (Vj ). It is an open covering of X and

Cˇ• (V, f∗ I) = Cˇ• (U, I). This is true because f∗ I(Vj0 ...jp ) = I(f −1 (Vj0 ...jp )) = I(f −1 (Vj0 ) ∩ . . . ∩ f −1 (Vjp )) = I(Uj0 ...jp ). Thus the first statement of the lemma follows from Lemma 18.11.1. The second statement follows from the first and Lemma 18.11.7.  The following lemma implies in particular that f∗ : Ab(X) → Ab(Y ) transforms injective abelian sheaves into injective abelian sheaves. Lemma 18.11.10. Let f : X → Y be a morphism of ringed spaces. Assume f is flat. Then f∗ I is an injective OY -module for any injective OX -module I. Proof. In this case the functor f ∗ transforms injections into injections. Hence the result follows from Modules, Lemma 15.17.2 and Homology, Lemma 10.22.1  18.12. The Leray spectral sequence Lemma 18.12.1. Let f : X → Y be a morphism of ringed spaces. There is a commutative diagram D+ (X)

RΓ(X,−)

Rf∗

 D+ (Y )

/ D+ (OX (X)) restriction

RΓ(Y,−)

 / D+ (OY (Y ))

More generally for any V ⊂ Y open and U = f −1 (V ) there is a commutative diagram / D+ (OX (U )) D+ (X) RΓ(U,−)

Rf∗

 D+ (Y )

restriction

RΓ(V,−)

 / D+ (OY (V ))

See also Remark 18.12.2 for more explanation. Proof. Let Γres : Mod(OX ) → Mod(OY (Y )) be the functor which associates to an OX -module F the global sections of F viewed as a OY (Y )-module via the map f ] : OY (Y ) → OX (X). Let restriction : Mod(OX (X)) → Mod(OY (Y )) be the restriction functor induced by f ] : OY (Y ) → OX (X). Note that restriction is exact so that its right derived functor is computed by simply applying the restriction functor, see Derived Categories, Lemma 11.16.8. It is clear that Γres = restriction ◦ Γ(X, −) = Γ(Y, −) ◦ f∗ We claim that Derived Categories, Lemma 11.21.1 applies to both compositions. For the first this is clear by our remarks above. For the second, it follows from Lemma 18.11.9 which implies that injective OX -modules are mapped to Γ(Y, −)acyclic sheaves on Y . 

18.12. THE LERAY SPECTRAL SEQUENCE

1159

Remark 18.12.2. Here is a down-to-earth explanation of the meaning of Lemma 18.12.1. It says that given f : X → Y and F ∈ Mod(OX ) and given an injective resolution F → I • we have RΓ(X, F) is represented by Γ(X, I • ) Rf∗ F is represented by f∗ I • RΓ(Y, Rf∗ F) is represented by Γ(Y, f∗ I • ) the last fact coming from Leray’s acyclicity lemma (Derived Categories, Lemma 11.16.7) and Lemma 18.11.9. Finally, it combines this with the trivial observation that Γ(X, I • ) = Γ(Y, f∗ I • ). to arrive at the commutativity of the diagram of the lemma. Lemma 18.12.3. Let X be a ringed space. Let F be an OX -module. (1) The cohomology groups H i (U, F) for U ⊂ X open of F computed as an OX -module, or computed as an abelian sheaf are identical. (2) Let f : X → Y be a morphism of ringed spaces. The higher direct images Ri f∗ F of F computed as an OX -module, or computed as an abelian sheaf are identical. There are similar statements in the case of bounded below complexes of OX -modules. Proof. Consider the morphism of ringed spaces (X, OX ) → (X, ZX ) given by the identity on the underlying topological space and by the unique map of sheaves of rings ZX → OX . Let F be an OX -module. Denote Fab the same sheaf seens as an ZX -module, i.e., seens as a sheaf of abelian groups. Let F → I • be an injective resolution. By Remark 18.12.2 we see that Γ(X, I • ) computes both RΓ(X, F) and RΓ(X, Fab ). This proves (1). To prove (2) we use (1) and Lemma 18.6.3. The result follows immediately.



Lemma 18.12.4 (Leray spectral sequence). Let f : X → Y be a morphism of ringed spaces. Let F • be a bounded below complex of OX -modules. There is a spectral sequence E2p,q = H p (Y, Rq f∗ (F • )) converging to H p+q (X, F • ). Proof. This is just the Grothendieck spectral sequence Derived Categories, Lemma 11.21.2 coming from the composition of functors Γres = Γ(Y, −)◦f∗ where Γres is as in the proof of Lemma 18.12.1. To see that the assumptions of Derived Categories, Lemma 11.21.2 are satisfied, see the proof of Lemma 18.12.1 or Remark 18.12.2.  Remark 18.12.5. The Leray spectral sequence, the way we proved it in Lemma 18.12.4 is a spectral sequence of Γ(Y, OY )-modules. However, it is quite easy to see that it is in fact a spectral sequence of Γ(X, OX )-modules. For example f gives rise to a morphism of ringed spaces f 0 : (X, OX ) → (Y, f∗ OX ). By Lemma 18.12.3 the terms Erp,q of the Leray spectral sequence for an OX -module F and f are identical with those for F and f 0 at least for r ≥ 2. Namely, they both agree with the terms of the Leray spectral sequence for F as an abelian sheaf. And since (f∗ OX )(Y ) = OX (X) we see the result. It is often the case that the Leray spectral sequence carries additional structure.

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Lemma 18.12.6. Let f : X → Y be a morphism of ringed spaces. Let F be an OX -module. (1) If Rq f∗ F = 0 for q > 0, then H p (X, F) = H p (Y, f∗ F) for all p. (2) If H p (Y, Rq f∗ F) = 0 for all q and p > 0, then H q (X, F) = H 0 (Y, Rq f∗ F) for all q. Proof. These are two simple conditions that force the Leray spectral sequence to converge. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves.  Lemma 18.12.7. Let f : X → Y and g : Y → Z be morphisms of ringed spaces. In this case Rg∗ ◦ Rf∗ = R(g ◦ f )∗ as functors from D+ (X) → D+ (Z). Proof. We are going to apply Derived Categories, Lemma 11.21.1. It is clear that g∗ ◦ f∗ = (g ◦ f )∗ , see Sheaves, Lemma 6.21.2. It remains to show that f∗ I is g∗ acyclic. This follows from Lemma 18.11.9 and the description of the higher direct images Ri g∗ in Lemma 18.6.3.  Lemma 18.12.8 (Relative Leray spectral sequence). Let f : X → Y and g : Y → Z be morphisms of ringed spaces. Let F be an OX -module. There is a spectral sequence with E2p,q = Rp g∗ (Rq f∗ F) converging to Rp+q (g ◦ f )∗ F. This spectral sequence is functorial in F, and there is a version for bounded below complexes of OX -modules. Proof. This is a Grothendieck spectral sequence for composition of functors and follows from Lemma 18.12.7 and Derived Categories, Lemma 11.21.2.  18.13. Functoriality of cohomology Lemma 18.13.1. Let f : X → Y be a morphism of ringed spaces. Let G • , resp. F • be a bounded below complex of OY -modules, resp. OX -modules. Let ϕ : G • → f∗ F • be a morphism of complexes. There is a canonical morphism G • −→ Rf∗ (F • ) in D+ (Y ). Moreover this construction is functorial in the triple (G • , F • , ϕ). Proof. Choose an injective resolution F • → I • . By definition Rf∗ (F • ) is represented by f∗ I • in K + (OY ). The composition G • → f∗ F • → f∗ I • is a morphism in K + (Y ) which turns into the morphism of the lemma upon applying the localization functor jY : K + (Y ) → D+ (Y ).  Let f : X → Y be a morphism of ringed spaces. Let G be an OY -module and let F be an OX -module. Recall that an f -map ϕ from G to F is a map ϕ : G → f∗ F, or what is the same thing, a map ϕ : f ∗ G → F. See Sheaves, Definition 6.21.7. Such an f -map gives rise to a morphism of complexes (18.13.1.1) +

ϕ : RΓ(Y, G) −→ RΓ(X, F)

in D (OY (Y )). Namely, we use the morphism G → Rf∗ F in D+ (Y ) of Lemma 18.13.1, and we apply RΓ(Y, −). By Lemma 18.12.1 we see that RΓ(X, F) =

18.13. FUNCTORIALITY OF COHOMOLOGY

1161

RΓ(Y, Rf∗ F) and we get the displayed arrow. We spell this out completely in Remark 18.13.2 below. In particular it gives rise to maps on cohomology ϕ : H i (Y, G) −→ H i (X, F).

(18.13.1.2)

Remark 18.13.2. Let f : X → Y be a morphism of ringed spaces. Let G be an OY -module. Let F be an OX -module. Let ϕ be an f -map from G to F. Choose a resolution F → I • by a complex of injective OX -modules. Choose resolutions G → J • and f∗ I → (J 0 )• by complexes of injective OY -modules. By Derived Categories, Lemma 11.17.6 there exists a map of complexes β such that the diagram / f∗ F

G

(18.13.2.1)

 J•

/ f∗ I •  / (J 0 )•

β

commutes. Applying global section functors we see that we get a diagram Γ(Y, f∗ I • )

Γ(X, I • )

qis

β

Γ(Y, J • )

 / Γ(Y, (J 0 )• )

The complex on the bottom left represents RΓ(Y, G) and the complex on the top right represents RΓ(X, F). The vertical arrow is a quasi-isomorphism by Lemma 18.12.1 which becomes invertible after applying the localization functor K + (OY (Y )) → D+ (OY (Y )). The arrow (18.13.1.1) is given by the composition of the horizontal map by the inverse of the vertical map. Lemma 18.13.3. Let f : X → Y be a morphism of ringed spaces. Let F be an OX -module. Let G be an OY -module. Let ϕ : fS∗ G → F be an f -map. Let S U : X = i∈I Ui be an open covering. Let V : Y = j∈J Vj be an open covering. S Assume that U is a refinement of f −1 V : X = j∈J f −1 (Vj ). In this case there exists a commutative diagram Cˇ• (U, F) O

/ RΓ(X, F) O

γ

Cˇ• (V, G)

/ RΓ(Y, G)

in D+ (OX (X)) where the horizontal arrows come from Lemma 18.11.2 and the right vertical arrow is Equation (18.13.1.1). In particular we get commutative diagrams of cohomology groups ˇ p (U, F) H O

/ H p (X, F) O

γ

ˇ p (V, G) H where the right vertical arrow is (18.13.1.2)

/ H p (Y, G)

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18. COHOMOLOGY OF SHEAVES

Proof. We first define the left vertical arrow. Namely, choose a map c : I → J such that Ui ⊂ f −1 (Vc(i) ) for all i ∈ I. In degree p we define the map by the rule γ(s)i0 ...ip = ϕ(s)c(i0 )...c(ip ) This makes sense because ϕ does indeed induce maps G(Vc(i0 )...c(ip ) ) → F(Ui0 ...ip ) by assumption. It is also clear that this defines a morphism of complexes. Choose injective resolutions F → I • on X and G → J • on Y . According to the proof of Lemma 18.11.2 we introduce the double complexes A•,• and B •,• with terms B p,q = Cˇp (V, J q ) and Ap,q = Cˇp (U, I q ). As in Remark 18.13.2 above we also choose an injective resolution f∗ I → (J 0 )• on Y and a morphism of complexes β : J → (J 0 )• making (18.13.2.1) commutes. We introduce some more double complexes, namely (B 0 )•,• and (B 00 )•, • with (B 0 )p,q = Cˇp (V, (J 0 )q ) and (B 00 )p,q = Cˇp (V, f∗ I q ). Note that there is an f -map of complexes from f∗ I • to I • . Hence it is clear that the same rule as above defines a morphism of double complexes γ : (B 00 )•,• −→ A•,• . Consider the diagram of complexes Cˇ• (U, F) O

/ sA• ok sγ

γ

Cˇ• (V, G)

Γ(X, I • )

qis

/ sB • O

β

/ s(B 0 )• o O

β

/ Γ(Y, (J 0 )• ) o

s(B 00 )• O

qis

Γ(Y, J • )

qis

Γ(Y, f∗ I • )

The two horizontal arrows with targets sA• and sB • are the ones explained in Lemma 18.11.2. The left upper shape (a pentagon) is commutative simply because (18.13.2.1) is commutative. The two lower squares are trivially commutative. It is also immediate from the definitions that the right upper shape (a square) is commutative. The result of the lemma now follows from the definitions and the fact that going around the diagram on the outer sides from Cˇ• (V, G) to Γ(X, I • ) either on top or on bottom is the same (where you have to invert any quasi-isomorphisms along the way).  18.14. The base change map We will need to know how to construct the base change map in some cases. Since we have not yet discussed derived pullback we only discuss this in the case of a base change by a flat morphism of ringed spaces. Before we state the result, let us discuss flat pullback on the derived category. Namely, suppose that g : X → Y is a flat morphism of ringed spaces. By Modules, Lemma 15.17.2 the functor g ∗ : Mod(OY ) → Mod(OX ) is exact. Hence it has a derived functor g ∗ : D+ (Y ) → D+ (X) which is computed by simply pulling back an representative of a given object in D+ (Y ), see Derived Categories, Lemma 11.16.8. Hence as indicated we indicate this functor by g ∗ rather than Lg ∗ .

18.15. COHOMOLOGY AND COLIMITS

Lemma 18.14.1. Let X0 f0

g0

1163

/X f

  g /S S0 be a commutative diagram of ringed spaces. Let F • be a bounded below complex of OX -modules. Assume both g and g 0 are flat. Then there exists a canonical base change map g ∗ Rf∗ F • −→ R(f 0 )∗ (g 0 )∗ F • in D+ (S 0 ). Proof. Choose injective resolutions F • → I • and (g 0 )∗ F • → J • . By Lemma 18.11.10 we see that (g 0 )∗ J • is a complex of injectives representing R(g 0 )∗ (g 0 )∗ F • . Hence by Derived Categories, Lemmas 11.17.6 and 11.17.7 the arrow β in the diagram / (g 0 )∗ J • (g 0 )∗ (g 0 )∗ F • O O adjunction

β

/ I• F• exists and is unique up to homotopy. Pushing down to S we get f∗ β : f∗ I • −→ f∗ (g 0 )∗ J • = g∗ (f 0 )∗ J • By adjunction of g ∗ and g∗ we get a map of complexes g ∗ f∗ I • → (f 0 )∗ J • . Note that this map is unique up to homotopy since the only choice in the whole process was the choice of the map β and everything was done on the level of complexes.  Remark 18.14.2. The “correct” version of the base change map is map Lg ∗ Rf∗ F • −→ R(f 0 )∗ L(g 0 )∗ F • . The construction of this map really involves dealing with unbounded complexes and having adjoint functors Lj ∗ , Rj∗ on unbounded complexes. We will deal with this later (insert future reference here). 18.15. Cohomology and colimits Let X be a ringed space. Let (Fi , ϕii0 ) be a directed system of sheaves of OX modules over the partially ordered set I, see Categories, Section 4.19. Since for each i there is a canonical map Fi → colimi Fi we get a canonical map colimi H p (X, Fi ) −→ H p (X, colimi Fi ) for every p ≥ 0. Of course there is a similar map for every open U ⊂ X. These maps are in general not isomorphisms, even for p = 0. In this section we generalize the results of Sheaves, Lemma 6.29.1. See also Modules, Lemma 15.11.6 (in the special case G = OX ). Lemma 18.15.1. Let X be a ringed space. Assume that the underlying topological space of X has the following properties: (1) there exists a basis of quasi-compact open subsets, and (2) the intersection of any two quasi-compact opens is quasi-compact.

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18. COHOMOLOGY OF SHEAVES

Then for any directed system (Fi , ϕii0 ) of sheaves of OX -modules and for any quasicompact open U ⊂ X the canonical map colimi H q (U, Fi ) −→ H q (U, colimi Fi ) is an isomorphism for every q ≥ 0. Proof. It is important in this proof to argue for all quasi-compact opens U ⊂ X at the same time. The result is true for i = 0 and any quasi-compact open U ⊂ X by Sheaves, Lemma 6.29.1 (combined with Topology, Lemma 5.18.2). Assume that we have proved the result for all q ≤ q0 and let us prove the result for q = q0 + 1. By our conventions on directed systems the index set I is directed, and any system of OX -modules (Fi , ϕii0 ) over I is directed. By Injectives, Lemma 17.9.1 the category of OX -modules has functorial injective embeddings. Thus for any system (Fi , ϕii0 ) there exists a system (Ii , ϕii0 ) with each Ii an injective OX -module and a morphism of systems given by injective OX -module maps Fi → Ii . Denote Qi the cokernel so that we have short exact sequences 0 → Fi → Ii → Qi → 0. We claim that the sequence 0 → colimi Fi → colimi Ii → colimi Qi → 0. is also a short exact sequence of OX -modules. We may check this on stalks. By Sheaves, Sections 6.28 and 6.29 taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma 7.8.9) we deduce the result. We claim that H q (U, colimi Ii ) = 0 for all quasi-compact open U ⊂ X and all q ≥ 1. Accepting this claim for the moment consider the diagram colimi H q0 (U, Ii )

/ colimi H q0 (U, Qi )

/ colimi H q0 +1 (U, Fi )

/0

 H q0 (U, colimi Ii )

 / H q0 (U, colimi Qi )

 / H q0 +1 (U, colimi Fi )

 /0

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves Ii are injective. The top row is exact by an application of Algebra, Lemma 7.8.9. Hence by the snake lemma we deduce the result for q = q0 + 1. It remains to show that the claim is true. We will use Lemma 18.11.8. Let B be the collection of all quasi-compact open subsets of X. This is a basis for the topology S on X by assumption. Let Cov be the collection of finite open coverings U : U = j=1,...,m Uj with each of U , Uj quasi-compact open in X. By the result for q = 0 we see that for U ∈ Cov we have Cˇ• (U, colimi Ii ) = colimi Cˇ• (U, Ii ) because all the multiple intersections Uj0 ...jp are quasi-compact. By Lemma 18.11.1 ˇ each of the complexes in the colimit of Cech complexes is acyclic in degree ≥ 1. ˇ Hence by Algebra, Lemma 7.8.9 we see that also the Cech complex Cˇ• (U, colimi Ii ) ˇ is acyclic in degrees ≥ 1. In other words we see that H p (U, colimi Ii ) = 0 for all p ≥ 1. Thus the assumptions of Lemma 18.11.8 are satisfied and the claim follows. 

18.16. VANISHING ON NOETHERIAN TOPOLOGICAL SPACES

1165

18.16. Vanishing on Noetherian topological spaces The aim is to prove a theorem of Grothendieck namely Lemma 18.16.5. [Gro57].

See

Lemma 18.16.1. Let i : Z → X be a closed immersion of topological spaces. For any abelian sheaf F on Z we have H p (Z, F) = H p (X, i∗ F). Proof. This is true because i∗ is exact (see Modules, Lemma 15.6.1), and hence Rp i∗ = 0 as a functor (Derived Categories, Lemma 11.16.8). Thus we may apply Lemma 18.12.6.  Lemma 18.16.2. Let X be an irreducible topological space. Then H p (X, A) = 0 for all p > 0 and any abelian group A. Proof. Recall that A is the constant sheaf as defined in Sheaves, Definition 6.7.4. It is clear that for any nonempty open U ⊂ X we have A(U ) = A as X is irreducible ˇ (and hence U is connected). We will show that theShigher Cech cohomology groups p ˇ H (U, A) are zero for any open covering U : U = i∈I Ui of an open U ⊂ X. Then the lemma will follow from Lemma 18.11.7. Recall that the value of an abelian sheaf on the empty open set is 0. Hence we may clearly assume Ui 6= ∅ for all i ∈ I. In this case we see that Ui ∩ Ui0 6= ∅ for all ˇ i, i0 ∈ I. Hence we see that the Cech complex is simply the complex Y Y Y A→ A→ A → ... i0 ∈I

(i0 ,i1 )∈I 2

(i0 ,i1 ,i2 )∈I 3

We have to see this has trivial higher cohomology groups. We can see this for example because this is the cech complex for the covering of a 1-point space and ˇ Cech cohomology agrees with cohomology on such a space. (You can also directly verify it by writing an explicit homotopy.)  18.16.3. Let X be a topological space. Let n ≥ 0 be an integer. Assume there exists a basis of quasi-compact open subsets, and the intersection of any two quasi-compact opens is quasi-compact. H p (X, F) = 0 for any abelian sheaf F which is a quotient of j! ZU for some open j : U → X. p Then H (X, F) = 0 for all p ≥ n and any abelian sheaf F on X. ` Proof. Let S = U ⊂X F(U ). For any finite subset A = {s1 , . . . , sn } ⊂ S, let FA be the subsheaf of F generated by all si (see Modules, Definition 15.4.5). Note that if A ⊂ A0 , then FA ⊂ FA0 . Hence {FA } forms a system over the partially ordered set of finite subsets of S. By Modules, Lemma 15.4.6 it is clear that Lemma (1) (2) (3)

colimA FA = F by looking at stalks. By Lemma 18.15.1 we have H p (X, F) = colimA H p (X, FA ) Hence it suffices to prove the vanishing for the abelian sheaves FA . In other words, it suffices to prove the result when F is generated by finitely many local sections. Suppose that F is gerated by the local sections s1 , . . . , sn . Let F 0 ⊂ F be the subsheaf generated by s1 , . . . , sn−1 . Then we have a short exact sequence 0 → F 0 → F → F/F 0 → 0

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18. COHOMOLOGY OF SHEAVES

From the long exact sequence of cohomology we see that it suffices to prove the vanishing for the abelian sheaves F 0 and F/F 0 which are generated by fewer than n local sections. Hence it suffices to prove the vanishing for sheaves generated by at most one local section. These sheaves are exactly the quotients of the sheaves  j! ZU mentioned in the lemma. Lemma 18.16.4. Let X be an irreducible topological space. Let H ⊂ Z be an abelian subsheaf of the constant sheaf. Then there exists a nonempty open U ⊂ X such that H|U = dZU for some d ∈ Z. Proof. Recall that Z(V ) = Z for any nonempty open V of X (see proof of Lemma 18.16.2). If H = 0, then the lemma holds with d = 0. If H = 6 0, then there exists a nonempty open U ⊂ X such that H(U ) 6= 0. Say H(U ) = nZ for some n ≥ 1. Hence we see that nZU ⊂ H|U ⊂ ZU . If the first inclusion is strict we can find a nonempty U 0 ⊂ U and an integer 1 ≤ n0 < n such that n0 ZU 0 ⊂ H|U 0 ⊂ ZU 0 . This process has to stop after a finite number of steps, and hence we get the lemma.  Lemma 18.16.5. Let X be a Noetherian topological space. If dim(X) ≤ n, then H p (X, F) = 0 for all p > n and any abelian sheaf F on X. Proof. We prove this lemma by induction on n. So fix n and assume the lemma holds for all Noetherian topological spaces of dimension < n. Let F be an abelian sheaf on X. Suppose U ⊂ X is an open. Let Z ⊂ X denote the closed complement. Denote j : U → X and i : Z → X the inclusion maps. Then there is a short exact sequence 0 → j! j ∗ F → F → i∗ i∗ F → 0 see Modules, Lemma 15.7.1. Note that j! j ∗ F is supported on the topological closure Z 0 of U , i.e., it is of the form i0∗ F 0 for some abelian sheaf F 0 on Z 0 , where i0 : Z 0 → X is the inclusion. We can use this to reduce to the case where X is irreducible. Namely, according to Topology, Lemma 5.6.2 X has finitely many irreducible components. If X has more than one irreducible component, then let Z ⊂ X be an irreducible component of X and set U = X \ Z. By the above, and the long exact sequence of cohomology, it suffices to prove the vanishing of H p (X, i∗ i∗ F) and H p (X, i0∗ F 0 ) for p > n. By Lemma 18.16.1 it suffices to prove H p (Z, i∗ F) and H p (Z 0 , F 0 ) for p > n. Since Z 0 and Z have fewer irreducible components we indeed reduce to the case of an irreducible X. If n = 0 and X = {∗}, then every sheaf is constant and higher cohomology groups vanish (for example by Lemma 18.16.2). Suppose X is irreducible of dimension n. By Lemma 18.16.3 we reduce to the case where F is generated by a single local section, i.e., to the case where there is an exact sequence 0 → H → j!0 ZV → F → 0 for some open j 0 : V → X. By Lemma 18.16.4 (applied to the restriction of H to V ) there exists a nonempty open U ⊂ V , and d ∈ Z such that H|U = dZU . Hence we see that F|U ∼ = Z/dZU . Let Z be the complement of U in X. Denote j : U → X

ˇ 18.17. THE ALTERNATING CECH COMPLEX

1167

and i : Z → X the inclusion maps. As in the first paragraph of the proof we obtain a short exact sequence 0 → j! Z/dZ → F → i∗ i∗ F → 0 OK, and now dim(Z) < n so by induction we have H p (X, i∗ i∗ F) = H p (Z, i∗ F) = 0 for all p ≥ n. Hence it suffices to prove the vanishing for sheaves of the form j! (AU ) where j : U → X is an open immersion and A is an abelian group. In this case we again look at the short exact sequence 0 → j! (AU ) → A → i∗ AZ → 0 By Lemma 18.16.2 we have the vanishing of H p (X, A) for all p ≥ 1. By induction we have H p (X, i∗ AZ ) = H p (Z, AZ ) = 0 for p ≤ n. Hence we win by the long exact cohomology sequence.  ˇ 18.17. The alternating Cech complex ˇ ˇ This section compares the Cech complex with the alternating Cech complex and some related complexes. S Let X be a topological space. Let U : U = i∈I Ui be an open covering. For p ≥ 0 set   s ∈ Cˇp (U, F) such that si0 ...ip = 0 if in = im for some n 6= m p ˇ Calt (U, F) = and si0 ...in ...im ...ip = −si0 ...im ...in ...ip in any case. p We omit the verification that the differential d of Equation (18.9.0.1) maps Cˇalt (U, F) p+1 ˇ into Calt (U, F). S Definition 18.17.1. Let X be a topological space. Let U : U = i∈I Ui be an • open covering. Let F be an abelian presheaf on X. The complex Cˇalt (U, F) is the ˇ alternating Cech complex associated to F and the open covering U.

Hence there is a canonical morphism of complexes • Cˇalt (U, F) −→ Cˇ• (U, F)

ˇ ˇ namely the inclusion of the alternating Cech complex into the usual Cech complex. S Suppose our covering U : U = i∈I Ui comes equipped with a total ordering < on I. In this case, set Y p Cˇord (U, F) = F(Ui0 ...ip ). p+1 (i0 ,...,ip )∈I

,i0 0. This is the complex ...

L

 Z[Mor (V, U C i0 ×U Ui1 ×U Ui2 )] i0 i1 i2

L

i0 i1

 Z[MorC (V, Ui0 ×U Ui1 )]

L

i0

 Z[MorC (V, Ui0 )]  0

For any morphism ϕ : V → U denote Morϕ (V, Ui ) = {ϕi : V → Ui | fi ◦ ϕi = ϕ}. We will use a similar notation for Morϕ (V, Ui0 ×U . . . ×U Uip ). Note that composing with the various projection maps between the fibred products Ui0 ×U . . . ×U Uip preserves these morphism sets. Hence we see that the complex above is the same as the complex ...

L L ϕ

i0 i1 i2

L L ϕ

 Z[Morϕ (V, Ui0 ×U Ui1 ×U Ui2 )]  Z[Mor (V, Ui0 ×U Ui1 )] ϕ i0 i1

L L ϕ

i0

 Z[Morϕ (V, Ui0 )]  0

19.10. CECH COHOMOLOGY AS A FUNCTOR ON PRESHEAVES

1197

Next, we make the remark that we have Morϕ (V, Ui0 ×U . . . ×U Uip ) = Morϕ (V, Ui0 ) × . . . × Morϕ (V, Uip ) ` Using this and the fact that Z[A] ⊕ Z[B] = Z[A B] we see that the complex becomes ...

L

ϕZ

L

  Mor (V, Ui0 ) × Morϕ (V, Ui2 ) ϕ i0 i1 i2

`

ϕZ

  Mor (V, Ui0 ) × Morϕ (V, Ui1 ) ϕ i0 i1

`

L

ϕ

Z

`

i0

  Morϕ (V, Ui0 )

 0 ` Finally, on setting Sϕ = i∈I Morϕ (V, Ui ) we see that we get M (. . . → Z[Sϕ × Sϕ × Sϕ ] → Z[Sϕ × Sϕ ] → Z[Sϕ ] → 0 → . . .) ϕ

Thus we have simplified our task. Namely, it suffices to show that for any nonempty set S the (extended) complex of free abelian groups Σ

. . . → Z[S × S × S] → Z[S × S] → Z[S] − → Z → 0 → ... is exact in all degrees. To see this fix an element s ∈ S, and use the homotopy n(s0 ,...,sp ) 7−→ n(s,s0 ,...,sp ) with obvious notations.



Lemma 19.10.5. Let C be a category. Let U = {fi : Ui → U }i∈I be a family of morphisms with fixed target. Let O be a presheaf of rings on C. The chain complex ZU ,• ⊗p,Z O is exact in positive degrees. Here ZU ,• is the cochain complex of Lemma 19.10.3, and the tensor product is over the constant presheaf of rings with value Z. Proof. Let V be an object of C. In the proof of Lemma 19.10.4 we saw that ZU ,• (V ) is isomorphic as a complex to a direct sum of complexes which are homotopic to Z placed in degree zero. Hence also ZU ,• (V ) ⊗Z O(V ) is isomorphic as a complex to a direct sum of complexes which are homotopic to O(V ) placed in degree zero. Or you can use Modules on Sites, Lemma 16.26.9, which applies since the presheaves ZU ,i are flat, and the proof of Lemma 19.10.4 shows that H0 (ZU ,• ) is a flat presheaf also.  Lemma 19.10.6. Let C be a category. Let U = {fi : Ui → U }i∈I be a family of ˇ p (U, −) are canonimorphisms with fixed target. The Cech cohomology functors H cally isomomorphic as a δ-functor to the right derived functors of the functor ˇ 0 (U, −) : PAb(C) −→ Ab. H

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19. COHOMOLOGY ON SITES

Moreover, there is a functorial quasi-isomorphism ˇ 0 (U, F) Cˇ• (U, F) −→ RH where the right hand side indicates the derived functor ˇ 0 (U, −) : D+ (PAb(C)) −→ D+ (Z) RH ˇ 0 (U, −). of the left exact functor H Proof. Note that the category of abelian presheaves has enough injectives, see ˇ 0 (U, −) is a left exact functor from the Injectives, Proposition 17.10.1. Note that H category of abelian presheaves to the category of Z-modules. Hence the derived functor and the right derived fuctor exist, see Derived Categories, Section 11.19. Let I be a injective abelian presheaf. In this case the functor HomPAb(C) (−, I) is exact on PAb(C). By Lemma 19.10.3 we have HomPAb(C) (ZU ,• , I) = Cˇ• (U, I). By Lemma 19.10.4 we have that ZU ,• is exact in positive degrees. Hence by the ˇ i (U, I) = 0 for all i > exactness of Hom into I mentioned above we see that H n ˇ 0. Thus the δ-functor (H , δ) (see Lemma 19.10.2) satisfies the assumptions of Homology, Lemma 10.9.4, and hence is a universal δ-functor. ˇ 0 (U, −) forms a uniBy Derived Categories, Lemma 11.19.4 also the sequence Ri H versal δ-functor. By the uniqueness of universal δ-functors, see Homology, Lemma ˇ 0 (U, −) = H ˇ i (U, −). This is enough for most applica10.9.5 we conclude that Ri H tions and the reader is suggested to skip the rest of the proof. Let F be any abelian presheaf on C. Choose an injective resolution F → I • in the category PAb(C). Consider the double complex A•,• with terms Ap,q = Cˇp (U, I q ). Consider the simple complex sA• associated to this double complex. There is a map of complexes Cˇ• (U, F) −→ sA• coming from the maps Cˇp (U, F) → Ap,0 = Cˇ• (U, I 0 ) and there is a map of complexes ˇ 0 (U, I • ) −→ sA• H ˇ 0 (U, I q ) → A0,q = Cˇ0 (U, I q ). Both of these maps are coming from the maps H quasi-isomorphisms by an application of Homology, Lemma 10.19.6. Namely, the columns of the double complex are exact in positive degrees because the Cech complex as a functor is exact (Lemma 19.10.1) and the rows of the double complex are exact in positive degrees since as we just saw the higher Cech cohomology groups of the injective presheaves I q are zero. Since quasi-isomorphisms become invertible in D+ (Z) this gives the last displayed morphism of the lemma. We omit the verification that this morphism is functorial. 

19.11. CECH COHOMOLOGY AND COHOMOLOGY

1199

19.11. Cech cohomology and cohomology The relationship between cohomology and Cech cohomology comes from the fact that the Cech cohomology of an injective abelian sheaf is zero. To see this we note that an injective abelian sheaf is an injective abelian presheaf and then we apply results in Cech cohomology in the preceding section. Lemma 19.11.1. Let C be a site. An injective abelian sheaf is also injective as an object in the category PAb(C). Proof. Apply Homology, Lemma 10.22.1 to the categories A = Ab(C), B = PAb(C), the inclusion functor and sheafification. (See Modules on Sites, Section 16.3 to see that all assumptions of the lemma are satisfied.)  Lemma 19.11.2. Let C be a site. Let U = {Ui → U }i∈I be a covering of C. Let I be an injective abelian sheaf, i.e., an injective object of Ab(C). Then  I(U ) if p = 0 p ˇ H (U, I) = 0 if p > 0 Proof. By Lemma 19.11.1 we see that I is an injective object in PAb(C). Hence we can apply Lemma 19.10.6 (or its proof) to see the vanishing of higher Cech cohomology group. For the zeroth see Lemma 19.9.2.  Lemma 19.11.3. Let C be a site. Let U = {Ui → U }i∈I be a covering of C. There is a transformation Cˇ• (U, −) −→ RΓ(U, −) of functors Ab(C) → D+ (Z). In particular this gives a transformation of functors ˇ p (U, F) → H p (U, F) for F ranging over Ab(C). H Proof. Let F be an abelian sheaf. Choose an injective resolution F → I • . Consider the double complex A•,• with terms Ap,q = Cˇp (U, I q ). Moreover, consider the associated simple complex sA• , see Homology, Definition 10.19.2. There is a map of complexes α : Γ(U, I • ) −→ sA• ˇ 0 (U, I q ) and a map of complexes coming from the maps I q (U ) → H β : Cˇ• (U, F) −→ sA• coming from the map F → I 0 . We can apply Homology, Lemma 10.19.6 to see that α is a quasi-isomorphism. Namely, Lemma 19.11.2 implies that the qth row of the double complex A•,• is a resolution of Γ(U, I q ). Hence α becomes invertible in D+ (Z) and the transformation of the lemma is the composition of β followed by the inverse of α. We omit the verification that this is functorial.  Lemma 19.11.4. Let C be a site. Consider the functor i : Ab(C) → PAb(C). It is a left exact functor with right derived functors given by Rp i(F) = H p (F) : U 7−→ H p (U, F) see discussion in Section 19.8.

1200

19. COHOMOLOGY ON SITES

Proof. It is clear that i is left exact. Choose an injective resolution F → I • . By definition Rp i is the pth cohomology presheaf of the complex I • . In other words, the sections of Rp i(F) over an open U are given by Ker(I n (U ) → I n+1 (U )) . Im(I n−1 (U ) → I n (U )) which is the definition of H p (U, F).



Lemma 19.11.5. Let C be a site. Let U = {Ui → U }i∈I be a covering of C. For any abelian sheaf F there is a spectral sequence (Er , dr )r≥0 with ˇ p (U, H q (F)) E p,q = H 2

converging to H p+q (U, F). This spectral sequence is functorial in F. Proof. This is a Grothendieck spectral sequence (see Derived Categories, Lemma 11.21.2) for the functors ˇ 0 (U, −) : PAb(C) → Ab. i : Ab(C) → PAb(C) and H ˇ 0 (U, i(F)) = F(U ) by Lemma 19.9.2. We have that i(I) is Cech Namely, we have H ˇ 0 (U, −) as functors ˇ p (U, −) = Rp H acyclic by Lemma 19.11.2. And we have that H on PAb(C) by Lemma 19.10.6. Putting everything together gives the lemma.  Lemma 19.11.6. Let C be a site. Let U = {Ui → U }i∈I be a covering. Let F ∈ Ob(Ab(C)). Assume that H i (Ui0 ×U . . . ×U Uip , F) = 0 for all i > 0, all p ≥ 0 ˇ p (U, F) = H p (U, F). and all i0 , . . . , ip ∈ I. Then H Proof. We will use the spectral sequence of Lemma 19.11.5. The assumptions mean that E2p,q = 0 for all (p, q) with q 6= 0. Hence the spectral sequence degenerates at E2 and the result follows.  Lemma 19.11.7. Let C be a site. Let 0→F →G→H→0 be a short exact sequence of abelian sheaves on C. Let U be an object of C. If there ˇ 1 (U, F) = 0, then the map exists a cofinal system of coverings U of U such that H G(U ) → H(U ) is surjective. Proof. Take an element s ∈ H(U ). Choose a covering U = {Ui → U }i∈I such that ˇ 1 (U, F) = 0 and (b) s|U is the image of a section si ∈ G(Ui ). Since we can (a) H i certainly find a covering such that (b) holds it follows from the assumptions of the lemma that we can find a covering such that (a) and (b) both hold. Consider the sections si0 i1 = si1 |Ui0 ×U Ui1 − si0 |Ui0 ×U Ui1 . ˇ 1 (U, F) we Since si lifts s we see that si0 i1 ∈ F(Ui0 ×U Ui1 ). By the vanishing of H can find sections ti ∈ F(Ui ) such that si0 i1 = ti1 |Ui0 ×U Ui1 − ti0 |Ui0 ×U Ui1 . Then clearly the sections si − ti satsify the sheaf condition and glue to a section of G over U which maps to s. Hence we win.  Lemma 19.11.8. (Variant of Cohomology, Lemma 18.11.7.) Let C be a site. Let CovC be the set of coverings of C (see Sites, Definition 9.6.2). Let B ⊂ Ob(C), and Cov ⊂ CovC be subsets. Let F be an abelian sheaf on C. Assume that

19.12. COHOMOLOGY OF MODULES

1201

(1) For every U ∈ Cov, U = {Ui → U }i∈I we have U, Ui ∈ B and every Ui0 ×U . . . ×U Uip ∈ B. (2) For every U ∈ B the coverings of U occuring in Cov is a cofinal system of coverings of U . ˇ p (U, F) = 0 for all p > 0. (3) For every U ∈ Cov we have H Then H p (U, F) = 0 for all p > 0 and any U ∈ B. Proof. Let F and Cov be as in the lemma. We will indicate this by saying “F has vanishing higher Cech cohomology for any U ∈ Cov”. Choose an embedding F → I into an injective abelian sheaf. By Lemma 19.11.2 I has vanishing higher Cech cohomology for any U ∈ Cov. Let Q = I/F so that we have a short exact sequence 0 → F → I → Q → 0. By Lemma 19.11.7 and our assumption (2) this sequence gives rise to an exact sequence 0 → F(U ) → I(U ) → Q(U ) → 0. for every U ∈ B. Hence for any U ∈ Cov we get a short exact sequence of Cech complexes 0 → Cˇ• (U, F) → Cˇ• (U, I) → Cˇ• (U, Q) → 0 since each term in the Cech complex is made up out of a product of values over elements of B by assumption (1). In particular we have a long exact sequence of Cech cohomology groups for any covering U ∈ Cov. This implies that Q is also an abelian sheaf with vanishing higher Cech cohomology for all U ∈ Cov. Next, we look at the long exact cohomology sequence 0

/ H 0 (U, F)

H 1 (U, F)

t

/ H 0 (U, I)

/ H 0 (U, Q)

/ H 1 (U, I)

/ H 1 (U, Q)

... s ... ... n for any U ∈ B. Since I is injective we have H (U, I) = 0 for n > 0 (see Derived Categories, Lemma 11.19.4). By the above we see that H 0 (U, I) → H 0 (U, Q) is surjective and hence H 1 (U, F) = 0. Since F was an arbitrary abelian sheaf with vanishing higher Cech cohomology for all U ∈ Cov we conclude that also H 1 (U, Q) = 0 since Q is another of these sheaves (see above). By the long exact sequence this in turn implies that H 2 (U, F) = 0. And so on and so forth.  19.12. Cohomology of modules Everything that was said for cohomology of abelian sheaves goes for cohomology of modules, since the two agree. Lemma 19.12.1. Let (C, O) be a ringed site. An injective sheaf of modules is also injective as an object in the category PMod(O). Proof. Apply Homology, Lemma 10.22.1 to the categories A = Mod(O), B = PMod(O), the inclusion functor and sheafification. (See Modules on Sites, Section 16.11 to see that all assumptions of the lemma are satisfied.) 

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Lemma 19.12.2. Let (C, O) be a ringed site. Consider the functor i : Mod(C) → PMod(C). It is a left exact functor with right derived functors given by Rp i(F) = H p (F) : U 7−→ H p (U, F) see discussion in Section 19.8. Proof. It is clear that i is left exact. Choose an injective resolution F → I • in Mod(O). By definition Rp i is the pth cohomology presheaf of the complex I • . In other words, the sections of Rp i(F) over an open U are given by Ker(I n (U ) → I n+1 (U )) . Im(I n−1 (U ) → I n (U )) which is the definition of H p (U, F).



Lemma 19.12.3. Let (C, O) be a ringed site. Let U = {Ui → U }i∈I be a covering of C. Let I be an injective O-module, i.e., an injective object of Mod(O). Then  I(U ) if p = 0 p ˇ H (U, I) = 0 if p > 0 Proof. Lemma 19.10.3 gives the first equality in the following sequence of equalities Cˇ• (U, I) = MorPAb(C) (ZU ,• , I) = MorPMod(Z) (ZU ,• , I) = MorPMod(O) (ZU ,• ⊗p,Z O, I) The third equality by Modules on Sites, Lemma 16.9.2. By Lemma 19.12.1 we see that I is an injective object in PMod(O). Hence HomPMod(O) (−, I) is an exact functor. By Lemma 19.10.5 we see the vanishing of higher Cech cohomology groups. For the zeroth see Lemma 19.9.2.  Lemma 19.12.4. Let C be a site. Let O be a sheaf of rings on C. Let F be an O-module, and denote Fab the underlying sheaf of abelian groups. Then we have H i (C, Fab ) = H i (C, F) and for any object U of C we also have H i (U, Fab ) = H i (U, F). Here the left hand side is cohomology computed in Ab(C) and the right hand side is cohomology computed in Mod(O). Proof. By Derived Categories, Lemma 11.19.4 the δ-funcor (F 7→ H p (U, F))p≥0 is universal. The functor Mod(O) → Ab(C), F 7→ Fab is exact. Hence (F 7→ H p (U, Fab ))p≥0 is a δ-functor also. Suppose we show that (F 7→ H p (U, Fab ))p≥0 is also universal. This will imply the second statement of the lemma by uniqueness of universal δ-functors, see Homology, Lemma 10.9.5. Since Mod(O) has enough injectives, it suffices to show that H i (U, Iab ) = 0 for any injective object I in Mod(O), see Homology, Lemma 10.9.4. Let I be an injective object of Mod(O). Apply Lemma 19.11.8 with F = I, B = C and Cov = CovC . Assumption (3) of that lemma holds by Lemma 19.12.3. Hence we see that H i (U, Iab ) = 0 for every object U of C. If C has a final object then this also implies the first equality. If not, then according to Sites, Lemma 9.25.5 we see that the ringed topos (Sh(C), O) is equivalent to a

19.13. LIMP SHEAVES

1203

ringed topos where the underlying site does have a final object. Hence the lemma follows.  Lemma 19.12.5. Cohomology and products. Let Fi be a family of abelian sheaves on a site C. Then there are canonical maps Y Y H p (U, Fi ) −→ H p (U, Fi ) i∈I

i∈I

for any object U of C. For p = 0 this map is an isomorphism and for p = 1 this map is injective. Q • Proof. Q • Choose injective resolutions Fi → Ii . Then F = Fi maps to the complex ( Ii ) which consists of injectives, see Homology, Lemma 10.20.3. Choose Q an injective resolution F → I • . Q There exists a map of complexes β : I • → ( Ii )• whichQinduces Q the identity on Fi , see Derived Categories, Lemma 11.17.6. Since Γ(U, Iip ) = Γ(U, Iip ) and since H p commutes with products (see Homology, Lemma 10.24.1) we obtain a canonical map Y Y Y H p (U, Fi ) = H p (Γ(U, I • )) −→ H p (Γ(U, ( Ii )• )) = H p (U, Fi ). To the assertion for H 1 , pick an element ξ ∈ H 1 (U, F) which maps to zero in Q prove 1 H (U, Fi ). By locality of cohomology, see Lemma 19.8.3, there exists a covering U = {Uj → U } such that ξ|Uj = 0 for all j. Hence ξ comes from an element of ˇ 1 (U, F) in the spectral sequence of Lemma 19.11.5. Since the edge maps ξˇ ∈ H 1 ˇ H Fi ) → H 1 (U, Fi ) are injective for all i, and since the image of ξ is zero in Q (U, 1 ˇ 1 (U, Fi ). However, since F = Q Fi H (U, Fi ) we see that the image ξˇi = 0 in H ˇ F) is the product of the complexes C(U, ˇ Fi ), hence by Homology, we see that C(U, Lemma 10.24.1 we conclude that ξˇ = 0 as desired.  19.13. Limp sheaves Let (C, O) be a ringed site. Let K be a sheaf of sets on C (we intentionally use a roman capital here to distinguish from abelian sheaves). Given an abelian sheaf F we denote F(K) = MorSh(C) (K, F). The functor F 7→ F(K) is a left exact functor Mod(O) → Ab hence we have its right derived functors. We will denote these H p (K, F) so that H 0 (K, F) = F(K). We mention two special cases. The first is the case where K = h# U for some object U of C. In this case H p (K, F) = H p (U, F), because MorSh(C) (h# U , F) = F(U ), see Sites, Section 9.12. The second is the case O = Z (the constant sheaf). In this case the cohomology groups are functors H p (K, −) : Ab(C) → Ab. Here is the analogue of Lemma 19.12.4. Lemma 19.13.1. Let (C, O) be a ringed site. Let K be a sheaf of sets on C. Let F be an O-module and denote Fab the underlying sheaf of abelian groups. Then H p (K, F) = H p (K, Fab ). Proof. Note that both H p (K, F) and H p (K, Fab ) depend only on the topos, not on the underlying site. Hence by Sites, Lemma 9.25.5 we may replace C by a “larger” site such that K = hU for some object U of C. In this case the result follows from Lemma 19.12.4. 

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Lemma 19.13.2. Let C be a site. Let K 0 → K be a surjective map of sheaves of sets on C. Set Kp0 = K 0 ×K . . . ×K K 0 (p + 1-factors). For every abelian sheaf F there is a spectral sequence with E1p,q = H q (Kp0 , F) converging to H p+q (K, F). Proof. After replacing C by a “larger” site as in Sites, Lemma 9.25.5 we may assume that K, K 0 are objects of C and that U = {K 0 → K} is a covering. Then we ˇ have the Cech to cohomology spectral sequence of Lemma 19.11.5 whose E1 page is as indicated in the statement of the lemma.  Lemma 19.13.3. Let C be a site. Let K be a sheaf of sets on C. Consider the morphism of topoi j : Sh(C/K) → Sh(C), see Sites, Lemma 9.26.3. Then j −1 preserves injectives and H p (K, F) = H p (C/K, j −1 F) for any abelian sheaf F on C. Proof. By Sites, Lemmas 9.26.1 and 9.26.3 the morphism of topoi j is equivalent to a localization. Hence this follows from Lemma 19.8.1.  Keeping in mind Lemma 19.13.1 we see that the following definition is the “correct one” also for sheaves of modules on ringed sites. Definition 19.13.4. Let C be a site. We say an abelian sheaf F is limp1 if for every sheaf of sets K we have H p (K, F) = 0 for all p ≥ 1. It is clear that being limp is an intrinsic property, i.e., preserved under equivalences of topoi. A limp sheaf has vanishing higher cohomology on all objects of the site, but in general the condition of being limp is strictly stronger. Here is a characterization of limp sheaves which is sometimes useful. Lemma 19.13.5. Let C be a site. Let F be an abelian sheaf. If (1) H p (U, F) = 0 for p > 0, and ˇ (2) for every surjection K 0 → K of sheaves of sets the extended Cech complex 0 → H 0 (K, F) → H 0 (K 0 , F) → H 0 (K 0 ×K K 0 , F) → . . . is exact, then F is limp (and the converse holds too). −1 Proof. By assumption (1) we` have H p (h# I) = 0 for all p > 0 and all objects U,g U of C. Note that if K = K is a coproduct of sheaves of sets on C then i Q H p (K, g −1 I) = H p (Ki , g −1 I). For any sheaf of sets K there exists a surjection a # K0 = hUi −→ K

see Sites, Lemma 9.12.4. Thus we conclude that: (*) for every sheaf of sets K there exists a surjection K 0 → K of sheaves of sets such that H p (K 0 , F) = 0 for p > 0. We claim that (*) and condition (2) imply that F is limp. Note that conditions (*) and (2) only depend on F as an object of the topos Sh(C) and not on the underlying site. (We will not use property (1) in the rest of the proof.) We are going to prove by induction on n ≥ 0 that (*) and (2) imply the following induction hypothesis IHn : H p (K, F) = 0 for all 0 < p ≤ n and all sheaves of sets K. Note that IH0 holds. Assume IHn . Pick a sheaf of sets K. Pick a surjection K 0 → K such that H p (K 0 , F) = 0 for all p > 0. We have a spectral sequence with E1p,q = H q (Kp0 , F) 1This is probably nonstandard notation. Please email [email protected] if you know the correct terminology.

19.14. THE LERAY SPECTRAL SEQUENCE

1205

convering to H p+q (K, F), see Lemma 19.13.2. By IHn we see that E1p,q = 0 for 0 < q ≤ n and by assumption (2) we see that E2p,0 = 0 for p > 0. Finally, we have E10,q = 0 for q > 0 because H q (K 0 , F) = 0 by choice of K 0 . Hence we conclude that  H n+1 (K, F) = 0 because all the terms E2p,q with p + q = n + 1 are zero. 19.14. The Leray spectral sequence The key to proving the existence of the Leray spectral sequence is the following lemma. Lemma 19.14.1. Let f : (Sh(C), OC ) → (Sh(D), OD ) be a morphism of ringed topoi. Then for any injective object I in Mod(OC ) the pushforward f∗ I is limp. Proof. Let K be a sheaf of sets on D. By Modules on Sites, Lemma 16.7.2 we may replace C, D by “larger” sites such that f comes from a morphism of ringed sites induced by a continuous functor u : D → C such that K = hV for some object V of D. Thus we have to show that H q (V, f∗ I) is zero for q > 0 and all objects V of D when f is given by a morphism of ringed sites. Let V = {Vj → V } be any covering of D. Since u is continuous we see that U = {u(Vj ) → u(v)} is a covering of C. Then we ˇ have an equality of Cech complexes Cˇ• (V, f∗ I) = Cˇ• (U, I) by the definition of f∗ . By Lemma 19.12.3 we see that the cohomology of this complex is zero in positive degrees. We win by Lemma 19.11.8.  For flat morphisms the functor f∗ preserves injective modules. In particular the functor f∗ : Ab(C) → Ab(D) always transforms injective abelian sheaves into injective abelian sheaves. Lemma 19.14.2. Let f : (Sh(C), OC ) → (Sh(D), OD ) be a morphism of ringed topoi. If f is flat, then f∗ I is an injective OD -module for any injective OC -module I. Proof. In this case the functor f ∗ is exact, see Modules on Sites, Lemma 16.27.2. Hence the result follows from Homology, Lemma 10.22.1.  Lemma 19.14.3. Let (Sh(C), OC ) be a ringed topos. A limp sheaf is right acyclic for the following functors: (1) the functor H 0 (U, −) for any object U of C, (2) the functor F 7→ F(K) for any presheaf of sets K, (3) the functor Γ(C, −) of global sections, (4) the functor f∗ for any morphism f : (Sh(C), OC ) → (Sh(D), OD ) of ringed topoi. Proof. Part (2) is the definition of a limp sheaf. Part (1) is a consequence of (2) as pointed out in the discussion following the definition of limp sheaves. Part (3) is a special case of (2) where K = e is the final object of Sh(C). To prove (4) we may assume, by Modules on Sites, Lemma 16.7.2 that f is given by a morphism of sites. In this case we see that Ri f∗ , i > 0 of a limp sheaf are zero by the description of higher direct images in Lemma 19.8.4. 

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19. COHOMOLOGY ON SITES

Lemma 19.14.4 (Leray spectral sequence). Let f : (Sh(C), OC ) → (Sh(D), OD ) be a morphism of ringed topoi. Let F • be a bounded below complex of OC -modules. There is a spectral sequence E2p,q = H p (D, Rq f∗ (F • )) converging to H p+q (C, F • ). Proof. This is just the Grothendieck spectral sequence Derived Categories, Lemma 11.21.2 coming from the composition of functors Γ(C, −) = Γ(D, −)◦f∗ . To see that the assumptions of Derived Categories, Lemma 11.21.2 are satisfied, see Lemmas 19.14.1 and 19.14.3.  Lemma 19.14.5. Let f : (Sh(C), OC ) → (Sh(D), OD ) be a morphism of ringed topoi. Let F be an OC -module. (1) If Rq f∗ F = 0 for q > 0, then H p (C, F) = H p (D, f∗ F) for all p. (2) If H p (D, Rq f∗ F) = 0 for all q and p > 0, then H q (C, F) = H 0 (D, Rq f∗ F) for all q. Proof. These are two simple conditions that force the Leray spectral sequence to converge. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves.  Lemma 19.14.6 (Relative Leray spectral sequence). Let f : (Sh(C), OC ) → (Sh(D), OD ) and g : (Sh(D), OD ) → (Sh(E), OE ) be morphisms of ringed topoi. Let F be an OC module. There is a spectral sequence with E2p,q = Rp g∗ (Rq f∗ F) converging to Rp+q (g ◦ f )∗ F. This spectral sequence is functorial in F, and there is a version for bounded below complexes of OC -modules. Proof. This is a Grothendieck spectral sequence for composition of functors, see Derived Categories, Lemma 11.21.2 and Lemmas 19.14.1 and 19.14.3.  19.15. The base change map In this section we construct the base change map in some cases; the general case is treated in Remark 19.19.2. The discussion in this section avoids using derived pullback by restricting to the case of a base change by a flat morphism of ringed sites. Before we state the result, let us discuss flat pullback on the derived category. Suppose g : (Sh(C), OC ) → (Sh(D), OD ) is a flat morphism of ringed topoi. By Modules on Sites, Lemma 16.27.2 the functor g ∗ : Mod(OD ) → Mod(OC ) is exact. Hence it has a derived functor g ∗ : D(OC ) → D(OD ) which is computed by simply pulling back an representative of a given object in D(OD ), see Derived Categories, Lemma 11.16.8. It preserved the bounded (above, below) subcategories. Hence as indicated we indicate this functor by g ∗ rather than Lg ∗ .

19.16. COHOMOLOGY AND COLIMITS

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Lemma 19.15.1. Let (Sh(C 0 ), OC 0 )

g0

f0

 (Sh(D0 ), OD0 )

/ (Sh(C), OC ) f

g

 / (Sh(D), OD )

be a commutative diagram of ringed topoi. Let F • be a bounded below complex of OC -modules. Assume both g and g 0 are flat. Then there exists a canonical base change map g ∗ Rf∗ F • −→ R(f 0 )∗ (g 0 )∗ F • in D+ (OD0 ). Proof. Choose injective resolutions F • → I • and (g 0 )∗ F • → J • . By Lemma 19.14.2 we see that (g 0 )∗ J • is a complex of injectives representing R(g 0 )∗ (g 0 )∗ F • . Hence by Derived Categories, Lemmas 11.17.6 and 11.17.7 the arrow β in the diagram / (g 0 )∗ J • (g 0 )∗ (g 0 )∗ F • O O adjunction

β

/ I• F• exists and is unique up to homotopy. Pushing down to D we get f∗ β : f∗ I • −→ f∗ (g 0 )∗ J • = g∗ (f 0 )∗ J • By adjunction of g ∗ and g∗ we get a map of complexes g ∗ f∗ I • → (f 0 )∗ J • . Note that this map is unique up to homotopy since the only choice in the whole process was the choice of the map β and everything was done on the level of complexes.  19.16. Cohomology and colimits Let (C, O) be a ringed site. Let I → Mod(O), i 7→ Fi be a diagram over the index category I, see Categories, Section 4.13. For each i there is a canonical map Fi → colimi Fi which induces a map on cohomology. Hence we get a canonical map colimi H p (U, Fi ) −→ H p (U, colimi Fi ) for every p ≥ 0 and every object U of C. These maps are in general not isomorphisms, even for p = 0. To repeat the arguments given in the case of topological spaces we will say that an object U of a site C is quasi-compact if every covering of U in C can be refined by a finite covering. Lemma 19.16.1. Let C be a site. Let I → Sh(C), i 7→ Fi be a filtered diagram of sheaves of sets. Let U ∈ Ob(C). Consider the canonical map Ψ : colimi Fi (U ) −→ (colimi Fi ) (U ) With the terminology introduced above: (1) If all the transition maps are injective then Ψ is injective for any U . (2) If U is quasi-compact, then Ψ is injective. (3) If U is quasi-compact and all the transition maps are injective then Ψ is an isomorphism.

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19. COHOMOLOGY ON SITES

(4) If U has a cofinal system of coverings {Uj → U }j∈J with J finite and Uj ×U Uj 0 quasi-compact for all j, j 0 ∈ J, then Ψ is bijective. Proof. Assume all the transition maps are injective. In this case the presheaf F 0 : V 7→ colimi Fi (V ) is separated (see Sites, Definition 9.10.9). By Sites, Lemma 9.10.13 we have (F 0 )# = colimi Fi . By Sites, Theorem 9.10.10 we see that F 0 → (F 0 )# is injective. This proves (1). Assume U is quasi-compact. Suppose that s ∈ Fi (U ) and s0 ∈ Fi0 (U ) give rise to elements on the left hand side which have the same image under Ψ. Since U is quasi-compact this means there exists a finite covering {Uj → U }j=1,...,m and for each j an index ij ∈ I, ij ≥ i, ij ≥ i0 such that ϕiij (s) = ϕi0 ij (s0 ). Let i00 ∈ I be ≥ than all of the ij . We conclude that ϕii00 (s) and ϕi0 i00 (s) agree on Uj for all j and hence that ϕii00 (s) = ϕi0 i00 (s). This proves (2). Assume U is quasi-compact and all transition maps injective. Let s be an element of the target of Ψ. Since U is quasi-compact there exists a finite covering {Uj → U }j=1,...,m , for each j an index ij ∈ I and sj ∈ Fij (Uj ) such that s|Uj comes from sj for all j. Pick i ∈ I which is ≥ than all of the ij . By (1) the sections ϕij i (sj ) agree over Uj ×U Uj 0 . Hence they glue to a section s0 ∈ Fi (U ) which maps to s under Ψ. This proves (3). Assume the hypothesis of (4). Let s be an element of the target of Ψ. By assumption there exists a finite covering {Uj → U }j=1,...,m Uj , with Uj ×U Uj 0 quasi-compact for all j, j 0 ∈ J and for each j an index ij ∈ I and sj ∈ Fij (Uj ) such that s|Uj is the image of sj for all j. Since Uj ×U Uj 0 is quasi-compact we can apply (2) and we see that there exists an ijj 0 ∈ I, ijj 0 ≥ ij , ijj 0 ≥ ij 0 such that ϕij ijj0 (sj ) and ϕij0 ijj0 (sj 0 ) agree over Uj ×U Uj 0 . Choose an index i ∈ I wich is bigger or equal than all the ijj 0 . Then we see that the sections ϕij i (sj ) of Fi glue to a section of Fi over U . This section is mapped to the element s as desired.  The following lemma is the analogue of the previous lemma for cohomology. Lemma 19.16.2. Let C be a site. Let CovC be the set of coverings of C (see Sites, Definition 9.6.2). Let B ⊂ Ob(C), and Cov ⊂ CovC be subsets. Assume that (1) For every U ∈ Cov we have U = {Ui → U }i∈I with I finite, U, Ui ∈ B and every Ui0 ×U . . . ×U Uip ∈ B. (2) For every U ∈ B the coverings of U occuring in Cov is a cofinal system of coverings of U . Then the map colimi H p (U, Fi ) −→ H p (U, colimi Fi ) is an isomorphism for every p ≥ 0, every U ∈ B, and every filtered diagram I → Ab(C). Proof. To prove the lemma we will argue by induction on p. Note that we require in (1) the coverings U ∈ Cov to be finite, so that all the elements of B are quasicompact. Hence (2) and (1) imply that any U ∈ B satsifies the hypothesis of Lemma 19.16.1 (4). Thus we see that the result holds for p = 0. Now we assume the lemma holds for p and prove it for p + 1. Choose a filtered diagram F : I → Ab(C), i 7→ Fi . Since Ab(C) has functorial injective embeddings, see Injectives, Theorem 17.11.4, we can find a morphism of

19.17. FLAT RESOLUTIONS

1209

filtered diagrams F → I such that each Fi → Ii is an injective map of abelian sheaves into an injective abelian sheaf. Denote Qi the cokernel so that we have short exact sequences 0 → Fi → Ii → Qi → 0. Since colimits of sheaves are the sheafification of colimits on the level of preshease, since sheafification is exact, and since filtered colimits of abelian groups are exact (see Algebra, Lemma 7.8.9), we see the sequence 0 → colimi Fi → colimi Ii → colimi Qi → 0. is also a short exact sequence. We claim that H q (U, colimi Ii ) = 0 for all U ∈ B and all q ≥ 1. Accepting this claim for the moment consider the diagram colimi H p (U, Ii )

/ colimi H p (U, Qi )

/ colimi H p+1 (U, Fi )

/0

 H p (U, colimi Ii )

 / H p (U, colimi Qi )

 / H p+1 (U, colimi Fi )

 /0

The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves Ii are injective. The top row is exact by an application of Algebra, Lemma 7.8.9. Hence by the snake lemma we deduce the result for p + 1. It remains to show that the claim is true. We will use Lemma 19.11.8. By the result for p = 0 we see that for U ∈ Cov we have Cˇ• (U, colimi Ii ) = colimi Cˇ• (U, Ii ) because all the Uj0 ×U . . .×U Ujp are in B. By Lemma 19.11.2 each of the complexes in the colimit of Cech complexes is acyclic in degree ≥ 1. Hence by Algebra, Lemma 7.8.9 we see that also the Cech complex Cˇ• (U, colimi Ii ) is acyclic in degrees ≥ 1. ˇ p (U, colimi Ii ) = 0 for all p ≥ 1. Thus the assumptions In other words we see that H of Lemma 19.11.8. are satisfied and the claim follows.  19.17. Flat resolutions In this section we redo the arguments of Cohomology, Section 18.20 in the setting of ringed sites and ringed topoi. Lemma 19.17.1. Let (C, O) be a ringed site. Let G • be a complex of O-modules. The functor K(Mod(O)) −→ K(Mod(O)),

F • 7−→ Tot(F • ⊗O G • )

is an exact functor of triangulated categories. Proof. Omitted. Hint: See More on Algebra, Lemmas 12.5.1 and 12.5.2.



Definition 19.17.2. Let (C, O) be a ringed site. A complex K• of O-modules is called K-flat if for every acyclic complex F • of O-modules the complex Tot(F • ⊗O K• ) is acyclic.

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19. COHOMOLOGY ON SITES

Lemma 19.17.3. Let (C, O) be a ringed site. Let K• be a K-flat complex. Then the functor K(Mod(O)) −→ K(Mod(O)),

F • 7−→ Tot(F • ⊗O K• )

transforms quasi-isomorphisms into quasi-isomorphisms. Proof. Follows from Lemma 19.17.1 and the fact that quasi-isomorphisms are characterized by having acyclic cones.  Lemma 19.17.4. Let (C, O) be a ringed site. If K• , L• are K-flat complexes of O-modules, then Tot(K• ⊗O L• ) is a K-flat complex of O-modules. Proof. Follows from the isomorphism Tot(M• ⊗O Tot(K• ⊗O L• )) = Tot(Tot(M• ⊗O K• ) ⊗O L• ) and the definition.



Lemma 19.17.5. Let (C, O) be a ringed site. Let (K1• , K2• , K3• ) be a distinguished triangle in K(Mod(O)). If two out of three of Ki• are K-flat, so is the third. Proof. Follows from Lemma 19.17.1 and the fact that in a distinguished triangle in K(Mod(O)) if two out of three are acyclic, so is the third.  Lemma 19.17.6. Let (C, O) be a ringed space. A bounded above complex of flat O-modules is K-flat. Proof. Let K• be a bounded above complex of flat O-modules. Let L• be an acyclic complex of O-modules. Note that L• = colimm τ≤m L• where we take termwise colimits. Hence also Tot(K• ⊗O L• ) = colimm Tot(K• ⊗O τ≤m L• ) termwise. Hence to prove the complex on the left is acyclic it suffices to show each of the complexes on the right is acyclic. Since τ≤m L• is acyclic this reduces us to the case where L• is bounded above. In this case the spectral sequence of Homology, Lemma 10.19.5 has 0

E1p,q = H p (L• ⊗R Kq )

which is zero as Kq is flat and L• acyclic. Hence we win. Lemma 19.17.7. Let (C, O) be a ringed site. Let K-flat complexes. Then colimi Ki• is K-flat.

K1•

→

 K2•

→ . . . be a system of

Proof. Because we are taking termwise colimits it is clear that colimi Tot(F • ⊗O Ki• ) = Tot(F • ⊗O colimi Ki• ) Hence the lemma follows from the fact that filtered colimits are exact.



Lemma 19.17.8. Let (C, O) be a ringed site. For any complex G • of O-modules there exists a commutative diagram of complexes of O-modules K1•

/ K2•

/ ...

 τ≤1 G •

 / τ≤2 G •

/ ...

19.17. FLAT RESOLUTIONS

1211

with the following properties: (1) the vertical arrows are quasi-isomorphisms, (2) each Kn• is a bounded above complex whose terms are direct sums of O-modules of • the form jU ! OU , and (3) the maps Kn• → Kn+1 are termwise split injections whose cokernels are direct sums of O-modules of the form jU ! OU . Moreover, the map colim Kn• → G • is a quasi-isomorphism. Proof. The existence of the diagram and properties (1), (2), (3) follows immediately from Modules on Sites, Lemma 16.26.6 and Derived Categories, Lemma 11.27.1. The induced map colim Kn• → G • is a quasi-isomorphism because filtered colimits are exact.  Lemma 19.17.9. Let (C, O) be a ringed site. For any complex G • of O-modules there exists a K-flat complex K• and a quasi-isomorphism K• → G • . Proof. Choose a diagram as in Lemma 19.17.8. Each complex Kn• is a bounded above complex of flat modules, see Modules on Sites, Lemma 16.26.5. Hence Kn• is K-flat by Lemma 19.17.6. The induced map colim Kn• → G • is a quasi-isomorphism  by construction. Since colim Kn• is K-flat by Lemma 19.17.7 we win. Lemma 19.17.10. Let (C, O) be a ringed site. Let α : P • → Q• be a quasiisomorphism of K-flat complexes of O-modules. For every complex F • of O-modules the induced map Tot(idF • ⊗ α) : Tot(F • ⊗O P • ) −→ Tot(F • ⊗O Q• ) is a quasi-isomorphism. Proof. Choose a quasi-isomorphism K• → F • with K• a K-flat complex, see Lemma 19.17.9. Consider the commutative diagram Tot(K• ⊗O P • )

/ Tot(K• ⊗O Q• )

 Tot(F • ⊗O P • )

 / Tot(F • ⊗O Q• )

The result follows as by Lemma 19.17.3 the vertical arrows and the top horizontal arrow are quasi-isomorphisms.  Let (C, O) be a ringed site. Let F • be an object of D(O). Choose a K-flat resolution K• → F • , see Lemma 19.17.9. By Lemma 19.17.1 we obtain an exact functor of triangulated categories K(O) −→ K(O),

G • 7−→ Tot(G • ⊗O K• )

By Lemma 19.17.3 this functor induces a functor D(O) → D(O) simply because D(O) is the localization of K(O) at quasi-isomorphisms. By Lemma 19.17.10 the resulting functor (up to isomorphism) does not depend on the choice of the K-flat resolution. Definition 19.17.11. Let (C, O) be a ringed site. Let F • be an object of D(O). The derived tensor product • − ⊗L O F : D(O) −→ D(O)

is the exact functor of triangulated categories described above.

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19. COHOMOLOGY ON SITES

It is clear from our explicit constructions that there is a canonical isomorphism ∼ G • ⊗L F • F • ⊗L G • = O

•

O

•

• for G and F in D(O). Hence when we write F • ⊗L O G we will usually be agnostic about which variable we are using to define the derived tensor product with.

19.18. Derived pullback Let f : (Sh(C), O) → (Sh(C 0 ), O0 ) be a morphism of ringed topoi. We can use K-flat resolutions to define a derived pullback functor Lf ∗ : D(O0 ) → D(O) However, we have to be a little careful since we haven’t yet proved the pullback of a flat module is flat in complete generality, see Modules on Sites, Section 16.33. In this section, we will use the hypothesis that our sites have enough points, but once we improve the result of the aforementioned section, all of the results in this section will hold without the assumption on the existence of points. Lemma 19.18.1. Let f : Sh(C) → Sh(C 0 ) be a morphism of topoi. Let O0 be a sheaf of rings on C 0 . Assume C has enough points. For any complex of O0 -modules G • , there exists a quasi-isomorphism K• → G • such that K• is a K-flat complex of O0 -modules and f −1 K• is a K-flat complex of f −1 O0 -modules. Proof. In the proof of Lemma 19.17.9 we find a quasi-isomorphism K• = colimi Ki• → G • where each Ki• is a bounded above complex of flat O0 -modules. By Modules on Sites, Lemma 16.33.3 applied to the morphism of ringed topoi (Sh(C), f −1 O0 ) → (Sh(C 0 ), O0 ) we see that f −1 Fi• is a bounded above complex of flat f −1 O0 -modules. Hence f −1 K• = colimi f −1 Ki• is K-flat by Lemmas 19.17.6 and 19.17.7.  Remark 19.18.2. It is straightforward to show that the pullback of a K-flat complex is K-flat for a morphism of ringed topoi with enough points; this slightly improves the result of Lemma 19.18.1. However, in applications it seems rather that the explicit form of the K-flat complexes constructed in Lemma 19.17.9 is what is useful (as in the proof above) and not the plain fact that they are K-flat. Note for example that the terms of the complex constructed are each direct sums of modules of the form jU ! OU , see Lemma 19.17.8. Lemma 19.18.3. Let f : (Sh(C), O) → (Sh(C 0 ), O0 ) be a morphism of ringed topoi. Assume C has enough points. There exists an exact functor Lf ∗ : D(O0 ) −→ D(O) of triangulated categories so that Lf ∗ K• = f ∗ K• for any complex as in Lemma 19.18.1 in particular for any bounded above complex of flat O0 -modules. Proof. To see this we use the general theory developed in Derived Categories, Section 11.14. Set D = K(O0 ) and D0 = D(O). Let us write F : D → D0 the exact functor of triangulated categories defined by the rule F (G • ) = f ∗ G • . We let S be the set of quasi-isomorphisms in D = K(O0 ). This gives a situation as in Derived Categories, Situation 11.14.1 so that Derived Categories, Definition 11.14.2 applies. We claim that LF is everywhere defined. This follows from Derived Categories, Lemma 11.14.15 with P ⊂ Ob(D) the collection of complexes K• such that f −1 K• is a K-flat complex of f −1 O0 -modules: (1) follows from Lemma 19.18.1 and to see

19.19. COHOMOLOGY OF UNBOUNDED COMPLEXES

1213

(2) we have to show that for a quasi-isomorphism K1• → K2• between elements of P the map f ∗ K1• → f ∗ K2• is a quasi-isomorphism. To see this write this as f −1 K1• ⊗f −1 O0 O −→ f −1 K2• ⊗f −1 O0 O The functor f −1 is exact, hence the map f −1 K1• → f −1 K2• is a quasi-isomorphism. The complexes f −1 K1• and f −1 K2• are K-flat complexes of f −1 O0 -modules by our choice of P. Hence Lemma 19.17.10 guarantees that the displayed map is a quasiisomorphism. Thus we obtain a derived functor LF : D(O0 ) = S −1 D −→ D0 = D(O) see Derived Categories, Equation (11.14.9.1). Finally, Derived Categories, Lemma 11.14.15 also guarantees that LF (K• ) = F (K• ) = f ∗ K• when K• is in P. Since the proof of Lemma 19.18.1 shows that bounded above complexes of flat modules are in P we win.  Lemma 19.18.4. Let f : (Sh(C), O) → (Sh(D), O0 ) be a morphism of ringed topoi. Assume C has enough points. There is a canonical bifunctorial isomorphism • ∗ • L ∗ • Lf ∗ (F • ⊗L O 0 G ) = Lf F ⊗O Lf G

for F • , G • ∈ Ob(D(O0 )). Proof. We may assume that F • and G • are K-flat complexes of O0 -modules. In this • • • case F • ⊗L O 0 G is just the total complex associated to the double complex F ⊗O 0 G . • • By Lemma 19.17.4 Tot(F ⊗O0 G ) is K-flat also. Hence the isomorphism of the lemma comes from the isomorphism Tot(f ∗ F • ⊗O f ∗ G • ) −→ f ∗ Tot(F • ⊗O0 G • ) whose constituents are the isomorphisms f ∗ F p ⊗O f ∗ G q → f ∗ (F p ⊗O0 G q ) of Modules on Sites, Lemma 16.24.1.  19.19. Cohomology of unbounded complexes Let (C, O) be a ringed site. The category Mod(O) is a Grothendieck abelian category: it has all colimits, filtered colimits are exact, and it has a generator, namely M jU ! OU , U ∈Ob(C)

see Modules on Sites, Section 16.14 and Lemmas 16.26.5 and 16.26.6. By Injectives, Theorem 17.16.6 for every complex F • of O-modules there exists an injective quasiisomorphism F • → I • to a K-injective complex of O-modules. Hence we can define RΓ(C, F • ) = Γ(C, I • ) and similarly for any left exact functor, see Derived Categories, Lemma 11.28.5. For any morphism of ringed topoi f : (Sh(C), O) → (Sh(D), O0 ) we obtain Rf∗ : D(O) −→ D(O0 ) on the unbounded derived categories. Lemma 19.19.1. Let f : (Sh(C), O) → (Sh(D), O0 ) be a morphism of ringed topoi. Assume C has enough points. The functor Rf∗ defined above and the functor Lf ∗ defined in Lemma 19.18.3 are adjoint: HomD(O) (Lf ∗ G • , F • ) = HomD(O0 ) (G • , Rf∗ F • ) bifunctorially in F • ∈ Ob(D(O)) and G • ∈ Ob(D(O0 )).

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19. COHOMOLOGY ON SITES

Proof. This is formal from the results obtained above. Choose a K-flat resolution K• → G • and a K-injective resolution F • → I • . Then HomD(O) (Lf ∗ G • , F • ) = HomD(O) (f ∗ K• , I • ) = HomK(Mod(O)) (f ∗ K• , I • ) by our definition of Lf ∗ and because I • is K-injective, see Derived Categories, Lemma 11.28.2. On the other hand HomD(O0 ) (G • , Rf∗ F • ) = HomD(O0 ) (K• , f∗ I • ) by our definition of Rf∗ . By definition of morphisms in D(O0 ) this is equal to colims:H• →K• HomK(Mod(O0 )) (H• , f∗ I • ) where the colimit is over all quasi-isomorphisms s : H• → K• of complexes of O0 -modules. Since every complex has a left K-flat resolution it suffices to look at quasi-isomorphisms s : (K0 )• → K• where (K0 )• is K-flat as well. In this case we have HomK(Mod(O0 )) ((K0 )• , f∗ I • ) = HomK(Mod(O0 )) (f ∗ (K0 )• , I • ) = HomK(Mod(O0 )) (f ∗ K• , I • ) The first equality because f ∗ and f∗ are adjoint functors and the second because I • is K-injective and because f ∗ (K0 )• → f ∗ K• is a quasi-isomorphism (by virtue of the fact that Lf ∗ is well defined).  Remark 19.19.2. The construction of unbounded derived functor Lf ∗ and Rf∗ allows one to construct the base change map in full generality. Namely, suppose that / (Sh(C), OC ) (Sh(C 0 ), OC 0 ) 0 g

f0

 (Sh(D0 ), OD0 )

f

g

 / (Sh(D), OD )

is a commutative diagram of ringed topoi. Let F • be a complex of OC -modules. Then there exists a canonical base change map Lg ∗ Rf∗ F • −→ R(f 0 )∗ L(g 0 )∗ F • in D(OD0 ). Namely, this map is adjoint to a map L(f 0 )∗ Lg ∗ Rf∗ F • → L(g 0 )∗ F • Since L(f 0 )∗ Lg ∗ = L(g 0 )∗ Lf ∗ we see this is the same as a map L(g 0 )∗ Lf ∗ Rf∗ F • → L(g 0 )∗ F • which we can take to be L(g 0 )∗ of the adjunction map Lf ∗ Rf∗ F • → F • . 19.20. Producing K-injective resolutions Let (C, O) be a ringed site. Let F • be a complex of O-modules. The category Mod(O) has enough injectives, hence we can use Derived Categories, Lemma 11.27.3 produce a diagram / τ≥−2 F • / τ≥−1 F • ...

...

 / I2•

 / I1•

in the category of complexes of O-modules such that (1) the vertical arrows are quasi-isomorphisms, (2) In• is a bounded below complex of injectives,

19.20. PRODUCING K-INJECTIVE RESOLUTIONS

1215

• (3) the arrows In+1 → In• are termwise split surjections. The category of O-modules has limits (they are computed on the level of presheaves), hence we can form the termwise limit I • = limn In• . By Derived Categories, Lemmas 11.28.3 and 11.28.6 this is a K-injective complex. In general the canonical map

(19.20.0.1)

F • → I•

may not be a quasi-isomorphism. In the following lemma we describe some conditions under which it is. Lemma 19.20.1. In the situation described above. Denote Hi = H i (F • ) the ith cohomology sheaf. Let B ⊂ Ob(C) be a subset. Let d ∈ N. Assume (1) every object of C has a covering whose members are elements of B, (2) for every U ∈ B we have H p (U, Hq ) = 0 for p > d2. Then (19.20.0.1) is a quasi-isomorphism. Proof. Let U ∈ B. Note that H m (I • (U )) is the cohomology of limn Inm−2 (U ) → limn Inm−1 (U ) → limn Inm (U ) → limn Inm+1 (U ) m in the third spot from the left. Note that the transition maps In+1 (U ) → Inm (U ) • • are always surjective because the maps In+1 → In are termwise split surjections. By construction there are distingushed triangles • H−n [n] → In• → In−1 → H−n [n + 1]

in D(O). As In• is a bounded below complex of injectives we have H m (U, In• ) = H m (In• (U )). By assumption (2) we see that if m > d − n then H m (U, H−n [n]) = H n+m (U, H−n ) = 0 • (U )) and similarly H m (U, H−n [n+1]) = 0. This implies that H m (In• (U )) → H m (In−1 is an isomorphism for m > d−n. In other words, the cohomologies of the complexes In• (U ) are eventually constant in every cohomological degree. Thus we may apply Homology, Lemma 10.23.7 to conclude that

H m (I • (U )) = lim H m (In• (U )). Using the eventual stabilization once again we see that • H m (I • (U )) = H m (Imax(1,−m+d) (U ))

for every U ∈ B. We want to show that the map Hm → H m (I • ) is an isomorphism for all m. The sheaf H m (I • ) is the sheafification of the presheaf U 7→ H m (I • (U )). We have seen above that this presheaf equals the presheaf • U 7−→ H m (Imax(1,−m+d) (U ))

when U runs through the elements of B. Since every object of C has a covering whose members are elements of B we see that it suffices to prove the sheafification • of U 7−→ H m (Imax(1,−m+d) (U )) is Hm . On the other hand, this sheafification is m • • equal to H (Imax(1,−m+d) ). Since τ≥− max(1,−m+d) F • → Imax(1,−m+d) is a quasiisomorphism we win.  2In fact, analyzing the proof we see that it suffices if there exists a function d : Z → Z∪{+∞} such that H p (U, Hq ) = 0 for p > d(q) where q + d(q) → −∞ as q → −∞

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19. COHOMOLOGY ON SITES

The construction above can be used in the following setting. Let C be a category. Let Cov(C) ⊃ Cov0 (C) be two ways to endow C with the structure of a site. Denote τ the topology corresponding to Cov(C) and τ 0 the topology corresponding to Cov0 (C). Then the identity functor on C defines a morphism of sites  : Cτ −→ Cτ 0 where ∗ is the identity functor on underlying presheaves and where −1 is the τ sheafification of a τ 0 -sheaf (hence clearly exact). Let O be a sheaf of rings for the τ -topology. Then O is also a sheaf for the τ 0 -topology and  becomes a morphism of ringed sites  : (Cτ , Oτ ) −→ (Cτ 0 , Oτ 0 ) In this situation we can sometimes point out subcategories of D(Oτ ) and D(Oτ 0 ) which are identified by the functors ∗ and R∗ . Lemma 19.20.2. With  : (Cτ , Oτ ) −→ (Cτ 0 , Oτ 0 ) as above. Let B ⊂ Ob(C) be a subset. Let A ⊂ PMod(O) be a full subcategory. Assume (1) every object of A is a sheaf for the τ -topology, (2) A is a weak Serre subcategory of Mod(Oτ ), (3) every object of C has a τ 0 -covering whose members are elements of B, and (4) for every U ∈ B we have Hτp (U, F) = 0, p > 0 for all F ∈ A. Then A is a weak Serre subcategory of Mod(Oτ ) and there is an equivalence of triangulated categories DA (Oτ ) = DA (Oτ 0 ) given by ∗ and R∗ . Proof. Note that for A ∈ A we can think of A as a sheaf in either topology and (abusing notation) that ∗ A = A and ∗ A = A. Consider an exact sequence A0 → A1 → A2 → A3 → A4 in Mod(Oτ 0 ) with A0 , A1 , A3 , A4 in A. We have to show that A2 is an element of A, see Homology, Definition 10.7.1. Apply the exact functor ∗ = −1 to conclude that ∗ A2 is an object of A. Consider the map of sequences / A2 / A3 / A4 / A1 A0  A0

 / A1

 /  ∗  ∗ A2

 / A3

 / A4

to conclude that A2 = ∗ ∗ A2 is an object of A. At this point it makes sense to talk about the derived categories DA (Oτ ) and DA (Oτ 0 ), see Derived Categories, Section 11.12. Since ∗ is exact and preserves A, it is clear that we obtain a functor ∗ : DA (Oτ 0 ) → DA (Oτ ). We claim that R∗ is a quasi-inverse. Namely, let F • be an object of DA (Oτ ). Construct a map F • → I • = lim In• as in (19.20.0.1). By Lemma 19.20.1 and assumption (4) we see that F • → I • is a quasi-isomorphism. Then R∗ F • = ∗ I • = limn ∗ In• For every U ∈ B we have H m (∗ In• (U )) = H m (In• (U )) =



H m (F • )(U ) 0

if m ≥ −n if m < n

by the assumed vanishing of (4), the spectral sequence Derived Categories, Lemma 11.20.3, and the fact that τ≥−n F • → In• is a quasi-isomorphism. The maps

19.22. DERIVED LOWER SHRIEK

1217

• ∗ In+1 → ∗ In• are termwise split surjections as ∗ is a functor. Hence we can apply Homology, Lemma 10.23.7 to the sequence of complexes

limn ∗ Inm−2 (U ) → limn ∗ Inm−1 (U ) → limn ∗ Inm (U ) → limn ∗ Inm+1 (U ) to conclude that H m (∗ I • (U )) = H m (F • )(U ) for U ∈ B. Sheafifying and using property (3) this proves that H m (∗ I • ) is isomorphic to ∗ H m (F • ), i.e., is an object of A. Thus R∗ indeed gives rise to a functor R∗ : DA (Oτ ) −→ DA (Oτ 0 ) •

For F ∈ DA (Oτ ) the adjunction map ∗ R∗ F • → F • is a quasi-isomorphism as we’ve seen above that the cohomology sheaves of R∗ F • are ∗ H m (F • ). For G • ∈ DA (Oτ 0 ) the adjunction map G • → R∗ ∗ G • is a quasi-isomorphism for the same reason, i.e., because the cohomology sheaves of R∗ ∗ G • are isomorphic to ∗ H m (∗ G) = H m (G • ).  19.21. Spectral sequences for Ext In this section we collect various spectral sequences that come up when considering the Ext functors. For any pair of complexes G • , F • of complexes of modules on a ringed site (C, O) we denote ExtnO (G • , F • ) = HomD(O) (G • , F • [n]) according to our general conventions in Derived Categories, Section 11.26. Example 19.21.1. Let (C, O) be a ringed site. Let K• be a bounded above complex of O-modules. Let F be an O-module. Then there is a spectral sequence with E2 page • E2i,j = ExtiO (H −j (K• ), F) ⇒ Exti+j O (K , F) and another spectral sequence with E1 -page • E1i,j = ExtjO (K−i , F) ⇒ Exti+j O (K , F).

To construct these spectral sequences choose an injective resolution F → I • and consider the two spectral sequences coming from the double complex HomO (K• , I • ), see Homology, Section 10.19. 19.22. Derived lower shriek In this section we study some situations where besides Lf ∗ and Rf∗ there also a derived functor Lf! . Lemma 19.22.1. Let u : C → D be a continuous and cocontinuous functor of sites which induces a morphism of topoi g : Sh(C) → Sh(D). Let OD be a sheaf of rings and set OC = g −1 OD . The functor g! : Mod(OC ) → Mod(OD ) (see Modules on Sites, Lemma 16.35.1) has a left derived functor Lg! : D(OC ) −→ D(OD ) which is left adjoint to g ∗ . Proof. We are going to use Derived Categories, Proposition 11.27.2 to construct Lg! . To do this we have to verify assumptions (1), (2), (3), (4), and (5) of that proposition. First, since g! is a left adjoint we see that it is right exact and commutes with all colimits, so (5) holds. Conditions (3) and (4) hold because the category of modules on a ringed site is a Grothendieck abelian category. Let P ⊂ Ob(Mod(OC ))

1218

19. COHOMOLOGY ON SITES

be the collection of OC -modules which are direct sums of modules of the form jU ! OU . Here U ∈ Ob(C) and jU ! is the extension by zero associated to the localization morphism jU : C/U → C. Every OC -module is a quotient of an object of P, see Modules on Sites, Lemma 16.26.6. Thus (1) holds. Finally, we have to prove (2). Let K• be a bounded above acyclic complex of OC -modules with Kn ∈ P for all n. We have to show that g! K• is exact. To do this it suffices to show, for every injective OD -module I that HomD(OD ) (g! K• , I[n]) = 0 for all n ∈ Z. Since I is injective we have HomD(OD ) (g! K• , I[n]) = HomK(OD ) (g! K• , I[n]) = H n (HomOD (g! K• , I)) = H n (HomOC (K• , g −1 I)) the last equality by the adjointness of g! and g −1 . The vanishing of this group would be clear if g −1 I were an injective OC -module. But g −1 I isn’t necessarily an injective OC -module as g! isn’t exact in general. We do know that ExtpOC (jU ! OU , g −1 I) = H p (U, g −1 I) = 0 for p ≥ 1 Namely, the first equality follows from HomOC (jU ! OU , H) = H(U ) and taking derived functors. The vanishing of H p (U, g −1 I) for all U ∈ Ob(C) comes from the ˇ ˇ p (U, g −1 I) via Lemma 19.11.8. vanishing of all higher Cech cohomology groups H ˇ p (U, g −1 I) = H ˇ p (u(U), I). Namely, for a covering U = {Ui → U }i∈I in C we have H ˇ Since I is an injective O-module these Cech cohomology groups vanish, see Lemma 19.12.3. Since each K−q is a direct sum of modules of the form jU ! OU we see that ExtpOC (K−q , g −1 I) = 0 for p ≥ 1 and all q Let us use the spectral sequence (see Example 19.21.1) • −1 E1p,q = ExtpOC (K−q , g −1 I) ⇒ Extp+q I) = 0. OC (K , g

Note that the spectral sequence abuts to zero as K• is acyclic (hence vanishes in the derived category, hence produces vanishing ext groups). By the vanishing of higher exts proved above the only nonzero terms on the E1 page are the terms E10,q = HomOC (K−q , g −1 I). We conclude that the complex HomOC (K• , g −1 I) is acyclic as desired. Thus the left derived functor Lg! exists. We still have to show that it is left adjoint to g −1 = g ∗ = Rg ∗ = Lg ∗ , i.e., that we have (19.22.1.1)

HomD(OC ) (H• , g −1 E • ) = HomD(OD ) (Lg! H• , E • )

This is actually a formal consequence of the discussion above. Choose a quasiisomorphism K• → H• such that K• computes Lg! . Moreover, choose a quasiisomorphism E • → I • into a K-injective complex of OD -modules I • . Then the RHS of (19.22.1.1) is HomK(OD ) (g! K• , I • )

19.23. OTHER CHAPTERS

1219

On the other hand, by the definition of morphisms in the derived category the LHS of (19.22.1.1) is HomD(OC ) (K• , g −1 I • ) = colims:L• →K• HomK(OC ) (L• , g −1 I • ) = colims:L• →K• HomK(OD ) (g! L• , I • ) by the adjointness of g! and g ∗ on the level of sheaves of modules. The colimit is over all quasi-isomorphisms with target K• . Since for every complex L• there exists a quasi-isomorphism (K0 )• → L• such that (K0 )• computes Lg! we see that we may as well take the colimit over quasi-isomorphisms of the form s : (K0 )• → K• where (K0 )• computes Lg! . In this case HomK(OD ) (g! K• , I • ) −→ HomK(OD ) (g! (K0 )• , I • ) is an isomorphism as g! (K0 )• → g! K• is a quasi-isomorphism and I • is K-injective. This finishes the proof.  Remark 19.22.2. Warning! Let u : C → D, g, OD , and OC be as in Lemma 19.22.1. In general it is not the case that the diagram D(OC )

Lg!

f orget

 D(C)

/ D(OD ) f orget

Lg!Ab

 / D(D)

commutes where the functor Lg!Ab is the one constructed in Lemma 19.22.1 but using the constant sheaf Z as the structure sheaf on both C and D. In general it isn’t even the case that g! = g!Ab (see Modules on Sites, Remark 16.35.2), but this phenomenon can occur even if g! = g!Ab ! In general all we can say is that there exists a natural transformation Lg!Ab ◦ f orget −→ f orget ◦ Lg! 19.23. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves

(19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36)

Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes

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(37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55)

´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap

(56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 20

Hypercoverings 20.1. Introduction Let C be a site, see Sites, Definition 9.6.2. Let X be an object of C. Given an abelian sheaf F on C we would like to compute its cohomology groups H i (X, F). According to our general definitions (insert future reference here) this cohomology group is computed by choosing an injective resolution 0 → F → I0 → I1 → . . . and setting H i (X, F) = H i (Γ(X, I 0 ) → Γ(X, I 1 ) → Γ(X, I 2 ) → . . .) We will have to do quite a bit of work to prove that we may also compute these cohomology groups without choosing an injective resolution. Also, we will only do this in case the site C has fibre products. A hypercovering in a site is a generalization of a covering. See [AGV71, Expos´e V, Sec. 7]. A hypercovering is a special case of a simplicial augmentation where one has cohomological descent, see [AGV71, Expos´e Vbis]. A nice manuscript on cohomological descent is the text by Brian Conrad, see http://math. stanford.edu/~conrad/papers/hypercover.pdf. Brian’s text follows the exposition in [AGV71, Expos´e Vbis], and in particular discusses a more general kind of hypercoverings, such as proper hypercoverings of schemes used to compute ´etale cohomology for example. A proper hypercovering can be seen as a hypercovering in the category of schemes endowed with a different topology than the ´etale topology, but still they can be used to compute the ´etale cohomology. 20.2. Hypercoverings In order to start we make the following definition. The letters “SR” stand for Semi-Representable. Definition 20.2.1. Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. We denote SR(C, X) the category of semi-representable objects defined as follows (1) objects are families of morphisms {Ui → X}i∈I , and (2) morphisms {Ui → X}i∈I → {Vj → X}j∈J are given by a map α : I → J and for each i ∈ I a morphism fi : Ui → Vα(i) over X. This definition is different from the one in [AGV71, Expos´e V, Sec. 7], but it seems flexible enough to do all the required arguments. Note that this is a “big” 1221

1222

20. HYPERCOVERINGS

category. We will later “bound” the size of the index sets I that we need and we can then redefine SR(C, X) to become a category. Definition 20.2.2. Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. We denote F the functor which associates a sheaf to a semi-representable object. In a formula F : SR(C, X) −→

PSh(C)

{Ui → X}i∈I

qi∈I hUi

7−→

where hU denotes the representable presheaf associated to the object U . Given a morphism U → X we obtain a morphism hU → hX of representable presheaves. Thus it makes more sense to think of F as a functor into the category of presheaves of sets over hX , namely PSh(C)/hX . Lemma 20.2.3. Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. The category SR(C, X) has coproducts and finite limits. Moreover, the functor F commutes with coproducts and fibre products, and transforms products into fibre products over hX . In other words, it commutes with finite limits as a functor into PSh(C)/hX . Proof. It is clear that the coproduct of {Ui → X}i∈I and {Vj → X}j∈J is {Ui → X}i∈I q {Vj → X}j∈J and similarly for coproducts of families of families of morphisms with target X. The object {X → X} is a final object of SR(C, X). Suppose given a morphism (α, fi ) : {Ui → X}i∈I → {Vj → X}j∈J and a morphism (β, gk ) : {Wk → X}k∈K → {Vj → X}j∈J . The fibred product of these morphisms is given by {Ui ×fi ,Vj ,gk Wk → X}(i,j,k)∈I×J×K

such that k=α(i)=β(j)

The fibre products exist by the assumption that C has fibre products. Thus SR(C, X) has finite limits, see Categories, Lemma 4.16.4. The statements on the functor F are clear from the constructions above.  Definition 20.2.4. Let C be a site with fibred products. Let X be an object of C. Let f = (α, fi ) : {Ui → X}i∈I → {Vj → X}j∈J be a morphism in the category SR(C, X). We say that f is a covering if for every j ∈ J the family of morphisms {Ui → Vj }i∈I,α(i)=j is a covering for the site C. Lemma (1) (2) (3)

20.2.5. Let C be a site with fibred products. Let X ∈ Ob(C). A composition of coverings in SR(C, X) is a covering. A base change of coverings is a covering. If A → B and K → L are coverings, then A × K → B × L is a covering.

Proof. Immediate from the axioms of a site. (Number (3) is the composition A×K → B ×K → B ×L and hence a composition of basechanges of coverings.)  According to the results in the chapter on simplicial methods the coskelet of a truncated simplicial object of SR(C, X) exists. Hence the following definition makes sense. Definition 20.2.6. Let C be a site. Let X ∈ Ob(C) be an object of C. A hypercovering of X is a simplicial object K in the category SR(C, X) such that (1) The object K0 is a covering of X for the site C.

20.2. HYPERCOVERINGS

1223

(2) For every n ≥ 0 the canonical morphism Kn+1 −→ (coskn skn K)n+1 is a covering in the sense defined above. Condition (1) makes sense since each object of SR(C, X) is after all a family of morphisms with target X. It could also be formulated as saying that the morphism of K0 to the final object of SR(C, X) is a covering. Example 20.2.7. Let {Ui → X}i∈I be a covering of the site C. Set K0 = {Ui → X}i∈I . Then K0 is a 0-truncated simplicial object of SR(C, X). Hence we may form K = cosk0 K0 . Clearly K passes condition (1) of Definition 20.2.6. Since all the morphisms Kn+1 → (coskn skn K)n+1 are isomorphisms it also passes condition (2). Note that the terms Kn are the usual Kn = {Ui0 ×X Ui1 ×X . . . ×X Uin → X}(i0 ,i1 ,...,in )∈I n+1 Lemma 20.2.8. Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. The collection of all hypercoverings of X forms a set. Proof. Since C is a site, the set of all coverings of S forms a set. Thus we see that the collection of possible K0 forms a set. Suppose we have shown that the collection of all possible K0 , . . . , Kn form a set. Then it is enough to show that given K0 , . . . , Kn the collection of all possible Kn+1 forms a set. And this is clearly true since we have to choose Kn+1 among all possible coverings of (coskn skn K)n+1 .  Remark 20.2.9. The lemma does not just say that there is a cofinal system of choices of hypercoverings that is a set, but that really the hypercoverings form a set. The category of presheaves on C has finite (co)limits. Hence the functors coskn exists for presheaves of sets. Lemma 20.2.10. Let C be a site with fibre products. Let X ∈ Ob(C) be an object of C. Let K be a hypercovering of X. Consider the simplicial object F (K) of PSh(C), endowed with its augmentation to the constant simplicial presheaf hX . (1) The morphism of presheaves F (K)0 → hX becomes a surjection after sheafification. (2) The morphism (d10 , d11 ) : F (K)1 −→ F (K)0 ×hX F (K)0 becomes a surjection after sheafification. (3) For every n ≥ 1 the morphism F (K)n+1 −→ (coskn skn F (K))n+1 turns into a surjection after sheafification. Proof. We will use the fact that if {Ui → U }i∈I is a covering of the site C, then the morphism qi∈I hUi → hU becomes surjective after sheafification, see Sites, Lemma 9.12.5. Thus the first assertion follows immediately.

1224

20. HYPERCOVERINGS

For the second assertion, note that according to Simplicial, Example 14.17.2 the simplicial object cosk0 sk0 K has terms K0 × . . . × K0 . Thus according to the definition of a hypercovering we see that (d10 , d11 ) : K1 → K0 ×K0 is a covering. Hence (2) follows from the claim above and the fact that F transforms products into fibred products over hX . For the third, we claim that coskn skn F (K) = F (coskn skn K) for n ≥ 1. To prove this, denote temporarily F 0 the functor SR(C, X) → PSh(C)/hX . By Lemma 20.2.3 the functor F 0 commutes with finite limits. By our description of the coskn functor in Simplicial, Section 14.17 we see that coskn skn F 0 (K) = F 0 (coskn skn K). Recall that the category used in the description of (coskn U )m in Simplicial, Lemma 14.17.3 is the category (∆/[m])opp ≤n . It is an amusing exercise to show that (∆/[m])≤n is a nonempty connected category (see Categories, Definition 4.15.1) as soon as n ≥ 1. Hence, Categories, Lemma 4.15.2 shows that coskn skn F 0 (K) = coskn skn F (K). Whence the claim. Property (2) follows from this, because now we see that the morphism in (2) is the result of applying the functor F to a covering as in Definition 20.2.4, and the result follows from the first fact mentioned in this proof.  20.3. Acyclicity Let C be a site. For a presheaf of sets F we denote ZF the presheaf of abelian groups defined by the rule ZF (U ) = free abelian group on F(U ). We will sometimes call this the free abelian presheaf on F. Of course the construction F 7→ ZF is a functor and it is left adjoint to the forgetful functor PAb(C) → PSh(C). Of course the sheafification Z# F is a sheaf of abelian groups, and the functor F 7→ Z# is a left adjoint as well. We sometimes call Z# F F the free abelian sheaf on F. For an object X of the site C we denote ZX the free abelian presheaf on hX , and we denote Z# X its sheafification. Definition 20.3.1. Let C be a site. Let K be a simplicial object of PSh(C). By the above we get a simplicial object Z# K of Ab(C). We can take its associated complex # of abelian presheaves s(ZK ), see Simplicial, Section 14.21. The homology of K is the homology of the complex of abelian sheaves s(Z# K ). In other words, the ith homology Hi (K) of K is the sheaf of abelian groups Hi (K) = Hi (s(Z# K )). In this section we worry about the homology in case K is a hypercovering of an object X of C. Lemma 20.3.2. Let C be a site. Let F → G be a morphism of presheaves of sets. Denote K the simplicial object of PSh(C) whose nth term is the (n + 1)st fibre product of F over G, see Simplicial, Example 14.3.5. Then, if F → G is surjective after sheafification, we have  0 if i > 0 Hi (K) = Z# if i = 0 G The isomorphism in degree 0 is given by the morphsm H0 (K) → Z# G coming from # # the map (Z# ) = Z → Z . 0 K F G

20.3. ACYCLICITY

1225

Proof. Let G 0 ⊂ G be the image of the morphism F → G. Let U ∈ Ob(C). Set A = F(U ) and B = G 0 (U ). Then the simplicial set K(U ) is equal to the simplicial set with n-simplices given by A ×B A ×B . . . ×B A (n + 1 factors). By Simplicial, Lemma 14.28.4 the morphism K(U ) → B is a homotopy equivalence. Hence applying the functor “free abelian group on” to this we deduce that ZK (U ) −→ ZB is a homotopy equivalence. Note that s(ZB ) is the complex M M M M 0 1 0 ... → Z− → Z− → Z− → b∈B

b∈B

b∈B

b∈B

Z→0

see Simplicial, LemmaL14.21.3. Thus we see that Hi (s(ZK (U ))) = 0 for i > 0, L and H0 (s(ZK (U ))) = b∈B Z = s∈G 0 (U ) Z. These identifications are compatible with restriction maps. We conclude that Hi (s(ZK )) = 0 for i > 0 and H0 (s(ZK )) = ZG 0 , where here we compute homology groups in PAb(C). Since sheafification is an exact functor we deduce the result of the lemma. Namely, the exactness implies that H0 (s(ZK ))# = H0 (s(Z#  K )), and similarly for other indices. Lemma 20.3.3. Let C be a site. Let f : L → K be a morphism of simplicial objects of PSh(C). Let n ≥ 0 be an integer. Assume that (1) For i < n the morphism Li → Ki is an isomorphism. (2) The morphism Ln → Kn is surjective after sheafification. (3) The canonical map L → coskn skn L is an isomorphism. (4) The canonical map K → coskn skn K is an isomorphism. Then Hi (f ) : Hi (L) → Hi (K) is an isomorphism. Proof. This proof is exactly the same as the proof of Lemma 20.3.2 above. Namely, we first let Kn0 ⊂ Kn be the sub presheaf which is the image of the map Ln → Kn . Assumption (2) means that the sheafification of Kn0 is equal to the sheafification of Kn . Moreover, since Li = Ki for all i < n we see that get an n-truncated simplicial presheaf U by taking U0 = L0 = K0 , . . . , Un−1 = Ln−1 = Kn−1 , Un = 0 Kn0 . Denote K 0 = coskn U , a simplicial presheaf. Because we can construct Km as 0 # a finite limit, and since sheafification is exact, we see that (Km ) = Km . In other words, (K 0 )# = K # . We conclude, by exactness of sheafification once more, that Hi (K) = Hi (K 0 ). Thus it suffices to prove the lemma for the morphism L → K 0 , in other words, we may assume that Ln → Kn is a surjective morphism of presheaves! In this case, for any object U of C we see that the morphism of simplicial sets L(U ) −→ K(U ) satisfies all the assumptions of Simplicial, Lemma 14.28.3. Hence it is a homotopy equivalence, and thus ZL (U ) −→ ZK (U ) is a homotopy equivalence too. This for all U . The result follows.



Lemma 20.3.4. Let C be a site. Let K be a simplicial presheaf. Let G be a presheaf. Let K → G be an augmentation of K towards G. Assume that

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20. HYPERCOVERINGS

(1) The morphism of presheaves K0 → G becomes a surjection after sheafification. (2) The morphism (d10 , d11 ) : K1 −→ K0 ×G K0 becomes a surjection after sheafification. (3) For every n ≥ 1 the morphism Kn+1 −→ (coskn skn K)n+1 turns into a surjection after sheafification. Then Hi (K) = 0 for i > 0 and H0 (K) = Z# G. Proof. Denote K n = coskn skn K for n ≥ 1. Define K 0 as the simplicial object with terms (K 0 )n equal to the (n + 1)-fold fibred product K0 ×G . . . ×G K0 , see Simplicial, Example 14.3.5. We have morphisms K −→ . . . → K n → K n−1 → . . . → K 1 → K 0 . The morphisms K → K i , K j → K i for j ≥ i ≥ 1 come from the universal properties of the coskn functors. The morphism K 1 → K 0 is the canonical morphism from Simplicial, Remark 14.18.4. We also recall that K 0 → cosk1 sk1 K 0 is an isomorphism, see Simplicial, Lemma 14.18.3. By Lemma 20.3.2 we see that Hi (K 0 ) = 0 for i > 0 and H0 (K 0 ) = Z# G. Pick n ≥ 1. Consider the morphism K n → K n−1 . It is an isomorphism on terms of degree < n. Note that K n → coskn skn K n and K n−1 → coskn skn K n−1 are isomorphisms. Note that (K n )n = Kn and that (K n−1 )n = (coskn−1 skn−1 K)n . Hence by assumption, we have that (K n )n → (K n−1 )n is a morphism of presheaves which becomes surjective after sheafification. By Lemma 20.3.3 we conclude that Hi (K n ) = Hi (K n−1 ). Combined with the above this proves the lemma.  Lemma 20.3.5. Let C be a site with fibre products. Let X be an object of of C. Let K be a hypercovering of X. The homology of the simplicial presheaf F (K) is 0 in degrees > 0 and equal to Z# X in degree 0. Proof. Combine Lemmas 20.3.4 and 20.2.10.



20.4. Covering hypercoverings Here are some ways to construct hypercoverings. We note that since the category SR(C, X) has fibre products the category of simplicial objects of SR(C, X) has fibre products as well, see Simplicial, Lemma 14.7.2. Lemma 20.4.1. Let C be a site with fibre products. Let X be an object of C. Let K, L, M be simplicial objects of SR(C, X). Let a : K → L, b : M → L be morphisms. Assume (1) K is a hypercovering of X, (2) the morphism M0 → L0 is a covering, and

20.4. COVERING HYPERCOVERINGS

1227

(3) for all n ≥ 0 in the diagram / (coskn skn M )n+1 3

Mn+1 γ

* Ln+1 ×(coskn skn L)n+1 (coskn skn M )n+1  / (coskn skn L)n+1

 t Ln+1 the arrow γ is a covering. Then the fibre product K ×L M is a hypercovering of X.

Proof. The morphism (K ×L M )0 = K0 ×L0 M0 → K0 is a base change of a covering by (2), hence a covering, see Lemma 20.2.5. And K0 → {X → X} is a covering by (1). Thus (K ×L M )0 → {X → X} is a covering by Lemma 20.2.5. Hence K ×L M satisfies the first condition of Definition 20.2.6. We still have to check that Kn+1 ×Ln+1 Mn+1 = (K ×L M )n+1 −→ (coskn skn (K ×L M ))n+1 is a covering for all n ≥ 0. We abbreviate as follows: A = (coskn skn K)n+1 , B = (coskn skn L)n+1 , and C = (coskn skn M )n+1 . The functor coskn skn commutes with fibre products, see Simplicial, Lemma 14.17.13. Thus the right hand side above is equal to A ×B C. Consider the following commutative diagram Kn+1 ×Ln+1 Mn+1

/ Mn+1





Kn+1

/ Ln+1 o (

γ

& Ln+1 ×B C

A

*/ C  /* B

This diagram shows that Kn+1 ×Ln+1 Mn+1 = (Kn+1 ×B C) ×(Ln+1 ×B C),γ Mn+1 Now, Kn+1 ×B C → A ×B C is a base change of the covering Kn+1 → A via the morphism A ×B C → A, hence is a covering. By assumption (3) the morphism γ is a covering. Hence the morphism (Kn+1 ×B C) ×(Ln+1 ×B C),γ Mn+1 −→ Kn+1 ×B C is a covering as a base change of a covering. The lemma follows as a composition of coverings is a covering.  Lemma 20.4.2. Let C be a site with fibre products. Let X be an object of C. If K, L are hypercoverings of X, then K × L is a hypercovering of X. Proof. You can either verify this directly, or use Lemma 20.4.1 above and check that L → {X → X} has property (3). 

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Let C be a site with fibre products. Let X be an object of C. Since the category SR(C, X) has coproducts and finite limits, it is permissible to speak about the objects U × K and Hom(U, K) for certain simplicial sets U (for example those with finitely many nondegenerate simplices) and any simplicial object K of SR(C, X). See Simplicial, Sections 14.12 and 14.15. Lemma 20.4.3. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let k ≥ 0 be an integer. Let u : Z → Kk be a covering in in SR(C, X). Then there exists a morphism of hypercoverings f : L → K such that Lk → Kk factors through u. Proof. Denote Y = Kk . There is a canonical morphism K → Hom(∆[k], Y ) corresponding to idY via Simplicial, Lemma 14.15.5. We will use the description of Hom(∆[k], Y ) and Hom(∆[k], Z) given in that lemma. In particular there is a morphism Hom(∆[k], Y ) → Hom(∆[k], Z) which on degree n terms is the morphism Y Y Y −→ Z. α:[k]→[n]

α:[k]→[n]

Set L = K ×Hom(∆[n],Y ) Hom(∆[n], Z). The morphism Lk → Kk sits in to a commutative diagram prid

Lk

/Q

 Kk

/Q

α:[k]→[n]

Y



[k]

prid

α:[k]→[n] Z

[k]

/Y  /Z

Since the composition of the two bottom arrows is the identity we conclude that we have the desired factorization. We still have to show that L is a hypercovering of X. To see this we will use Lemma 20.4.1. Condition (1) is satisfied by assumption. For (2), the morphism Hom(∆[k], Y )0 → Hom(∆[k], Z)0 is a covering because it is a product of coverings, see Lemma 20.2.5. For (3) suppose first that n ≥ 1. In this case by Simplicial, Lemma 14.19.12 we have Hom(∆[k], Y ) = coskn skn Hom(∆[k], Y ) and similarly for Z. Thus condition (3) for n > 0 is clear. For n = 0, the diagram of condition (3) of Lemma 20.4.1 is, according to Simplicial, Lemma 14.19.13, the diagram Q / Z ×Z α:[k]→[1] Z

Q



α:[k]→[1]

Y

 / Y ×Y

with obvious horizontal arrows. Thus the morphism γ is the morphism Y Y Y Z −→ Z× Y α:[k]→[1]

α:[k]→[1] not onto

α:[k]→[1] onto

which is a product of coverings and hence a covering according to Lemma 20.4.1 once again. 

20.5. ADDING SIMPLICES

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Lemma 20.4.4. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let n ≥ 0 be an integer. Let u : F → F (Kn ) be a morphism of presheaves which becomes surjective on sheafification. Then there exists a morphism of hypercoverings f : L → K such that F (fn ) : F (Ln ) → F (Kn ) factors through u. Proof. Write Kn = {Ui → X}i∈I . Thus the map u is a morphism of presheaves of sets u : F → qhui . The assumption on u means that for every i ∈ I there exists a covering {Uij → Ui }j∈Ii of the site C and a morphism of presheaves tij : hUij → F such that u ◦ tij is the map hUij → hUi coming from the morphism Uij → Ui . Set J = qi∈I Ii , and let α : J → I be the obvious map. For j ∈ J denote Vj = Uα(j)j . Set Z = {Vj → X}j∈J . Finally, consider the morphism u0 : Z → Kn given by α : J → I and the morphisms Vj = Uα(j)j → Uα(j) above. Clearly, this is a covering in the category SR(C, X), and by construction F (u0 ) : F (Z) → F (Kn ) factors through u. Thus the result follows from Lemma 20.4.3 above.  20.5. Adding simplices In this section we prove some technical lemmas which we will need later. Let C be a site with fibre products. Let X be an object of C. As we pointed out in Section 20.4 above, the objects U × K and Hom(U, K) for certain simplicial sets U and any simplicial object K of SR(C, X) are defined. See Simplicial, Sections 14.12 and 14.15. Lemma 20.5.1. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let U ⊂ V be simplicial sets, with Un , Vn finite nonempty for all n. Assume that U has finitely many nondegenerate simplices. Suppose n ≥ 0 and x ∈ Vn , x 6∈ Un are such that (1) Vi = Ui for i < n, (2) Vn = Un ∪ {x}, (3) any z ∈ Vj , z 6∈ Uj for j > n is degenerate. Then the morphism Hom(V, K)0 −→ Hom(U, K)0 of SR(C, X) is a covering. Proof. If n = 0, then it follows easily that V = U q ∆[0] (see below). In this case Hom(V, K)0 = Hom(U, K)0 × K0 . The result, in this case, then follows from Lemma 20.2.5. Let a : ∆[n] → V be the morphism associated to x as in Simplicial, Lemma 14.11.3. Let us write ∂∆[n] = i(n−1)! skn−1 ∆[n] for the (n − 1)-skeleton of ∆[n]. Let b : ∂∆[n] → U be the restriction of a to the (n − 1) skeleton of ∆[n]. By Simplicial, Lemma 14.19.7 we have V = U q∂∆[n] ∆[n]. By Simplicial, Lemma 14.15.6 we get that / Hom(U, K)0 Hom(V, K)0  Hom(∆[n], K)0

 / Hom(∂∆[n], K)0

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is a fibre product square. Thus it suffices to show that the bottom horizontal arrow is a covering. By Simplicial, Lemma 14.19.11 this arrow is identified with Kn → (coskn−1 skn−1 K)n and hence is a covering by definition of a hypercovering.



Lemma 20.5.2. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let U ⊂ V be simplicial sets, with Un , Vn finite nonempty for all n. Assume that U and V have finitely many nondegenerate simplices. Then the morphism Hom(V, K)0 −→ Hom(U, K)0 of SR(C, X) is a covering. Proof. By Lemma 20.5.1 above, it suffices to prove a simple lemma about inclusions of simplicial sets U ⊂ V as in the lemma. And this is exactly the result of Simplicial, Lemma 14.19.8.  20.6. Homotopies Let C be a site with fibre products. Let X be an object of C. Let L be a simplicial object of SR(C, X). According to Simplicial, Lemma 14.15.4 there exists an object Hom(∆[1], L) in the category Simp(SR(C, X)) which represents the functor T 7−→ MorSimp(SR(C,X)) (∆[1] × T, L) There is a canonical morphism Hom(∆[1], L) → L × L coming from ei : ∆[0] → ∆[1] and the identification Hom(∆[0], L) = L. Lemma 20.6.1. Let C be a site with fibre products. Let X be an object of C. Let L be a simplicial object of SR(C, X). Let n ≥ 0. Consider the commutative diagram (20.6.1.1)

Hom(∆[1], L)n+1

/ (coskn skn Hom(∆[1], L))n+1

 (L × L)n+1

 / (coskn skn (L × L))n+1

coming from the morphism defined above. We can identify the terms in this diagram as follows, where ∂∆[n+1] = in! skn ∆[n+1] is the n-skeleton of the (n+1)-simplex: Hom(∆[1], L)n+1

=

Hom(∆[1] × ∆[n + 1], L)0

(coskn skn Hom(∆[1], L))n+1

=

Hom(∆[1] × ∂∆[n + 1], L)0

(L × L)n+1

=

Hom((∆[n + 1] q ∆[n + 1], L)0

(coskn skn (L × L))n+1

=

Hom(∂∆[n + 1] q ∂∆[n + 1], L)0

and the morphism between these objects of SR(C, X) come from the commutative diagram of simplicial sets (20.6.1.2)

∆[1] × ∆[n + 1] o O

∆[1] × ∂∆[n + 1] O

∆[n + 1] q ∆[n + 1] o

∂∆[n + 1] q ∂∆[n + 1]

20.6. HOMOTOPIES

1231

Moreover the fibre product of the bottom arrow and the right arrow in (20.6.1.1) is equal to Hom(U, L)0 where U ⊂ ∆[1] × ∆[n + 1] is the smallest simplicial subset such that both ∆[n + 1] q ∆[n + 1] and ∆[1] × ∂∆[n + 1] map into it. Proof. The first and third equalities are Simplicial, Lemma 14.15.4. The second and fourth follow from the cited lemma combined with Simplicial, Lemma 14.19.11. The last assertion follows from the fact that U is the push-out of the bottom and right arrow of the diagram (20.6.1.2), via Simplicial, Lemma 14.15.6. To see that U is equal to this push-out it suffices to see that the intersection of ∆[n + 1] q ∆[n + 1] and ∆[1] × ∂∆[n + 1] in ∆[1] × ∆[n + 1] is equal to ∂∆[n + 1] q ∂∆[n + 1]. This we leave to the reader.  Lemma 20.6.2. Let C be a site with fibre products. Let X be an object of C. Let K, L be hypercoverings of X. Let a, b : K → L be morphisms of hypercoverings. There exists a morphism of hypercoverings c : K 0 → K such that a ◦ c is homotopic to b ◦ c. Proof. Consider the following commutative diagram K0

def

K ×(L×L) Hom(∆[1], L) c

(  K

(a,b)

/ Hom(∆[1], L)  / L×L

By the functorial property of Hom(∆[1], L) the composition of the horizontal morphisms corresponds to a morphism K 0 ∆[1] → L which defines a homotopy between c ◦ a and c ◦ b. Thus if we can show that K 0 is a hypercovering of X, then we obtain the lemma. To see this we will apply Lemma 20.4.1 to the pair of morphisms K → L × L and Hom(∆[1], L) → L × L. Condition (1) of Lemma 20.4.1 is statisfied. Condition (2) of Lemma 20.4.1 is true because Hom(∆[1], L)0 = L1 , and the morphism (d10 , d11 ) : L1 → L0 × L0 is a covering of SR(C, X) by our assumption that L is a hypercovering. To prove condition (3) of Lemma 20.4.1 we use Lemma 20.6.1 above. According to this lemma the morphism γ of condition (3) of Lemma 20.4.1 is the morphism Hom(∆[1] × ∆[n + 1], L)0 −→ Hom(U, L)0 where U ⊂ ∆[1] × ∆[n + 1]. According to Lemma 20.5.2 this is a covering and hence the claim has been proven.  Remark 20.6.3. Note that the crux of the proof is to use Lemma 20.5.2. This lemma is completely general and does not care about the exact shape of the simplicial sets (as long as they have only finitely many nondegenerate simplices). It seems altogether reasonable to expect a result of the following kind: Given any morphism a : K × ∂∆[k] → L, with K and L hypercoverings, there exists a morphism of hypercoverings c : K 0 → K and a morphism g : K 0 × ∆[k] → L such that g|K 0 ×∂∆[k] = a ◦ (c × id∂∆[k] ). In other words, the category of hypercoverings is in a suitable sense contractible.

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20.7. Cech cohomology associated to hypercoverings Let C be a site with fibre products. Let X be an object of C. Consider a presheaf of abelian groups F on the site C. It defines a functor F : SR(C, X)opp

−→

{Ui → X}i∈I

7−→

Ab Y i∈I

F(Ui )

Thus a simplicial object K of SR(C, X) is turned into a cosimplicial object F(K) of Ab. In this situation we define ˇ i (K, F) = H i (s(F(K))). H Recall that s(F(K)) is the cochain complex associated to the cosimplicial abelian group F(K), see Simplicial, Section 14.23. In this section we prove analogues of some of the results for Cech cohomology of open coverings proved in Cohomology, Sections 18.9, 18.10 and 18.11. Lemma 20.7.1. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let F be a sheaf of abelian groups on C. Then ˇ 0 (K, F) = F(X). H Proof. We have ˇ 0 (K, F) = Ker(F(K0 ) −→ F(K1 )) H Write K0 = {Ui → X}. It is a covering in the site C. As well, we have that K1 → K0 × K0 is a covering in SR(C, X). Hence we may write K1 = qi0 ,i1 ∈I {Vi0 i1 j → X} so that the morphism K1 → K0 × K0 is given by coverings {Vi0 i1 j → Ui0 ×X Ui1 } of the site C. Thus we can further identify Y Y ˇ 0 (K, F) = Ker( H F(Ui ) −→ F(Vi0 i1 j )) i

i0 i1 j

ˇ 0 (K, F) = H 0 (X, F). with obvious map. The sheaf property of F implies that H



In fact this property characterizes the abelian sheaves among all abelian presheaves on C of course. The analogue of Cohomology, Lemma 20.7.2 in this case is the following. Lemma 20.7.2. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let I be an injective sheaf of abelian groups on C. Then  I(X) if p = 0 p ˇ H (K, I) = 0 if p > 0 Proof. Observe that for any object Z = {Ui → X} of SR(C, X) and any abelian sheaf F on C we have Y F(Z) = F(Ui ) Y = MorPSh(C) (hUi , F) =

MorPSh(C) (F (Z), F)

=

MorPAb(C) (ZF (Z) , F)

=

MorAb(C) (Z# F (Z) , F)

Thus we see, for any simplicial object K of SR(C, X) that we have (20.7.2.1)

s(F(K)) = HomAb(C) (s(Z# K ), F)

20.7. CECH COHOMOLOGY ASSOCIATED TO HYPERCOVERINGS

1233

see Definition 20.3.1 for notation. Now, we know that s(Z# K ) is quasi-isomorphic to Z# if K is a hypercovering, see Lemma 20.3.5. We conclude that if I is an injective X abelian sheaf, and K a hypercovering, then the complex s(I(K)) is acyclic except possibly in degree 0. In other words, we have ˇ i (K, I) = 0 H for i > 0. Combined with Lemma 20.7.1 the lemma is proved.



Next we come to the analogue of Cohomology, Lemma 20.7.3. To state it we need to introduce a little more notation. Let C be a site with fibre products. Let F be a sheaf of abelian groups on C. The symbol H i (F) indicates the presheaf of abelian groups on C which is defined by the rule H i (F) : U 7−→ H i (U, F) where U ranges over the objects of C. Lemma 20.7.3. Let C be a site with fibre products. Let X be an object of C. Let K be a hypercovering of X. Let F be a sheaf of abelian groups on C. There is a map s(F(K)) −→ RΓ(X, F) in D+ (Ab) functorial in F, which induces natural transformations ˇ i (K, −) −→ H i (X, −) H as functors Ab(C) → Ab. Moreover, there is a spectral sequence (Er , dr )r≥0 with ˇ p (K, H q (F)) E2p,q = H converging to H p+q (X, F). This spectral sequence is functorial in F and in the hypercovering K. Proof. We could prove this by the same method as employed in the corresponding lemma in the chapter on cohomology. Instead let us prove this by a double complex argument. Choose an injective resolution F → I • in the category of abelian sheaves on C. Consider the double complex A•,• with terms Ap,q = I q (Kp ) p,q where the differential dp,q → Ap+1,q is the one coming from the differential 1 : A p p+1 : Ap,q → Ap,q+1 is the one coming from the I → I and the differential dp,q 2 p differential on the complex s(I (K)) associated to the cosimplicial abelian group I p (K) as explained above. As usual we denote sA• the simple complex associated to the double complex A•,• . We will use the two spectral sequences (0 Er , 0 dr ) and (00 Er , 00 dr ) associated to this double complex, see Homology, Section 10.19.

By Lemma 20.7.2 the complexes s(I p (K)) are acyclic in positive degrees and have H 0 equal to I p (X). Hence by Homology, Lemma 10.19.6 and its proof the spectral sequence (0 Er , 0 dr ) degenerates, and the natural map I • (X) −→ sA• is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that H n (sA• ) = H n (X, F).

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The map s(F(K)) −→ RΓ(X, F) of the lemma is the composition of the natural map s(F(K)) → sA• followed by the inverse of the displayed quasi-isomorphism above. This works because I • (X) is a representative of RΓ(X, F). Consider the spectral sequence (00 Er , 00 dr )r≥0 . By Homology, Lemma 10.19.3 we see that p 00 p,q E2 = HII (HIq (A•,• )) In other words, we first take cohomology with respect to d1 which gives the groups 00 p,q E1 = H p (F)(Kq ). Hence it is indeed the case (by the description of the differˇ p (K, H q (F)). And by the other spectral sequence above ential 00 d1 ) that 00 E2p,q = H we see that this one converges to H n (X, F) as desired. We omit the proof of the statements regarding the functoriality of the above constructions in the abelian sheaf F and the hypercovering K.  20.8. Cohomology and hypercoverings Let C be a site with fibre products. Let X be an object of C. Let F be a sheaf of abelian groups on C. Let K, L be hypercoverings of X. If a, b : K → L are homotopic maps, then F(a), F(b) : F(K) → F(L) are homotopic maps, see Simplicial, Lemma 14.26.3. Hence have the same effect on cohomology groups of the associated cochain complexes, see Simplicial, Lemma 14.26.5. We are going to use this to define the colimit over all hypercoverings. Let us temporarily denote HC(C, X) the category of hypercoverings of X. We have seen that this is a category and not a “big” category, see Lemma 20.2.8. This will be the index category for our diagram, see Categories, Section 4.13 for notation. Consider the diagram ˇ i (−, F) : HC(C, X) −→ Ab. H By Lemma 20.4.2 and Lemma 20.6.2, and the remark on homotopies above, this diagram is directed, see Categories, Definition 4.17.1. Thus the colimit i ˇ i (K, F) ˇ HC H (X, F) = colimK∈HC(C,X) H has a particularly simple discription (see location cited). Theorem 20.8.1. Let C be a site with fibre products. Let X be an object of C. Let i ≥ 0. The functors Ab(C) −→ F F

Ab

7−→ H i (X, F) i ˇ HC 7−→ H (X, F)

are canonically isomorphic. Proof using spectral sequences. Suppose that ξ ∈ H p (X, F) for some p ≥ 0. ˇ p (X, F) → H p (X, F) of Lemma Let us show that ξ is in the image of the map H 20.7.3 for some hypercovering K of X. This is true if p = 0 by Lemma 20.7.1. If p = 1, choose a Cech hypercovering K of X as in Example 20.2.7 starting with a covering K0 = {Ui → X} in the site C such that ξ|Ui = 0, see Cohomology on Sites, Lemma 19.8.3. It follows immediately from the spectral sequence in Lemma 20.7.3 that ξ comes from an ˇ 1 (K, F) in this case. In general, choose any hypercovering K of X element of H

20.8. COHOMOLOGY AND HYPERCOVERINGS

1235

such that ξ maps to zero in H p (F)(K0 ) (using Example 20.2.7 and Cohomology on Sites, Lemma 19.8.3 again). By the spectral sequence of Lemma 20.7.3 the ˇ p (K, F) is a sequence of elements obstruction for ξ to come from an element of H q p−q ˇ ξ1 , . . . , ξp−1 with ξq ∈ H (K, H (F)) (more precisely the images of the ξq in certain subquotients of these groups). We can inductively replace the hypercovering K by refinements such that the obstructions ξ1 , . . . , ξp−1 restrict to zero (and not just the images in the subquotients – so no subtlety here). Indeed, suppose we have already managed to reach the sitˇ p−q (K, H q (F)) is the class uation where ξq+1 , . . . , ξp−1 are zero. Note that ξq ∈ H of some element Y ξ˜q ∈ H q (F)(Kp−q ) = H q (Ui , F) if Kp−q = {Ui → X}i∈I . Let ξq,i be the component of ξ˜q in H q (Ui , F). As q ≥ 1 we can use Cohomology on Sites, Lemma 19.8.3 yet again to choose coverings {Ui,j → Ui } of the site such that each restriction ξq,i |Ui,j = 0. Consider the object Z = {Ui,j → X} of the category SR(C, X) and its obvious morphism u : Z → Kp−q . It is clear that u is a covering, see Definition 20.2.4. By Lemma 20.4.3 there exists a morphism L → K of hypercoverings of X such that Lp−q → Kp−q factors through u. Then clearly the image of ξq in H q (F)(Lp−q ). is zero. Since the spectral sequence of Lemma 20.7.3 is functorial this means that after replacing K by L we reach the situation where ξq , . . . , ξp−1 are all zero. Continuing like this we end up with a hypercovering where they are all zero and hence ξ is in the image of the map ˇ p (X, F) → H p (X, F). H ˇ p (K, F) and that the image of ξ Suppose that K is a hypercovering of X, that ξ ∈ H p p ˇ under the map H (X, F) → H (X, F) of Lemma 20.7.3 is zero. To finish the proof of the theorem we have to show that there exists a morphism of hypercoverings ˇ p (L, F). By the spectral sequence of L → K such that ξ restricts to zero in H Lemma 20.7.3 the vanishing of the image of ξ in H p (X, F) means that there exist ˇ p−1−q (K, H q (F)) (more precisely the images of elements ξ1 , . . . , ξp−2 with ξq ∈ H these in certain subquotients) such that the images dp−1−q,q ξq (in the spectral q+1 sequence) add up to ξ. Hence by exacly the same mechanism as above we can find a morphism of hypercoverings L → K such that the restrictions of the elements ˇ p−1−q (L, H q (F)) are zero. Then it follows that ξ is zero ξq , q = 1, . . . , p − 2 in H since the morphism L → K induces a morphism of spectral sequences according to Lemma 20.7.3.  Proof without using spectral sequences. We have seen the result for i = 0, see Lemma 20.7.1. We know that the functors H i (X, −) form a universal δ-functor, see Derived Categories, Lemma 11.19.4. In order to prove the theorem it suffices ˇ i (X, −) forms a δ-functor. Namely we to show that the sequence of functors H HC know that Cech cohomology is zero on injective sheaves (Lemma 20.7.2) and then we can apply Homology, Lemma 10.9.4. Let 0→F →G→H→0 ˇ p (X, H). Choose be a short exact sequence of abelian sheaves on C. Let ξ ∈ H HC a hypercovering K of X and an element σ ∈ H(Kp ) representing ξ in cohomology.

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There is a corresponding exact sequence of complexes 0 → s(F(K)) → s(G(K)) → s(H(K)) but we are not assured that there is a zero on the right also and this is the only thing that prevents us from defining δ(ξ) by a simple application of the snake lemma. Recall that Y H(Kp ) = H(Ui ) Q if Kp = {Ui → X}. Let σ = σi with σi ∈ H(Ui ). Since G → H is a surjection of sheaves we see that there exist coverings {Ui,j → Ui } such that σi |Ui,j is the image of some element τi,j ∈ G(Ui,j ). Consider the object Z = {Ui,j → X} of the category SR(C, X) and its obvious morphism u : Z → Kp . It is clear that u is a covering, see Definition 20.2.4. By Lemma 20.4.3 there exists a morphism L → K of hypercoverings of X such that Lp → Kp factors through u. After replacing K by L we may therefore assume that σ is the image of an element τ ∈ G(Kp ). Note that d(σ) = 0, but not necessarily d(τ ) = 0. Thus d(τ ) ∈ F(Kp+1 ) is a cocycle. In ˇ p+1 (X, F). this situation we define δ(ξ) as the class of the cocycle d(τ ) in H HC At this point there are several things to verify: (a) δ(ξ) does not depend on the choice of τ , (b) δ(ξ) does not depend on the choice of the hypercovering L → K such that σ lifts, and (c) δ(ξ) does not depend on the initial hypercovering and σ chosen to represent ξ. We omit the verification of (a), (b), and (c); the independence of the choices of the hypercoverings really comes down to Lemmas 20.4.2 and 20.6.2. We also omit the verification that δ is functorial with respect to morphisms of short exact sequences of abelian sheaves on C. Finally, we have to verify that with this definition of δ our short exact sequence of abelian sheaves above leads to a long exact sequence of Cech cohomology groups. First we show that if δ(ξ) = 0 (with ξ as above) then ξ is the image of some ˇ p (X, G). Namely, if δ(ξ) = 0, then, with notation as above, we element ξ 0 ∈ H HC ˇ p+1 (X, F). Hence there exists a morphism of see that the class of d(τ ) is zero in H HC hypercoverings L → K such that the restriction of d(τ ) to an element of F(Lp+1 ) is equal to d(υ) for some υ ∈ F(Lp ). This implies that τ |Lp + υ form a cocycle, and ˇ p (L, G) which maps to ξ as desired. determine a class ξ 0 ∈ H ˇ p+1 (X, F) maps to zero in H ˇ p+1 (X, G), then it is We omit the proof that if ξ 0 ∈ H HC HC ˇ p (X, H). equal to δ(ξ) for some ξ ∈ H  HC 20.9. Hypercoverings of spaces The theory above is mildly interesting even in the case of topological spaces. In this case we can work out what is a hypercovering and see what the result actually says. Let X be a topological space. Consider the site TX of Sites, Example 9.6.4. Recall that an object of TX is simply an open of X and that morphisms of TX correspond simply to inclusions. So what is a hypercovering of X for the site TX ? Let us first unwind Definition 20.2.1. An object of SR(C, X) is simply given by a set I and for each i ∈ I an open Ui ⊂ X. Let us denote this by {Ui }i∈I since there can be no confusion about the morphism Ui → X. A morphism {Ui }i∈I → {Vj }j∈J between two such objects is given by a map of sets α : I → J such that Ui ⊂ Vα(i)

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for all i ∈ I. When is such a S morphism a covering? This is the case if and only if for every j ∈ J we have Vj = i∈I, α(i)=j Ui (and is a covering in the site TX ). Using the above we get the following description of a hypercovering in the site TX . A hypercovering of X in TX is given by the following data (1) a simplicial set I (see Simplicial, Section 14.11), and (2) for each n ≥ 0 and every i ∈ In an open set Ui ⊂ X. We will denote such a collection of data by the notation (I, {Ui }). In order for this to be a hypercovering of X we require the following properties • for i ∈ In and 0 ≤ a ≤ n + 1 we have Ui ⊂ Udna (i) , • for i ∈ In and 0 ≤ a ≤ n we have Ui = Usna (i) , • we have [ Ui , (20.9.0.1) X= i∈I0

• for every i0 , i1 ∈ I0 , we have [ (20.9.0.2) Ui0 ∩ Ui1 =

i∈I1 , d10 (i)=i0 , d11 (i)=i1

Ui ,

• for every n ≥ 1 and every (i0 , . . . , in+1 ) ∈ (In )n+2 such that dnb−1 (ia ) = dna (ib ) for all 0 ≤ a < b ≤ n + 1 we have [ (20.9.0.3) Ui0 ∩ . . . ∩ Uin+1 = Ui , n+1 i∈In+1 , da

(i)=ia , a=0,...,n+1

• each of the open coverings (20.9.0.1), (20.9.0.2), and (20.9.0.3) is an element of Cov(TX ) (this is a set theoretic condition, bounding the size of the index sets of the coverings). Condititions (20.9.0.1) and (20.9.0.2) should be familiar from the chapter on sheaves on spaces for example, and condition (20.9.0.3) is the natural generalization. Remark 20.9.1. One feature of this description is that if one of the multiple intersections Ui0 ∩ . . . ∩ Uin+1 is empty then the covering on the right hand side may be the empty covering. Thus it is not automatically the case that the maps In+1 → (coskn skn I)n+1 are surjective. This means that the geometric realization of I may be an interesting (non-contractible) space. In fact, let In0 ⊂ In be the subset consisting of those simplices i ∈ In such that Ui 6= ∅. It is easy to see that I 0 ⊂ I is a subsimplicial set, and that (I 0 , {Ui }) is a hypercovering. Hence we can always refine a hypercovering to a hypercovering where none of the opens Ui is empty. Remark 20.9.2. Let us repackage this information in yet another way. Namely, suppose that (I, {Ui }) is a hypercovering of the topological space X. Given this data we can construct a simplicial toplogical space U• by setting a Un = Ui , i∈In

and where for given ϕ : [n] → [m] we let morphisms U (ϕ) : Un → Um be the morphism coming from the inclusions Ui ⊂ Uϕ(i) for i ∈ In . This simplicial topological space comes with an augmentation  : U• → X. With this morphism the simplicial space U• becomes a hypercovering of X along which one has cohomological descent in the sense of [AGV71, Expos´e Vbis]. In other words, H n (U• , ∗ F) = H n (X, F).

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(Insert future reference here to cohomology over simplicial spaces and cohomological descent formulated in those terms.) Suppose that F is an abelian sheaf on X. In this case the spectral sequence of Lemma 20.7.3 becomes the spectral sequence with E1 -term E1p,q = H q (Up , ∗q F) ⇒ H p+q (U• , ∗ F) = H p+q (X, F) comparing the total cohomology of ∗ F to the cohomology groups of F over the pieces of U• . (Insert future reference to this spectral sequence here.) In topology we often want to find hypercoverings of X which have the property that all the Ui come from a given basis for the topology of X and that all the coverings (20.9.0.2) and (20.9.0.3) are from a given cofinal collection of coverings. Here are two example lemmas. Lemma 20.9.3. Let X be a topological space. Let B be a basis for the topology of X. There exists a hypercovering (I, {Ui }) of X such that each Ui is an element of B. Proof. Let n ≥ 0. Let us say that an n-truncated hypercovering of X is given by an n-truncated simplicial set I and for each i ∈ Ia , 0 ≤ a ≤ n an open Ui of X such that the conditions defining a hypercovering hold whenever they make sense. In other words we require the inclusion relations and covering conditions only when all simplices that occur in them are a-simplices with a ≤ n. The lemma follows if we can prove that given a n-truncated hypercovering (I, {Ui }) with all Ui ∈ B we can extend it to an (n + 1)-truncated hypercovering without adding any a-simplices for a ≤ n. This we do as follows. First we consider the (n + 1)-truncated simplicial set I 0 defined by I 0 = skn+1 (coskn I). Recall that   (i0 , . . . , in+1 ) ∈ (In )n+2 such that 0 In+1 = dnb−1 (ia ) = dna (ib ) for all 0 ≤ a < b ≤ n + 1 0 If i0 ∈ In+1 is degenerate, say i0 = sna (i) then we set Ui0 = Ui (this is forced on us 0 anyway by the second condition). We also set Ji0 = {i0 } in this case. If i0 ∈ In+1 is 0 nondegerate, say i = (i0 , . . . , in+1 ), then we choose a set Ji0 and an open covering [ (20.9.3.1) Ui0 ∩ . . . ∩ Uin+1 = Ui , i∈Ji0

with Ui ∈ B for i ∈ Ji0 . Set In+1 =

a 0 i0 ∈In+1

Ji0

0 There is a canonical map π : In+1 → In+1 which is a bijection over the set of 0 degenerate simplices in In+1 by construction. For i ∈ In+1 we define dn+1 (i) = a n dn+1 (π(i)). For i ∈ I we define s (i) ∈ I as the unique simplex lying over n n+1 a a 0 the degenerate simplex sna (i) ∈ In+1 . We omit the verification that this defines an (n + 1)-truncated hypercovering of X. 

Lemma 20.9.4. Let X be a topological space. Let B be a basis for the topology of X. Assume that (1) X is quasi-compact, (2) each U ∈ B is quasi-compact open, and (3) the intersection of any two quasi-compact opens in X is quasi-compact. Then there exists a hypercovering (I, {Ui }) of X with the following properties

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(1) each Ui is an element of the basis B, (2) each of the In is a finite set, and in particular (3) each of the coverings (20.9.0.1), (20.9.0.2), and (20.9.0.3) is finite. Proof. This follows directly from the construction in the proof of Lemma 20.9.3 if we choose finite coverings by elements of B in (20.9.3.1). Details omitted.  20.10. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology

(39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 21

Schemes 21.1. Introduction In this document we define schemes. A basic reference is [DG67]. 21.2. Locally ringed spaces Recall that we defined ringed spaces in Sheaves, Section 6.25. Briefly, a ringed space is a pair (X, OX ) consisting of a topological space X and a sheaf of rings OX . A morphism of ringed spaces f : (X, OX ) → (Y, OY ) is given by a continuous map f : X → Y and an f -map of sheaves of rings f ] : OY → OX . You can think of f ] as a map OY → f∗ OX , see Sheaves, Definition 6.21.7 and Lemma 6.21.8. A good geometric example of this to keep in mind is C ∞ -manifolds and morphisms ∞ of C ∞ -manifolds. Namely, if M is a C ∞ -manifold, then the sheaf CM of smooth functions is a sheaf of rings on M . And any map f : M → N of manifolds is ∞ the composition h ◦ f is a local smooth if and only if for every local section h of CN ∞ section of CM . Thus a smooth map f gives rise in a natural way to a morphism of ringed spaces ∞ ∞ f : (M, CM ) −→ (N, CN ) see Sheaves, Example 6.25.2. It is instructive to consider what happens to stalks. ∞ is the Namely, let m ∈ M with image f (m) = n ∈ N . Recall that the stalk CM,m ring of germs of smooth functions at m, see Sheaves, Example 6.11.4. The algebra of germs of functions on (M, m) is a local ring with maximal ideal the functions ∞ ∞ ∞ maps → CM,m . The map on stalks f ] : CN,n which vanish at m. Similarly for CN,n the maximal ideal into the maximal ideal, simply because f (m) = n. In algebraic geometry we study schemes. On a scheme the sheaf of rings is not determined by an intrinsic property of the space. The spectrum of a ring R (see Algebra, Section 7.16) endowed with a sheaf of rings constructed out of R (see below), will be our basic building block. It will turn out that the stalks of O on Spec(R) are the local rings of R at its primes. There are two reasons to introduce locally ringed spaces in this setting: (1) There is in general no mechanism that assigns to a continuous map of spectra a map of the corresponding rings. This is why we add as an extra datum the map f ] . (2) If we consider morphisms of these spectra in the category of ringed spaces, then the maps on stalks may not be local homomorphisms. Since our geometric intuition says it should we introduce locally ringed spaces as follows. Definition 21.2.1. Locally ringed spaces. (1) A locally ringed space (X, OX ) is a pair consisting of a topological space X and a sheaf of rings OX all of whose stalks are local rings. 1241

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(2) Given a locally ringed space (X, OX ) we say that OX,x is the local ring of X at x. We denote mX,x or simply mx the maximal ideal of OX,x . Moreover, the residue field of X at x is the residue field κ(x) = OX,x /mx . (3) A morphism of locally ringed spaces (f, f ] ) : (X, OX ) → (Y, OY ) is a morphism of ringed spaces such that for all x ∈ X the induced ring map OY,f (x) → OX,x is a local ring map. We will usually suppress the sheaf of rings OX in the notation when discussing locally ringed spaces. We will simply refer to “the locally ringed space X”. We will by abuse of notation think of X also as the underlying topological space. Finally we will denote the corresponding sheaf of rings OX as the structure sheaf of X. In addition, it is customary to denote the maximal ideal of the local ring OX,x by mX,x or simply mx . We will say “let f : X → Y be a morphism of locally ringed spaces” thereby surpressing the structure sheaves even further. In this case, we will by abuse of notation think of f : X → Y also as the underlying continuous map of topological spaces. The f -map corresponding to f will customarily be denoted f ] . The condition that f is a morphism of locally ringed spaces can then be expressed by saying that for every x ∈ X the map on stalks fx] : OY,f (x) −→ OX,x maps the maximal ideal mY,f (x) into mX,x . Let us use these notational conventions to show that the collection of locally ringed spaces and morphisms of locally ringed spaces forms a category. In order to see this we have to show that the composition of morphisms of locally ringed spaces is a morphism of locally ringed spaces. OK, so let f : X → Y and g : Y → Z be morphism of locally ringed spaces. The composition of f and g is defined in Sheaves, Definition 6.25.3. Let x ∈ X. By Sheaves, Lemma 6.21.10 the composition g]

f]

OZ,g(f (x)) −→ OY,f (x) −→ OX,x is the associated map on stalks for the morphism g ◦ f . The result follows since a composition of local ring homomorphisms is a local ring homomorphism. A pleasing feature of the definition is the fact that the functor Locally ringed spaces −→ Ringed spaces reflects isomorphisms (plus more). Here is a less abstract statement. Lemma 21.2.2. Let X, Y be locally ringed spaces. If f : X → Y is an isomorphism of ringed spaces, then f is an isomorphism of locally ringed spaces. Proof. This follows trivially from the corresponding fact in algebra: Suppose A, B are local rings. Any isomorphism of rings A → B is a local ring homomorphism.  21.3. Open immersions of locally ringed spaces Definition 21.3.1. Let f : X → Y be a morphism of locally ringed spaces. We say that f is an open immersion if f is a homeomorphism of X onto an open subset of Y , and the map f −1 OY → OX is an isomorphism. The following construction is parallel to Sheaves, Definition 6.31.2 (3).

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Example 21.3.2. Let X be a locally ringed space. Let U ⊂ X be an open subset. Let OU = OX |U be the restriction of OX to U . For u ∈ U the stalk OU,u is equal to the stalk OX,u , and hence is a local ring. Thus (U, OU ) is a locally ringed space and the morphism j : (U, OU ) → (X, OX ) is an open immersion. Definition 21.3.3. Let X be a locally ringed space. Let U ⊂ X be an open subset. The locally ringed space (U, OU ) of Example 21.3.2 above is the open subspace of X associated to U . Lemma 21.3.4. Let f : X → Y be an open immersion of locally ringed spaces. Let j : V = f (X) → Y be the open subspace of Y associated to the image of f . There is a unique isomorphism f 0 : X ∼ = V of locally ringed spaces such that f = j ◦ f 0 . Proof. Omitted.



From now on we do not distinguish between open subsets and their associated subspaces. Lemma 21.3.5. Let f : X → Y be a morphism of locally ringed spaces. Let U ⊂ X, and V ⊂ Y be open subsets. Suppose that f (U ) ⊂ V . There exists a unique morphism of locally ringed spaces f |U : U → V such that the following diagram is a commutative square of locally ringed spaces

f |U

U

/X

 V

 /Y

f

Proof. Omitted.



In the following we will use without further mention the following fact which follows from the lemma above. Given any morphism f : Y → X of locally ringed spaces, and any open subset U ⊂ X such that f (Y ) ⊂ U , then there exists a unique morphism of locally ringed spaces Y → U such that the composition Y → U → X is equal to f . In fact, we will even by abuse of notation write f : Y → U since this rarely gives rise to confusion. 21.4. Closed immersions of locally ringed spaces We follow our conventions introduced in Modules, Definition 15.13.1. Definition 21.4.1. Let i : Z → X be a morphism of locally ringed spaces. We say that i is an closed immersion if: (1) The map i is a homeomorphism of Z onto a closed subset of X. (2) The map OX → i∗ OZ is surjective; let I denote the kernel. (3) The OX -module I is locally generated by sections. Lemma 21.4.2. Let f : Z → X be a morphism of locally ringed spaces. In order S for f to be a closed immersion it suffices if there exists an open covering X = Ui such that each f : f −1 Ui → Ui is a closed immersion. Proof. Omitted.



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Example 21.4.3. Let X be a locally ringed space. Let I ⊂ OX be a sheaf of ideals which is locally generated by sections as a sheaf of OX -modules. Let Z be the support of the sheaf of rings OX /I. This is a closed subset of X, by Modules, Lemma 15.5.3. Denote i : Z → X the inclusion map. By Modules, Lemma 15.6.1 there is a unique sheaf of rings OZ on Z with i∗ OZ = OX /I. For any z ∈ Z the local ring OZ,z is equal to the quotient ring OX,x /Ix and nonzero, hence a local ring. Thus i : (Z, OZ ) → (X, OX ) is a closed immersion of locally ringed spaces. Definition 21.4.4. Let X be a locally ringed space. Let I be a sheaf of ideals on X which is locally generated by sections. The locally ringed space (Z, OZ ) of Example 21.4.3 above is the closed subspace of X associated to the sheaf of ideals I. Lemma 21.4.5. Let f : X → Y be a closed immersion of locally ringed spaces. Let I be the kernel of the map OY → f∗ OX . Let i : Z → Y be the closed subspace of Y associated to I. There is a unique isomorphism f 0 : X ∼ = Z of locally ringed spaces such that f = i ◦ f 0 . Proof. Omitted.



Lemma 21.4.6. Let X, Y be a locally ringed spaces. Let I ⊂ OX be a locally generated sheaf of ideals. Let i : Z → X be the associated closed subspace. A morphism f : Y → X factors through Z if and only if the map f ∗ I → f ∗ OX = OY is zero. If this is the case the morphism g : Y → Z such that f = i ◦ g is unique. Proof. Clearly if f factors as Y → Z → X then the map f ∗ I → OY is zero. Conversely suppose that f ∗ I → OY is zero. Pick any y ∈ Y , and consider the ring map fy] : OX,f (y) → OY,y . Since the composition Iy → OX,f (y) → OY,y is zero by assumption and since fy] (1) = 1 we see that 1 6∈ Iy , i.e., Iy 6= OX,f (y) . We conclude that f (Y ) ⊂ Z = Supp(OX /I). Hence f = i ◦ g where g : Y → Z is continuous. Consider the map f ] : OX → f∗ OY . The assumption f ∗ I → OY is zero implies that the composition I → OX → f∗ OY is zero by adjointness of f∗ and f ∗ . In other words, we obtain a morphism of sheaves of rings f ] : OX /I → f∗ OY . Note that f∗ OY = i∗ g∗ OY and that OX /I = i∗ OZ . By Sheaves, Lemma 6.32.4 we obtain a unique morphism of sheaves of rings g ] : OZ → g∗ OY whose pushforward under i is f ] . We omit the verification that (g, g ] ) defines a morphism of locally ringed spaces and that f = i ◦ g as a morphism of locally ringed spaces. The uniqueness of (g, g ] ) was pointed out above.  Lemma 21.4.7. Let f : X → Y be a morphism of locally ringed spaces. Let I ⊂ OY be a sheaf of ideals which is locally generated by functions. Let i : Z → Y be the closed subspace associated to the sheaf of ideals I. Let J be the image of the map f ∗ I → f ∗ OY = OX . Then this ideal is locally generated by sections. Moreover, let i0 : Z 0 → X be the associated closed subspace of X. There exists a unique morphism of locally ringed spaces f 0 : Z 0 → Z such that the following diagram is a commutative square of locally ringed spaces Z0

i0

f0

 Z

i

/X  /Y

f

Moreover, this diagram is a fibre square in the category of locally ringed spaces.

21.5. AFFINE SCHEMES

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Proof. The ideal J is locally generated by sections by Modules, Lemma 15.8.2. The rest of the lemma follows from the characterization, in Lemma 21.4.6 above, of what it means for a morphism to factor through a closed subscheme.  21.5. Affine schemes Let R be a ring. Consider the topological space Spec(R) associated to R, see Algebra, Section 7.16. We will endow this space with a sheaf of rings OSpec(R) and the resulting pair (Spec(R), OSpec(R) ) will be an affine scheme. Recall that Spec(R) has a basis of open sets D(f ), f ∈ R which we call standard opens, see Algebra, Definition 7.16.3. In addition, the intersection of two standard opens is another: D(f ) ∩ D(g) = D(f g), f, g ∈ R. Lemma 21.5.1. Let R be a ring. Let f ∈ R. (1) If g ∈ R and D(g) ⊂ D(f ), then (a) f is invertible in Rg , (b) g e = af for some e ≥ 1 and a ∈ R, (c) there is a canonical ring map Rf → Rg , and (d) there is a canonical Rf -module map Mf → Mg for any R-module M . (2) Any open covering Sn of D(f ) can be refined to a finite open covering of the form D(f ) = i=1 D(gi ). S (3) If g1 , . . . , gn ∈ R, then D(f ) ⊂ D(gi ) if and only if g1 , . . . , gn generate the unit ideal in Rf . Proof. Recall that D(g) = Spec(Rg ) (see Algebra, Lemma 7.16.6). Thus (a) holds because f maps to an element of Rg which is not contained in any prime ideal, and hence invertible, see Algebra, Lemma 7.16.2. Write the inverse of f in Rg as a/g d . This means g d − af is annihilated by a power of g, whence (b). For (c), the map Rf → Rg exists by (a) from the universal property of localization, or we can define it by mapping b/f n to an b/g ne . The equality Mf = M ⊗R Rf can be used to obtain the map on modules, or we can define Mf → Mg by mapping x/f n to an x/g ne . Recall that D(f ) is quasi-compact, see Algebra, Lemma 7.27.1. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology. The third statement follows directly from Algebra, Lemma 7.16.2.



In Sheaves, Section 6.30 we defined the notion of a sheaf on a basis, and we showed that it is essentially equivalent to the notion of a sheaf on the space, see Sheaves, Lemmas 6.30.6 and 6.30.9. Moreover, we showed in Sheaves, Lemma 6.30.4 that it is sufficient to check the sheaf condition on a cofinal system of open coverings for each standard open. By the lemma above it suffices to check on the finite coverings by standard opens. Definition 21.5.2. Let R be a ring. Sn (1) A standard open covering of Spec(R) is a covering Spec(R) = i=1 D(fi ), where f1 , . . . , fn ∈ R. (2) Suppose that D(f ) ⊂ Spec(R) is a standard open. A standard open covSn ering of D(f ) is a covering D(f ) = i=1 D(gi ), where g1 , . . . , gn ∈ R.

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f on the basis Let R be a ring. Let M be an R-module. We will define a presheaf M of standard opens. Suppose that U ⊂ Spec(R) is a standard open. If f, g ∈ R are such that D(f ) = D(g), then by Lemma 21.5.1 above there are canonical maps Mf → Mg and Mg → Mf which are mutually inverse. Hence we may choose any f such that U = D(f ) and define f(U ) = Mf . M Note that if D(g) ⊂ D(f ), then by Lemma 21.5.1 above we have a canonical map f(D(f )) = Mf −→ Mg = M f(D(g)). M Clearly, this defines a presheaf of abelian groups on the basis of standard opens. If e is a presheaf of rings on the basis of standard opens. M = R, then R f at a point x ∈ Spec(R). Suppose that x corresponds Let us compute the stalk of M to the prime p ⊂ R. By definition of the stalk we see that fx = colimf ∈R,f 6∈p Mf M Here the set {f ∈ R, f 6∈ p} is partially ordered by the rule f ≥ f 0 ⇔ D(f ) ⊂ D(f 0 ). If f1 , f2 ∈ R\p, then we have f1 f2 ≥ f1 in this ordering. Hence by Algebra, Lemma 7.9.9 we conclude that fx = Mp . M Next, we check the sheaf condition for the standard open coverings. If D(f ) = S n i=1 D(gi ), then the sheaf condition for this covering is equivalent with the exactness of the sequence M M 0 → Mf → Mgi → Mgi gj . Note that D(gi ) = D(f gi ), and hence we can rewrite this sequence as the sequence M M 0 → Mf → Mf gi → Mf gi g j . In addition, by Lemma 21.5.1 above we see that g1 , . . . , gn generate the unit ideal in Rf . Thus we may apply Algebra, Lemma 7.21.2 to the module Mf over Rf and the elements g1 , . . . , gn . We conclude that the sequence is exact. By the remarks f is a sheaf on the basis of standard opens. made above, we see that M Thus we conclude from the material in Sheaves, Section 6.30 that there exists a e on the standard opens. Note unique sheaf of rings OSpec(R) which agrees with R that by our computation of stalks above, the stalks of this sheaf of rings are all local rings. Similarly, for any R-module M there exists a unique sheaf of OSpec(R) -modules F f on the standard opens, see Sheaves, Lemma 6.30.12. which agrees with M Definition 21.5.3. Let R be a ring. (1) The structure sheaf OSpec(R) of the spectrum of R is the unique sheaf of e on the basis of standard opens. rings OSpec(R) which agrees with R (2) The locally ringed space (Spec(R), OSpec(R) ) is called the spectrum of R and denoted Spec(R). f to all opens of Spec(R) is (3) The sheaf of OSpec(R) -modules extending M called the sheaf of OSpec(R) -modules associated to M . This sheaf is def as well. noted M

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We summarize the results obtained so far. f be the sheaf of Lemma 21.5.4. Let R be a ring. Let M be an R-module. Let M OSpec(R) -modules associated to M . We have Γ(Spec(R), OSpec(R) ) = R. f) = M as an R-module. We have Γ(Spec(R), M For every f ∈ R we have Γ(D(f ), OSpec(R) ) = Rf . f) = Mf as an Rf -module. For every f ∈ R we have Γ(D(f ), M f are Whenever D(g) ⊂ D(f ) the restriction mappings on OSpec(R) and M the maps Rf → Rg and Mf → Mg from Lemma 21.5.1. (6) Let p be a prime of R, and let x ∈ Spec(R) be the corresponding point. We have OSpec(R),x = Rp . (7) Let p be a prime of R, and let x ∈ Spec(R) be the corresponding point. We have Fx = Mp as an Rp -module.

(1) (2) (3) (4) (5)

Moreover, all these identifications are functorial in the R module M . In particular, f is an exact functor from the category of R-modules to the the functor M 7→ M category of OSpec(R) -modules. Proof. Assertions (1) - (7) are clear from the discussion above. The exactness f follows from the fact that the functor M 7→ Mp is exact of the functor M 7→ M and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 15.3.1.  Definition 21.5.5. An affine scheme is a locally ringed space isomorphic as a locally ringed space to Spec(R) for some ring R. A morphism of affine schemes is a morphism in the category of locally ringed spaces. It turns out that affine schemes play a special role among all locally ringed spaces, which is what the next section is about. 21.6. The category of affine schemes Note that if Y is an affine scheme, then its points are in canonical 1 − 1 bijection with prime ideals in Γ(Y, OY ). Lemma 21.6.1. Let X be a locally ringed space. Let Y be an affine scheme. Let f ∈ Mor(X, Y ) be a morphism of locally ringed spaces. Given a point x ∈ X consider the ring maps f]

Γ(Y, OY ) −→ Γ(X, OX ) → OX,x Let p ⊂ Γ(Y, OY ) denote the inverse image of mx . Let y ∈ Y be the corresponding point. Then f (x) = y. Proof. Consider the commutative diagram Γ(X, OX ) O

/ OX,x O

Γ(Y, OY )

/ OY,f (x)

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(see the discussion of f -maps below Sheaves, Definition 6.21.7). Since the right vertical arrow is local we see that mf (x) is the inverse image of mx . The result follows.  Lemma 21.6.2. Let X be a locally ringed space. Let f ∈ Γ(X, OX ). The set D(f ) = {x ∈ X | image f 6∈ mx } is open. Moreover f |D(f ) has an inverse. Proof. This is a special case of Modules, Lemma 15.21.7, but we also give a direct proof. Suppose that U ⊂ X and V ⊂ X are two open subsets such that f |U has an inverse g and f |V has an inverse h. Then clearly g|U ∩V = h|U ∩V . Thus it suffices to show that f is invertible in an open neighbourhood of any x ∈ D(f ). This is clear because f 6∈ mx implies that f ∈ OX,x has an inverse g ∈ OX,x which means there is some open neighbourhood x ∈ U ⊂ X so that g ∈ OX (U ) and g · f |U = 1.  Lemma 21.6.3. In Lemma 21.6.2 above, if X is an affine scheme, then the open D(f ) agrees with the standard open D(f ) defined previously (in Algebra, Definition 7.16.1). Proof. Omitted.



Lemma 21.6.4. Let X be a locally ringed space. Let Y be an affine scheme. The map Mor(X, Y ) −→ Hom(Γ(Y, OY ), Γ(X, OX )) which maps f to f ] (on global sections) is bijective. Proof. Since Y is affine we have (Y, OY ) ∼ = (Spec(R), OSpec(R) ) for some ring R. During the proof we will use facts about Y and its structure sheaf which are direct consequences of things we know about the spectrum of a ring, see e.g. Lemma 21.5.4. Motivated by the lemmas above we construct the inverse map. Let ψY : Γ(Y, OY ) → Γ(X, OX ) be a ring map. First, we define the corresponding map of spaces Ψ : X −→ Y by the rule of Lemma 21.6.1. In other words, given x ∈ X we define Ψ(x) to be the point of Y corresponding to the prime in Γ(Y, OY ) which is the inverse image ψY

of mx under the composition Γ(Y, OY ) −−→ Γ(X, OX ) → OX,x . We claim that the map Ψ : X → Y is continuous. The standard opens D(g), for g ∈ Γ(Y, OY ) are a basis for the toppology of Y . Thus it suffices to prove that Ψ−1 (D(g)) is open. By construction of Ψ the inverse image Ψ−1 (D(g)) is exactly the set D(ψY (g)) ⊂ X which is open by Lemma 21.6.2. Hence Ψ is continuous. Next we construct a Ψ-map of sheaves from OY to OX . By Sheaves, Lemma 6.30.14 it suffices to define ring maps ψD(g) : Γ(D(g), OY ) → Γ(Ψ−1 (D(g)), OX ) compatible with restriction maps. We have a canonical isomorphism Γ(D(g), OY ) = Γ(Y, OY )g , because Y is an affine scheme. Because ψY (g) is invertible on D(ψY (g)) we see that there is a canonical map Γ(Y, OY )g −→ Γ(Ψ−1 (D(g)), OX ) = Γ(D(ψY (g)), OX ) extending the map ψY by the universal property of localization. Note that there is no choice but to take the canonical map here! And we take this, combined with the

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canonical identification Γ(D(g), OY ) = Γ(Y, OY )g , to be ψD(g) . This is compatible with localization since the restriction mapping on the affine schemes are defined in terms of the universal properties of localization also, see Lemmas 21.5.4 and 21.5.1. Thus we have defined a morphism of ringed spaces (Ψ, ψ) : (X, OX ) → (Y, OY ) recovering ψY on global sections. To see that it is a morphism of locally ringed spaces we have to show that the induced maps on local rings ψx : OY,Ψ(x) −→ OX,x are local. This follows immediately from the commutative diagram of the proof of Lemma 21.6.1 and the definition of Ψ. Finally, we have to show that the constructions (Ψ, ψ) 7→ ψY and the construction ψY 7→ (Ψ, ψ) are inverse to each other. Clearly, ψY 7→ (Ψ, ψ) 7→ ψY . Hence the only thing to prove is that given ψY there is at most one pair (Ψ, ψ) giving rise to it. The uniqueness of Ψ was shown in Lemma 21.6.1 and given the uniqueness of Ψ the uniqueness of the map ψ was pointed out during the course of the proof above.  Lemma 21.6.5. The category of affine schemes is equivalent to the opposite of the category of rings. The equivalence is given by the functor that associates to an affine scheme the global sections of its structure sheaf. Proof. This is now clear from Definition 21.5.5 and Lemma 21.6.4.



Lemma 21.6.6. Let Y be an affine scheme. Let f ∈ Γ(Y, OY ). The open subspace D(f ) is an affine scheme. Proof. We may assume that Y = Spec(R) and f ∈ R. Consider the morphism of affine schemes φ : U = Spec(Rf ) → Spec(R) = Y induced by the ring map R → Rf . By Algebra, Lemma 7.16.6 we know that it is a homeomorphism onto D(f ). On the other hand, the map f −1 OY → OU is an isomorphism on stalks, hence an isomorphism. Thus we see that φ is an open immersion. We conclude that D(f ) is isomorphic to U by Lemma 21.3.4.  Lemma 21.6.7. The category of affine schemes has finite products, and fibre products. In other words, it has finite limits. Moreover, the products and fibre products in the category of affine schemes are the same as in the category of locally ringed spaces. In a formula, we have (in the category of locally ringed spaces) Spec(R) × Spec(S) = Spec(R ⊗Z S) and given ring maps R → A, R → B we have Spec(A) ×Spec(R) Spec(B) = Spec(A ⊗R B). Proof. This is just an application of Lemma 21.6.4. First of all, by that lemma, the affine scheme Spec(Z) is the final object in the category of locally ringed spaces. Thus the first displayed formula follows from the second. To prove the second note that for any locally ringed space X we have Mor(X, Spec(A ⊗R B))

=

Hom(A ⊗R B, OX (X))

=

Hom(A, OX (X)) ×Hom(R,OX (X)) Hom(B, OX (X))

=

Mor(X, Spec(A)) ×Mor(X,Spec(R)) Mor(X, Spec(B))

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which proves the formula. See Categories, Section 4.6 for the relevant definitions.  Lemma 21.6.8. Let X be a locally ringed space. Assume X = U q V with U and V open and such that U , V are affine schemes. Then X is an affine scheme. Proof. Set R = Γ(X, OX ). Note that R = OX (U ) × OX (V ) by the sheaf property. By Lemma 21.6.4 there is a canonical morphism of locally ringed spaces X → Spec(R). By Algebra, Lemma 7.19.2 we see that as a topological space Spec(OX (U )) q Spec(OX (V )) = Spec(R) with the maps coming from the ring homomorphisms R → OX (U ) and R → OX (V ). This of course means that Spec(R) is the coproduct in the category of locally ringed spaces as well. By assumption the morphism X → Spec(R) induces an isomorphism of Spec(OX (U )) with U and similarly for V . Hence X → Spec(R) is an isomorphism.  21.7. Quasi-Coherent sheaves on affines Recall that we have defined the abstract notion of a quasi-coherent sheaf in Modules, Definition 15.10.1. In this section we show that any quasi-coherent sheaf on an affine f associated to an R-module M . scheme Spec(R) corresponds the the sheaf M Lemma 21.7.1. Let (X, OX ) = (Spec(R), OSpec(R) ) be an affine scheme. Let M be f associated an R-module. There exists a canonical isomorphism between the sheaf M to the R-module M (Definition 21.5.3) and the sheaf FM associated to the Rmodule M (Modules, Definition 15.10.6). This isomorphism is functorial in M . In f are quasi-coherent. Moreover, they are characterized by particular, the sheaves M the following mapping property f, F) = HomR (M, Γ(X, F)) HomO (M X

f → F corresponds to its effect for any sheaf of OX -modules F. Here a map α : M on global sections. f corresponding Proof. By Modules, Lemma 15.10.5 we have a morphism FM → M f to the map M → Γ(X, M ) = M . Let x ∈ X correspond to the prime p ⊂ R. The induced map on stalks are the maps OX,x ⊗R M → Mp which are isomorphisms f is an isomorphism. The because Rp ⊗R M = Mp . Hence the map FM → M mapping property follows from the mapping property of the sheaves FM .  Lemma 21.7.2. Let (X, OX ) = (Spec(R), OSpec(R) ) be an affine scheme. There are canonical isomorphisms f ⊗O N e , see Modules, Section 15.15. (1) M^ ⊗R N ∼ =M X n n n (M ) ∼ ^ f), Sym f), and ∧^ f), see (2) T^ (M ) ∼ (M ) ∼ = Tn (M = Symn (M = ∧n (M Modules, Section 15.18. f, N e) ∼ (3) if M is a finitely presented R-module, then Hom OX (M = Hom^ R (M, N ), see Modules, Section 15.19. f ⊗O N e we have to give a map on global Proof. To give a map M^ ⊗R N into M X f e sections M ⊗R N → Γ(X, M ⊗OX N ) which exists by definition of the tensor product of sheaves of modules. To see that this map is an isomorphism it suffices to check that it is an isomorphism on stalks. And this follows from the description of the f (as a functor) and Modules, Lemma 15.15.1. stalks of M

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The proof of (2) is similar, using Modules, Lemma 15.18.2. f has a global For (3) note that if M is finitely presented as an R-module then M finite presentation as an OX -module. Hence Modules, Lemma 15.19.3 applies.  Lemma 21.7.3. Let (X, OX ) = (Spec(S), OSpec(S) ), (Y, OY ) = (Spec(R), OSpec(R) ) be affine schemes. Let ψ : (X, OX ) → (Y, OY ) be a morphism of affine schemes, corresponding to the ring map ψ ] : R → S (see Lemma 21.6.5). f = S^ (1) We have ψ ∗ M ⊗R M functorially in the R-module M . e =N g (2) We have ψ∗ N R functorially in the S-module N . Proof. The first assertion follows from the identification in Lemma 21.7.1 and the result of Modules, Lemma 15.10.7. The second assertion follows from the fact that ψ −1 (D(f )) = D(ψ ] (f )) and hence e (D(f )) = N e (D(ψ ] (f ))) = Nψ] (f ) = (NR )f = N g ψ∗ N R (D(f )) as desired.



f to a standard Lemma 21.7.3 above says in particular that if you restrict the sheaf M g affine open subspace D(f ), then you get Mf . We will use this from now on without further mention. Lemma 21.7.4. Let (X, OX ) = (Spec(R), OSpec(R) ) be an affine scheme. Let F be a quasi-coherent OX -module. Then F is isomorphic to the sheaf associated to the R-module Γ(X, F). Proof. Let F be a quasi-coherent OX -module. Since every standard open D(f ) is quasi-compact we see that X is a locally quasi-compact, i.e., every point has a fundamental system of quasi-compact neighbourhoods, see Topology, Definition 5.18.1. Hence by Modules, Lemma 15.10.8 for every prime p ⊂ R corresponding to x ∈ X there exists an open neighbourhood x ∈ U ⊂ X such that F|U is isomorphic to the quasi-coherent sheaf associated to some OX (U )-module M . In other words, we get an open covering by U ’s with this property. By Lemma 21.5.1 for example we can refine S this covering to a standard open covering. Thus we get a covering Spec(R) = D(fi ) and Rfi -modules Mi and isomorphisms ϕi : F|D(fi ) → FMi for some Rfi -module Mi . On the overlaps we get isomorphisms FMi |D(fi fj )

ϕ−1 i |D(fi fj )

/ F|D(f f ) i j

ϕj |D(fi fj )

/ FM |D(f f ) . j i j

Let us denote these ψij . It is clear that we have the cocycle condition ψjk |D(fi fj fk ) ◦ ψij |D(fi fj fk ) = ψik |D(fi fj fk ) on triple overlaps. Recall that each of the open subspaces D(fi ), D(fi fj ), D(fi fj fk ) is an affine fi by Lemma 21.7.1 scheme. Hence the sheaves FMi are isomorphic to the sheaves M above. In particular we see that FMi (D(fi fj )) = (Mi )fj , etc. Also by Lemma 21.7.1 above we see that ψij corresponds to a unique Rfi fj -module isomorphism ψij : (Mi )fj −→ (Mj )fi

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namely, the effect of ψij on sections over D(fi fj ). Moreover these then satisfy the cocycle condition that / (Mk )fi fj 9

ψik

(Mi )fj fk ψij

% (Mj )fi fk

ψjk

commutes (for any triple i, j, k). Now Algebra, Lemma 7.22.4 shows that there exist an R-module M such that f. At this point Mi = Mfi compatible with the morphisms ψij . Consider FM = M f it is a formality to show that M is isomorphic to the quasi-coherent sheaf F we f give rise to isomorphic sets of started out with. Namely, the sheaves F and M S glueing data of sheaves of OX -modules with respect to the covering X = D(fi ), see Sheaves, Section 6.33 and in particular Lemma 6.33.4. Explicitly, in the current situation, this boils down to the following argument: Let us construct an R-module map M −→ Γ(X, F). Namely, given m ∈ M we get mi = m/1 ∈ Mfi = Mi by construction of M . By construction of Mi this corresponds to a section si ∈ F(Ui ). (Namely, ϕ−1 i (mi ).) We claim that si |D(fi fj ) = sj |D(fi fj ) . This is true because, by construction of M , we have ψij (mi ) = mj , and by the construction of the ψij . By the sheaf condition of F this collection of sections gives rise to a unique section s of F over X. We leave it to the reader to show that m 7→ s is a R-module map. By Lemma 21.7.1 we obtain an associated OX -module map f −→ F. M By construction this map reduces to the isomorphisms ϕ−1 i on each D(fi ) and hence is an isomorphism.  Lemma 21.7.5. Let (X, OX ) = (Spec(R), OSpec(R) ) be an affine scheme. The f and F 7→ Γ(X, F) define quasi-inverse equivalences of categories functors M 7→ M / QCoh(OX ) o Mod-R between the category of quasi-coherent OX -modules and the category of R-modules. Proof. See Lemmas 21.7.1 and 21.7.4 above.



From now on we will not distinghuish between quasi-coherent sheaves on affine f. schemes and sheaves of the form M Lemma 21.7.6. Let X = Spec(R) be an affine scheme. Kernels and cokernels of maps of quasi-coherent OX -modules are quasi-coherent. Proof. This follows from the exactness of the functor e since by Lemma 21.7.1 we f→N e comes from an R-module map ϕ : M → N . (So know that any map ψ : M ^ and Coker(ψ) = Coker(ϕ).) ^ we have Ker(ψ) = Ker(ϕ)  Lemma 21.7.7. Let X = Spec(R) be an affine scheme. The direct sum of an arbitrary collection of quasi-coherent sheaves on X is quasi-coherent. The same holds for colimits.

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Proof. Suppose Fi , i ∈ I is a collection of quasi-coherent sheaves on X. By Lemma fi for some R-module Mi . Set M = L Mi . 21.7.5 above we can write Fi = M f. For each standard open D(f ) we have Consider the sheaf M M  M f(D(f )) = Mf = M Mi = Mi,f . f

f is the direct sum of the sheaves Hence we see that the quasi-coherent OX -module M Fi . A similar argument works for general colimits.  Lemma 21.7.8. Let (X, OX ) = (Spec(R), OSpec(R) ) be an affine scheme. Suppose that 0 → F1 → F2 → F3 → 0 is a short exact sequence of sheaves OX -modules. If two out of three are quasicoherent then so is the third. Proof. This is clear in case both F1 and F2 are quasi-coherent because the functor f is exact, see Lemma 21.5.4. Similarly in case both F2 and F3 are quasiM 7→ M f1 and F3 = M f3 are quasi-coherent. Set coherent. Now, suppose that F1 = M M2 = Γ(X, F2 ). We claim it suffices to show that the sequence 0 → M1 → M2 → M3 → 0 is exact. Namely, if this is the case, then (by using the mapping property of Lemma 21.7.1) we get a commutative diagram 0

f1 /M

f2 /M

f3 /M

/0

0

 / F1

 / F2

 / F3

/0

and we win by the snake lemma. The “correct” argument here would be to show first that H 1 (X, F) = 0 for any quasi-coherent sheaf F. This is actually not all that hard, but it is perhaps better to postpone this till later. Instead we use a small trick. Pick m ∈ M3 = Γ(X, F3 ). Consider the following set I = {f ∈ R | the element f m comes from M2 }. Clearly this is an ideal. It suffices to show 1 ∈ I. Hence it suffices to show that for any prime p there exists an f ∈ I, f 6∈ p. Let x ∈ X be the point corresponding to p. Because surjectivity can be checked on stalks there exists an open neighbourhood U of x such that m|U comes from a local section s ∈ F2 (U ). In fact we may assume that U = D(f ) is a standard open, i.e., f ∈ R, f 6∈ p. We will show that for some N  0 we have f N ∈ I, which will finish the proof. Take any point z ∈ V (f ), say corresponding to the prime q ⊂ R. We can also find a g ∈ R, g 6∈ q such that m|D(g) lifts to some s0 ∈ F2 (D(g)). Consider the difference s|D(f g) − s0 |D(f g) . This is an element m0 of F1 (D(f g)) = (M1 )f g . For some integer n = n(z) the element f n m0 comes from some m01 ∈ (M1 )g . We see that f n s extends to a section σ of F2 on D(f ) ∪ D(g) because it agrees with the restriction of f n s0 + m01 on D(f ) ∩ D(g) = D(f g). Moreover, σ maps to the restriction of f n m to D(f ) ∪ D(g).

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Since V (f ) is quasi-compact, there exists a finite list of elements g1 , . . . , gm ∈ R S such that V (f ) ⊂ D(gj ), an integer n > 0 and sections σj ∈ F2 (D(f ) ∪ D(gj )) such that σj |D(f ) = f n s and σj maps to the section f n m|D(f )∪D(gj ) of F3 . Consider the differences σj |D(f )∪D(gj gk ) − σk |D(f )∪D(gj gk ) . These correspond to sections of F1 over D(f ) ∪ D(gj gk ) which are zero on D(f ). In particular their images in F1 (D(gj gk )) = (M1 )gj gk are zero in (M1 )gj gk f . Thus some high power of f kills each and every one of these. In other words, the elements f N σj , for some N  0 satisfySthe glueing condition of the sheaf property and give rise to a section σ of F2 over (D(f ) ∪ D(gj )) = X as desired.  21.8. Closed subspaces of affine schemes Example 21.8.1. Let R be a ring. Let I ⊂ R be an ideal. Consider the morphism of affine schemes i : Z = Spec(R/I) → Spec(R) = X. By Algebra, Lemma 7.16.7 this is a homeomorphism of Z onto a closed subset of X. Moreover, if I ⊂ p ⊂ R is a prime corresponding to a point x = i(z), x ∈ X, z ∈ Z, then on stalks we get the map OX,x = Rp −→ Rp /IRp = OZ,z Thus we see that i is a closed immersion of locally ringed spaces, see Definition 21.4.1. Clearly, this is (isomorphic) to the closed subspace associated to the quasie as in Example 21.4.3. coherent sheaf of ideals I, Lemma 21.8.2. Let (X, OX ) = (Spec(R), OSpec(R) ) be an affine scheme. Let i : Z → X be any closed immersion of locally ringed spaces. Then there exists an unique ideal I ⊂ R such that the morphism i : Z → X can be identified with the closed immersion Spec(R/I) → Spec(R) constructed in Example 21.8.1 above. Proof. This is kind of silly! Namely, by Lemma 21.4.5 we can identify Z → X with the closed subspace associated to a sheaf of ideals I ⊂ OX as in Definition 21.4.4 and Example 21.4.3. By our conventions this sheaf of ideals is locally generated by sections as a sheaf of OXL -modules. Hence the quotient sheaf OX /I is locally on X the cokernel of a map j∈J OU → OU . Thus by definition, OX /I is quasicoherent. By our results in Section 21.7 it is of the form Se for some R-module S. e → Se is surjective we see by Lemma 21.7.8 that also I Moreover, since OX = R e Of course I ⊂ R and S = R/I and everything is is quasi-coherent, say I = I. clear.  21.9. Schemes Definition 21.9.1. A scheme is a locally ringed space with the property that every point has an open neighbourhood which is an affine scheme. A morphism of schemes is a morphism of locally ringed spaces. The category of schemes will be denoted Sch. Let X be a scheme. We will use the following (very slight) abuse of language. We will say U ⊂ X is an affine open, or an open affine if the open subspace U is an affine scheme. We will often write U = Spec(R) to indicate that U is isomorphic to Spec(R) and moreover that we will identify (temporarily) U and Spec(R).

21.10. IMMERSIONS OF SCHEMES

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Lemma 21.9.2. Let X be a scheme. Let j : U → X be an open immersion of locally ringed spaces. Then U is a scheme. In particular, any open subspace of X is a scheme. Proof. Let U ⊂ X. Let u ∈ U . Pick an affine open neighbourhood u ∈ V ⊂ X. Because standard opens of V form a basis of the topology on V we see that there exists a f ∈ OV (V ) such that D(f ) ⊂ U . And D(f ) is an affine scheme by Lemma 21.6.6. This proves that every point of U has an open neighbourhood which is affine.  Clearly the lemma (or its proof) shows that any scheme X has a basis (see Topology, Section 5.3) for the topology consisting of affine opens. Example 21.9.3. Let k be a field. An example of a scheme which is not affine is given by the open subspace U = Spec(k[x, y]) \ {(x, y)} of the affine scheme X = Spec(k[x, y]). It is covered by two affines, namely D(x) = Spec(k[x, y, 1/x]) and D(y) = Spec(k[x, y, 1/y]) whose intersection is D(xy) = Spec(k[x, y, 1/xy]). By the sheaf property for OU there is an exact sequence 0 → Γ(U, OU ) → k[x, y, 1/x] × k[x, y, 1/y] → k[x, y, 1/xy] We conclude that the map k[x, y] → Γ(U, OU ) (coming from the morphism U → X) is an isomorphism. Therefore U cannot be affine since if it was then by Lemma 21.6.5 we would have U ∼ = X. 21.10. Immersions of schemes In Lemma 21.9.2 we saw that any open subspace of a scheme is a scheme. Below we will prove that the same holds for a closed subspace of a scheme. Note that the notion of a quasi-coherent sheaf of OX -modules is defined for any ringed space X in particular when X is a scheme. By our efforts in Section 21.7 f for some we know that such a sheaf is on any affine open U ⊂ X of the form M OX (U )-module M . Lemma 21.10.1. Let X be a scheme. Let i : Z → X be a closed immersion of locally ringed spaces. (1) The locally ringed space Z is a scheme, (2) the kernel I of the map OX → i∗ OZ is a quasi-coherent sheaf of ideals, (3) for any affine open U = Spec(R) of X the morphism i−1 (U ) → U can be identified with Spec(R/I) → Spec(R) for some ideal I ⊂ R, and e (4) we have I|U = I. In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of X is a scheme. Proof. Let i : Z → X be a closed immersion. Let z ∈ Z be a point. Choose any affine open neighbourhood i(z) ∈ U ⊂ X. Say U = Spec(R). By Lemma 21.8.2 we know that i−1 (U ) → U can be identified with the morphism of affine schemes Spec(R/I) → Spec(R). First of all this implies that z ∈ i−1 (U ) ⊂ Z is an affine e In neighbourhood of z. Thus Z is a scheme. Second this implies that I|U is I. other words for every point x ∈ i(Z) there exists an open neighbourhood such that I is quasi-coherent in that neighbourhood. Note that I|X\i(Z) ∼ = OX\i(Z) . Thus the restriction of the sheaf of ideals is quasi-coherent on X \ i(Z) also. We conclude that I is quasi-coherent. 

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Definition 21.10.2. Let X be a scheme. (1) A morphism of schemes is called an open immersion if it is an open immersion of locally ringed spaces (see Definition 21.3.1). (2) An open subscheme of X is an open subspace of X which is a scheme by Lemma 21.9.2 above. (3) A morphism of schemes is called a closed immersion if it is a closed immersion of locally ringed spaces (see Definition 21.4.1). (4) A closed subscheme of X is a closed subspace of X which is a scheme by Lemma 21.10.1 above. (5) A morphism of schemes f : X → Y is called an immersion, or a locally closed immersion if it can be factored as j ◦i where i is a closed immersion and j is an open immersion. It follows from the lemmas in Sections 21.3 and 21.4 that any open (resp. closed) immersion of schemes is isomorphic to the inclusion of an open (resp. closed) subscheme of the target. We will define locally closed subschemes below. Remark 21.10.3. If f : X → Y is an immersion of schemes, then it is in general not possible to factor f as an open immersion followed by a closed immersion. See Morphisms, Example 24.3.4. Lemma 21.10.4. Let f : Y → X be an immersion of schemes. Then f is a closed immersion if and only if f (Y ) ⊂ X is a closed subset. Proof. If f is a closed immersion then f (Y ) is closed by definition. Conversely, suppose that f (Y ) is closed. By definition there exists an open subscheme U ⊂ X such that f is the composition of a closed immersion i : Y → U and the open immersion j : U → X. Let I ⊂ OU be the quasi-coherent sheaf of ideals associated to the closed immerion i. Note that I|U \i(Y ) = OU \i(Y ) = OX\i(Y ) |U \i(Y ) . Thus we may glue (see Sheaves, Section 6.33) I and OX\i(Y ) to a sheaf of ideals J ⊂ OX . Since every point of X has a neighbourhood where J is quasi-coherent, we see that J is quasi-coherent (in particular locally generated by sections). By construction OX /J is supported on U and equal to OU /I. Thus we see that the closed subspaces associated to I and J are canonically isomorphic, see Example 21.4.3. In particular the closed subspace of U associated to I is isomorphic to a closed subspace of X. Since Y → U is identified with the closed subspace associated to I, see Lemma 21.4.5, we conclude that Y → U → X is a closed immersion.  Let f : Y → X be an immersion. Let Z = f (Y ) \ f (Y ) which is a closed subset of X. Let U = X \ Z. The lemma implies that U is the biggest open subspace of X such that f : Y → X factors through a closed immersion into U . If we define a locally closed subscheme of X as a pair (Z, U ) consisting of a closed subscheme Z of an open subscheme U of X such that in addition Z ∪ U = X. We usually just say “let Z be a locally closed subscheme of X” since we may recover U from the morphism Z → X. The above then shows that any immersion f : Y → X factors uniquely as Y → Z → X where Z is a locally closed subspace of X and Y → Z is an isomorphism. The interest of this is that the collection of locally closed subschemes of X forms a set. We may define a partial ordering on this set, which we call inclusion for obvious reasons. To be explicit, if Z → X and Z 0 → X are two locally closed subschemes of X, then we say that Z is contained in Z 0 simply if the morphism Z → X factors

21.11. ZARISKI TOPOLOGY OF SCHEMES

1257

through Z 0 . If it does, then of course Z is identified with a unique locally closed subscheme of Z 0 , and so on. 21.11. Zariski topology of schemes See Topology, Section 5.1 for some basic material in topology adapted to the Zariski topology of schemes. Lemma 21.11.1. Let X be a scheme. Any irreducible closed subset of X has a unique generic point. In other words, X is a sober topological space, see Topology, Definition 5.5.4. Proof. Let Z ⊂ X be an irreducible closed subset. For every affine open U ⊂ X, U = Spec(R) we know that Z ∩ U = V (I) for a unique radical ideal I ⊂ R. Note that Z ∩ I is either empty or irreducible. In the second case (which occurs for at least one U ) we see that I = p is a prime ideal, which is a generic point ξ of Z ∩ U . It follows that Z = {ξ}, in other words ξ is a generic point of Z. If ξ 0 was a second generic point, then ξ 0 ∈ Z ∩ U and it follows immediately that ξ 0 = ξ.  Lemma 21.11.2. Let X be a scheme. The collection of affine opens of X forms a basis for the topology on X. Proof. This follows from the discussion on open subschemes in Section 21.9.



Remark 21.11.3. In general the intersection of two affine opens in X is not affine open. See Example 21.14.3. Lemma 21.11.4. The underlying topological space of any scheme is locally quasicompact, see Topology, Definition 5.18.1. Proof. This follows from Lemma 21.11.2 above and the fact that the spectrum of ring is quasi-compact, see Algebra, Lemma 7.16.10.  Lemma 21.11.5. Let X be a scheme. Let U, V be affine opens of X, and let x ∈ U ∩ V . There exists an affine open neighbourhood W of x such that W is a standard open of both U and V . Proof. Write U = Spec(A) and V = Spec(B). Say x corresponds to the prime p ⊂ A and the prime q ⊂ B. We may choose a f ∈ A, f 6∈ p such that D(f ) ⊂ U ∩V . Note that any standard open of D(f ) is a standard open of Spec(A) = U . Hence we may assume that U ⊂ V . In other words, now we may think of U as an affine open of V . Next we choose a g ∈ B, g 6∈ q such that D(g) ⊂ U . In this case we see that D(g) = D(gA ) where gA ∈ A denotes the image of g ∈ A. Thus the lemma is proved.  S Lemma 21.11.6. Let X be a scheme. Let X = i Ui be an affine open covering. SLet V ⊂ X be an affine open. There exists a standard open covering V = j=1,...,m Vj (see Definition 21.5.2) such that each Vj is a standard open in one of the Ui . Proof. Pick v ∈ V . Then v ∈ Ui for some i. By Lemma 21.11.5 above there exists an open v ∈ Wv ⊂ V ∩ Ui such that Wv is a standard open in both V and Ui . Since V is quasi-compact the lemma follows.  Lemma 21.11.7. Let X be a scheme whose underlying topological space is a finite discrete set. Then X is affine.

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Proof. Say X = {x1 , . . . , xn }. Then Ui = {xi } is an open neighbourhood of xi . By Lemma 21.11.2 it is affine. Hence X is a finite disjoint union of affine schemes, and hence is affine by Lemma 21.6.8.  Example 21.11.8. There exists a scheme without closed points. Namely, let R be a local domain whose spectrum looks like (0) = p0 ⊂ p1 ⊂ p2 ⊂ . . . ⊂ m. Then the open subscheme Spec(R) \ {m} does not have a closed point. To see that such a ring R exists, we use that given any totally ordered group (Γ, ≥) there exists a valuation ring A with valuation group (Γ, ≥), see [Kru32]. See Algebra, P Section 7.47 for notation. We take Γ = Zx1 ⊕ Zx2 ⊕ Zx3 ⊕ . . . and we define i ai xi ≥ 0 if and only if the first nonzero ai is > 0, or all ai = 0. So x1 ≥ x2 ≥ x3 ≥ . . . ≥ 0. The subsets xi +Γ≥0 are prime ideals of (Γ, ≥), see Algebra, notation above Lemma 7.47.11. These together with ∅ and Γ≥0 are the only prime ideals. Hence A is an example of a ring with the given structure of its spectrum, by Algebra, Lemma 7.47.11. 21.12. Reduced schemes Definition 21.12.1. Let X be a scheme. We say X is reduced if every local ring OX,x is reduced. Lemma 21.12.2. A scheme X is reduced if and only if OX (U ) is a reduced ring for all U ⊂ X open. Proof. Assume that X is reduced. Let f ∈ OX (U ) be a section such that f n = 0. Then the image of f in OU,u is zero for all u ∈ U . Hence f is zero, see Sheaves, Lemma 6.11.1. Conversely, assume that OX (U ) is reduced for all opens U . Pick any nonzero element f ∈ OX,x . Any representative (U, f ∈ O(U )) of f is nonzero and hence not nilpotent. Hence f is not nilpotent in OX,x .  Lemma 21.12.3. An affine scheme Spec(R) is reduced if and only if R is reduced. Proof. The direct implication follows immediately from Lemma 21.12.2 above. In the other direction it follows since any localization of a reduced ring is reduced, and in particular the local rings of a reduced ring are reduced.  Lemma 21.12.4. Let X be a scheme. Let T ⊂ X be a closed subset. There exists a unique closed subscheme Z ⊂ X with the following properties: (a) the underlying topological space of Z is equal to T , and (b) Z is reduced. Proof. Let I ⊂ OX be the sub presheaf defined by the rule I(U ) = {f ∈ OX (U ) | f (t) = 0 for all t ∈ T ∩ U } Here we use f (t) to indicate the image of f in the residue field κ(t) of X at t. Because of the local nature of the condition it is clear that I is a sheaf of ideals. Moreover, let U = Spec(R) be an affine open. We may write T ∩ U = V (I) for a unique radical ideal I ⊂ R. Given a prime p ∈ V (I) corresponding to t ∈ T ∩ U and an element f ∈ R we have f (t) = 0 ⇔ f ∈ p. Hence I(U ) = ∩p∈V (I) p = I by Algebra, Lemma 7.16.2. Moreover, for any standard open D(g) ⊂ Spec(R) = U we have I(D(g)) = Ig by the same reasoning. Thus Ie and I|U agree (as ideals) on a basis of opens and hence are equal. Therefore I is a quasi-coherent sheaf of ideals. At this point we may define Z as the closed subspace associated to the sheaf of ideals I. For every affine open U = Spec(R) of X we see that Z ∩ U = Spec(R/I)

21.13. POINTS OF SCHEMES

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where I is a radical ideal and hence Z is reduced (by Lemma 21.12.3 above). By construction the underlying closed subset of Z is T . Hence we have found a closed subscheme with properties (a) and (b). Let Z 0 ⊂ X be a second closed subscheme with properties (a) and (b). For every affine open U = Spec(R) of X we see that Z 0 ∩ U = Spec(R/I 0 ) for some ideal I 0 ⊂ R. By Lemma 21.12.3 the ring R/I 0 is reduced and hence I 0 is radical. Since V (I 0 ) = T ∩ U = V (I) we deduced that I = I 0 by Algebra, Lemma 7.16.2. Hence Z 0 and Z are defined by the same sheaf of ideals and hence are equal.  Definition 21.12.5. Let X be a scheme. Let i : Z → X be the inclusion of a closed subset. A scheme structure on Z is given by a closed subscheme Z 0 of X whose underlying closed is equal to Z. We often say “let (Z, OZ ) be a scheme structure on Z” to indicate this. The reduced induced scheme structure on Z is the one constructed in Lemma 21.12.4. The reduction Xred of X is the reduced induced scheme structure on X itself. Often when we say “let Z ⊂ X be an irreducible component of X” we think of Z as a reduced closed subscheme of X using the reduced induced scheme structure. Lemma 21.12.6. Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let Y be a reduced scheme. A morphism f : Y → X factors through Z if and only if f (Y ) ⊂ Z (set theoretically). In particular, any morphism Y → X factors as Y → Xred → X. Proof. Assume f (Y ) ⊂ Z (set theoretically). Let I ⊂ OX be the ideal sheaf of Z. For any affine opens U ⊂ X, Spec(B) = V ⊂ Y with f (V ) ⊂ U and any g ∈ I(U ) the pullback b = f ] (g) ∈ Γ(V, OTY ) = B maps to zero in the residue field of any y ∈ V . In other words g ∈ p⊂B p. This implies b = 0 as B is reduced (Lemma 21.12.2, and Algebra, Lemma 7.16.2). Hence f factors through Z by Lemma 21.4.6.  21.13. Points of schemes Given a scheme X we can define a functor hX : Schopp −→ Sets,

T 7−→ Mor(T, X).

See Categories, Example 4.3.4. This is called the functor of points of X. A fun part of scheme theory is to find descriptions of the internal geometry of X in terms of this functor hX . In this section we find a simple way to describe points of X. Let X be a scheme. Let R be a local ring with maximal ideal m ⊂ R. Suppose that f : Spec(R) → X is a morphism of schemes. Let x ∈ X be the image of the closed point m ∈ Spec(R). Then we obtain a local homomorphism of local rings f ] : OX,x −→ OSpec(R),m = R. Lemma 21.13.1. Let X be a scheme. Let R be a local ring. The construction above gives a bijective correspondence between morphisms Spec(R) → X and pairs (x, ϕ) consisting of a point x ∈ X and a local homomorphism of local rings ϕ : OX,x → R. Proof. Let A be a ring. For any ring homomorphism ψ : A → R there exists a unique prime ideal p ⊂ A and a factorization A → Ap → R where the last map is a local homomorphism of local rings. Namely, p = ψ −1 (m). Via Lemma 21.6.4 this proves that the lemma holds if X is an affine scheme.

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Let X be a general scheme. Any x ∈ X is contained in an open affine U ⊂ X. By the affine case we conclude that every pair (x, ϕ) occurs as the end product of the construction above the lemma. To finish the proof it suffices to show that any morphism f : Spec(R) → X has image contained in any affine open containing the image x of the closed point of Spec(R). In fact, let x ∈ V ⊂ X be any open neighbourhood containing x. Then f −1 (V ) ⊂ Spec(R) is an open containing the unique closed point and hence equal to Spec(R).  As a special case of the lemma above we obtain for every point x of a scheme X a canonical morphism (21.13.1.1)

Spec(OX,x ) −→ X

corresponding to the identity map on the local ring of X at x. We may reformulate the lemma above as saying that for any morphism f : Spec(R) → X there exists a unique point x ∈ X such that f factors as Spec(R) → Spec(OX,x ) → X where the first map comes from a local homomorphism OX,x → R. In case we have a morphism of schemes f : X → S, and a point x mapping to a point s ∈ S we obtain a commutative diagram Spec(OX,x )

/X

 Spec(OS,s )

 /S

where the left vertical map corresponds to the local ring map fx] : OX,x → OS,s . Lemma 21.13.2. Let X be a scheme. Let x, x0 ∈ X be points of X. Then x0 ∈ X is a generalization of x if and only if x0 is in the image of the canonical morphism Spec(OX,x ) → X. Proof. A continuous map preserves the relation of specialization/generalization. Since every point of Spec(OX,x ) is a generalization of the closed point we see every point in the image of Spec(OX,x ) → X is a generalization of x. Conversely, suppose that x0 is a generalization of x. Choose an affine open neighbourhood U = Spec(R) of x. Then x0 ∈ U . Say p ⊂ R and p0 ⊂ R are the primes corresponding to x and x0 . Since x0 is a generalization of x we see that p0 ⊂ p. This means that p0 is in the image of the morphism Spec(OX,x ) = Spec(Rp ) → Spec(R) = U ⊂ X as desired.  Now, let us discuss morphisms from spectra of fields. Let (R, m, κ) be a local ring with maximal ideal m and residue field κ. Let K be a field. A local homomorphism R → K by definition factors as R → κ → K, i.e., is the same thing as a morphism κ → K. Thus we see that morphisms Spec(K) −→ X correspond to pairs (x, κ(x) → K). We may define a partial ordering on morphisms of spectra of fields to X by saying that Spec(K) → X dominates Spec(L) → X if Spec(K) → X factors through Spec(L) → X. This suggests the following notion:

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Let us temporarily say that two morphisms p : Spec(K) → X and q : Spec(L) → X are equivalent if there exists a third field Ω and a commutative diagram Spec(Ω)

/ Spec(L)

 Spec(K)

 /X

q p

Of course this immediately implies that the unique points of all three of the schemes Spec(K), Spec(L), and Spec(Ω) map to the same x ∈ X. Thus a diagram (by the remarks above) corresponds to a point x ∈ X and a commutative diagram ΩO o

LO

Ko

κ(x)

of fields. This defines an equivalence relation, because given any set of extensions κ ⊂ Ki there exists some field extension κ ⊂ Ω such that all the field extensions Ki are contained in the extension Ω. Lemma 21.13.3. Let X be a scheme. Points of X correspond bijectively to equivalence classes of morphisms from spectra of fields into X. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element Spec(κ(x)) → X. Proof. Follows from the discussion above.



Of course the morphisms Spec(κ(x)) → X factor through the canonical morphisms Spec(OX,x ) → X. And the content of Lemma 21.13.2 is in this setting that the morphism Spec(κ(x0 )) → X factors as Spec(κ(x0 )) → Spec(OX,x ) → X whenever x0 is a generalization of x. In case we have a morphism of schemes f : X → S, and a point x mapping to a point s ∈ S we obtain a commutative diagram Spec(κ(x))

/ Spec(OX,x )

/X

 Spec(κ(s))

 / Spec(OS,s )

 / S.

21.14. Glueing schemes Let I be a set. For each i ∈ I let (Xi , Oi ) be a locally ringed space. (Actually the construction that follows works equally well for ringed spaces.) For each pair i, j ∈ I let Uij ⊂ Xi be an open subspace. For each pair i, j ∈ I, let ϕij : Uij → Uji be an isomorphism of locally ringed spaces. For convenience we assume that Uii = Xi and ϕii = idXi . For each triple i, j, k ∈ I assume that (1) we have ϕ−1 ij (Uji ∩ Ujk ) = Uij ∩ Uik , and

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(2) the diagram Uij ∩ Uik

/ Uki ∩ Ukj 8

ϕik ϕij

&

ϕjk

Uji ∩ Ujk is commutative. Let us call a collection (I, (Xi )i∈I , (Uij )i,j∈I , (ϕij )i,j∈I ) satisfying the conditions above a glueing data. Lemma 21.14.1. Given any glueing data of locally ringed spaces there exists a locally ringed space X and open subspaces Ui ⊂ X together with isomorphisms ϕi : Xi → Ui of locally ringed spaces such that (1) ϕi (Uij ) = Ui ∩ Uj , and (2) ϕij = ϕ−1 j |Ui ∩Uj ◦ ϕi |Uij . The locally ringed space X is characterized by the following mapping properties: Given a locally ringed space Y we have Mor(X, Y )

=

{(fi )i∈I | fi : Xi → Y, fj ◦ ϕij = fi |Uij }

7→ (f |Ui ◦ ϕi )i∈I   S open covering Y = i∈I Vi and (gi : Vi → Xi )i∈I such that Mor(Y, X) = gi−1 (Uij ) = Vi ∩ Vj and gj |Vi ∩Vj = ϕij ◦ gi |Vi ∩Vj f

g

7→ Vi = g −1 (Ui ), gi = g|Vi

Proof. We construct X in stages. As a set we take a X = ( Xi )/ ∼ . Here given x ∈ Xi and x0 ∈ Xj we say x ∼ x0 if and only if x ∈ Uij , x0 ∈ Uji and ϕij (x) = x0 . This is an equivalence relation since if x ∈ Xi , x0 ∈ Xj , x00 ∈ Xk , and x ∼ x0 and x0 ∼ x00 , then x0 ∈ Uji ∩ Ujk , hence by condition (1) of a glueing data also x ∈ Uij ∩ Uik and x00 ∈ Uki ∩ Ukj and by condition (2) we see that ϕik (x) = x00 . (Reflexivity and symmetry follows from our assumptions that Uii = Xi and ϕii = idXi .) Denote ϕi : Xi → X the natural maps. Denote Ui = ϕi (Xi ) ⊂ X. Note that ϕi : Xi → Ui is a bijection. The topology on X is defined by the rule that U ⊂ X is open if and only if ϕ−1 i (U ) is open for all i. We leave it to the reader to verify that this does indeed define a topology. Note that in particular Ui is open since ϕ−1 j (Ui ) = Uji which is open in Xj for all j. Moreover, for any open set W ⊂ Xi the image ϕi (W ) ⊂ Ui is open −1 because ϕ−1 j (ϕi (W )) = ϕji (W ∩Uij ). Therefore ϕi : Xi → Ui is a homeomorphism. To obtain a locally ringed space we have to construct the sheaf of rings OX . We do this by glueing the sheaves of rings OUi := ϕi,∗ OXi . Namely, in the commutative diagram / Uji Uij ϕij ϕi |Uij

# { Ui ∩ Uj

ϕj |Uji

21.14. GLUEING SCHEMES

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the arrow on top is an isomorphism of ringed spaces, and hence we get unique isomorphisms of sheaves of rings OUi |Ui ∩Uj −→ OUj |Ui ∩Uj . These satisfy a cocycle condition as in Sheaves, Section 6.33. By the results of that section we obtain a sheaf of rings OX on X such that OX |Ui is isomorphic to OUi compatibly with the glueing maps displayed above. In particular (X, OX ) is a locally ringed space since the stalks of OX are equal to the stalks of OXi at corresponding points. The proof of the mapping properties is omitted.



Lemma 21.14.2. In Lemma 21.14.1 above, assume that all Xi are schemes. Then the resulting locally ringed space X is a scheme. Proof. This is clear since each of the Ui is a scheme and hence every x ∈ X has an affine neighbourhood.  It is customary to think of Xi as an open subspace of X via the isomorphisms ϕi . We will do this in the next two examples. Example 21.14.3. (Affine space with zero doubled.) Let k be a field. Let n ≥ 1. Let X1 = Spec(k[x1 , . . . , xn ]), let X2 = Spec(k[y1 , . . . , yn ]). Let 01 ∈ X1 be the point corresponding to the maximal ideal (x1 , . . . , xn ) ⊂ k[x1 , . . . , xn ]. Let 02 ∈ X2 be the point corresponding to the maximal ideal (y1 , . . . , yn ) ⊂ k[y1 , . . . , yn ]. Let U12 = X1 \ {01 } and let U21 = X2 \ {02 }. Let ϕ12 : U12 → U21 be the isomorphism coming from the isomorphism of k-algebras k[y1 , . . . , yn ] → k[x1 , . . . , xn ] mapping yi to xi (which induces X1 ∼ = X2 mapping 01 to 02 ). Let X be the scheme obtained from the glueing data (X1 , X2 , U12 , U21 , ϕ12 , ϕ21 = ϕ−1 12 ). Via the slight abuse of notation introduced above the example we think of Xi ⊂ X as open subschemes. There is a morphism f : X → Spec(k[t1 , . . . , tn ]) which on Xi corresponds to k algebra map k[t1 , . . . , tn ] → k[x1 , . . . , xn ] (resp. k[t1 , . . . , tn ] → k[y1 , . . . , yn ]) mapping ti to xi (resp. ti to yi ). It is easy to see that this morphism identifies k[t1 , . . . , tn ] with Γ(X, OX ). Since f (01 ) = f (02 ) we see that X is not affine. Note that X1 and X2 are affine opens of X. But, if n = 2, then X1 ∩ X2 is the scheme described in Example 21.9.3 and hence not affine. Thus in general the intersection of affine opens of a scheme is not affine. (This fact holds more generally for any n > 1.) Another curious feature of this example is the following. If n > 1 there are many irreducible closed subsets T ⊂ X (take the closure of any non closed point in X1 for example). But unless T = {01 }, or T = {02 } we have 01 ∈ T ⇔ 02 ∈ T . Proof omitted. Example 21.14.4. (Projective line.) Let k be a field. Let X1 = Spec(k[x]), let X2 = Spec(k[y]). Let 0 ∈ X1 be the point corresponding to the maximal ideal (x) ⊂ k[x]. Let ∞ ∈ X2 be the point corresponding to the maximal ideal (y) ⊂ k[y]. Let U12 = X1 \ {0} = D(x) = Spec(k[x, 1/x]) and let U21 = X2 \ {∞} = D(y) = Spec(k[y, 1/y]). Let ϕ12 : U12 → U21 be the isomorphism coming from the isomorphism of k-algebras k[y, 1/y] → k[x, 1/x] mapping y to 1/x. Let P1k be the scheme obtained from the glueing data (X1 , X2 , U12 , U21 , ϕ12 , ϕ21 = ϕ−1 12 ). Via the slight abuse of notation introduced above the example we think of Xi ⊂ P1k as open

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subschemes. In this case we see that Γ(P1k , O) = k because the only polynomials g(x) in x such that g(1/y) is also a polynomial in y are constant polynomials. Since P1k is infinite we see that P1k is not affine. We claim that there exists an affine open U ⊂ P1k which contains both 0 and ∞. Namely, let U = P1k \ {1}, where 1 is the point of X1 corresponding to the maximal ideal (x − 1) and also the point of X2 corresponding to the maximal ideal (y − 1). Then it is easy to see that s = 1/(x − 1) = y/(1 − y) ∈ Γ(U, OU ). In fact you can show that Γ(U, OU ) is equal to the polynomial ring k[s] and that the corresponding morphism U → Spec(k[s]) is an isomorphism of schemes. Details omitted. 21.15. A representability criterion In this section we reformulate the glueing lemma of Section 21.14 in terms of functors. We recall some of the material from Categories, Section 4.3. Recall that given a scheme X we can define a functor hX : Schopp −→ Sets,

T 7−→ Mor(T, X).

This is called the functor of points of X. Let F be a contravariant functor from the category of schemes to the category of sets. In a formula F : Schopp −→ Sets. We will use the same terminology as in Sites, Section 9.2. Namely, given a scheme T , an element ξ ∈ F (T ), and a morphism f : T 0 → T we will denote f ∗ ξ the element F (f )(ξ), and sometimes we will even use the notation ξ|T 0 Definition 21.15.1. (See Categories, Definition 4.3.6.) Let F be a contravariant functor from the category of schemes to the category of sets (as above). We say that F is representable by a scheme or representable if there exists a scheme X such that hX ∼ = F. Suppose that F is representable by the scheme X and that s : hX → F is an isomorphism. By Categories, Yoneda Lemma 4.3.5 the pair (X, s : hX → F ) is unique up to unique isomorphism if it exists. Moreover, the Yoneda lemma says that given any contravariant functor F as above and any scheme Y , we have a bijection MorFun(Schopp ,Sets) (hY , F ) −→ F (Y ), s 7−→ s(idY ). Here is the reverse construction. Given any ξ ∈ F (Y ) the transformation of functors sξ : hY → F associates to any morphism f : T → Y the element f ∗ ξ ∈ F (T ). In particular, in the case that F is representable, there exists a scheme X and an element ξ ∈ F (X) such that the corresponding morphism hX → F is an isomorphism. In this case we also say the pair (X, ξ) represents F . The element ξ ∈ F (X) is often called the “universal family” for reasons that will become more clear when we talk about algebraic stacks (insert future reference here). For the moment we simply observe that the fact that if the pair (X, ξ) represents F , then every element ξ 0 ∈ F (T ) for any T is of the form ξ 0 = f ∗ ξ for a unique morphism f : T → X. Example 21.15.2. Consider the rule which associates to every scheme T the set F (T ) = Γ(T, OT ). We can turn this into a contravariant functor by using for a morphism f : T 0 → T the pullback map f ] : Γ(T, OT ) → Γ(T 0 , OT 0 ). Given a ring

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R and an element t ∈ R there exists a unique ring homomorphism Z[x] → R which maps x to t. Thus, using Lemma 21.6.4, we see that Mor(T, Spec(Z[x])) = Hom(Z[x], Γ(T, OT )) = Γ(T, OT ). This does indeed give an isomorphism hSpec(Z[x]) → F . What is the “universal family” ξ? To get it we have to apply the identifications above to idSpec(Z[x]) . Clearly under the identifications above this gives that ξ = x ∈ Γ(Spec(Z[x]), OSpec(Z[x]) ) = Z[x] as expected. Definition 21.15.3. Let F be a contravariant functor on the category of schemes with values in sets. (1) We say that F satisfies the sheaf property forSthe Zariski topology if for every scheme T and every open covering T = i∈I Ui , and for any collection of elements ξi ∈ F (Ui ) such that ξi |Ui ∩Uj = ξj |Ui ∩Uj there exists a unique element ξ ∈ F (T ) such that ξi = ξ|Ui in F (Ui ). (2) A subfunctor H ⊂ F is a rule that associates to every scheme T a subset H(T ) ⊂ F (T ) such that the maps F (f ) : F (T ) → F (T 0 ) maps H(T ) into H(T 0 ) for all morphisms of schemes f : T 0 → T . (3) Let H ⊂ F be a subfunctor. We say that H ⊂ F is representable by open immersions if for all pairs (T, ξ), where T is a scheme and ξ ∈ F (T ) there exists an open subscheme Uξ ⊂ T with the following property: (∗) A morphism f : T 0 → T factors through Uξ if and only if f ∗ ξ ∈ H(T 0 ). (4) Let I be a set. For each i ∈ I let Hi ⊂ F be a subfunctor. We say that the collection (Hi )i∈I coversSF if and only if for every ξ ∈ F (T ) there exists an open covering T = Ui such that ξ|Ui ∈ Hi (Ui ). Lemma 21.15.4. Let F be a contravariant functor on the category of schemes with values in the category of sets. Suppose that (1) F satisfies the sheaf property for the Zariski topology, (2) there exists a set I and a collection of subfunctors Fi ⊂ F such that (a) each Fi is representable, (b) each Fi ⊂ F is representable by open immersions, and (c) the collection (Fi )i∈I covers F . Then F is representable. Proof. Let Xi be a scheme representing Fi and let ξi ∈ Fi (Xi ) ⊂ F (Xi ) be the “universal family”. Because Fj ⊂ F is representable by open immersions, there exists an open Uij ⊂ Xi such that T → Xi factors through Uij if and only if ξi |T ∈ Fj (T ). In particular ξi |Uij ∈ Fj (Uij ) and therefore we obtain a canonical morphism ϕij : Uij → Xj such that ϕ∗ij ξj = ξi |Uij . By defintion of Uji this implies that ϕij factors through Uji . Since (ϕij ◦ ϕji )∗ ξj = ϕ∗ji (ϕ∗ij ξj ) = ϕ∗ji ξi = ξj we conclude that ϕij ◦ϕji = idUji because the pair (Xj , ξj ) represents Fj . In particular the maps ϕij : Uij → Uji are isomorphisms of schemes. Next we have to show that ϕ−1 ij (Uji ∩ Ujk ) = Uij ∩ Uik . This is true because (a) Uji ∩ Ujk is the largest open of Uji such that ξj restricts to an element of Fk , (b) Uij ∩ Uik is the largest open of Uij such that ξi restricts to an element of Fk , and (c) ϕ∗ij ξj = ξi . Moreover, the cocycle condition in Section 21.14 follows because both ϕjk |Uji ∩Ujk ◦ϕij |Uij ∩Uik and ϕik |Uij ∩Uik pullback ξk to the element ξi . Thus S we may apply Lemma 21.14.2 to obtain a scheme X with an open covering X = Ui and isomorphisms ϕi : Xi → Ui

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∗ with properties as in Lemma 21.14.1. Let ξi0 = (ϕ−1 i ) ξi . The conditions of Lemma 0 0 21.14.1 imply that ξi |Ui ∩Uj = ξj |Ui ∩Uj . Therefore, by the condition that F satisfies the sheaf condition in the Zariski topology we see that there exists an element ξ 0 ∈ F (X) such that ξi = ϕ∗i ξ 0 |Ui for all i. Since ϕi is an isomorphism we also get that (Ui , ξ 0 |Ui ) represents the functor Fi .

We claim that the pair (X, ξ 0 ) represents the functor F . To show this, let T be a scheme and let ξ ∈ F (T ). We will construct a unqiue morphism g : T → X such that g ∗ ξ 0 = ξ. Namely, by S the condition that the subfunctors Fi cover T there exists an open covering T = Vi such that for each i the restriction ξ|Vi ∈ Fi (Vi ). Moreover, since each of the inclusions Fi ⊂ F are representable by open immersions we may assume that each Vi ⊂ T is maximal open with this property. Because, (Ui , ξU0 i ) represents the functor Fi we get a unique morphism gi : Vi → Ui such that gi∗ ξ 0 |Ui = ξ|Vi . On the overlaps Vi ∩ Vj the morphisms gi and gj agree, for example because they both pull back ξ 0 |Ui ∩Uj ∈ Fi (Ui ∩ Uj ) to the same element. Thus the morphisms gi glue to a unique morphism from T → X as desired.  Remark 21.15.5. Suppose the functor F is defined on all locally ringed spaces, and if conditions of Lemma 21.15.4 are replaced by the following: (1) F satisfies the sheaf property on the category of locally ringed spaces, (2) there exists a set I and a collection of subfunctors Fi ⊂ F such that (a) each Fi is representable by a scheme, (b) each Fi ⊂ F is representable by open immersions on the category of locally ringed spaces, and (c) the collection (Fi )i∈I covers F as a functor on the category of locally ringed spaces. We leave it to the reader to spell this out further. Then the end result is that the functor F is representable in the category of locally ringed spaces and that the representing object is a scheme. 21.16. Existence of fibre products of schemes A very basic question is whether or not products and fibre products exist on the category of schemes. We first prove abstractly that products and fibre products exist, and in the next section we show how we may think in a reasonable way about fibre products of schemes. Lemma 21.16.1. The category of schemes has a final object, products and fibre products. In other words, the category of schemes has finite limits, see Categories, Lemma 4.16.4. Proof. Please skip this proof. It is more important to learn how to work with the fibre product which is explained in the next section. By Lemma 21.6.4 the scheme Spec(Z) is a final object in the category of locally ringed spaces. Thus it suffices to prove that fibred products exist. Let f : X → S and g : Y → S be morphisms of schemes. We have to show that the functor F : Schopp

−→

Sets

T

7−→

Mor(T, X) ×Mor(T,S) Mor(T, Y )

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is representable. We claim that Lemma 21.15.4 applies to the functor F . If we prove this then the lemma is proved. First we show that F satisfies the sheaf S property in the Zariski topology. Namely, suppose that T is a scheme, T = i∈I Ui is an open covering, and ξi ∈ F (Ui ) such that ξi |Ui ∩Uj = ξj |Ui ∩Uj for all pairs i, j. By definition ξi corresponds to a pair (ai , bi ) where ai : Ui → X and bi : Ui → Y are morphisms of schemes such that f ◦ ai = g ◦ bi . The glueing condition says that ai |Ui ∩Uj = aj |Ui ∩Uj and bi |Ui ∩Uj = bj |Ui ∩Uj . Thus by glueing the morphisms ai we obtain a morphism of locally ringed spaces (i.e., a morphism of schemes) a : T → X and similarly b : T → Y (see for example the mapping property of Lemma 21.14.1). Moreover, on the members of an open covering the compositions f ◦a and g◦b agree. Therefore f ◦ a = g ◦ b and the pair (a, b) defines an element of F (T ) which restricts to the pairs (ai , bi ) on each Ui . The sheaf condition is verified. Next, we construct the family of subfunctors. Choose an open covering by open S affines S = i∈I Ui . For every i ∈ I choose open coverings by open affines S S S S f −1 (Ui ) = j∈Ji Vj and g −1 (Ui ) = k∈Ki Wk . Note that X = i∈I j∈Ji Vj is an open covering and similarly for Y . For any i ∈ I and each pair (j, k) ∈ Ji × Ki we have a commutative diagram Wk  / Ui

Vj

!

X

Y  /S

where all the skew arrows are open immersions. For such a triple we get a functor Fi,j,k : Schopp

−→

Sets

T

7−→

Mor(T, Vj ) ×Mor(T,Ui ) Mor(T, Wj ).

There is an obvious transformation of functors Fi,j,k → F (coming from the huge commutative diagram above) which is injective, so we may think of Fi,j,k as a subfunctor of F . We check condition (2)(a) of Lemma 21.15.4. This follows directly from Lemma 21.6.7. (Note that we use here that the fibre products in the category of affine schemes are also fibre products in the whole category of locally ringed spaces.) We check condition (2)(b) of Lemma 21.15.4. Let T be a scheme and let ξ ∈ F (T ). In other words, ξ = (a, b) where a : T → X and b : T → Y are morphisms of schemes such that f ◦ a = g ◦ b. Set Vi,j,k = a−1 (Vj ) ∩ b−1 (Wk ). For any further morphism h : T 0 → T we have h∗ ξ = (a◦h, b◦h). Hence we see that h∗ ξ ∈ Fi,j,k (T 0 ) if and only if a(h(T 0 )) ⊂ Vj and b(h(T 0 )) ⊂ Wk . In other words, if and only if h(T 0 ) ⊂ Vi,j,k . This proves condition (2)(b). We check condition (2)(c) of Lemma 21.15.4. Let T be a scheme and let ξ = −1 −1 (a, b) ∈ F (T ) as above. S Set Vi,j,k = a (Vj ) ∩ b (Wk ) as above. Condition (2)(c) just means that T = Vi,j,k which is evident. Thus the lemma is proved and fibre products exist. 

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Remark 21.16.2. Using Remark 21.15.5 you can show that the fibre product of morphisms of schemes exists in the category of locally ringed spaces and is a scheme. 21.17. Fibre products of schemes Here is a review of the general definition, even though we have already shown that fibre products of schemes exist. Definition 21.17.1. Given morphisms of schemes f : X → S and g : Y → S the fibre product is a scheme X ×S Y together with projection morphisms p : X ×S Y → X and q : X ×S Y → Y sitting into the following commutative diagram X ×S Y

/Y

q

p

 X

 /S

f

g

which is universal among all diagrams of this sort, see Categories, Definition 4.6.1. In other words, given any solid commutative diagram of morphisms of schemes T ( X ×S Y

/* Y

 X

 /S

there exists a unique dotted arrow making the diagram commute. We will prove some lemmas which will tell us how to think about fibre products. Lemma 21.17.2. Let f : X → S and g : Y → S be morphisms of schemes with the same target. If X, Y, S are all affine then X ×S Y is affine. Proof. Suppose that X = Spec(A), Y = Spec(B) and S = Spec(R). By Lemma 21.6.7 the affine scheme Spec(A ⊗R B) is the fibre product X ×S Y in the category of locally ringed spaces. Hence it is a fortiori the fibre product in the category of schemes.  Lemma 21.17.3. Let f : X → S and g : Y → S be morphisms of schemes with the same target. Let X ×S Y , p, q be the fibre product. Suppose that U ⊂ S, V ⊂ X, W ⊂ Y are open subschemes such that f (V ) ⊂ U and g(W ) ⊂ U . Then the canonical morphism V ×U W → X ×S Y is an open immersion which identifies V ×U W with p−1 (V ) ∩ q −1 (W ). Proof. Let T be a scheme Suppose a : T → V and b : T → W are morphisms such that f ◦ a = g ◦ b as morphisms into U . Then they agree as morphisms into S. By the universal property of the fibre product we get a unique morphism T → X ×S Y . Of course this morphism has image contained in the open p−1 (V ) ∩ q −1 (W ). Thus p−1 (V ) ∩ q −1 (W ) is a fibre product of V and W over U . The result follows from the uniqueness of fibre products, see Categories, Section 4.6. 

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In particular this shows that V ×U W = V ×S W in the situation of the lemma. Moreover, if U, V, W are all affine, then we know that V ×U W is affine. And of course we may cover X ×S Y by such affine opens V ×U W . We formulate this as a lemma. Lemma 21.17.4. Let f : S X → S and g : Y → S be morphisms of schemes with the same target. Let S = Ui be any affine open covering of S. For each i ∈ I, S let f −1 (Ui ) = j∈Ji Vj be an affine open covering of f −1 (Ui ) and let g −1 (Ui ) = S −1 (Ui ). Then k∈Ki Wk be an affine open covering of f [ [ Vj ×Ui Wk X ×S Y = i∈I

j∈Ji , k∈Ki

is an affine open covering of X ×S Y . Proof. See discussion above the lemma.



In other words, we might have used the previous lemma as a way of construction the fibre product directly by glueing the affine schemes. (Which is of course exactly what we did in the proof of Lemma 21.16.1 anyway.) Here is a way to describe the set of points of a fibre product of schemes. Lemma 21.17.5. Let f : X → S and g : Y → S be morphisms of schemes with the same target. Points z of X ×S Y are in bijective correspondence to quadruples (x, y, s, p) where x ∈ X, y ∈ Y , s ∈ S are points with f (x) = s, g(y) = s and p is a prime ideal of the ring κ(x) ⊗κ(s) κ(y). The residue field of z corresponds to the residue field of the prime p. Proof. Let z be a point of X ×S Y and let us construct a triple as above. Recall that we may think of z as a morphism Spec(κ(z)) → X ×S Y , see Lemma 21.13.3. This morphism corresponds to morphisms a : Spec(κ(z)) → X and b : Spec(κ(z)) → Y such that f ◦ a = g ◦ b. By the same lemma again we get points x ∈ X, y ∈ Y lying over the same point s ∈ S as well as field maps κ(x) → κ(z), κ(y) → κ(z) such that the compositions κ(s) → κ(x) → κ(z) and κ(s) → κ(y) → κ(z) are the same. In other words we get a ring map κ(x) ⊗κ(s) κ(y) → κ(z). We let p be the kernel of this map. Conversely, given a quadruple (x, y, s, p) we get a commutative solid diagram X ×S Y i Spec(κ(x) ⊗κ(s) κ(y)/p)

/ Spec(κ(y))

 Spec(κ(x))

 / Spec(κ(s))

 X

/+ Y

$/  S

see the discussion in Section 21.13. Thus we get the dotted arrow. The corrsponding point z of X ×S Y is the image of the generic point of Spec(κ(x) ⊗κ(s) κ(y)/p). We omit the verification that the two constructions are inverse to each other. 

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Lemma 21.17.6. Let f : X → S and g : Y → S be morphisms of schemes with the same target. (1) If f : X → S is a closed immersion, then X ×S Y → Y is a closed immersion. Moreover, if X → S corresponds to the quasi-coherent sheaf of ideals I ⊂ OS , then X ×S Y → Y corresponds to the sheaf of ideals Im(g ∗ I → OY ). (2) If f : X → S is an open immersion, then X ×S Y → Y is an open immersion. (3) If f : X → S is an immersion, then X ×S Y → Y is an immersion. Proof. Assume that X → S is a closed immersion corresponding to the quasicoherent sheaf of ideals I ⊂ OS . By Lemma 21.4.7 the closed subspace Z ⊂ Y defined by the sheaf of ideals Im(g ∗ I → OY ) is the fibre product in the category of locally ringed spaces. By Lemma 21.10.1 Z is a scheme. Hence Z = X ×S Y and the first statement follows. The second follows from Lemma 21.17.3 for example. The third is a combination of the first two.  Definition 21.17.7. Let f : X → Y be a morphism of schemes. Let Z ⊂ Y be a closed subscheme of Y . The inverse image f −1 (Z) of the closed subscheme Z is the closed subscheme Z ×Y X of X. See Lemma 21.17.6 above. We may occasionally also use this terminology with locally closed and open subschemes. 21.18. Base change in algebraic geometry One motivation for the introduction of the language of schemes is that it gives a very precise notion of what it means to define a variety over a particular field. For example a variety X over Q is synonymous (insert future reference here) with X → Spec(Q) which is of finite type, separated, irreducible and reduced1. In any case, the idea is more generally to work with schemes over a given base scheme, often denoted S. We use the language: “let X be a scheme over S” to mean simply that X comes equipped with a morphism X → S. In diagrams we will try to picture the structure morphism X → S as a downward arrow from X to S. We are often more interested in the properties of X relative to S rather than the internal geometry of X. For example, we would like to know things about the fibres of X → S, what happens to X after base change, etc, etc. We introduce some of the language that is customarily used. Of course this language is just a special case of thinking about the category of objects over a given object in a category, see Categories, Example 4.2.13. Definition 21.18.1. Let S be a scheme. (1) We say X is a scheme over S to mean that X comes equipped with a morphism of schemes X → S. The morphism X → S is sometimes called the structure morphism. (2) If R is a ring we say X is a scheme over R instead of X is a scheme over Spec(R). 1Of course algebraic geometers still quibble over whether one should require X to be geometrically irreducible over Q.

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(3) A morphism f : X → Y of schemes over S is a morphism of schemes such that the composition X → Y → S of f with the structure morphism of Y is equal to the structure morphism of X. (4) We denote MorS (X, Y ) the set of all morphisms from X to Y over S. (5) Let X be a scheme over S. Let S 0 → S be a morphism of schemes. The base change of X is the scheme XS 0 = S 0 ×S X over S 0 . (6) Let f : X → Y be a morphism of schemes over S. Let S 0 → S be a morphism of schemes. The base change of f is the induced morphism f 0 : XS 0 → YS 0 (namely the morphsm idS 0 ×idS f ). (7) Let R be a ring. Let X be a scheme over R. Let R → R0 be a ring map. The base change XR0 is the scheme Spec(R0 ) ×Spec(R) X over R0 . Here is a typical result. Lemma 21.18.2. Let S be a scheme. Let f : X → Y be an immersion (resp. closed immersion, resp. open immersion) of schemes over S. Then any base change of f is an immersion (resp. closed immersion, resp. open immersion). Proof. We can think of the base change of f via the morphism S 0 → S as the top left vertical arrow in the following commutative diagram: XS 0

/X

 YS 0

 /Y

 S0

  /S

The diagram implies XS 0 ∼ = YS 0 ×Y X, and the lemma follows from Lemma 21.17.6.  In fact this type of result is so typical that there is a piece of language to express it. Here it is. Definition 21.18.3. Properties and base change. (1) Let P be a property of schemes over a base. We say that P is preserved under arbitrary base change, or simply that preserved under base change if whenever X/S has P, any base change XS 0 /S 0 has P. (2) Let P be a property of morphisms of schemes over a base. We say that P is preserved under arbitrary base change, or simply that preserved under base change if whenever f : X → Y over S has P, any base change f 0 : XS 0 → YS 0 over S 0 has P. At this point we can say that “being a closed immersion” is preserved under arbitrary base change. Definition 21.18.4. Let f : X → S be a morphism of schemes. Let s ∈ S be a point. The scheme theoretic fibre Xs of f over s, or simply the fibre of f over s is

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the scheme fitting in the following fibre product diagram Xs = Spec(κ(s)) ×S X

/X

 Spec(κ(s))

 /S

We think of the fibre Xs always as a scheme over κ(s). Lemma 21.18.5. Let f : X → S be a morphism of schemes. Consider the diagrams /X /X Spec(OS,s ) ×S X Xs  Spec(κ(s))

 /S

 Spec(OS,s )

 /S

In both cases the top horizontal arrow is a homeomorphism onto its image. Proof. Choose an open affine U ⊂ S that contains s. The bottom horizontal morphisms factor through U , see Lemma 21.13.1 for example. Thus we may assume that S is affine. If X is also affine, then the result follows from Algebra, Remark 7.16.8. In the general case the result follows by covering X by open affines.  21.19. Quasi-compact morphisms A scheme is quasi-compact if its underlying topological space is quasi-compact. There is a relative notion which is defined as follows. Definition 21.19.1. A morphism of schemes is called quasi-compact if the underlying map of topological spaces is quasi-compact, see Topology, Definition 5.9.1. Lemma 21.19.2. Let f : X → S be a morphism of schemes. The following are equivalent (1) f : X → S is quasi-compact, (2) the inverse image of every affine open is quasi-compact, and S (3) there exists some affine open covering S = i∈I Ui such that f −1 (Ui ) is quasi-compact for all i. S Proof. Suppose we are given a covering X = i∈I Ui as in (3). First, let U ⊂ S be any affine open. For any u ∈ U we can find an index i(u) ∈ I such that u ∈ Ui(u) . By Lemma 21.11.5 we can find an affine open Wu ⊂ U ∩Ui(u) which is standard open in both U and Ui(u) . By compactness we can find finitely many points u1 , . . . , un ∈ U S Sn such that U = j=1 Wuj . For each j write f −1 Ui(uj ) = k∈Kj Vjk as a finite union of affine opens. Since Wuj ⊂ Ui(u) is a standard open we see that f −1 (Wuj ) ∩ Vjk is a standard open of Vjk , see Algebra, Lemma 7.16.4. Hence f −1 (Wuj ) ∩ Vjk is affine, and so f −1 (Wuj ) is a finite union of affines. This proves that the inverse image of any affine open is a finite union of affine opens. Next, assume that the inverse image of every affine open is a finite union of affine opens. Let K ⊂ X be any quasi-compact open. Since X has a basis of the topology consisting of affine opens we see that K is a finite union of affine opens. Hence the inverse image of K is a finite union of affine opens. Hence f is quasi-compact.

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Finally, assume that f is quasi-compact. In this case the argument of the previous paragraph shows that the inverse image of any affine is a finite union of affine opens.  Lemma 21.19.3. Being quasi-compact is a property of morphisms of schemes over a base which is preserved under arbitrary base change. Proof. Omitted.



Lemma 21.19.4. The composition of quasi-compact morphisms is quasi-compact. Proof. Omitted.



Lemma 21.19.5. A closed immersion is quasi-compact. Proof. Follows from the definitions and Topology, Lemma 5.9.3.



Example 21.19.6. An open immersion is in general not quasi-compact. The standard example of this is the open subspace U ⊂ X, where X = Spec(k[x1 , x2 , x3 , . . .]), where U is X \ {0}, and where 0 is the point of X corresponding to the maximal ideal (x1 , x2 , x3 , . . .). Lemma 21.19.7. Let f : X → S be a quasi-compact morphism of schemes. The following are equivalent (1) f (X) ⊂ S is closed, and (2) f (X) ⊂ S is stable under specialization. Proof. We have (1) ⇒ (2) by Topology, Lemma 5.14.2. Assume (2). Let U ⊂ S be an affine open. It suffices to prove that f (X) ∩ U is closed. Since U ∩ f (X) is stable under specializations, we have reduced to the case where S is affine. Because −1 f is quasi-compact we deduce Sn that X = f (S) is quasi-compact as S is affine. Thus we may write X = i=1 Ui with Ui ⊂ X open affine. Say S = Spec(R) and Ui = Spec(Ai ) for some R-algebra Ai . Then f (X) = Im(Spec(A1 × . . . × An ) → Spec(R)). Thus the lemma follows from Algebra, Lemma 7.37.5.  Lemma 21.19.8. Let f : X → S be a quasi-compact morphism of schemes. Then f is closed if and only if specializations lift along f , see Topology, Definition 5.14.3. Proof. According to Topology, Lemma 5.14.6 if f is closed then specializations lift along f . Conversely, suppose that specializations lift along f . Let Z ⊂ X be a closed subset. We may think of Z as a scheme with the reduced induced scheme structure, see Definition 21.12.5. Since Z ⊂ X is closed the restriction of f to Z is still quasi-compact. Moreover specializations lift along Z → S as well, see Topology, Lemma 5.14.4. Hence it suffices to prove f (X) is closed if specializations lift along f . In particular f (X) is stable under specializations, see Topology, Lemma 5.14.5. Thus f (X) is closed by Lemma 21.19.7.  21.20. Valuative criterion for universal closedness In Topology, Section 5.12 there is a discussion of proper maps as closed maps of topological spaces all of whose fibres are quasi-compact, or as maps such that all base changes are closed maps. Here is the corresponding notion in algebraic geometry. Definition 21.20.1. A morphism of schemes f : X → S is said to be universally closed if every base change f 0 : XS 0 → S 0 is closed.

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In fact the adjective “universally” is often used in this way. In other words, given a property P of morphisms the we say that “X → S is universally P” if and only if every base change XS 0 → S 0 has P. Please take a look at Morphisms, Section 24.42 for a more detailed discussion of the properties of universally closed morphisms. In this section we restrict the discussion to the relationship between universal closed morphisms and morphisms satisfying the existence part of the valuative criterion. Lemma 21.20.2. Let f : X → S be a morphism of schemes. (1) If f is universally closed then specializations lift along any base change of f , see Topology, Definition 5.14.3. (2) If f is quasi-compact and specializations lift along any base change of f , then f is universally closed. Proof. Part (1) is a direct consequence of Topology, Lemma 5.14.6. Part (2) follows from Lemmas 21.19.8 and 21.19.3.  Definition 21.20.3. Let f : X → S be a morphism of schemes. We say f satisfies the existence part of the valuative criterion if given any commutative solid diagram Spec(K)

/X ;

 Spec(A)

 /S

where A is a valuation ring with field of fractions K, the dotted arrow exists. We say f satisfies the uniqueness part of the valuative criterion if there is at most one dotted arrow given any diagram as above (without requiring existence of course). A valuation ring is a local domain maximal among the relation of domination in its fraction field, see Algebra, Definition 7.47.1. Hence the spectrum of a valuation ring has a unique generic point η and a unique closed point 0, and of course we have the specialization η 0. The significance of valuation rings is that any specialization of points in any scheme is the image of η 0 under some morphism from the spectrum of some valuation ring. Here is the precise result. Lemma 21.20.4. Let S be a scheme. Let s0 S. Then

s be a specialization of points of

(1) there exists a valuation ring A and a morphism Spec(A) → S such that the generic point η of Spec(A) maps to s0 and the special point maps to s, and (2) given a field extension κ(s0 ) ⊂ K we may arrange it so that the extension κ(s0 ) ⊂ κ(η) induced by f is isomorphic to the given extension. Proof. Let s0 s be a specialization in S, and let κ(s0 ) ⊂ K be an extension of fields. By Lemma 21.13.2 and the discussion following Lemma 21.13.3 this leads to ring maps OS,s → κ(s0 ) → K. Let A ⊂ K be any valuation ring whose field of fractions is K and which dominates the image of OS,s → K, see Algebra, Lemma 7.47.2. The ring map OS,s → A induces the morphism f : Spec(A) → S, see Lemma 21.13.1. This morphism has all the desired properties by construction. 

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Lemma 21.20.5. Let f : X → S be a morphism of schemes. The following are equivalent (1) Specializations lift along any base change of f (2) The morphism f satisfies the existence part of the valuative criterion. Proof. Assume (1) holds. Let a solid diagram as in Definition 21.20.3 be given. In order to find the dotted arrow we may replace X → S by XSpec(A) → Spec(A) since after all the assumption is stable under base change. Thus we may assume S = Spec(A). Let x0 ∈ X be the image of Spec(K) → X, so that we have κ(x0 ) ⊂ K, see Lemma 21.13.3. By assumption there exists a specialization x0 x in X such that x maps to the closed point of S = Spec(A). We get a local ring map A → OX,x and a ring map OX,x → κ(x0 ), see Lemma 21.13.2 and the discussion following Lemma 21.13.3. The composition A → OX,x → κ(x0 ) → K is the given injection A → K. Since A → OX,x is local, the image of OX,x → K dominates A and hence is equal to A, by Algebra, Definition 7.47.1. Thus we obtain a ring map OX,x → A and hence a morphism Spec(A) → X (see Lemma 21.13.1 and discussion following it). This proves (2). Conversely, assume (2) holds. It is immediate that the existence part of the valuative criterion holds for any base change XS 0 → S 0 of f by considering the following commutative diagram /5 X / XS 0 Spec(K) :  Spec(A)

 / S0

 /S

Namely, the more horizontal dotted arrow will lead to the other one by definition of the fibre product. OK, so it clearly suffices to show that specializations lift along f . Let s0 s be a specialization in S, and let x0 ∈ X be a point lying over s0 . Apply Lemma 21.20.4 to s0 s and the extension of fields κ(s0 ) ⊂ κ(x0 ) = K. We get a commutative diagram 4/ X

Spec(K)  Spec(A)

/ Spec(OS,s )

 /S

and by condition (2) we get the dotted arrow. The image x of the closed point of Spec(A) in X will be a solution to our problem, i.e., x is a specialization of x0 and maps to s.  Proposition 21.20.6 (Valuative criterion of universal closedness). Let f be a quasi-compact morphism of schemes. Then f is universally closed if and only if f satisfies the existence part of the valuative criterion. Proof. This is a formal consequence of Lemmas 21.20.2 and 21.20.5 above.



Example 21.20.7. Let k be a field. Consider the structure morphism p : P1k → Spec(k) of the projective line over k, see Example 21.14.4. Let us use the valuative criterion above to prove that p is universally closed. By construction P1k is covered

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by two affine opens and hence p is quasi-compact. Let a commutative diagram Spec(K)  Spec(A)

/ P1 k

ξ

ϕ

 / Spec(k)

be given, where A is a valuation ring and K is its field of fractions. Recall that P1k is gotten by glueing Spec(k[x]) to Spec(k[y]) by glueing D(x) to D(y) via x = y −1 (or more symmetrically xy = 1). To show there is a morphism Spec(A) → P1k fitting diagonally into the diagram above we may assume that ξ maps into the open Spec(k[x]) (by symmetry). This gives the following commutative diagram of rings KO o ] k[x] O ξ Ao

ϕ]

k

By Algebra, Lemma 7.47.3 we see that either ξ ] (x) ∈ A or ξ ] (x)−1 ∈ A. In the first case we get a ring map k[x] → A, λ 7→ ϕ] (λ), x 7→ ξ ] (x) fitting into the diagram of rings above, and we win. In the second case we see that we get a ring map k[y] → A, λ 7→ ϕ] (λ), y 7→ ξ ] (x)−1 . This gives a morphism Spec(A) → Spec(k[y]) → P1k which fits diagonally into the initial commutative diagram of this example (check omitted). 21.21. Separation axioms A topological space X is Hausdorff if and only if the diagonal ∆ ⊂ X × X is a closed subset. The analogue in algebraic geometry is, given a scheme X over a base scheme S, to consider the diagonal morphism ∆X/S : X −→ X ×S X. This is the unique morphism of schemes such that pr1 ◦ ∆X/S = idX and pr2 ◦ ∆X/S = idX (it exists in any category with fibre products). Lemma 21.21.1. The diagonal morphism of a morphism between affines is closed. Proof. The diagonal morphism associated to the morphism Spec(S) → Spec(R) is the morphism on spectra corresponding to the ring map S ⊗R S → S, a ⊗ b 7→ ab. This map is clearly surjective, so S ∼ = S ⊗R S/J for some ideal J ⊂ S ⊗R S. Hence ∆ is a closed immersion according to Example 21.8.1  Lemma 21.21.2. Let X be a scheme over S. The diagonal morphism ∆X/S is an immersion. Proof. Recall that if V ⊂ X is affine open and maps into U ⊂ S affine open, then V ×U V is affine open in X ×S X, see Lemmas 21.17.2 and 21.17.3. Consider the open subscheme W of X ×S X which is the union of these affine opens V ×U V . By Lemma 21.4.2 it is enough to show that each morphism ∆−1 X/S (V ×U V ) → V ×U V

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is a closed immersion. Since V = ∆−1 X/S (V ×U V ) we are just checking that ∆V /U is a closed immersion, which is Lemma 21.21.1.  Definition 21.21.3. Let f : X → S be a morphism of schemes. (1) We say f is separated if the diagonal morphism ∆X/S is a closed immersion. (2) We say f is quasi-separated if the diagonal morphism ∆X/S is a quasicompact morphism. (3) We say a scheme Y is separated if the morphism Y → Spec(Z) is separated. (4) We say a scheme Y is quasi-separated if the morphism Y → Spec(Z) is quasi-separated. By Lemmas 21.21.2 and 21.10.4 we see that ∆X/S is a closed immersion if an only if ∆X/S (X) ⊂ X ×S X is a closed subset. Moreover, by Lemma 21.19.5 we see that a separated morphism is quasi-separated. The reason for introducing quasi-separated morphisms is that nonseparated morphisms come up naturally in studying algebraic varieties (especially when doing moduli, algebraic stacks, etc). But most often they are still quasi-separated. Example 21.21.4. Here is an example of a non-quasi-separated morphism. Suppose X = X1 ∪ X2 → S = Spec(k) with X1 = X2 = Spec(k[t1 , t2 , t3 , . . .]) glued along the complement of {0} = {(t1 , t2 , t3 , . . .)} (glued as in Example 21.14.3). In this case the inverse image of the affine scheme X1 ×S X2 under ∆X/S is the scheme Spec(k[t1 , t2 , t3 , . . .]) \ {0} which is not quasi-compact. Lemma 21.21.5. Let X, Y be schemes over S. Let a, b : X → Y be morphisms of schemes over S. There exists a largest locally closed subscheme Z ⊂ X such that a|Z = b|Z . In fact Z is the equalizer of (a, b). Moreover, if Y is separated over S, then Z is a closed subscheme. Proof. The equalizer of (a, b) is for categorical reasons the fibre product Z in the following diagram /X Z = Y ×(Y ×S Y ) X  Y

(a,b)

∆Y /S

 / Y ×S Y

Thus the lemma follows from Lemmas 21.18.2, 21.21.2 and Definition 21.21.3.



Lemma 21.21.6. An affine scheme is separated. A morphism of affine schemes is separated. Proof. See Lemma 21.21.1.



Lemma 21.21.7. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is quasi-separated. (2) For every pair of affine opens U, V ⊂ X which map into a common affine open of S the intersection U ∩ V is a finite S union of affine opens of X. (3) There exists an affine open covering S = i∈I Ui and for each i an affine S open covering f −1 Ui = j∈Ii Vj such that for each i and each pair j, j 0 ∈ Ii the intersection Vj ∩ Vj 0 is a finite union of affine opens of X.

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Proof. S S Let us prove that (3) implies (1). By Lemma 21.17.4 the covering−1X ×S X = 0 i j,j 0 Vj ×Ui Vj is an affine open covering of X ×S X. Moreover, ∆X/S (Vj ×Ui Vj 0 ) = Vj ∩ Vj 0 . Hence the implication follows from Lemma 21.19.2. The implication (1) ⇒ (2) follows from the fact that under the hypotheses of (1) the fibre product U ×S V is an affine open of X ×S X. The implication (2) ⇒ (3) is trivial.  Lemma 21.21.8. Let f : X → S be a morphism of schemes. (1) If f is separated then for every pair of affine opens (U, V ) of X which map into a common affine open of S we have (a) the intersection U ∩ V is affine. (b) the ring map OX (U ) ⊗Z OX (V ) → OX (U ∩ V ) is surjective. (2) If any pair of points x1 , x2 ∈ X lying over a common point s ∈ S are contained in affine opens x1 ∈ U , x2 ∈ V which map into a common affine open of S such that (a), (b) hold, then f is separated. Proof. Assume f separated. Suppose (U, V ) is a pair as in (1). Let W = Spec(R) be an affine open of S containing both f (U ) and g(V ). Write U = Spec(A) and V = Spec(B) for R-algebras A and B. By Lemma 21.17.3 we see that U ×S V = U ×W V = Spec(A ⊗R B) is an affine open of X ×S X. Hence, by Lemma 21.10.1 we see that ∆−1 (U ×S V ) → U ×S V can be identified with Spec(A ⊗R B/J) for some ideal J ⊂ A ⊗R B. Thus U ∩ V = ∆−1 (U ×S V ) is affine. Assertion (1)(b) holds because A ⊗Z B → (A ⊗R B)/J is surjective. Assume the hypothesis formulated in (2) holds. Clearly the collection of affine opens U ×S V for pairs (U, V ) as in (2) form an affine open covering of X ×S X (see e.g. Lemma 21.17.4). Hence it suffices to show that each morphism U ∩ V = ∆−1 X/S (U ×S V ) → U ×S V is a closed immersion, see Lemma 21.4.2. By assumption (a) we have U ∩ V = Spec(C) for some ring C. After choosing an affine open W = Spec(R) of S into which both U and V map and writing U = Spec(A), V = Spec(B) we see that the assumption (b) means that the composition A ⊗Z B → A ⊗R B → C is surjective. Hence A ⊗R B → C is surjective and we conclude that Spec(C) → Spec(A ⊗R B) is a closed immersion.  Example 21.21.9. Let k be a field. Consider the structure morphism p : P1k → Spec(k) of the projective line over k, see Example 21.14.4. Let us use the lemma above to prove that p is separated. By construction P1k is covered by two affine opens U = Spec(k[x]) and V = Spec(k[y]) with intersection U ∩V = Spec(k[x, y]/(xy−1)) (using obvious notation). Thus it suffices to check that conditions (2)(a) and (2)(b) of Lemma 21.21.8 hold for the pairs of affine opens (U, U ), (U, V ), (V, U ) and (V, V ). For the pairs (U, U ) and (V, V ) this is trivial. For the pair (U, V ) this amounts to proving that U ∩ V is affine, which is true, and that the ring map k[x] ⊗Z k[y] −→ k[x, y]/(xy − 1) is surjective. This is clear because any element in the right hand side can be written as a sum of a polynomial in x and a polynomial in y.

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Lemma 21.21.10. Let f : X → T and g : Y → T be morphisms of schemes with the same target. Let h : T → S be a morphism of schemes. Then the induced morphism i : X ×T Y → X ×S Y is an immersion. If T → S is separated, then i is a closed immersion. If T → S is quasi-separated, then i is a quasi-compact morphism. Proof. By general category theory the following diagram / X ×S Y X ×T Y  T

∆T /S

 / T ×S T

is a fibre product diagram. The lemma follows from Lemmas 21.21.2, 21.17.6 and 21.19.3.  Lemma 21.21.11. Let g : X → Y be a morphism of schemes over S. The morphism i : X → X ×S Y is an immersion. If Y is separated over S it is a closed immersion. If Y is quasi-separated over S it is quasi-compact. Proof. This is a special case of Lemma 21.21.10 applied to the morphism X = X ×Y Y → X ×S Y .  Lemma 21.21.12. Let f : X → S be a morphism of schemes. Let s : S → X be a section of f (in a formula f ◦ s = idS ). Then s is an immersion. If f is separated then s is a closed immersion. If f is quasi-separated, then s is quasi-compact. Proof. This is a special case of Lemma 21.21.11 applied to g = s so the morphism i = s : S → S ×S X.  Lemma (1) (2) (3) (4) (5) (6)

21.21.13. Permanence properties. A composition of separated morphisms is separated. A composition of quasi-separated morphisms is quasi-separated. The base change of a separated morphism is separated. The base change of a quasi-separated morphism is quasi-separated. A (fibre) product of separated morphisms is separated. A (fibre) product of quasi-separated morphisms is quasi-separated.

Proof. Let X → Y → Z be morphisms. Assume that X → Y and Y → Z are separated. The composition X → X ×Y X → X ×Z X is closed because the first one is by assumption and the second one by Lemma 21.21.10. The same argument works for “quasi-separated” (with the same references). Let f : X → Y be a morphism of schemes over a base S. Let S 0 → S be a morphism of schemes. Let f 0 : XS 0 → YS 0 be the base change of f . Then the diagonal morphism of f 0 is a morphism ∆f 0 : XS 0 = S 0 ×S X −→ XS 0 ×YS0 XS 0 = S 0 ×S (X ×Y X) which is easily seen to be the base change of ∆f . Thus (3) and (4) follow from the fact that closed immersions and quasi-compact morphisms are preserved under arbitrary base change (Lemmas 21.17.6 and 21.19.3).

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If f : X → Y and g : U → V are morphisms of schemes over a base S, then f × g is the composition of X ×S U → X ×S V (a base change of g) and X ×S V → Y ×S V (a base change of f ). Hence (5) and (6) follow from (1) – (4).  Lemma 21.21.14. Let f : X → Y and g : Y → Z be morphisms of schemes. If g ◦ f is separated then so is f . If g ◦ f is quasi-separated then so is f . Proof. Assume that g ◦f is separated. Consider the factorization X → X ×Y X → X ×Z X of the diagonal morphism of g ◦ f . By Lemma 21.21.10 the last morphism is an immersion. By assumption the image of X in X ×Z X is closed. Hence it is also closed in X ×Y X. Thus we see that X → X ×Y X is a closed immersion by Lemma 21.10.4. Assume that g ◦ f is quasi-separated. Let V ⊂ Y be an affine open which maps into an affine open of Z. Let U1 , U2 ⊂ X be affine opens which map into V . Then U1 ∩ U2 is a finite union of affine opens because U1 , U2 map into a common affine open of Z. Since we may cover Y by affine opens like V we deduce the lemma from Lemma 21.21.7.  Lemma 21.21.15. Let f : X → Y and g : Y → Z be morphisms of schemes. If g ◦ f is quasi-compact and g is quasi-separated then f is quasi-compact. Proof. This is true because f equals the composition (1, f ) : X → X ×Z Y → Y . The first map is quasi-compact by Lemma 21.21.12 because it is a section of the quasi-separated morphism X ×Z Y → X (a base change of g, see Lemma 21.21.13). The second map is quasi-compact as it is the base change of f , see Lemma 21.19.3. And compositions of quasi-compact morphisms are quasi-compact, see Lemma 21.19.4.  You may have been wondering whether the condition of only considering pairs of affine opens whose image is contained in an affine open is really necessary to be able to conclude that their intersection is affine. Often it isn’t! Lemma 21.21.16. Let f : X → S be a morphism. Assume f is separated and S is a separated scheme. Suppose U ⊂ X and V ⊂ X are affine. Then U ∩ V is affine (and a closed subscheme of U × V ). Proof. In this case X is separated by Lemma 21.21.13. Hence U ∩ V is affine by applying Lemma 21.21.8 to the morphism X → Spec(Z).  On the other hand, the following example shows that we cannot expect the image of an affine to be contained in an affine. Example 21.21.17. Consider the nonaffine scheme U = Spec(k[x, y]) \ {(x, y)} of Example 21.9.3. On the other hand, consider the scheme GL2,k = Spec(k[a, b, c, d, 1/ad − bc]). There is a morphism GL2,k → U corresponding to the ring map x 7→ a, y 7→ b. It is easy to see that this is a surjective morphism, and hence the image is not contained in any affine open of U . In fact, the affine scheme GL2,k also surjects onto P1k , and P1k does not even have an immersion into any affine scheme.

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21.22. Valuative criterion of separatedness Lemma 21.22.1. Let f : X → S be a morphism of schemes. If f is separated, then f satisfies the uniqueness part of the valuative criterion. Proof. Let a diagram as in Definition 21.20.3 be given. Suppose there are two morphisms a, b : Spec(A) → X fitting into the diagram. Let Z ⊂ Spec(A) be the equalizer of a and b. By Lemma 21.21.5 this is a closed subscheme of Spec(A). By assumption it contains the generic point of Spec(A). Since A is a domain this implies Z = Spec(A). Hence a = b as desired.  Lemma 21.22.2 (Valuative criterion separatedness). Let f : X → S be a morphism. Assume (1) the morphism f is quasi-separated, and (2) the morphism f satisfies the uniqueness part of the valuative criterion. Then f is separated. Proof. By assumption (1) and Proposition 21.20.6 we see that it suffices to prove the morphism ∆X/S : X → X ×S X satisfies the existence part of the valuative criterion. Let a solid commutative diagram Spec(K)

9/ X

 Spec(A)

 / X ×S X

be given. The lower right arrow corresponds to a pair of morphisms a, b : Spec(A) → X over S. By (2) we see that a = b. Hence using a as the dotted arrow works.  21.23. Monomorphisms Definition 21.23.1. A morphism of schemes is called a monomorphism if it is a monomorphism in the category of schemes, see Categories, Definition 4.23.1. Lemma 21.23.2. Let j : X → Y be a morphism of schemes. Then j is a monomorphism if and only if the diagonal morphism ∆X/Y : X → X ×Y X is an isomorphism. Proof. This is true in any category with fibre products.



Lemma 21.23.3. A monomorphism of schemes is separated. Proof. This is true because an isomorphism is a closed immersion, and Lemma 21.23.2 above.  Lemma 21.23.4. A composition of monomorphisms is a monomorpism. Proof. True in any category.



Lemma 21.23.5. The base change of a monomorphism is a monomorphism. Proof. True in any category with fibre products. Lemma 21.23.6. Let j : X → Y be a morphism of schemes. If (1) j is injective on points, and (2) for any x ∈ X the ring map jx] : OY,j(x) → OX,x is surjective,



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then j is a monomorphism. Proof. Let a, b : Z → X be two morphisms of schemes such that j ◦ a = j ◦ b. Then (1) implies a = b as underlying maps of topological spaces. For any z ∈ Z we ] ] as maps OY,j(a(z)) → OZ,z . The surjectivity of the maps = b]z ◦ jb(z) have a]z ◦ ja(z) jx] forces a]z = b]z , ∀z ∈ Z. This implies that a] = b] . Hence we conclude a = b as morphisms of schemes as desired.  Lemma 21.23.7. An immersion of schemes is a monomorphism. In particular, any immersion is separated. Proof. We can see this by checking that the criterion of Lemma 21.23.6 applies. More elegantly perhaps, we can use that Lemmas 21.3.5 and 21.4.6 imply that open and closed immersions are monomorphisms and hence any immersion (which is a composition of such) is a monomorphism.  Lemma 21.23.8. Let f : X → S be a separated morphism. Any locally closed subscheme Z ⊂ X is separated over S. Proof. Follows from Lemma 21.23.7 and the fact that a composition of separated morphisms is separated (Lemma 21.21.13).  Example 21.23.9. The morphism Spec(Q) → Spec(Z) is a monomorphism. This is true because Q ⊗Z Q = Q. More generally, for any scheme S and any point s ∈ S the canonical morphism Spec(OS,s ) −→ S is a monomorphism. Lemma 21.23.10. Let k1 , . . . , kn be fields. For any monomorphism of schemes ∼ X →QSpec(k1 × . . . × kn ) there exists a subset I ⊂ {1, . . . , n} such that ` X = Spec( i∈I ki ) as schemes over Spec(k1 ×. . .×kn ). More generally, if X = i∈I Spec(ki ) is a disjoint union of spectra of fields`and Y → X is a monomorphism, then there exists a subset J ⊂ I such that Y = i∈J Spec(ki ). Proof. First reduce to the case n = 1 (or #I = 1) by taking the inverse images of the open and closed subschemes Spec(ki ). In this case X has only one point hence is affine. The corresponding algebra problem is this: If k → R is an algebra map with R ⊗k R ∼ = k. This holds for dimension reasons. See also Algebra, = R, then R ∼ Lemma 7.100.8  21.24. Functoriality for quasi-coherent modules Let X be a scheme. We denote QCoh(OX ) or QCoh(X) the category of quasicoherent OX -modules as defined in Modules, Definition 15.10.1. We have seen in Section 21.7 that the category QCoh(OX ) has a lot of good properties when X is affine. Since the property of being quasi-coherent is local on X, these properties are inherited by the category of quasi-coherent sheaves on any scheme X. We enumerate them here. (1) A sheaf of OX -modules F is quasi-coherent if and only if the restriction of f for some R-module F to each affine open U = Spec(R) is of the form M M. (2) A sheaf of OX -modules F is quasi-coherent if and only if the restriction of F to each of the members of an affine open covering is quasi-coherent.

21.24. FUNCTORIALITY FOR QUASI-COHERENT MODULES

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(3) Any direct sum of quasi-coherent sheaves is quasi-coherent. (4) Any colimit of quasi-coherent sheaves is quasi-coherent. (5) The kernel and cokernel of a morphism of quasi-coherent sheaves is quasicoherent. (6) Given a short exact sequence of OX -modules 0 → F1 → F2 → F3 → 0 if two out of three are quasi-coherent so is the third. (7) Given a morphism of schemes f : Y → X the pullback of a quasi-coherent OX -module is a quasi-coherent OY -module. See Modules, Lemma 15.10.4. (8) Given two quasi-coherent OX -modules the tensor product is quasi-coherent, see Modules, Lemma 15.15.5. (9) Given a quasi-coherent OX -module F the tensor, symmetric and exterior algebras on F are quasi-coherent, see Modules, Lemma 15.18.6. (10) Given two quasi-coherent OX -modules F, G such that F is of finite presentation, then the internal hom Hom OX (F, G) is quasi-coherent, see Modules, Lemma 15.19.4 and (5) above. On the other hand, it is in general not the case that the pushforward of a quasicoherent module is quasi-coherent. Here is a case where it this does hold. Lemma 21.24.1. Let f : X → S be a morphism of schemes. If f is quasicompact and quasi-separated then f∗ transforms quasi-coherent OX -modules into quasi-coherent OS -modules. Proof. The question is local on S and hence we Sn may assume that S is affine. Because X is quasi-compact we may write X = i=1 i with each Ui open affine. SU nij Because f is quasi-separated we may write Ui ∩Uj = k=1 Uijk for some affine open Uijk , see Lemma 21.21.7. Denote fi : Ui → S and fijk : Uijk → S the restrictions of f . For any open V of S and any sheaf F on X we have f∗ F(V )

= F(f −1 V ) M  M = Ker F(f −1 V ∩ Ui ) → F(f −1 V ∩ Uijk ) i i,j,k M  M = Ker fi,∗ (F|Ui )(V ) → fijk,∗ (F|Uijk ) (V ) i i,j,k M  M = Ker fi,∗ (F|Ui ) → fijk,∗ (F|Uijk ) (V ) i

i,j,k

In other words there is a short exact sequence of sheaves M M 0 → f∗ F → fi,∗ Fi → fijk,∗ Fijk where Fi , Fijk denotes the restriction of F to the corresponding open. If F is a quasi-coherent OX -modules then Fi , Fijk is a quasi-coherent OUi , OUijk -module. Hence by Lemma 21.7.3 we see that the second and third term of the exact sequence are quasi-coherent OS -modules. Thus we conclude that f∗ F is a quasi-coherent OS module.  Using this we can characterize (closed) immersions of schemes as follows. Lemma 21.24.2. Let f : X → Y be a morphism of schemes. Suppose that (1) f induces a homeomorphism of X with a closed subset of Y , and (2) f ] : OY → f∗ OX is surjective. Then f is a closed immersion of schemes.

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Proof. Assume (1) and (2). By (1) the morphism f is quasi-compact (see Topology, Lemma 5.9.3). Conditions (1) and (2) imply conditions (1) and (2) of Lemma 21.23.6. Hence f : X → Y is a monomorphism. In particular, f is separated, see Lemma 21.23.3. Hence Lemma 21.24.1 above applies and we conclude that f∗ OX is a quasi-coherent OY -module. Therefore the kernel of OY → f∗ OX is quasi-coherent by Lemma 21.7.8. Since a quasi-coherent sheaf is locally generated by sections (see Modules, Definition 15.10.1) this implies that f is a closed immersion, see Definition 21.4.1.  We can use this lemma to prove the following lemma. Lemma 21.24.3. A composition of immersions of schemes is an immersion, a composition of closed immersions of schemes is a closed immersion, and a composition of open immersions of schemes is an open immersion. Proof. This is clear for the case of open immersions since an open subspace of an open subspace is also an open subspace. Suppose a : Z → Y and b : Y → X are closed immersions of schemes. We will verify that c = b ◦ a is also a closed immersion. The assumption implies that a and b are homeomorphisms onto closed subsets, and hence also c = b ◦ a is a homeomorphism onto a closed subset. Moreover, the map OX → c∗ OZ is surjective since it factors as the composition of the surjective maps OX → b∗ OY and b∗ OY → b∗ a∗ OZ (surjective as b∗ is exact, see Modules, Lemma 15.6.1). Hence by Lemma 21.24.2 above c is a closed immersion. Finally, we come to the case of immersions. Suppose a : Z → Y and b : Y → X are immersions of schemes. This means there exist open subschemes V ⊂ Y and U ⊂ X such that a(Z) ⊂ V , b(Y ) ⊂ U and a : Z → V and b : Y → U are closed immersions. Since the topology on Y is induced from the topology on U we can find an open U 0 ⊂ U such that V = b−1 (U 0 ). Then we see that Z → V = b−1 (U 0 ) → U 0 is a composition of closed immersions and hence a closed immersion. This proves that Z → X is an immersion and we win.  21.25. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules

(16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30)

Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes

21.25. OTHER CHAPTERS

(31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52)

Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory

(53) (54) (55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

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Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 22

Constructions of Schemes 22.1. Introduction In this chapter we introduce ways of constructing schemes out of others. A basic reference is [DG67]. 22.2. Relative glueing The following lemma is relevant in case we are trying to construct a scheme X over S, and we already know how to construct the restriction of X to the affine opens of S. The actual result is completely general and works in the setting of (locally) ringed spaces, allthough our proof is written in the language of schemes. Lemma 22.2.1. Let S be a scheme. Let B be a basis for the topology of S. Suppose given the following data: (1) For every U ∈ B a scheme fU : XU → U over U . (2) For every pair U, V ∈ B such that V ⊂ U a morphism ρU V : XV → XU . Assume that −1 (a) each ρU V induces an isomorphism XV → fU (V ) of schemes over V , V U (b) whenever W, V, U ∈ B, with W ⊂ V ⊂ U we have ρU W = ρV ◦ ρW . Then there exists a unique scheme f : X → S over S and isomorphisms iU : f −1 (U ) → XU over U such that for V ⊂ U ⊂ S affine open the composition XV

i−1 V

/ f −1 (V )

inclusion

/ f −1 (U )

iU

/ XU

is the morphism ρU V. Proof. To prove this we will use Schemes, Lemma 21.15.4. First we define a contravariant functor F from the category of schemes to the category of sets. Namely, for a scheme T we set   (g, {hU }U ∈B ), g : T → S, hU : g −1 (U ) → XU , . F (T ) = fU ◦ hU = g|g−1 (U ) , hU |g−1 (V ) = ρU V ◦ hV ∀ V, U ∈ B, V ⊂ U The restriction mapping F (T ) → F (T 0 ) given a morphism T 0 → T is just gotten by composition. For any W ∈ B we consider the subfunctor FW ⊂ F consisting of those systems (g, {hU }) such that g(T ) ⊂ W . First we show F satisfies S the sheaf property for the Zariski topology. Suppose that T is a scheme, T = Vi is an open covering, and ξi ∈ F (Vi ) is an element such that ξi |Vi ∩Vj = ξj |Vi ∩Vj . Say ξi = (gi , {fi,U }). Then we immediately see that the morphisms gi glue S to a unique global morphism g : T → S. Moreover, it is clear that g −1 (U ) = gi−1 (U ). Hence the morphisms hi,U : gi−1 (U ) → XU glue to a 1287

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22. CONSTRUCTIONS OF SCHEMES

unique morphism hU : U → XU . It is easy to verify that the system (g, {fU }) is an element of F (T ). Hence F satisfies the sheaf property for the Zariski topology. Next we verify that each FW , W ∈ B is representable. Namely, we claim that the transformation of functors FW −→ Mor(−, XW ), (g, {hU }) 7−→ hW is an isomorphism. To see this suppose that T is a scheme and α : T → XW is a morphism. Set g = fW ◦ α. For any U ∈ B such that U ⊂ W we can define −1 hU : g −1 (U ) → XU be the composition (ρW ◦ α|g−1 (U ) . This works because U ) −1 −1 the image α(g (U )) is contained in fW (U ) and condition (a) of the lemma. It is clear that fU ◦ hU = g|g−1 (U ) for such a U . Moreover, if also V ∈ B and V ⊂ U ⊂ W , then ρU V ◦ hV = hU |g −1 (V ) by property (b) of the lemma. We still have to define hU for an arbitrary element U ∈ B. Since S B is a basis for the topology on S we can find an open covering U ∩ WS = Ui with Ui ∈ B. Since g maps into W we have g −1 (U ) = g −1 (U ∩ W ) = g −1 (Ui ). Consider the −1 morphisms hi = ρU (Ui ) → XU . It is a simple matter to use condition Ui ◦ hUi : g (b) of the lemma to prove that hi |g−1 (Ui )∩g−1 (Uj ) = hj |g−1 (Ui )∩g−1 (Uj ) . Hence these morphisms glue to give the desired morphism hU : g −1 (U ) → XU . We omit the (easy) verification that the system (g, {hU }) is an element of FW (T ) which maps to α under the displayed arrow above. Next, we verify each FW ⊂ F is representable by open immersions. This is clear from the definitions. Finally we have to verify the collection (FW )W ∈B covers F . This is clear by construction and the fact that B is a basis for the topology of S. Let X be a scheme representing the functor F . Let (f, {iU }) ∈ F (X) be a “universal family”. Since each FW is representable by XW (via the morphism of functors displayed above) we see that iW : f −1 (W ) → XW is an isomorphism as desired. The lemma is proved.  Lemma 22.2.2. Let S be a scheme. Let B be a basis for the topology of S. Suppose given the following data: (1) For every U ∈ B a scheme fU : XU → U over U . (2) For every U ∈ B a quasi-coherent sheaf FU over XU . (3) For every pair U, V ∈ B such that V ⊂ U a morphism ρU V : XV → XU . ∗ (4) For every pair U, V ∈ B such that V ⊂ U a morphism θVU : (ρU V ) FU → FV . Assume that −1 (a) each ρU V induces an isomorphism XV → fU (V ) of schemes over V , U (b) each θV is an isomorphism, U V (c) whenever W, V, U ∈ B, with W ⊂ V ⊂ U we have ρU W = ρV ◦ ρW , U V (d) whenever W, V, U ∈ B, with W ⊂ V ⊂ U we have θW = θW ◦ (ρVW )∗ θVU . Then there exists a unique scheme f : X → S over S together with a unique quasicoherent sheaf F on X and isomorphisms iU : f −1 (U ) → XU and θU : i∗U FU → F|f −1 (U ) over U such that for V ⊂ U ⊂ S affine open the composition XV

i−1 V

/ f −1 (V )

inclusion

/ f −1 (U )

iU

/ XU

22.3. RELATIVE SPECTRUM VIA GLUEING

1289

is the morphism ρU V , and the composition θU |f −1 (V )

θ −1

V −1 ∗ −1 ∗ ∗ ∗ (22.2.2.1) (ρU V ) FU = (iV ) ((iU FU )|f −1 (V ) ) −−−−−−→ (iV ) (F|f −1 (V ) ) −−→ FV

is equal to θVU . Proof. By Lemma 22.2.1 we get the scheme X over S and the isomorphisms iU . Set FU0 = i∗U FU for U ∈ B. This is a quasi-coherent Of −1 (U ) -module. The maps i∗ θ U

V V ∗ ∗ 0 FU0 |f −1 (V ) = i∗U FU |f −1 (V ) = i∗V (ρU V ) FU −−−→ iV FV = FV

0 0 define isomorphisms (θ0 )U V : FU |f −1 (V ) → FV whenever V ⊂ U are elements of B. Condition (d) says exactly that this is compatible in case we have a triple of elements W ⊂ V ⊂ U of B. This allows us to get well defined isomorphisms

ϕ12 : FU0 1 |f −1 (U1 ∩U2 ) −→ FU0 2 |f −1 (U1 ∩U2 ) S whenever U1 , U2 ∈ B by covering the intersection U1 ∩ U2 = Vj by elements Vj of B and taking  −1 2 1 ϕ12 |Vj = (θ0 )U ◦ (θ0 )U Vj Vj . We omit the verification that these maps do indeed glue to a ϕ12 and we omit the verification of the cocycle condition of a glueing datum for sheaves (as in Sheaves, Section 6.33). By Sheaves, Lemma 6.33.2 we get our F on X. We omit the verification of (22.2.2.1).  Remark 22.2.3. There is a functoriality property for the constructions explained in Lemmas 22.2.1 and 22.2.2. Namely, suppose given two collections of data (fU : U XU → U, ρU V ) and (gU : YU → U, σV ) as in Lemma 22.2.1. Suppose for every U ∈ B given a morphism hU : XU → YU over U compatible with the restrictions U ρU V and σV . Functoriality means that this gives rise to a morphism of schemes h : X → Y over S restricting back to the morphisms hU , where f : X → S is obtained from the datum (fU : XU → U, ρU V ) and g : Y → S is obtained from the datum (gU : YU → U, σVU ). U Similarly, suppose given two collections of data (fU : XU → U, FU , ρU V , θV ) and U U (gU : YU → U, GU , σV , ηV ) as in Lemma 22.2.2. Suppose for every U ∈ B given a U morphism hU : XU → YU over U compatible with the restrictions ρU V and σV , and ∗ U U a morphism τU : hU GU → FU compatible with the maps θV and ηV . Functoriality means that these give rise to a morphism of schemes h : X → Y over S restricting back to the morphisms hU , and a morphism h∗ G → F restricting back to the maps U hU where (f : X → S, F) is obtained from the datum (fU : XU → U, FU , ρU V , θV ) U U and where (g : Y → S, G) is obtained from the datum (gU : YU → U, GU , σV , ηV ).

We omit the verifications and we omit a suitable formulation of “equivalence of categories” between relative glueing data and relative objects. 22.3. Relative spectrum via glueing Situation 22.3.1. Here S is a scheme, and A is a quasi-coherent OS -algebra. In this section we outline how to construct a morphism of schemes SpecS (A) −→ S

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by glueing the spectra Spec(Γ(U, A)) where U ranges over the affine opens of S. We first show that the spectra of the values of A over affines form a suitable collection of schemes, as in Lemma 22.2.1. Lemma 22.3.2. In Situation 22.3.1. Suppose U ⊂ U 0 ⊂ S are affine opens. Let A = A(U ) and A0 = A(U 0 ). The map of rings A0 → A induces a morphism Spec(A) → Spec(A0 ), and the diagram Spec(A)

/ Spec(A0 )

 U

 / U0

is cartesian. Proof. Let R = OS (U ) and R0 = OS (U 0 ). Note that the map R ⊗R0 A0 → A is an isomorphism as A is quasi-coherent (see Schemes, Lemma 21.7.3 for example). The result follows from the description of the fibre product of affine schemes in Schemes, Lemma 21.6.7.  In particular the morphism Spec(A) → Spec(A0 ) of the lemma is an open immersion. Lemma 22.3.3. In Situation 22.3.1. Suppose U ⊂ U 0 ⊂ U 00 ⊂ S are affine opens. Let A = A(U ), A0 = A(U 0 ) and A00 = A(U 00 ). The composition of the morphisms Spec(A) → Spec(A0 ), and Spec(A0 ) → Spec(A00 ) of Lemma 22.3.2 gives the morphism Spec(A) → Spec(A00 ) of Lemma 22.3.2. Proof. This follows as the map A00 → A is the composition of A00 → A0 and A0 → A (because A is a sheaf).  Lemma 22.3.4. In Situation 22.3.1. There exists a morphism of schemes π : SpecS (A) −→ S with the following properties: (1) for every affine open U ⊂ S there exists an isomorphism iU : π −1 (U ) → Spec(A(U )), and (2) for U ⊂ U 0 ⊂ S affine open the composition Spec(A(U ))

i−1 U

/ π −1 (U )

inclusion

/ π −1 (U 0 )

iU 0

/ Spec(A(U 0 ))

is the open immersion of Lemma 22.3.2 above. Proof. Follows immediately from Lemmas 22.2.1, 22.3.2, and 22.3.3.



22.4. Relative spectrum as a functor We place ourselves in Situation 22.3.1. So S is a scheme and A is a quasi-coherent sheaf of OS -algebras. (This means that A is a sheaf of OS -algebras which is quasicoherent as an OS -module.) For any f : T → S the pullback f ∗ A is a quasi-coherent sheaf of OT -algebras. We are going to consider pairs (f : T → S, ϕ) where f is a morphism of schemes and ϕ : f ∗ A → OT is a morphism of OT -algebras. Note that this is the same as giving a f −1 OS -algebra homomorphism ϕ : f −1 A → OT , see Sheaves, Lemma 6.20.2. This

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is also the same as giving a OS -algebra map ϕ : A → f∗ OT , see Sheaves, Lemma 6.24.7. We will use all three ways of thinking about ϕ, without further mention. Given such a pair (f : T → S, ϕ) and a morphism a : T 0 → T we get a second pair (f 0 = f ◦ a, ϕ0 = a∗ ϕ) which we call the pullback of (f, ϕ). One way to describe ϕ0 = a∗ ϕ is as the composition A → f∗ OT → f∗0 OT 0 where the second map is f∗ a] with a] : OT → a∗ OT 0 . In this way we have defined a functor (22.4.0.1)

F : Schopp

−→

Sets

T

7−→

F (T ) = {pairs (f, ϕ) as above}

Lemma 22.4.1. In Situation 22.3.1. Let F be the functor associated to (S, A) above. Let g : S 0 → S be a morphism of schemes. Set A0 = g ∗ A. Let F 0 be the functor associated to (S 0 , A0 ) above. Then there is a canonical isomorphism ∼ hS 0 ×h F F0 = S

of functors. Proof. A pair (f 0 : T → S 0 , ϕ0 : (f 0 )∗ A0 → OT ) is the same as a pair (f, ϕ : f ∗ A → OT ) together with a factorization of f as f = g ◦ f 0 . Namely with this notation we have (f 0 )∗ A0 = (f 0 )∗ g ∗ A = f ∗ A. Hence the lemma.  Lemma 22.4.2. In Situation 22.3.1. Let F be the functor associated to (S, A) above. If S is affine, then F is representable by the affine scheme Spec(Γ(S, A)). e Proof. Write S = Spec(R) and A = Γ(S, A). Then A is an R-algebra and A = A. The ring map R → A gives rise to a canonical map funiv : Spec(A) −→ S = Spec(R). ∗ We have funiv A = A^ ⊗R A by Schemes, Lemma 21.7.3. Hence there is a canonical map ∗ e = OSpec(A) ϕuniv : funiv A = A^ ⊗R A −→ A coming from the A-module map A ⊗R A → A, a ⊗ a0 7→ aa0 . We claim that the pair (funiv , ϕuniv ) represents F in this case. In other words we claim that for any scheme T the map

Mor(T, Spec(A)) −→ {pairs (f, ϕ)}, a 7−→ (a∗ funiv , a∗ ϕ) is bijective. Let us construct the inverse map. For any pair (f : T → S, ϕ) we get the induced ring map A = Γ(S, A)

f∗

/ Γ(T, f ∗ A)

ϕ

/ Γ(T, OT )

This induces a morphism of schemes T → Spec(A) by Schemes, Lemma 21.6.4. The verification that this map is inverse to the map displayed above is omitted.  Lemma 22.4.3. In Situation 22.3.1. The functor F is representable by a scheme. Proof. We are going to use Schemes, Lemma 21.15.4. First we check that F satisfies the sheaf S property for the Zariski topology. Namely, suppose that T is a scheme, that T = i∈I Ui is an open covering, and that (fi , ϕi ) ∈ F (Ui ) such that (fi , ϕi )|Ui ∩Uj = (fj , ϕj )|Ui ∩Uj . This implies that the morphisms fi : Ui → S glue to a morphism of schemes f : T → S such that f |Ii = fi , see

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Schemes, Section 21.14. Thus fi∗ A = f ∗ A|Ui and by assumption the morphisms ϕi agree on Ui ∩ Uj . Hence by Sheaves, Section 6.33 these glue to a morphism of OT -algebras f ∗ A → OT . This proves that F satisfies the sheaf condition with respect to the Zariski topology. S Let S = i∈I Ui be an affine open covering. Let Fi ⊂ F be the subfunctor consisting of those pairs (f : T → S, ϕ) such that f (T ) ⊂ Ui . We have to show each Fi is representable. This is the case because Fi is identified with the functor associated to Ui equipped with the quasi-coherent OUi -algebra A|Ui , by Lemma 22.4.1. Thus the result follows from Lemma 22.4.2. Next we show that Fi ⊂ F is representable by open immersions. Let (f : T → S, ϕ) ∈ F (T ). Consider Vi = f −1 (Ui ). It follows from the definition of Fi that given a : T 0 → T we gave a∗ (f, ϕ) ∈ Fi (T 0 ) if and only if a(T 0 ) ⊂ Vi . This is what we were required to show. Finally, we have to show that the collection (F Si )i∈I covers F . Let (f : T → S, ϕ) ∈ −1 F (T ). Consider V = f (U ). Since S = i i∈I Ui is an open covering of S we S i see that T = i∈I Vi is an open covering of T . Moreover (f, ϕ)|Vi ∈ Fi (Vi ). This finishes the proof of the lemma.  Lemma 22.4.4. In Situation 22.3.1. The scheme π : SpecS (A) → S constructed in Lemma 22.3.4 and the scheme representing the functor F are canonically isomorphic as schemes over S. Proof. Let X → S be the scheme representing the functor F . Consider the sheaf of OS -algebras R = π∗ OSpec (A) . By construction of SpecS (A) we have isomorphisms S A(U ) → R(U ) for every affine open U ⊂ S; this follows from Lemma 22.3.4 part (1). For U ⊂ U 0 ⊂ S open these isomorphisms are compatible with the restriction mappings; this follows from Lemma 22.3.4 part (2). Hence by Sheaves, Lemma 6.30.13 these isomorphisms result from an isomorphism of OS -algebras ϕ : A → R. Hence this gives an element (SpecS (A), ϕ) ∈ F (SpecS (A)). Since X represents the functor F we get a corresponding morphism of schemes can : SpecS (A) → X over S. Let U ⊂ S be any affine open. Let FU ⊂ F be the subfunctor of F corresponding to pairs (f, ϕ) over schemes T with f (T ) ⊂ U . Clearly the base change XU represents FU . Moreover, FU is represented by Spec(A(U )) = π −1 (U ) according to Lemma 22.4.2. In other words XU ∼ = π −1 (U ). We omit the verification that this identification is brought about by the base change of the morphism can to U .  Definition 22.4.5. Let S be a scheme. Let A be a quasi-coherent sheaf of OS algebras. The relative spectrum of A over S, or simply the spectrum of A over S is the scheme constructed in Lemma 22.3.4 which represents the functor F (22.4.0.1), see Lemma 22.4.4. We denote it π : SpecS (A) → S. The “universal family” is a morphism of OS -algebras A −→ π∗ OSpec (A) S

The following lemma says among other things that forming the relative spectrum commutes with base change. Lemma 22.4.6. Let S be a scheme. Let A be a quasi-coherent sheaf of OS -algebras. Let π : SpecS (A) → S be the relative spectrum of A over S.

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(1) For every affine open U ⊂ S the inverse image f −1 (U ) is affine. (2) For every morphism g : S 0 → S we have S 0 ×S SpecS (A) = SpecS 0 (g ∗ A). (3) The universal map A −→ π∗ OSpec

S

(A)

is an isomorphism of OS -algebras. Proof. Part (1) comes from the description of the relative spectrum by glueing, see Lemma 22.3.4. Part (2) follows immediately from Lemma 22.4.1. Part (3) follows because it is local on S and it is clear in case S is affine by Lemma 22.4.2 for example.  Lemma 22.4.7. Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. By Schemes, Lemma 21.24.1 the sheaf f∗ OX is a quasi-coherent sheaf of OS -algebras. There is a canonical morphism can : X −→ SpecS (f∗ OX ) of schemes over S. For any affine open U ⊂ S the restriction can|f −1 (U ) is identified with the canonical morphism f −1 (U ) −→ Spec(Γ(f −1 (U ), OX )) coming from Schemes, Lemma 21.6.4. Proof. The morphism comes, via the definition of Spec as the scheme representing the functor F , from the canonical map ϕ : f ∗ f∗ OX → OX (which by adjointness of push and pull corresponds to id : f∗ OX → f∗ OX ). The statement on the restriction to f −1 (U ) follows from the description of the relative spectrum over affines, see Lemma 22.4.2.  22.5. Affine n-space As an application of scheme S as follows. sheaf of OS -algebras OS -modules it is just

the relative spectrum we define affine n-space over a base For any integer n ≥ 0 we can consider the quasi-coherent OS [T1 , . . . , Tn ]. It is quasi-coherent because as a sheaf of the direct sum of copies of OS indexed by multi-indices.

Definition 22.5.1. Let S be a scheme and n ≥ 0. The scheme AnS = SpecS (OS [T1 , . . . , Tn ]) over S is called affine n-space over S. If S = Spec(R) is affine then we also call this affine n-space over R and we denote it AnR . Note that AnR = Spec(R[T1 , . . . , Tn ]). For any morphism g : S 0 → S of schemes we have g ∗ OS [T1 , . . . , Tn ] = OS 0 [T1 , . . . , Tn ] and hence AnS 0 = S 0 ×S AnS is the base change. Therefore an alternative definition of affine n-space is the formula AnS = S ×Spec(Z) AnZ . Also, a morphism from an S-scheme f : X → S to AnS is given by a homomorphism of OS -algebras OS [T1 , . . . , Tn ] → f∗ OX . This is clearly the same thing as giving the images of the Ti . In other words, a morphism from X to AnS over S is the same as giving n elements h1 , . . . , hn ∈ Γ(X, OX ).

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22.6. Vector bundles Let S be a scheme. Let E be a quasi-coherent sheaf of OS -modules. By Modules, Lemma 15.18.6 the symmetric algebra Sym(E) of E over OS is a quasi-coherent sheaf of OS -algebras. Hence it makes sense to apply the construction of the previous section to it. Definition 22.6.1. Let S be a scheme. Let E be a quasi-coherent OS -module1. The vector bundle associated to E is V(E) = SpecS (Sym(E)). The vector bundle associated to E comes with a bit of extra structure. Namely, we have a grading M π∗ OV(E) = Symn (E). n≥0

which turns π∗ OV(E) into a graded OS -algebra. Conversely, we can recover E from the degree 1 part of this. Thus we define an abstract vector bundle as follows. Definition 22.6.2. Let S be a scheme. A vector bundle π : V → S over S is an affine morphism of schemes L such that π∗ OV is endowed with the structure of a graded OS -algebra π∗ OV = n≥0 En such that E0 = OS and such that the maps Symn (E1 ) −→ En are isomorphisms for all n ≥ 0. A morphism of vector bundles over S is a morphism f : V → V 0 such that the induced map f ∗ : π∗0 OV 0 −→ π∗ OV is compatible with the given gradings. An example of a vector bundle over S is affine n-space AnS over S, see Definition 22.5.1. This is true because OS [T1 , . . . , Tn ] = Sym(OS⊕n ). Lemma 22.6.3. The category of vector bundles over a scheme S is anti-equivalent to the category of quasi-coherent OS -modules. Proof. Omitted. Hint: In one direction one uses the functor SpecS (−) and in the other the functor (π : V → S) (π∗ OV )1 (degree 1 part).  22.7. Cones In algebraic geometry cones correspond to graded algebras. By our conventions a L graded ring or algebra A comes with a grading A = d≥0 Ad by the nonnegative integers, see Algebra, Section 7.53. Definition 22.7.1. Let S be a scheme. Let A be a quasi-coherent graded OS algebra. Assume that OS → A0 is an isomorphism2. The cone associated to A or the affine cone associated to A is C(A) = SpecS (A). 1The reader may expect here the condition that E is finite locally free. We do not do so in order to be consistent with [DG67, II, Definition 1.7.8]. 2Often one imposes the assumption that A is generated by A over O . We do not assume 1 S this in order to be consisten with [DG67, II, (8.3.1)].

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The cone associated to a graded sheaf of OS -algebras comes with a bit of extra structure. Namely, we obtain a grading M π∗ OC(A) = An n≥0

Thus we can define an abstract cone as follows. Definition 22.7.2. Let S be a scheme. A cone π : C → S over S is an affine morphism of schemes L such that π∗ OC is endowed with the structure of a graded OS -algebra π∗ OC = n≥0 An such that A0 = OS . A morphism of cones from 0 0 π : C → S to π : C → S is a morphism f : C → C 0 such that the induced map f ∗ : π∗0 OC 0 −→ π∗ OC is compatible with the given gradings. Any vector bundle is an example of a cone. In fact the category of vector bundles over S is a full subcategory of the category of cones over S. 22.8. Proj of a graded ring Let S be a graded ring. Consider the topological space Proj(S) associated to S, see Algebra, Section 7.54. We will endow this space with a sheaf of rings OProj(S) such that the resulting pair (Proj(R), OProj(R) ) will be a scheme. Recall that Proj(S) has a basis of open sets D+ (f ), f ∈ Sd , d ≥ 1 which we call standard opens, see Algebra, Section 7.54. This terminology will always imply that f is homogeneous of positive degree even if we forget to mention it. In addition, the intersection of two standard opens is another: D+ (f ) ∩ D+ (g) = D+ (f g), for f, g ∈ S homogeneous of positive degree. Lemma 22.8.1. Let S be a graded ring. Let f ∈ S homogeneous of positive degree. (1) If g ∈ S homogeneous of positive degree and D+ (g) ⊂ D+ (f ), then (a) f is invertible in Sg , and f deg(g) /g deg(f ) is invertible in S(g) , (b) g e = af for some e ≥ 1 and a ∈ S homogeneous, (c) there is a canonical S-algebra map Sf → Sg , (d) there is a canonical S0 -algebra map S(f ) → S(g) compatible with the map Sf → Sg , (e) the map S(f ) → S(g) induces an isomorphism ∼ S(g) , (S(f ) )gdeg(f ) /f deg(g) = (f) these maps induce a commutative diagram of topological spaces D+ (g) o

{Z-graded primes of Sg }

/ Spec(S(g) )

 D+ (f ) o

 {Z-graded primes of Sf }

 / Spec(S(f ) )

where the horizontal maps are homeomorphisms and the vertical maps are open immersions, (g) there are a compatible canonical Sf and S(f ) -module maps Mf → Mg and M(f ) → M(g) for any graded S-module M , and (h) the map M(f ) → M(g) induces an isomorphism (M(f ) )gdeg(f ) /f deg(g) ∼ = M(g) .

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(2) Any open covering Sn of D+ (f ) can be refined to a finite open covering of the form D+ (f ) = i=1 D+ (gi ). (3) Let g1 , . . . , gn ∈ S be homogeneous of positive degree. Then D+ (f ) ⊂ S deg(f ) deg(f ) D+ (gi ) if and only if g1 /f deg(g1 ) , . . . , gn /f deg(gn ) generate the unit ideal in S(f ) . Proof. Recall that D+ (g) = Spec(S(g) ) with identification given by the ring maps S → Sg ← S(g) , see Algebra, Lemma 7.54.3. Thus f deg(g) /g deg(f ) is an element of S(g) which is not contained in any prime ideal, and hence invertible, see Algebra, Lemma 7.16.2. We conclude that (a) holds. Write the inverse of f in Sg as a/g d . We may replace a by its homogeneous part of degree d deg(g) − deg(f ). This means g d −af is annihilated by a power of g, whence g e = af for some a ∈ S homogeneous of degree e deg(g) − deg(f ). This proves (b). For (c), the map Sf → Sg exists by (a) from the universal property of localization, or we can define it by mapping b/f n to an b/g ne . This clearly induces a map of the subrings S(f ) → S(g) of degree zero elements as well. We can similarly define Mf → Mg and M(f ) → M(g) by mapping x/f n to an x/g ne . The statements writing S(g) resp. M(g) as principal localizations of S(f ) resp. M(f ) are clear from the formulas above. The maps in the commutative diagram of topological spaces correspond to the ring maps given above. The horizontal arrows are homeomorphisms by Algebra, Lemma 7.54.3. The vertical arrows are open immersions since the left one is the inclusion of an open subset. The open D+ (f ) is quasi-compact because it is homeomorphic to Spec(S(f ) ), see Algebra, Lemma 7.27.1. Hence the second statement follows directly from the fact that the standard opens form a basis for the topology. The third statement follows directly from Algebra, Lemma 7.16.2.



In Sheaves, Section 6.30 we defined the notion of a sheaf on a basis, and we showed that it is essentially equivalent to the notion of a sheaf on the space, see Sheaves, Lemmas 6.30.6 and 6.30.9. Moreover, we showed in Sheaves, Lemma 6.30.4 that it is sufficient to check the sheaf condition on a cofinal system of open coverings for each standard open. By the lemma above it suffices to check on the finite coverings by standard opens. Definition 22.8.2. Let S be a graded ring. Suppose that D+ (f ) ⊂ Proj(S) is Sna standard open. A standard open covering of D+ (f ) is a covering D+ (f ) = i=1 D+ (gi ), where g1 , . . . , gn ∈ S are homogeneous of positive degree. Let S be a graded ring. Let M be a graded S-module. We will define a presheaf f on the basis of standard opens. Suppose that U ⊂ Proj(S) is a standard open. M If f, g ∈ S are homogeneous of positive degree such that D+ (f ) = D+ (g), then by Lemma 22.8.1 above there are canonical maps M(f ) → M(g) and M(g) → M(f ) which are mutually inverse. Hence we may choose any f such that U = D+ (f ) and define f(U ) = M(f ) . M Note that if D+ (g) ⊂ D+ (f ), then by Lemma 22.8.1 above we have a canonical map f(D+ (f )) = M(f ) −→ M(g) = M f(D+ (g)). M

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Clearly, this defines a presheaf of abelian groups on the basis of standard opens. If M = S, then Se is a presheaf of rings on the basis of standard opens. And for f is a presheaf of S-modules e general M we see that M on the basis of standard opens. f at a point x ∈ Proj(S). Suppose that x corresponds Let us compute the stalk of M to the homogeneous prime ideal p ⊂ S. By definition of the stalk we see that fx = colimf ∈S ,d>0,f 6∈p M(f ) M d Here the set {f ∈ Sd , d > 0, f 6∈ p} is partially ordered by the rule f ≥ f 0 ⇔ D+ (f ) ⊂ D+ (f 0 ). If f1 , f2 ∈ S \ p are homogeneous of positive degree, then we have f1 f2 ≥ f1 in this ordering. In Algebra, Section 7.54 we defined M(p) as the ring whose elements are fractions x/f with x, f homogeneous, deg(x) = deg(f ), f 6∈ p. Since p ∈ Proj(S) there exists at least one f0 ∈ S homogeneous of positive degree with f0 6∈ p. Hence x/f = f0 x/f f0 and we see that we may always assume the denominator of an element in M(p) has positive degree. From these remarks it follows easily that fx = M(p) . M Next, we check the sheaf condition for the standard open coverings. If D+ (f ) = S n i=1 D+ (gi ), then the sheaf condition for this covering is equivalent with the exactness of the sequence M M 0 → M(f ) → M(gi ) → M(gi gj ) . Note that D+ (gi ) = D+ (f gi ), and hence we can rewrite this sequence as the sequence M M 0 → M(f ) → M(f gi ) → M(f gi gj ) . deg(f )

deg(f )

/f deg(gn ) generate the unit By Lemma 22.8.1 we see that g1 /f deg(g1 ) , . . . , gn ideal in S(f ) , and that the modules M(f gi ) , M(f gi gj ) are the principal localizations of the S(f ) -module M(f ) at these elements and their products. Thus we may apply Algebra, Lemma 7.21.2 to the module M(f ) over S(f ) and the elements deg(f )

deg(f )

/f deg(gn ) . We conclude that the sequence is exact. By g1 /f deg(g1 ) , . . . , gn f is a sheaf on the basis of standard opens. the remarks made above, we see that M Thus we conclude from the material in Sheaves, Section 6.30 that there exists a unique sheaf of rings OProj(S) which agrees with Se on the standard opens. Note that by our computation of stalks above and Algebra, Lemma 7.54.5 the stalks of this sheaf of rings are all local rings. Similarly, for any graded S-module M there exists a unique sheaf of OProj(S) f on the standard opens, see Sheaves, Lemma modules F which agrees with M 6.30.12. Definition 22.8.3. Let S be a graded ring. (1) The structure sheaf OProj(S) of the homogeneous spectrum of S is the unique sheaf of rings OProj(S) which agrees with Se on the basis of standard opens. (2) The locally ringed space (Proj(S), OProj(S) ) is called the homogeneous spectrum of S and denoted Proj(S).

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f to all opens of Proj(S) is (3) The sheaf of OProj(S) -modules extending M called the sheaf of OProj(S) -modules associated to M . This sheaf is def as well. noted M We summarize the results obtained so far. f be Lemma 22.8.4. Let S be a graded ring. Let M be a graded S-module. Let M the sheaf of OProj(S) -modules associated to M . (1) For every f ∈ S homogeneous of positive degree we have Γ(D+ (f ), OProj(S) ) = S(f ) . f) = (2) For every f ∈ S homogeneous of positive degree we have Γ(D+ (f ), M M(f ) as an S(f ) -module. f (3) Whenever D+ (g) ⊂ D+ (f ) the restriction mappings on OProj(S) and M are the maps S(f ) → S(g) and M(f ) → M(g) from Lemma 22.8.1. (4) Let p be a homogeneous prime of S not containing S+ , and let x ∈ Proj(S) be the corresponding point. We have OProj(S),x = S(p) . (5) Let p be a homogeneous prime of S not containing S+ , and let x ∈ Proj(S) be the corresponding point. We have Fx = M(p) as an S(p) -module. e and a canonical S0 (6) There is a canonical ring map S0 −→ Γ(Proj(S), S) f module map M0 −→ Γ(Proj(S), M ) compatible with the descriptions of sections over standard opens above and stalks above. Moreover, all these identifications are functorial in the graded S-module M . In f is an exact functor from the category of graded particular, the functor M 7→ M S-modules to the category of OProj(S) -modules. Proof. Assertions (1) - (5) are clear from the discussion above. We see (6) since there are canonical maps M0 → M(f ) , x 7→ x/1 compatible with the restriction f follows from the fact maps described in (3). The exactness of the functor M 7→ M that the functor M 7→ M(p) is exact (see Algebra, Lemma 7.54.5) and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 15.3.1.  f is generally far from Remark 22.8.5. The map from M0 to the global sections of M being an isomorphism. A trivial example is to take S = k[x, y, z] with 1 = deg(x) = deg(y) = deg(z) (or any number of variables) and to take M = S/(x100 , y 100 , z 100 ). f = 0, but M0 = k. It is easy to see that M Lemma 22.8.6. Let S be a graded ring. Let f ∈ S be homogeneous of positive degree. Suppose that D(g) ⊂ Spec(S(f ) ) is a standard open. Then there exists a h ∈ S homogeneous of positive degree such that D(g) corresponds to D+ (h) ⊂ D+ (f ) via the homeomorphism of Algebra, Lemma 7.54.3. In fact we can take h such that g = h/f n for some n. Proof. Write g = h/f n for some h homogeneous of positive degree and some n ≥ 1. If D+ (h) is not contained in D+ (f ) then we replace h by hf and n by n + 1. Then h has the required shape and D+ (h) ⊂ D+ (f ) corresponds to D(g) ⊂ Spec(S(f ) ).  Lemma 22.8.7. Let S be a graded ring. The locally ringed space Proj(S) is a scheme. The standard opens D+ (f ) are affine opens. For any graded S-module M f is a quasi-coherent sheaf of OProj(S) -modules. the sheaf M

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Proof. Consider a standard open D+ (f ) ⊂ Proj(S). By Lemmas 22.8.1 and 22.8.4 we have Γ(D+ (f ), OProj(S) ) = S(f ) , and we have a homeomorphism ϕ : D+ (f ) → Spec(S(f ) ). For any standard open D(g) ⊂ Spec(S(f ) ) we may pick a h ∈ S+ as in Lemma 22.8.6. Then ϕ−1 (D(g)) = D+ (h), and by Lemmas 22.8.4 and 22.8.1 we see Γ(D+ (h), OProj(S) ) = S(h) = (S(f ) )hdeg(f ) /f deg(h) = (S(f ) )g = Γ(D(g), OSpec(S(f ) ) ). Thus the restriction of OProj(S) to D+ (f ) corresponds via the homeomorphism ϕ exactly to the sheaf OSpec(S(f ) ) as defined in Schemes, Section 21.5. We conclude that D+ (f ) is an affine scheme isomorphic to Spec(S(f ) ) via ϕ and hence that Proj(S) is a scheme. f is a quasi-coherent sheaf of OProj(S) In exactly the same way we show that M modules. Namely, the argument above will show that   f|D (f ) ∼ ] M = ϕ∗ M (f ) + f is quasi-coherent. which shows that M



Lemma 22.8.8. Let S be a graded ring. The scheme Proj(S) is separated. Proof. We have to show that the canonical morphism Proj(S) → Spec(Z) is separated. We will use Schemes, Lemma 21.21.8. Thus it suffices to show given any pair of standard opens D+ (f ) and D+ (g) that D+ (f ) ∩ D+ (g) = D+ (f g) is affine (clear) and that the ring map S(f ) ⊗Z S(g) −→ S(f g) is surjective. Any element s in S(f g) is of the form s = h/(f n g m ) with h ∈ S homogeneous of degree n deg(f ) + m deg(g). We may multiply h by a suitable monomial f i g j and assume that n = n0 deg(g), and m = m0 deg(f ). Then we can 0 0 0 0 rewrite s as s = h/f (n +m ) deg(g) · f m deg(g) /g m deg(f ) . So s is indeed in the image of the displayed arrow.  Lemma 22.8.9. Let S be a graded ring. The scheme Proj(S) is quasi-compact if and only ifpthere exist finitely many homogeneous elements f1 , . . . , fn ∈ S+ such that S+ ⊂ (f1 , . . . , fn ). Proof. Given such a collection of elements the standard affine opens D+ (fi ) cover Proj(S) by Algebra, Lemma 7.54.3. Conversely, if Proj(S) is quasi-compact, then we may cover p it by finitely many standard opens D+ (fi ), i = 1, . . . , n and we see that S+ ⊂ (f1 , . . . , fn ) by the lemma referenced above.  Lemma 22.8.10. Let S be a graded ring. The scheme Proj(S) has a canonical morphism towards the affine scheme Spec(S0 ), agreeing with the map on topological spaces coming from Algebra, Definition 7.54.1. e resp. M f gives a sheaf of S0 Proof. We saw above that our construction of S, algebras, resp. S0 -modules. Hence we get a morphism by Schemes, Lemma 21.6.4. This morphism, when restricted to D+ (f ) comes from the canonical ring map S0 → S(f ) . The maps S → Sf , S(f ) → Sf are S0 -algebra maps, see Lemma 22.8.1. Hence if the homogeneous prime p ⊂ S corresponds to the Z-graded prime p0 ⊂ Sf and the (usual) prime p00 ⊂ S(f ) , then each of these has the same inverse image in S0 . 

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Lemma 22.8.11. Let S be a graded ring. If S is finitely generated as an algebra over S0 , then the morphism Proj(S) → Spec(S0 ) satisfies the existence and uniqueness parts of the valuative criterion, see Schemes, Definition 21.20.3. Proof. The uniqueness part follows from the fact that Proj(S) is separated (Lemma 22.8.8 and Schemes, Lemma 21.22.1). Choose xi ∈ S+ homogeneous, i = 1, . . . , n which generate S over S0 . Let di = deg(xi ) and set d = lcm{di }. Suppose we are given a diagram / Proj(S) Spec(K)  Spec(A)

 / Spec(S0 )

as in Schemes, Definition 21.20.3. Denote v : K ∗ → Γ the valuation of A, see Algebra, Definition 7.47.8. We may choose an f ∈ S+ homogeneous such that Spec(K) maps into D+ (f ). Then we get a commutative diagram of ring maps KO o

ϕ

Ao

S(f ) O S0 deg(f )

Let i0 ∈ {1, . . . , n} be an index minimizing the valuation (d/di )v(ϕ(xi /f di )) where we temporarily use the convention that the valuation of zero is bigger than any element of the value group. For convenience set x0 = xi0 and d0 = di0 . Since the open sets D+ (xi ) cover Proj(S) we see that ϕ(x0 ) 6= 0. This means that the ring map ϕ factors though a map ϕ0 : S(f x0 ) → K. We see that deg(f )

deg(f )v(ϕ0 (xdi 0 /xd0i )) = d0 v(ϕ(xi

deg(f )

/f di )) − di v(ϕ(x0

/f d0 )) ≥ 0

by our choice of i0 . This implies that the S0 -algebra S(x0 ) , which is generated by the elements xdi 0 /xd0i over S0 , maps into A via ϕ0 . The corresponding morphism of schemes Spec(A) → Spec(S(x0 ) ) = D+ (x0 ) ⊂ Proj(S) provides the morphism fitting into the first commutative diagram of this proof.  We saw in the proof of Lemma 22.8.11 that, under the hypotheses of that lemma, the morphism Proj(S) → Spec(S0 ) is quasi-compact as well. Hence (by Schemes, Proposition 21.20.6) we see that Proj(S) → Spec(S0 ) is universally closed in the situation of the lemma. We give two examples showing these results do not hold without some assumption on the graded ring S. Example 22.8.12. Let C[X1 , X2 , X3 , . . .] be the graded C-algebra with each Xi in degree 0. Consider the ring map C[X1 , X2 , X3 , . . .] −→ C[tα ; α ∈ Q≥0 ] which maps Xi to t1/i . The right hand side becomes a valuation ring A upon localization at the ideal m = (tα ; α > 0). This gives a morphism from Spec(f.f.(A)) to Proj(C[X1 , X2 , X3 , . . .]) which does not extend to a morphism defined on all of Spec(A). The reason is that the image of Spec(A) would be contained in one of the D+ (Xi ) but then Xi+1 /Xi would map to an element of A which it doesn’t since it maps to t1/(i+1)−1/i .

22.9. QUASI-COHERENT SHEAVES ON PROJ

1301

Example 22.8.13. Let R = C[t] and S = R[X1 , X2 , X3 , . . .]/(Xi2 − tXi+1 ). The grading is such that R = S0 and deg(Xi ) = 2i−1 . Note that if p ∈ Proj(S) then t 6∈ p (otherwise p has to contain all of the Xi which is not allowed for an element of the homogeneous spectrum). Thus we see that D+ (Xi ) = D+ (Xi+1 ) for all i. Hence Proj(S) is quasi-compact; in fact it is affine since it is equal to D+ (X1 ). It is easy to see that the image of Proj(S) → Spec(R) is D(t). Hence the morphism Proj(S) → Spec(R) is not closed. Thus the valuative criterion cannot apply because it would imply that the morphism is closed (see Schemes, Proposition 21.20.6 ). Example 22.8.14. Let A be a ring. Let S = A[T ] as a graded A algebra with T in degree 1. Then the canonical morphism Proj(S) → Spec(A) (see Lemma 22.8.10) is an isomorphism. 22.9. Quasi-coherent sheaves on Proj Let S be a graded ring. Let M be a graded S-module. We saw in the previous f on Proj(S) and a section how to construct a quasi-coherent sheaf of modules M map f) M0 −→ Γ(Proj(S), M f. The degree 0 part of the nth of the degree 0 part of M to the global sections of M twist M (n) of the graded module M (see Algebra, Section 7.53) is equal to Mn . Hence we can get maps ^ Mn −→ Γ(Proj(S), M (n)). We would like to be able to perform this operation for any quasi-coherent sheaf F on Proj(S). We will do this by tensoring with the nth twist of the structure sheaf, see Definition 22.10.1. In order to relate the two notions we will use the following lemma. Lemma 22.9.1. Let S be a graded ring. Let (X, OX ) = (Proj(S), OProj(S) ) be the scheme of Lemma 22.8.7. Let f ∈ S+ be homogeneous. Let x ∈ X be a point corresponding to the homogeneous prime p ⊂ S. Let M , N be graded S-modules. There is a canonical map of OProj(S) -modules f ⊗O N e −→ M^ M ⊗S N X which induces the canonical map M(f ) ⊗S(f ) N(f ) → (M ⊗S N )(f ) on sections over D+ (f ) and the canonical map M(p) ⊗S(p) N(p) → (M ⊗S N )(p) on stalks at x. Moreover, the following diagram

is commutative.

M0 ⊗S0 N0

/ (M ⊗S N )0

 f ⊗O N e) Γ(X, M X

 / Γ(X, M^ ⊗R N )

1302

22. CONSTRUCTIONS OF SCHEMES

Proof. To construct a morphism as displayed is the same as constructing a OX bilinear map f×N e −→ M^ M ⊗R N see Modules, Section 15.15. It suffices to define this on sections over the opens D+ (f ) compatible with restriction mappings. On D+ (f ) we use the S(f ) -bilinear map M(f ) × N(f ) → (M ⊗S N )(f ) , (x/f n , y/f m ) 7→ (x ⊗ y)/f n+m . Details omitted.  Remark 22.9.2. In general the map constructed in Lemma 22.9.1 above is not an isomorphism. Here is an example. Let k be a field. Let S = k[x, y, z] with k in degree 0 and deg(x) = 1, deg(y) = 2, deg(z) = 3. Let M = S(1) and N = S(2), see Algebra, Section 7.53 for notation. Then M ⊗S N = S(3). Note that Sz

=

k[x, y, z, 1/z]

S(z)

=

k[x3 /z, xy/z, y 3 /z 2 ] ∼ = k[u, v, w]/(uw − v 3 )

M(z)

=

S(z) · x + S(z) · y 2 /z ⊂ Sz

N(z)

=

S(z) · y + S(z) · x2 ⊂ Sz

S(3)(z)

= S(z) · z ⊂ Sz

Consider the maximal ideal m = (u, v, w) ⊂ S(z) . It is not hard to see that both M(z) /mM(z) and N(z) /mN(z) have dimension 2 over κ(m). But S(3)(z) /mS(3)(z) has dimension 1. Thus the map M(z) ⊗ N(z) → S(3)(z) is not an isomorphism. 22.10. Invertible sheaves on Proj Recall from Algebra, Section 7.53 the construction of the twisted module M (n) associated to a graded module over a graded ring. Definition 22.10.1. Let S be a graded ring. Let X = Proj(S). ] This is called the nth twist of the structure (1) We define OX (n) = S(n). sheaf of Proj(S). (2) For any sheaf of OX -modules F we set F(n) = F ⊗OX OX (n). We are going to use Lemma 22.9.1 to construct some canonical maps. Since S(n)⊗S S(m) = S(n + m) we see that there are canonical maps OX (n) ⊗OX OX (m) −→ OX (n + m).

(22.10.1.1)

These maps are not isomorphisms in general, see the example in Remark 22.9.2. The same example shows that OX (n) is not an invertible sheaf on X in general. Tensoring with an arbitrary OX -module F we get maps OX (n) ⊗OX F(m) −→ F(n + m).

(22.10.1.2)

The maps (22.10.1.1) on global sections give a map of graded rings M (22.10.1.3) S −→ Γ(X, OX (n)). n≥0

And for an arbitrary OX -module F the maps (22.10.1.2) give a graded module structure M M M (22.10.1.4) Γ(X, OX (n)) × Γ(X, F(m)) −→ Γ(X, F(m)) n≥0

m∈Z

m∈Z

22.10. INVERTIBLE SHEAVES ON PROJ

1303

and via (22.10.1.3) also a S-module structure. More generally, given any graded S-module M we have M (n) = M ⊗S S(n). Hence we get maps (22.10.1.5)

^ f(n) = M f ⊗O OX (n) −→ M M (n). X

On global sections we get a map of graded S-modules M ^ (22.10.1.6) M −→ Γ(X, M (n)). n∈Z

Here is an important fact which follows basically immediately from the definitions. Lemma 22.10.2. Let S be a graded ring. Set X = Proj(S). Let f ∈ S be homogeneous of degree d > 0. The sheaves OX (nd)|D+ (f ) are invertible, and in fact trivial for all n ∈ Z (see Modules, Definition 15.21.1). The maps (22.10.1.1) restricted to D+ (f ) OX (nd)|D+ (f ) ⊗OD+ (f ) OX (m)|D+ (f ) −→ OX (nd + m)|D+ (f ) and the maps (22.10.1.5) restricted to D+ (f ) ^ f(nd)|D (f ) = M f|D (f ) ⊗O M OX (nd)|D+ (f ) −→ M (nd)|D+ (f ) + + D+ (f ) are isomorphisms for all n, m ∈ Z. Proof. The (not graded) S-module maps S → S(n), and M → M (n), given by x 7→ f n/d x become isomorphisms after inverting f . The first shows that S(f ) ∼ = S(n)(f ) which gives an isomorphism OD+ (f ) ∼ = OX (n)|D+ (f ) . The second shows that the map S(n)(f ) ⊗S(f ) M(f ) → M (n)(f ) is an isomorphism.  Lemma 22.10.3. Let S be a graded ring generated as an S0 -algebra by the elements of S1 . Set X = Proj(S). In this case the sheaves OX (n) are all invertible, and all the maps (22.10.1.1) and (22.10.1.5) are isomorphisms. In particular, these maps induce isomorphisms OX (n) ∼ = OX (1)⊗n

and

^ f ⊗O OX (1)⊗n . M (n) = M X

In fact the lemma holds more generally if X is covered by the standard opens D+ (f ) with f ∈ S1 . Proof. Under the assumptions of the lemma X is covered by the open subsets D+ (f ) with f ∈ S1 and the lemma is a consequence of Lemma 22.10.2 above.  Lemma 22.10.4. Let S be a graded ring. Set X = Proj(S). Fix d ≥ 1 an integer. The following open subsets of X are equal: (1) The largest open subset W = Wd ⊂ X such that each OX (dn)|W is invertible and all the multiplication maps OX (nd)|W ⊗OW OX (md)|W → OX (nd + md)|W (see 22.10.1.1) are isomorphisms. (2) The union of the open subsets D+ (f g) with f, g ∈ S homogeneous and deg(f ) = deg(g) + d. ^ f(nd)|W = M f|W ⊗O OX (nd)|W → M Moreover, all the maps M (nd)|W (see 22.10.1.5) W are isomorphisms. Proof. If x ∈ D+ (f g) with deg(f ) = deg(g) + d then on D+ (f g) the sheaves OX (dn) are generated by the element (f /g)n = f 2n /(f g)n . This implies x is in the open subset W defined in (1) by arguing as in the proof of Lemma 22.10.2.

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22. CONSTRUCTIONS OF SCHEMES

Conversely, suppose that OX (d) is free of rank 1 in an open neighbourhood V of x ∈ X and all the multiplication maps OX (nd)|V ⊗OV OX (md)|V → OX (nd+md)|V are isomorphisms. We may choose h ∈ S+ homogeneous such that D+ (h) ⊂ V . By the definition of the twists of the structure sheaf we conclude there exists an element s of (Sh )d such that sn is a basis of (Sh )nd as a module over S(h) for all n ∈ Z. We may write s = f /hm for some m ≥ 1 and f ∈ Sd+m deg(h) . Set g = hm so s = f /g. Note that x ∈ D(g) by construction. Note that g d ∈ (Sh )−d deg(g) . By assumption we can write this as a multiple of sdeg(g) = f deg(g) /g deg(g) , say g d = a/g e · f deg(g) /g deg(g) . Then we conclude that g d+e+deg(g) = af deg(g) and hence also x ∈ D+ (f ). So x is an element of the set defined in (2). The existence of the generating section s = f /g over the affine open D+ (f g) whose powers freely generate the sheaves of modules OX (nd) easily implies that the mul^ f(nd)|W = M f|W ⊗O OX (nd)|W → M tiplication maps M (nd)|W (see 22.10.1.5) are W isomorphisms. Compare with the proof of Lemma 22.10.2.  Recall from Modules, Lemma 15.21.7 that given an invertible sheaf L on a locally ringed space X, and given a global section s of L the set Xs = {x ∈ X | s 6∈ mx Lx } is open. Lemma 22.10.5. Let S be a graded ring. Set X = Proj(S). Fix d ≥ 1 an integer. Let W = Wd ⊂ X be the open subscheme defined in Lemma 22.10.4. Let n ≥ 1 and f ∈ Snd . Denote s ∈ Γ(W, OW (nd)) the section which is the image of f via (22.10.1.3) restricted to W . Then Ws = D+ (f ) ∩ W. Proof. Let D+ (ab) ⊂ W be a standard affine open with a, b ∈ S homogeneous and deg(a) = deg(b) + d. Note that D+ (ab) ∩ D+ (f ) = D+ (abf ). On the other hand the restriction of s to D+ (ab) corresponds to the element f /1 = bn f /an (a/b)n ∈ (Sab )nd . We have seen in the proof of Lemma 22.10.4 that (a/b)n is a generator for OW (nd) over D+ (ab). We conclude that Ws ∩ D+ (ab) is the principal open associated to bn f /an ∈ OX (D+ (ab)). Thus the result of the lemma is clear.  The following lemma states the properties that we will later use to characterize schemes with an ample invertible sheaf. Lemma 22.10.6. Let S be a graded ring. Let X = Proj(S). Let Y ⊂ X be a quasi-compact open subscheme. Denote OY (n) the restriction of OX (n) to Y . There exists an integer d ≥ 1 such that (1) the subscheme Y is contained in the open Wd defined in Lemma 22.10.4, (2) the sheaf OY (dn) is invertible for all n ∈ Z, (3) all the maps OY (nd)⊗OY OY (m) −→ OY (nd+m) of Equation (22.10.1.1) are isomorphisms, ^ f(nd)|Y = M f|Y ⊗O OX (nd)|Y → M (4) all the maps M (n)|Y (see 22.10.1.5) Y are isomorphisms, (5) given f ∈ Snd denote s ∈ Γ(Y, OY (nd)) the image of f via (22.10.1.3) restricted to Y , then D+ (f ) ∩ Y = Ys , (6) a basis for the topology on Y is given by the collection of opens Ys , where s ∈ Γ(Y, OY (nd)), n ≥ 1, and (7) a basis for the topology of Y is given by those opens Ys ⊂ Y , for s ∈ Γ(Y, OY (nd)), n ≥ 1 which are affine.

22.11. FUNCTORIALITY OF PROJ

1305

Proof. Since Y is quasi-compact there exist finitely many homogeneous fi ∈ S+ , i = 1, . . . , n such that the standard opens D+ (fi ) give an open covering of Y . Let d/d di = deg(fi ) and set d = d1 . . . dn . Note that D+ (fi ) = D+ (fi i ) and hence we see immediately that Y ⊂ Wd , by characterization (2) in Lemma 22.10.4 or by (1) using Lemma 22.10.2. Note that (1) implies (2), (3) and (4) by Lemma 22.10.4. (Note that (3) is a special case of (4).) Assertion (5) follows from Lemma 22.10.5. Assertions (6) and (7) follow because the open subsets D+ (f ) form a basis for the topology of X and are affine.  22.11. Functoriality of Proj A graded ring map ψ : A → B does not always give rise to a morphism of associated projective homogeneous spectra. The reason is that the inverse image ψ −1 (q) of a homogeneous prime q ⊂ B may contain the irrelevant prime A+ even if q does not contain B+ . The correct result is stated as follows. Lemma 22.11.1. Let A, B be two graded rings. Set X = Proj(A) and Y = Proj(B). Let ψ : A → B be a graded ring map. Set [ U (ψ) = D+ (ψ(f )) ⊂ Y. f ∈A+ homogeneous

Then there is a canonical morphism of schemes rψ : U (ψ) −→ X and a map of Z-graded OU (ψ) -algebras  M M ∗ θ = θψ : rψ OX (d) −→ d∈Z

d∈Z

OU (ψ) (d).

The triple (U (ψ), rψ , θ) is characterized by the following properties: (1) For every d ≥ 0 the diagram Ad

/ Bd

ψ

 Γ(X, OX (d))

θ

/ Γ(U (ψ), OY (d)) o

 Γ(Y, OY (d))

is commutative. −1 (2) For any f ∈ A+ homogeneous we have rψ (D+ (f )) = D+ (ψ(f )) and the restriction of rψ to D+ (ψ(f )) corresponds to the ring map A(f ) → B(ψ(f )) induced by ψ. Proof. Clearly condition (2) uniquely determines the morphism of schemes and the open subset U (ψ). Pick f ∈ Ad with d ≥ 1. Note that OX (n)|D+ (f ) corresponds to the A(f ) -module (Af )n and that OY (n)|D+ (ψ(f )) corresponds to the B(ψ(f )) -module (Bψ(f ) )n . In other words θ when restricted to D+ (ψ(f )) corresponds to a map of Z-graded B(ψ(f )) -algebras Af ⊗A(f ) B(ψ(f )) −→ Bψ(f ) Condition (1) determines the images of all elements of A. Since f is an invertible element which is mapped to ψ(f ) we see that 1/f m is mapped to 1/ψ(f )m . It easily follows from this that θ is uniquely determined, namely it is given by the rule a/f m ⊗ b/ψ(f )e 7−→ ψ(a)b/ψ(f )m+e .

1306

22. CONSTRUCTIONS OF SCHEMES

To show existence we remark that the proof of uniqueness above gave a well defined prescription for the morphism r and the map θ when restricted to every standard open of the form D+ (ψ(f )) ⊂ U (ψ) into D+ (f ). Call these rf and θf . Hence we only need to verify that if D+ (f ) ⊂ D+ (g) for some f, g ∈ A+ homogeneous, then the restriction of rg to D+ (ψ(f )) matches rf . This is clear from the formulas given for r and θ above.  Lemma 22.11.2. Let A, B, and C be graded rings. Set X = Proj(A), Y = Proj(B) and Z = Proj(C). Let ϕ : A → B, ψ : B → C be graded ring maps. Then we have U (ψ ◦ ϕ) = rϕ−1 (U (ψ))

and

rψ◦ϕ = rϕ ◦ rψ |U (ψ◦ϕ) .

In addition we have ∗ θψ ◦ rψ θϕ = θψ◦ϕ

with obvious notation. Proof. Omitted. Lemma sume Ad (1) (2) (3)



22.11.3. With hypotheses and notation as in Lemma 22.11.1 above. As→ Bd is surjective for all d  0. Then U (ψ) = Y , rψ : Y → X is a closed immersion, and ∗ OX (n) → OY (n) are surjective but not isomorphisms in the maps θ : rψ general (even if A → B is surjective).

Proof. Part (1) follows from the definition of U (ψ) and the fact that D+ (f ) = D+ (f n ) for any n > 0. For f ∈ A+ homogeneous we see that A(f ) → B(ψ(f )) is surjective because any element of B(ψ(f )) can be represented by a fraction b/ψ(f )n with n arbitrarily large (which forces the degree of b ∈ B to be large). This proves (2). The same argument shows the map Af → Bψ(f ) is surjective which proves the surjectivity of θ. For an example where this map is not an isomorphism consider the graded ring A = k[x, y] where k is a field and deg(x) = 1, deg(y) = 2. Set I = (x), so that B = k[y]. Note that OY (1) = 0 ∗ OY (1) is not zero. (There are less silly in this case. But it is easy to see that rψ examples.)  Lemma sume Ad (1) (2) (3)

22.11.4. With hypotheses and notation as in Lemma 22.11.1 above. As→ Bd is an isomorphism for all d  0. Then U (ψ) = Y , rψ : Y → X is an isomorphism, and ∗ the maps θ : rψ OX (n) → OY (n) are isomorphisms.

Proof. We have (1) by Lemma 22.11.3. Let f ∈ A+ be homogeneous. The assumption on ψ implies that Af → Bf is an isomorphism (details omitted). Thus it is clear that rψ and θ restrict to isomorphisms over D+ (f ). The lemma follows.  Lemma 22.11.5. With hypotheses and notation as in Lemma 22.11.1 above. Assume Ad → Bd is surjective for d  0 and that A is generated by A1 over A0 . Then (1) U (ψ) = Y , (2) rψ : Y → X is a closed immersion, and

22.12. MORPHISMS INTO PROJ

1307

∗ (3) the maps θ : rψ OX (n) → OY (n) are isomorphisms.

Proof. By Lemmas 22.11.4 and 22.11.2 we may replace B by the image of A → B without changing X or the sheaves OX (n). Thus we may assume that A → B is surjective. By Lemma 22.11.3 we get (1) and (2) and surjectivity in (3). By Lemma 22.10.3 we see that both OX (n) and OY (n) are invertible. Hence θ is an isomorphism.  Lemma 22.11.6. With hypotheses and notation as in Lemma 22.11.1 above. Assume there exists a ring map R → A0 and a ring map R → R0 such that B = R0 ⊗R A. Then (1) U (ψ) = Y , (2) the diagram Y = Proj(B)  Spec(R0 )

rψ

/ Proj(A) = X  / Spec(R)

is a fibre product square, and ∗ (3) the maps θ : rψ OX (n) → OY (n) are isomorphisms. Proof. This follows immediately by looking at what happens over the standard opens D+ (f ) for f ∈ A+ .  Lemma 22.11.7. With hypotheses and notation as in Lemma 22.11.1 above. Assume there exists a g ∈ A0 such that ψ induces an isomorphism Ag → B. Then U (ψ) = Y , rψ : Y → X is an open immersion which induces an isomorphism of Y with the inverse image of D(g) ⊂ Spec(A0 ). Moreover the map θ is an isomorphism. Proof. This is a special case of Lemma 22.11.6 above.



22.12. Morphisms into Proj Let S be a graded ring. Let X = Proj(S) be the homogeneous spectrum of S. Let d ≥ 1 be an integer. Consider the open subscheme [ (22.12.0.1) Ud = D+ (f ) ⊂ X = Proj(S) f ∈Sd S Note that d|d0 ⇒ Ud ⊂ Ud0 and X = d Ud . Neither X nor Ud need be quasicompact, see Algebra, Lemma 7.54.3. Let us write OUd (n) = OX (n)|Ud . By Lemma 22.10.2 we know that OUd (nd), n ∈ Z is an invertible OUd -module and that all the multiplication maps OUd (nd) ⊗OUd OX (m) → OUd (nd + m) of (22.10.1.1) are isomorphisms. In particular we have OUd (nd) ∼ = OUd (d)⊗n . The graded ring map (22.10.1.3) on global sections combined with restriction to Ud give a homomorphism of graded rings (22.12.0.2)

ψ d : S (d) −→ Γ∗ (Ud , OUd (d)).

For the notation S (d) , see Algebra, Section 7.53. For the notation Γ∗ see Modules, Definition 15.21.4. Moreover, since Ud is covered by the opens D+ (f ), f ∈ Sd we (d) see that OUd (d) is globally generated by the sections in the image of ψ1d : S1 = Sd → Γ(Ud , OUd (d)), see Modules, Definition 15.4.1.

1308

22. CONSTRUCTIONS OF SCHEMES

Let Y be a scheme, and let ϕ : Y → X be a morphism of schemes. Assume the image ϕ(Y ) is contained in the open subscheme Ud of X. By the discussion following Modules, Definition 15.21.4 we obtain a homomorphism of graded rings Γ∗ (Ud , OUd (d)) −→ Γ∗ (Y, ϕ∗ OX (d)). The composition of this and ψ d gives a graded ring homomorphism ψϕd : S (d) −→ Γ∗ (Y, ϕ∗ OX (d))

(22.12.0.3)

which has the property that the invertible sheaf ϕ∗ OX (d) is globally generated by the sections in the image of (S (d) )1 = Sd → Γ(Y, ϕ∗ OX (d)). Lemma 22.12.1. Let S be a graded ring, and X = Proj(S). Let d ≥ 1 and Ud ⊂ X as above. Let Y be a scheme. Let L be an invertible sheaf on Y . Let ψ : S (d) → Γ∗ (Y, L) be a graded ring homomorphism such that L is generated by the sections in the image of ψ|Sd : Sd → Γ(Y, L). Then there exists a morphism ϕ : Y → X such that ϕ(Y ) ⊂ Ud and an isomorphism α : ϕ∗ OUd (d) → L such that ψϕd agrees with ψ via α: Γ∗ (Y, L) o O

α

Γ∗ (Y, ϕ∗ OUd (d)) o i

ψ

S (d) o

id

ϕ∗

Γ∗ (Ud , OUd (d)) O ψd

d ψϕ

S (d)

commutes. Moreover, the pair (ϕ, α) is unique. Proof. Pick f ∈ Sd . Denote s = ψ(f ) ∈ Γ(Y, L). On the open set Ys where s does not vanish multiplication by s induces an isomorphism OYs → L|Ys , see Modules, Lemma 15.21.7. We will denote the inverse of this map x 7→ x/s, and similarly for powers of L. Using this we define a ring map ψ(f ) : S(f ) → Γ(Ys , O) by mapping the fraction a/f n to ψ(a)/sn . By Schemes, Lemma 21.6.4 this corresponds to a morphism ϕf : Ys → Spec(S(f ) ) = D+ (f ). We also introduce the isomorphism αf : ϕ∗f OD+ (f ) (d) → L|Ys which maps the pullback of the trivializing section f over D+ (f ) to the trivializing section s over Ys . With this choice the commutativity of the diagram in the lemma holds with Y replace by Ys , ϕ replaced by ϕf , and α replaced by αf ; verification omitted. Suppose that f 0 ∈ Sd is a second element, and denote s0 = ψ(f 0 ) ∈ Γ(Y, L). Then Ys ∩Ys0 = Yss0 and similarly D+ (f )∩D+ (f 0 ) = D+ (f f 0 ). In Lemma 22.10.6 we saw that D+ (f 0 ) ∩ D+ (f ) is the same as the set of points of D+ (f ) where the section 0 0 of OX (d) defined by f 0 does not vanish. Hence ϕ−1 f (D+ (f ) ∩ D+ (f )) = Ys ∩ Ys = −1 0 0 ϕf 0 (D+ (f )∩D+ (f )). On D+ (f )∩D+ (f ) the fraction f /f 0 is an invertible section of the structure sheaf with inverse f 0 /f . Note that ψ(f 0 ) (f /f 0 ) = ψ(f )/s0 = s/s0 and ψ(f ) (f 0 /f ) = ψ(f 0 )/s = s0 /s. We claim there is a unique ring map S(f f 0 ) → Γ(Yss0 , O) making the following diagram commute Γ(Ys , O) O

/ Γ(Yss0 , O) o O

ψ(f 0 )

ψ(f )

S(f )

Γ(Ys,0 O) O

/ S(f f 0 ) o

S(f 0 )

It exists because we may use the rule x/(f f 0 )n 7→ ψ(x)/(ss0 )n , which “works” by the formulas above. Uniqueness follows as Proj(S) is separated, see Lemma 22.8.8

22.12. MORPHISMS INTO PROJ

1309

and its proof. This shows that the morphisms ϕf and ϕf 0 agree over Ys ∩ Ys0 . The restrictions of αf and αf 0 agree over Ys ∩ Ys0 because the regular functions s/s0 and ψ(f 0 ) (f ) agree. This proves that the morphisms ψf glue to a global morphism from Y into Ud ⊂ X, and that the maps αf glue to an isomorphism satsifying the conditions of the lemma. We still have to show the pair (ϕ, α) is unique. Suppose (ϕ0 , α0 ) is a second such pair. Let f ∈ Sd . By the commutativity of the diagrams in the lemma we have that the inverse images of D+ (f ) under both ϕ and ϕ0 are equal to Yψ(f ) . Since the opens D+ (f ) are a basis for the topology on X, and since X is a sober topological space (see Schemes, Lemma 21.11.1) this means the maps ϕ and ϕ0 are the same on underlying topological spaces. Let us use s = ψ(f ) to trivialize the invertible sheaf L over Yψ(f ) . By the commutativity of the diagrams we have that α⊗n (ψϕd (x)) = ψ(x) = (α0 )⊗n (ψϕd 0 (x)) for all x ∈ Snd . By construction of ψϕd and ψϕd 0 we have ψϕd (x) = ϕ] (x/f n )ψϕd (f n ) over Yψ(f ) , and similarly for ψϕd 0 . by the commutativity of the diagrams of the lemma we deduce that ϕ] (x/f n ) = (ϕ0 )] (x/f n ). This proves that ϕ and ϕ0 induce the same morphism from Yψ(f ) into the affine scheme D+ (f ) = Spec(S(f ) ). Hence ϕ and ϕ0 are the same as morphisms. Finally, it remains to show that the commutativity of the diagram of the lemma singles out, given ϕ, a unique α. We omit the verification.  We continue the discussion from above the lemma. Let S be a graded ring. Let Y be a scheme. We will consider triples (d, L, ψ) where (1) d ≥ 1 is an integer, (2) L is an invertible OY -module, and (3) ψ : S (d) → Γ∗ (Y, L) is a graded ring homomorphism such that L is generated by the global sections ψ(f ), with f ∈ Sd . Given a morphism h : Y 0 → Y and a triple (d, L, ψ) over Y we can pull it back to the triple (d, h∗ L, h∗ ◦ ψ). Given two triples (d, L, ψ) and (d, L0 , ψ 0 ) with the same integer d we say they are strictly equivalent if there exists an isomorphism β : L → L0 such that β ◦ ψ = ψ 0 as graded ring maps S (d) → Γ∗ (Y, L0 ). For each integer d ≥ 1 we define Fd : Schopp

−→

Sets,

Y

7−→

{strict equivalence classes of triples (d, L, ψ) as above}

with pullbacks as defined above. Lemma 22.12.2. Let S be a graded ring. Let X = Proj(S). The open subscheme Ud ⊂ X (22.12.0.1) represents the functor Fd and the triple (d, OUd (d), ψ d ) defined above is the universal family (see Schemes, Section 21.15). Proof. This is a reformulation of Lemma 22.12.1



Lemma 22.12.3. Let S be a graded ring generated as an S0 -algebra by the elements of S1 . In this case the scheme X = Proj(S) represents the functor which associates to a scheme Y the set of pairs (L, ψ), where (1) L is an invertible OY -module, and (2) ψ : S → Γ∗ (Y, L) is a graded ring homomorphism such that L is generated by the global sections ψ(f ), with f ∈ S1 up to strict equivalence as above.

1310

22. CONSTRUCTIONS OF SCHEMES

Proof. Under the assumptions of the lemma we have X = U1 and the lemma is a reformulation of Lemma 22.12.2 above.  We end this section with a discussion of a functor corresponding to Proj(S) for a general graded ring S. We advise the reader to skip the rest of this section. Fix an arbitrary graded ring S. Let T be a scheme. We will say two triples (d, L, ψ) and (d0 , L0 , ψ 0 ) over T with possibly different integers d, d0 are equivalent if there 0 exists an isomorphism β : L⊗d → (L0 )⊗d of invertible sheaves over T such that 0 0 β ◦ ψ|S (dd0 ) and ψ 0 |S (dd0 ) agree as graded ring maps S (dd ) → Γ∗ (Y, (L0 )⊗dd ). Lemma 22.12.4. Let S be a graded ring. Set X = Proj(S). Let T be a scheme. Let (d, L, ψ) and (d0 , L0 , ψ 0 ) be two triples over T . The following are equivalent: (1) Let n = lcm(d, d0 ). Write n = ad = a0 d0 . There exists an isomorphism 0 β : L⊗a → (L0 )⊗a with the property that β ◦ ψ|S (n) and ψ 0 |S (n) agree as graded ring maps S (n) → Γ∗ (Y, (L0 )⊗n ). (2) The triples (d, L, ψ) and (d0 , L0 , ψ 0 ) are equivalent. (3) For some positive integer n = ad = a0 d0 there exists an isomorphism 0 β : L⊗a → (L0 )⊗a with the property that β ◦ ψ|S (n) and ψ 0 |S (n) agree as graded ring maps S (n) → Γ∗ (Y, (L0 )⊗n ). (4) The morphisms ϕ : T → X and ϕ0 : T → X assocated to (d, L, ψ) and (d0 , L0 , ψ 0 ) are equal. Proof. Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Also (3) implies (4) by the uniqueness statement in Lemma 22.12.1. Thus we have to prove that (4) implies (1). Assume (4), in other words ϕ = ϕ0 . Note that this implies that we may write L = ϕ∗ OX (d) and L0 = ϕ∗ OX (d0 ). Moreover, via these identifications we have that the graded ring maps ψ and ψ 0 correspond to the restriction of the canonical graded ring map M S −→ Γ(X, OX (n)) n≥0

(d)

(d0 )

to S and S composed with pullback by ϕ (by Lemma 22.12.1 again). Hence taking β to be the isomorphism 0

(ϕ∗ OX (d))⊗a = ϕ∗ OX (n) = (ϕ∗ OX (d0 ))⊗a works.



Let S be a graded ring. Let X = Proj(S). Over the open subscheme scheme Ud ⊂ X = Proj(S) (22.12.0.1) we have the triple (d, OUd (d), ψ d ). Clearly, if d|d0 0 the triples (d, OUd (d), ψ d ) and (d0 , OUd0 (d0 ), ψ d ) are equivalent when restricted to the open Ud (which is a subset of Ud0 ). This, combined with Lemma 22.12.1 shows that morphisms Y → X correspond roughly to equivalence classes of triples over Y . This is not quite true since if Y is not quasi-compact, then there may not be a single triple which works. Thus we have to be slightly careful in defining the corresponding functor. Here is one possible way to do this. Suppose d0 = ad. Consider the transformation of functors Fd → Fd0 which assigns to the triple (d, L, ψ) over T the triple (d0 , L⊗a , ψ|S (d0 ) ). One of the implications of Lemma 22.12.4 is that the transformation Fd → Fd0 is injective! For a quasi-compact scheme T we define [ F (T ) = Fd (T ) d∈N

22.13. PROJECTIVE SPACE

1311

with transition maps as explained above. This clearly defines a contravariant functor on the category of quasi-compact schemes with values in sets. For a general scheme T we define F (T ) = limV ⊂T

quasi-compact open

F (V ).

In other words, an element ξ of F (T ) corresponds to a compatible system of choices of elements ξV ∈ F (V ) where V ranges over the quasi-compact opens of T . We omit the definition of the pullback map F (T ) → F (T 0 ) for a morphism T 0 → T of schemes. Thus we have defined our functor F : Schopp

−→

Sets

Lemma 22.12.5. Let S be a graded ring. Let X = Proj(S). The functor F defined above is representable by the scheme X. Proof. We have seen above that the functor Fd corresponds to the open subscheme Ud ⊂ X. Moreover the transformation of functors Fd → Fd0 (if d|d0 ) defined above corresponds to the inclusion morphism Ud → Ud0 (see discussion above). Hence to show that F is represented by X it suffices to show that T → X for a quasi-compact scheme T ends up in some Ud , and that for a general scheme T we have Mor(T, X) = limV ⊂T

quasi-compact open

Mor(V, X).

These verifications are omitted.



22.13. Projective space Projective space is one of the fundamental objects studied in algebraic geometry. In this section we just give its construction as Proj of a polynomial ring. Later we will discover many of its beautiful properties. Lemma 22.13.1. Let S = Z[T0 , . . . , Tn ] with deg(Ti ) = 1. The scheme PnZ = Proj(S) represents the functor which associates to a scheme Y the pairs (L, (s0 , . . . , sn )) where (1) L is an invertible OY -module, and (2) s0 , . . . , sn are global sections of L which generate L up to the following equivalence: (L, (s0 , . . . , sn )) ∼ (N , (t0 , . . . , tn )) ⇔ there exists an isomorphism β : L → N with β(si ) = ti for i = 0, . . . , n. Proof. This is a special case of Lemma 22.12.3 above. Namely, for any graded ring A we have Morgradedrings (Z[T0 , . . . , Tn ], A) ψ

=

A1 × . . . × A1

7→ (ψ(T0 ), . . . , ψ(Tn ))

and the degree 1 part of Γ∗ (Y, L) is just Γ(Y, L).



Definition 22.13.2. The scheme PnZ = Proj(Z[T0 , . . . , Tn ]) is called projective nspace over Z. Its base change PnS to a scheme S is called projective n-space over S. If R is a ring the base change to Spec(R) is denoted PnR and called projective n-space over R.

1312

22. CONSTRUCTIONS OF SCHEMES

Given a scheme Y over S and a pair (L, (s0 , . . . , sn )) as in Lemma 22.13.1 the induced morphism to PnS is denoted ϕ(L,(s0 ,...,sn )) : Y −→ PnS This makes sense since the pair defines a morphism into PnZ and we already have the structure morphism into S so combined we get a morphism into PnS = PnZ × S. Note that this is the S-morphism characterized by L = ϕ∗(L,(s0 ,...,sn )) OPnR (1)

and si = ϕ∗(L,(s0 ,...,sn )) Ti

where we think of Ti as a global section of OPnS (1) via (22.10.1.3). Lemma 22.13.3. Projective n-space over Z is covered by n + 1 standard opens [ PnZ = D+ (Ti ) i=0,...,n

where each D+ (Ti ) is isomorphic to AnZ affine n-space over Z. Proof. This is true because Z[T0 , . . . , Tn ]+ = (T0 , . . . , Tn ) and since    Tn T0 ∼ ,..., Spec Z = AnZ Ti Ti in an obvious way.



Lemma 22.13.4. Let S be a scheme. The structure morphism PnS → S is (1) (2) (3) (4)

separated, quasi-compact, satisfies the existence and uniqueness parts of the valuative criterion, and universally closed.

Proof. All these properties are stable under base change (this is clear for the last two and for the other two see Schemes, Lemmas 21.21.13 and 21.19.3). Hence it suffices to prove them for the morphism PnZ → Spec(Z). Separatedness is Lemma 22.8.8. Quasi-compactness follows from Lemma 22.13.3. Existence and uniqueness of the valuative criterion follow from Lemma 22.8.11. Universally closed follows from the above and Schemes, Proposition 21.20.6.  Remark 22.13.5. What’s missing in the list of properties above? Well to be sure the property of being of finite type. The reason we do not list this here is that we have not yet defined the notion of finite type at this point. (Another property which is missing is “smoothness”. And I’m sure there are many more you can think of.) We finish this section with two simple lemmas. These lemmas are special cases of more general results later, but perhaps it makes sense to prove these directly here now. Lemma 22.13.6. Let R be a ring. Let Z ⊂ PnR be a closed subscheme. Let
Id = Ker R[T0 , . . . , Tn ]d −→ Γ(Z, OPnR (d)|Z ) L Then I = Id ⊂ R[T0 , . . . , Tn ] is a graded ideal and Z = Proj(R[T0 , . . . , Tn ]/I).

22.13. PROJECTIVE SPACE

1313

Proof. It is clear that I is a graded ideal. Set Z 0 = Proj(R[T0 , . . . , Tn ]/I). By Lemma 22.11.5 we see that Z 0 is a closed subscheme of PnR . To see the equality Z = Z 0 it suffices to check on an standard affine open D+ (Ti ). By renumbering the homogeneous coordinates we may assume i = 0. Say Z ∩ D+ (T0 ), resp. Z 0 ∩ D+ (T0 ) is cut out by the ideal J, resp. J 0 of R[T1 /T0 , . . . , Tn /T0 ]. Then J 0 is the ideal deg(F ) generated by the elements F/T0 where F ∈ I is homogeneous. Suppose the degree of F ∈ I is d. Since F vanishes as a section of OPnR (d) restricted to Z we see that F/T0d is an element of J. Thus J 0 ⊂ J. Conversely, suppose that f ∈ J. If f has total degree d in T1 /T0 , . . . , Tn /T0 , then we can write f = F/T0d for some F ∈ R[T0 , . . . , Tn ]d . Pick i ∈ {1, . . . , n}. Then Z ∩ D+ (Ti ) is cut out by some ideal Ji ⊂ R[T0 /Ti , . . . , Tn /Ti ]. Moreover,     Tn T0 Tn T1 Tn T0 Tn T1 ,..., , ,..., = Ji · R ,..., , ,..., J ·R T0 T0 Ti Ti T0 T0 Ti Ti The left hand side is the localization of J with respect to the element Ti /T0 and the right hand side is the localization of Ji with respect to the element T0 /Ti . It follows that T0di F/Tid+di is an element of Ji for some di sufficiently large. This max(di ) proves that T0 F is an element of I, because its restriction to each standard affine open D+ (Ti ) vanishes on the closed subscheme Z ∩ D+ (Ti ). Hence f ∈ J 0 and we conclude J ⊂ J 0 as desired.  The following lemma is a special case of the more general Properties, Lemma 23.26.3. Lemma 22.13.7. Let R be a ring. Let F be a quasi-coherent sheaf on PnR . For d ≥ 0 set Md = Γ(PnR , F ⊗OPn OPnR (d)) = Γ(PnR , F(d)) R L Then M = d≥0 Md is a graded R[T0 , . . . , Rn ]-module and there is a canonical f. isomorphism F = M Proof. The multiplication maps R[T0 , . . . , Rn ]e × Md −→ Md+e come from the natural isomorphisms OPnR (e) ⊗OPn F(d) −→ F(e + d) R

f → F. On each of see Equation (22.10.1.4). Let us construct the map c : M f) = (M [1/Ti ])0 where the the standard affines Ui = D+ (Ti ) we see that Γ(Ui , M subscript 0 means degree 0 part. An element of this can be written as m/Tid with m ∈ Md . Since Ti is a generator of O(1) over Ui we can always write m|Ui = mi ⊗Tid where mi ∈ Γ(Ui , F) is a unique section. Thus a natural guess is c(m/Tid ) = mi . A small argument, which is omitted here, shows that this gives a well defined map f → F if we can show that c:M (Ti /Tj )d mi |Ui ∩Uj = mj |Ui ∩Uj in M [1/Ti Tj ]. But this is clear since on the overlap the generators Ti and Tj of O(1) differ by the invertible function Ti /Tj . Injectivity of c. We may check for injectivity over the affine opens Ui . Let i ∈ f) such that c(m/T d ) = 0. {0, . . . , n} and let s be an element s = m/Tid ∈ Γ(Ui , M i

1314

22. CONSTRUCTIONS OF SCHEMES

By the description of c above this means that mi = 0, hence m|Ui = 0. Hence Tie m = 0 in M for some e. Hence s = m/Tid = Tie /Tie+d = 0 as desired. Surjectivity of c. We may check for surjectivity over the affine opens Ui . By renumbering it suffices to check it over U0 . Let s ∈ F(U0 ). Let us write F|Ui = fi for some R[T0 /Ti , . . . , T0 /Ti ]-module Ni , which is possible because F is quasiN coherent. So s corresponds to an element x ∈ N0 . Then we have that (Ni )Tj /Ti ∼ = (Nj )Ti /Tj (where the subscripts mean “principal localization at”) as modules over the ring   T0 Tn T0 Tn R ,..., , ,..., . Ti Ti Tj Tj This means that for some large integer d there exist elements si ∈ Ni , i = 1, . . . , n such that s = (Ti /T0 )d si on U0 ∩ Ui . Next, we look at the difference tij = si − (Tj /Ti )d sj on Ui ∩ Uj , 0 < i < j. By our choice of si we know that tij |U0 ∩Ui ∩Uj = 0. Hence there exists a large integer e such that (T0 /Ti )e tij = 0. Set s0i = (T0 /Ti )e si , and s00 = s. Then we will have s0a = (Tb /Ta )e+d s0b on Ua ∩ Ub for all a, b. This is exactly the condition that the elements s0a glue to a global section m ∈ Γ(PnR , F(e+d)). And moreover c(m/T0e+d ) = s by construction. Hence c is surjective and we win.  22.14. Invertible sheaves and morphisms into Proj Let T be a scheme and let L be an invertible sheaf on T . For a section s ∈ Γ(T, L) we denote Ts the open subset of points where s does not vanish. See Modules, Lemma 15.21.7. We can view the following lemma as a slight generalization of Lemma 22.12.3. It also is a generalization of Lemma 22.11.1. Lemma 22.14.1. Let A be a graded ring. Set X = Proj(A). Let T be a scheme. Let L be an invertible OT -module. Let ψ : A → Γ∗ (T, L) be a homomorphism of graded rings. Set [ U (ψ) = Tψ(f ) f ∈A+ homogeneous

The morphism ψ induces a canonical morphism of schemes rL,ψ : U (ψ) −→ X together with a map of Z-graded OT -algebras M  M ∗ θ : rL,ψ OX (d) −→ d∈Z

d∈Z

L⊗d |U (ψ) .

The triple (U (ψ), rL,ψ , θ) is characterized by the following properties: −1 (1) For f ∈ A+ homogeneous we have rL,ψ (D+ (f )) = Tψ(f ) .

22.14. INVERTIBLE SHEAVES AND MORPHISMS INTO PROJ

1315

(2) For every d ≥ 0 the diagram Ad

ψ

(22.10.1.3)

 Γ(X, OX (d))

/ Γ(T, L⊗d ) restrict

θ

 / Γ(U (ψ), L⊗d )

is commutative. Moreover, for any d ≥ 1 and any open subscheme V ⊂ T such that the sections in ψ(Ad ) generate L⊗d |V the morphism rL,ψ |V agrees with the morphism ϕ : V → Proj(A) and the map θ|V agrees with the map α : ϕ∗ OX (d) → L⊗d |V where (ϕ, α) is the pair of Lemma 22.12.1 associated to ψ|A(d) : A(d) → Γ∗ (V, L⊗d ). Proof. Suppose that we have two triples (U, r : U → X, θ) and (U 0 , r0 : U 0 → X, θ0 ) satisfying (1) and (2). Property (1) implies that U = U 0 = U (ψ) and that r = r0 as maps of underlying topological spaces, since the opens D+ (f ) form a basis for the topology on X, and since X is a sober topological space (see Algebra, Section 7.54 and L Schemes, Lemma 21.11.1). Let f ∈ A+ be homogeneous. Note that Γ(D+ (f ), n∈Z OX (n)) = Af as a Z-graded algebra. Consider the two Z-graded ring maps M θ, θ0 : Af −→ Γ(Tψ(f ) , L⊗n ). We know that multiplication by f (resp. ψ(f )) is an isomorphism on the left (resp. right) hand side. We also know that θ(x/1) = θ0 (x/1) = ψ(x)|Tψ(f ) by (2) for all x ∈ A. Hence we deduce easily that θ = θ0 as desired. Considering the degree 0 parts we deduce that r] = (r0 )] , i.e., that r = r0 as morphisms of schemes. This proves the uniqueness. Now we come to existence. By the uniqueness just proved, it is enought to construct the pair (r, θ) locally on T . Hence we may assume that T = Spec(R) is affine, that L = OT and that for some f ∈ A+ homogeneous we have ψ(f ) generates ⊗ deg(f ) OT = OT . In other words, ψ(f ) = u ∈ R∗ is a unit. In this case the map ψ is a graded ring map A −→ R[x] = Γ∗ (T, OT ) which maps f to uxdeg(f ) . Clearly this extends (uniquely) to a Z-graded ring map θ : Af → R[x, x−1 ] by mapping 1/f to u−1 x− deg(f ) . This map in degree zero gives the ring map A(f ) → R which gives the morphism r : T = Spec(R) → Spec(A(f ) ) = D+ (f ) ⊂ X. Hence we have constructed (r, θ) in this special case. Let us show the last statement of the lemma. According to Lemma 22.12.1 the morphism constructed there is the unique one such that the displayed diagram in its statement commutes. The commutativity of the diagram in the lemma implies the commutativity when restricted to V and A(d) . Whence the result.  Remark 22.14.2. Assumptions as in Lemma 22.14.1 above. The image of the morphism rL,ψ need not be contained in the locus where the sheaf OX (1) is invertible. Here is an example. Let k be a field. Let S = k[A, B, C] graded by deg(A) = 1, deg(B) = 2, deg(C) = 3. Set X = Proj(S). Let T = P2k = Proj(k[X0 , X1 , X2 ]). Recall that L = OT (1) is invertible and that OT (n) = L⊗n . Consider the composition ψ of the maps S → k[X0 , X1 , X2 ] → Γ∗ (T, L).

1316

22. CONSTRUCTIONS OF SCHEMES

Here the first map is A 7→ X06 , B 7→ X13 , C 7→ X23 and the second map is (22.10.1.3). By the lemma this corresponds to a morphism rL,ψ : T → X = Proj(S) which is easily seen to be surjective. On the other hand, in Remark 22.9.2 we showed that the sheaf OX (1) is not invertible at all points of X. 22.15. Relative Proj via glueing Situation 22.15.1. Here S is a scheme, and A is a quasi-coherent graded OS algebra. In this section we outline how to construct a morphism of schemes ProjS (A) −→ S by glueing the homogeneous spectra Proj(Γ(U, A)) where U ranges over the affine opens of S. We first show that the homogeneous spectra of the values of A over affines form a suitable collection of schemes, as in Lemma 22.2.1. Lemma 22.15.2. In Situation 22.15.1. Suppose U ⊂ U 0 ⊂ S are affine opens. Let A = A(U ) and A0 = A(U 0 ). The map of graded rings A0 → A induces a morphism r : Proj(A) → Proj(A0 ), and the diagram Proj(A)

/ Proj(A0 )

 U

 / U0

is cartesian. Moreover there are canonical isomorphisms θ : r∗ OProj(A0 ) (n) → OProj(A) (n) compatible with multiplication maps. Proof. Let R = OS (U ) and R0 = OS (U 0 ). Note that the map R ⊗R0 A0 → A is an isomorphism as A is quasi-coherent (see Schemes, Lemma 21.7.3 for example). Hence the lemma follows from Lemma 22.11.6.  In particular the morphism Proj(A) → Proj(A0 ) of the lemma is an open immersion. Lemma 22.15.3. In Situation 22.15.1. Suppose U ⊂ U 0 ⊂ U 00 ⊂ S are affine opens. Let A = A(U ), A0 = A(U 0 ) and A00 = A(U 00 ). The composition of the morphisms r : Proj(A) → Proj(A0 ), and r0 : Proj(A0 ) → Proj(A00 ) of Lemma 22.15.2 gives the morphism r00 : Proj(A) → Proj(A00 ) of Lemma 22.15.2. A similar statement holds for the isomorphisms θ. Proof. This follows from Lemma 22.11.2 since the map A00 → A is the composition of A00 → A0 and A0 → A.  Lemma 22.15.4. In Situation 22.15.1. There exists a morphism of schemes π : ProjS (A) −→ S with the following properties: (1) for every affine open U ⊂ S there exists an isomorphism iU : π −1 (U ) → Proj(A) with A = A(U ), and (2) for U ⊂ U 0 ⊂ S affine open the composition Proj(A)

i−1 U

/ π −1 (U )

inclusion

/ π −1 (U 0 )

iU 0

/ Proj(A0 )

22.16. RELATIVE PROJ AS A FUNCTOR

1317

with A = A(U ), A0 = A(U 0 ) is the open immersion of Lemma 22.15.2 above. Proof. Follows immediately from Lemmas 22.2.1, 22.15.2, and 22.15.3.



Lemma 22.15.5. In Situation 22.15.1. The morphism π : ProjS (A) → S of Lemma 22.15.4 comes with the following additional L structure. There exists a quasicoherent Z-graded sheaf of OProj (A) -algebras n∈Z OProj (A) (n), and a morphism S S of graded OS -algebras   M ψ : A −→ π∗ OProj (A) (n) n≥0

S

uniquely determined by the following property: For every affine open U ⊂ S with A = A(U ) there is an isomorphism M  M  θU : i∗U OProj(A) (n) −→ OProj (A) (n) |π−1 (U ) n∈Z

n∈Z

S

of Z-graded Oπ−1 (U ) -algebras such that An

/ Γ(π −1 (U ), OProj (A) (n)) S 4

ψ (22.10.1.3)

' Γ(Proj(A), OProj(A) (n))

θU

is commutative. Proof. We are going to use Lemma 22.2.2 to glue the sheaves of Z-graded algebras L n∈Z OProj(A) (n) for A = A(U ), U ⊂ S affine open over the scheme ProjS (A). We have constructed the data necessary for this in Lemma 22.15.2 and we have checked condition (d) of Lemma 22.2.2 L in Lemma 22.15.3. Hence we get the sheaf of Zgraded OProj (A) -algebras n∈Z OProj (A) (n) together with the isomorphisms θU S S for all U ⊂ S affine open and all n ∈ Z. L For every affine open U ⊂ S with A = A(U ) we have a map A → Γ(Proj(A), n≥0 OProj(A) (n)). Hence the map ψ exists by functoriality of relative glueing, see Remark 22.2.3. The diagram of the lemma commutes L by construction. This characterizes the sheaf of Z-graded OProj (A) -algebras OProj (A) (n) because the proof of Lemma 22.11.1 shows that S S having these diagrams commute uniquely determines the maps θU . Some details omitted.  22.16. Relative Proj as a functor L We place ourselves in Situation 22.15.1. So S is a scheme and A = d≥0 Ad is a quasi-coherent graded OS -algebra. In this section we relativize the construction of Proj by constructing a functor which the relative homogeneous spectrum will represent. As a result we will construct a morphism of schemes ProjS (A) −→ S which above affine opens of S will look like the homogeneous spectrum of a graded ring. The discussion will be modeled after our discussion of the relative spectrum in Section 22.4. The easier method using glueing schemes of the form Proj(A), A = Γ(U, A), U ⊂ S affine open, is explained in Section 22.15, and the result in this section will be shown to be isomorphic to that one.

1318

22. CONSTRUCTIONS OF SCHEMES

L Fix for the moment an integer d ≥ 1. We denote A(d) = n≥0 And similarly to the notation in Algebra, Section 7.53. Let T be a scheme. Let us consider quadruples (d, f : T → S, L, ψ) over T where (1) d is the integer we fixed above, (2) f : T → S is a morphism of schemes, (3) L is an invertible L OT -module, and (4) ψ : f ∗ A(d) → n≥0 L⊗n is a homomorphism of graded OT -algebras such that f ∗ Ad → L is surjective. Given a morphism h : T 0 → T and a quadruple (d, f, L, ψ) over T we can pull it back to the quadruple (d, f ◦ h, h∗ L, h∗ ψ) over T 0 . Given two quadruples (d, f, L, ψ) and (d, f 0 , L0 , ψ 0 ) over T with the same integer d we say they are strictly equivalent if f = f 0 and there exists an isomorphism β : L → L0 such that β ◦ ψ = ψ 0 as L ∗ (d) 0 ⊗n graded OT -algebra maps f A → n≥0 (L ) . For each integer d ≥ 1 we define Fd : Schopp

−→

Sets,

T

7−→

{strict equivalence classes of (d, f : T → S, L, ψ) as above}

with pullbacks as defined above. Lemma 22.16.1. In Situation 22.15.1. Let d ≥ 1. Let Fd be the functor associated to (S, A) above. Let g : S 0 → S be a morphism of schemes. Set A0 = g ∗ A. Let Fd0 be the functor associated to (S 0 , A0 ) above. Then there is a canonical isomorphism ∼ hS 0 ×h Fd F0 = d

S

of functors. L 0 ⊗n Proof. A quadruple (d, f 0 : T → S 0 , L0 , ψ 0 : (f 0 )∗ (A0 )(d) → ) is the n≥0 (L ) L same as a quadruple (d, f, L, ψ : f ∗ A(d) → n≥0 L⊗n ) together with a factorization of f as f = g ◦ f 0 . Namely, the correspondence is f = g ◦ f 0 , L = L0 and ψ = ψ 0 via the identifications (f 0 )∗ (A0 )(d) = (f 0 )∗ g ∗ (A(d) ) = f ∗ A(d) . Hence the lemma.  Lemma 22.16.2. In Situation 22.15.1. Let Fd be the functor associated to (d, S, A) above. If S is affine, then Fd is representable by the open subscheme Ud (22.12.0.1) of the scheme Proj(Γ(S, A)). Proof. Write S = Spec(R) and A = Γ(S, A). Then A is a graded R-algebra and e To prove the lemma we have to identify the functor Fd with the functor A = A. triples Fd of triples defined in Section 22.12. Let (d, f : T L → S, L, ψ) be a quadruple. We may think of ψ as a OS -module map A(d) → n≥0 f∗ L⊗n . Since A(d) is quasi-coherent this is the same thing as L an R-linear homomorphism of graded rings A(d) → Γ(S, n≥0 f∗ L⊗n ). Clearly, L Γ(S, n≥0 f∗ L⊗n ) = Γ∗ (T, L). Thus we may associate to the quadruple the triple (d, L, ψ). Conversely, let (d, L, ψ) be a triple. The composition R → A0 → Γ(T, OT ) determines a morphism f : T → S = Spec(R), see Schemes, Lemma 21.6.4. With L this choice of f the map A(d) → Γ(S, n≥0 f∗ L⊗n ) is R-linear, and hence corresponds to a ψ which we can use for a quadruple (d, f : T → S, L, ψ). We omit the verification that this establishes an isomorphism of functors Fd = Fdtriples . 

22.16. RELATIVE PROJ AS A FUNCTOR

1319

Lemma 22.16.3. In Situation 22.15.1. The functor Fd is representable by a scheme. Proof. We are going to use Schemes, Lemma 21.15.4. First we check that Fd satisfies the sheaf property for the Zariski topology. Namely, S suppose that T is a scheme, that T = i∈I Ui is an open covering, and that (d, fi , Li , ψi ) ∈ Fd (Ui ) such that (d, fi , Li , ψi )|Ui ∩Uj and (d, fj , Lj , ψj )|Ui ∩Uj are strictly equivalent. This implies that the morphisms fi : Ui → S glue to a morphism of schemes f : T → S such that f |Ii = fi , see Schemes, Section 21.14. Thus fi∗ A(d) = f ∗ A(d) |Ui . It also implies there exist isomorphisms βij : Li |Ui ∩Uj → Lj |Ui ∩Uj such that βij ◦ ψi = ψj on Ui ∩ Uj . Note that the isomorphisms βij are uniquely determined by this requirement because the maps fi∗ Ad → Li are surjective. In particular we see that βjk ◦ βij = βik on Ui ∩ Uj ∩ Uk . Hence by Sheaves, Section 6.33 the invertible sheaves Li glue to an invertible OT -module L and the L ⊗n morphisms ψi glue to morphism of OT -algebras ψ : f ∗ A(d) → . This n≥0 L proves that Fd satisfies the sheaf condition with respect to the Zariski topology. S Let S = i∈I Ui be an affine open covering. Let Fd,i ⊂ Fd be the subfunctor consisting of those pairs (f : T → S, ϕ) such that f (T ) ⊂ Ui . We have to show each Fd,i is representable. This is the case because Fd,i is identified with the functor associated to Ui equipped with the quasi-coherent graded OUi algebra A|Ui ) by Lemma 22.16.1. Thus the result follows from Lemma 22.16.2. Next we show that Fd,i ⊂ Fd is representable by open immersions. Let (f : T → S, ϕ) ∈ Fd (T ). Consider Vi = f −1 (Ui ). It follows from the definition of Fd,i that given a : T 0 → T we gave a∗ (f, ϕ) ∈ Fd,i (T 0 ) if and only if a(T 0 ) ⊂ Vi . This is what we were required to show. Finally, we have to show that the collection (Fd,i )S i∈I covers Fd . Let (f : T → −1 S, ϕ) ∈ Fd (T ). Consider V = f (U ). Since S = i i i∈I Ui is an open covering of S S we see that T = i∈I Vi is an open covering of T . Moreover (f, ϕ)|Vi ∈ Fd,i (Vi ). This finishes the proof of the lemma.  At this point we can redo the material at the end of Section 22.12 in the current relative setting and define a functor which is representable by ProjS (A). To do this we introduce the notion of equivalence between two quadruples (d, f : T → S, L, ψ) and (d0 , f 0 : T → S, L0 , ψ 0 ) with possibly different values of the integers d, d0 . Namely, we say these are equivalent if f = f 0 , and there exists an isomorphism 0 β : L⊗d → (L0 )⊗d such that β ◦ ψ|f ∗ A(dd0 ) = ψ 0 |f ∗ A(dd0 ) . The following lemma implies that this defines an equivalence relation. (This is not a complete triviality.) Lemma 22.16.4. In Situation 22.15.1. Let T be a scheme. Let (d, f, L, ψ), (d0 , f 0 , L0 , ψ 0 ) be two quadruples over T . The following are equivalent: (1) Let m = lcm(d, d0 ). Write m = ad = a0 d0 . We have f = f 0 and there ex0 ists an isomorphism β : L⊗a → (L0 )⊗a with the property that β ◦ ψ|f ∗ A(m) L and ψ 0 |f ∗ A(m) agree as graded ring maps f ∗ A(m) → n≥0 (L0 )⊗mn . (2) The quadruples (d, f, L, ψ) and (d0 , f 0 , L0 , ψ 0 ) are equivalent. (3) We have f = f 0 and for some positive integer m = ad = a0 d0 there exists 0 an isomorphism β : L⊗a → (L0 )⊗a with the property that β ◦ ψ|f ∗ A(m) L and ψ 0 |f ∗ A(m) agree as graded ring maps f ∗ A(m) → n≥0 (L0 )⊗mn .

1320

22. CONSTRUCTIONS OF SCHEMES

Proof. Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Assume (3) for some integer m = ad = a0 d0 . Let m0 = lcm(d, d0 ) and write it as m0 = a0 d = a00 d0 . We are given an 0 isomorphism β : L⊗a → (L0 )⊗a with the property described in (3). We want to 0 find an isomorphism β0 : L⊗a0 → (L0 )⊗a0 having that property as well. Since by assumption the maps ψ : f ∗ Ad → L and ψ 0 : (f 0 )∗ Ad0 → L0 are surjective the same is true for the maps ψ : f ∗ Am0 → L⊗a0 and ψ 0 : (f 0 )∗ Am0 → (L0 )⊗a0 . Hence if β0 exists it is uniquely determined by the condition that β0 ◦ ψ = ψ 0 . This means that we may work locally on T . Hence we may assume that f = f 0 : T → S maps into an affine open, in other words we may assume that S is affine. In this case the result follows from the corresponding result for triples (see Lemma 22.12.4) and the fact that triples and quadruples correspond in the affine base case (see proof of Lemma 22.16.2).  Suppose d0 = ad. Consider the transformation of functors Fd → Fd0 which assigns to the quadruple (d, f, L, ψ) over T the quadruple (d0 , f, L⊗a , ψ|f ∗ A(d0 ) ). One of the implications of Lemma 22.16.4 is that the transformation Fd → Fd0 is injective! For a quasi-compact scheme T we define [ F (T ) = Fd (T ) d∈N

with transition maps as explained above. This clearly defines a contravariant functor on the category of quasi-compact schemes with values in sets. For a general scheme T we define F (T ) = limV ⊂T

quasi-compact open

F (V ).

In other words, an element ξ of F (T ) corresponds to a compatible system of choices of elements ξV ∈ F (V ) where V ranges over the quasi-compact opens of T . We omit the definition of the pullback map F (T ) → F (T 0 ) for a morphism T 0 → T of schemes. Thus we have defined our functor (22.16.4.1)

F : Schopp −→ Sets

Lemma 22.16.5. In Situation 22.15.1. The functor F above is representable by a scheme. Proof. Let Ud → SL be the scheme representing the functor Fd defined above. Let Ld , ψ d : πd∗ A(d) → n≥0 L⊗n be the universal object. If d|d0 , then we may cond ⊗d0 /d

sider the quadruple (d0 , πd , Ld , ψ d |A(d0 ) ) which determines a canonical morphism Ud → Ud0 over S. By construction this morphism corresponds to the transformation of functors Fd → Fd0 defined above. For every affine open Spec(R) = V ⊂ S setting A = Γ(V, A) we have a canonical identification of the base change Ud,V with the corresponding open subscheme of Proj(A), see Lemma 22.16.2. Moreover, the morphisms Ud,V → Ud0 ,V constructed above correspond to the inclusions of opens in Proj(A). Thus we conclude that Ud → Ud0 is an open immersion. This allows us to construct X by glueing the schemes Ud along the open immersions Ud → Ud0 . Technically, it is convenient to choose a sequence d1 |d2 |dS 3 | . . . such that every positive integer divides one of the di and to simply take X = Udi using the open immersions above. It is then a simple matter to prove that X represents the functor F . 

22.16. RELATIVE PROJ AS A FUNCTOR

1321

Lemma 22.16.6. In Situation 22.15.1. The scheme π : ProjS (A) → S constructed in Lemma 22.15.4 and the scheme representing the functor F are canonically isomorphic as schemes over S. Proof. Let X be the scheme representing the functor F . Note that X is a scheme over S since the functor F comes equipped with a natural transformation F → hS . Write Y = ProjS (A). We have to show that X ∼ = Y as S-schemes. We give two arguments. The first argument uses the construction of X as the union of the schemes Ud representing Fd in the proof of Lemma 22.16.5. Over each affine open of S we can identify X with the homogeneous spectrum of the sections of A over that open, since this was true for the opens Ud . Moreover, these identifications are compatible with further restrictions to smaller affine opens. On the other hand, Y was constructed by glueing these homogeneous spectra. Hence we can glue these isomorphisms to an isomorphism between X and ProjS (A) as desired. Details omitted. Here is the second argument. Lemma 22.15.5 shows that there exists a morphism of graded algebras M ψ : π ∗ A −→ OY (n) n≥0

over Y which on sections over affine opens of S agrees with (22.10.1.3). Hence for every y ∈ Y there exists an open neighbourhood V ⊂ Y of y and an integer d ≥ 1 such that for d|n the sheaf OY (n)|V is invertible and the multiplication maps OY (n)|V ⊗OV OY (m)|V → OY (n + m)|V are isomorphisms. Thus ψ restricted to the sheaf π ∗ A(d) |V gives an element of Fd (V ). Since the opens V cover Y we see “ψ” gives rise to an element of F (Y ). Hence a canonical morphism Y → X over S. Because this construction is completely canonical to see that it is an isomorphism we may work locally on S. Hence we reduce to the case S affine where the result is clear.  Definition 22.16.7. Let S be a scheme. Let A be a quasi-coherent sheaf of graded OS -algebras. The relative homogeneous spectrum of A over S, or the homogeneous spectrum of A over S, or the relative Proj of A over S is the scheme constructed in Lemma 22.15.4 which represents the functor F (22.16.4.1), see Lemma 22.16.6. We denote it π : ProjS (A) → S. The L relative Proj comes equipped with a quasi-coherent sheaf of Z-graded algebras n∈Z OProjS (A) (n) (the twists of the structure sheaf) and a “universal” homomorphism of graded algebras M  ψuniv : A −→ π∗ OProj (A) (n) n≥0

S

see Lemma 22.15.5. We may also think of this as a homomorphism M ψuniv : π ∗ A −→ OProj (A) (n) n≥0

S

if we like. The following lemma is a formulation of the universality of this object. Lemma 22.16.8. In Situation 22.15.1. Let (f : T → S, d, L, ψ) be a quadruple. Let rd,L,ψ : T → ProjS (A) be the associated S-morphism. There exists an isomorphism of Z-graded OT -algebras M  M ∗ θ : rd,L,ψ OProj (A) (nd) −→ L⊗n n∈Z

S

n∈Z

1322

22. CONSTRUCTIONS OF SCHEMES

such that the following diagram commutes A(d)


⊗n / f∗ L n∈Z L 5

ψ ψuniv

π∗

L '

n≥0

θ

OProj

S

 (A) (nd)

The commutativity of this diagram uniquely determines θ. Proof. Note that the quadruple (f : T → S, d, L, ψ) defines an element of Fd (T ). Let Ud ⊂ ProjS (A) be the locus where the sheaf OProj (A) (d) is invertible and S generated by the image of ψuniv : π ∗ Ad → OProj (A) (d). Recall that Ud represents S the functor Fd , see the proof of Lemma 22.16.5. Hence the result will follow if we can show the quadruple (Ud → S, d, OUd (d), ψuniv |A(d) ) is the universal family, i.e., the representing object in Fd (Ud ). We may do this after restricting to an affine open of S because (a) the formation of the Lfunctors Fd commutes with base change (see Lemma 22.16.1), and (b) the pair ( n∈Z OProj (A) (n), ψuniv ) is constructed S by glueing over affine opens in S (see Lemma 22.15.5). Hence we may assume that S is affine. In this case the functor of quadruples Fd and the functor of triples Fd agree (see proof of Lemma 22.16.2) and moreover Lemma 22.12.2 shows that (d, OUd (d), ψ d ) is the universal triple over Ud . Going backwards through the identifications in the proof of Lemma 22.16.2 shows that (Ud → S, d, OUd (d), ψuniv |A(d) ) is the universal quadruple as desired.  Lemma 22.16.9. Let S be a scheme and A be a quasi-coherent sheaf of graded OS -algebras. The morphism π : ProjS (A) → S is separated. Proof. To prove a morphism is separated we may work locally on the base, see Schemes, Section 21.21. By construction ProjS (A) is over any affine U ⊂ S isomorphic to Proj(A) with A = A(U ). By Lemma 22.8.8 we see that Proj(A) is separated. Hence Proj(A) → U is separated (see Schemes, Lemma 21.21.14) as desired.  Lemma 22.16.10. Let S be a scheme and A be a quasi-coherent sheaf of graded OS -algebras. Let g : S 0 → S be any morphism of schemes. Then there is a canonical isomorphism ProjS 0 (g ∗ A) −→ S 0 ×S ProjS (A) Proof. This follows from Lemma 22.16.1 and the construction of ProjS (A) in Lemma 22.16.5 as the union of the schemes Ud representing the functors Fd .  Lemma 22.16.11. Let S be a scheme. Let A be a quasi-coherent sheaf of graded OS -modules generated as an A0 -algebra by A1 . In this case the scheme X = ProjS (A) represents the functor F1 which associates to a scheme f : T → S over S the set of pairs (L, ψ), where (1) L is an invertible OT -module, and L (2) ψ : f ∗ A → n≥0 L⊗n is a graded OT -algebra homomorphism such that f ∗ A1 → L is surjective

22.17. QUASI-COHERENT SHEAVES ON RELATIVE PROJ

1323

up to strict equivalence as above. Moreover, in this case all the quasi-coherent sheaves OProj(A) (n) are invertible OProj(A) -modules and the multiplication maps induce isomorphsms OProj(A) (n) ⊗OProj(A) OProj(A) (m) = OProj(A) (n + m). Proof. Under the assumptions of the lemma the sheaves OProj(A) (n) are invertible and the multiplication maps isomorphisms by Lemma 22.16.5 and Lemma 22.12.3 over affine opens of S. Thus X actually represents the functor F1 , see proof of Lemma 22.16.5.  22.17. Quasi-coherent sheaves on relative Proj We briefly discuss how to deal with graded modules in the relative setting. We plave ourselves in Situation 22.15.1. So S is a scheme, and A is a quasi-coherent L graded OS -algebra. Let M = M n be a graded A-module, quasi-coherent n∈Z as an OS -module. We are going to describe the associated quasi-coherent sheaf of modules on ProjS (A). We first describe the value of this sheaf schemes T mapping into the relative Proj. Let T be a scheme. Let (d, f : T → S, L, ψ) be a quadruple over T , as in Section fT of OT -modules as follows 22.16. We define a quasi-coherent sheaf M  M  fT = f ∗ M(d) ⊗f ∗ A(d) (22.17.0.1) M L⊗n n∈Z

0

fT is the degree 0 part of the tensor product of the graded f ∗ A(d) -modules M(d) So M L fT depends on the quadruple even though and n∈Z L⊗n . Note that the sheaf M we suppressed this in the notation. This construction has the pleasing property fT 0 = g ∗ M fT where M fT 0 denotes that given any morphism g : T 0 → T we have M the quasi-coherent sheaf associated to the pullback quadruple (d, f ◦ g, g ∗ L, g ∗ ψ). Since all sheaves in (22.17.0.1) are quasi-coherent we can spell out the construction over an affine open Spec(C) = V ⊂ T which maps into an affine open Spec(R) = U ⊂ S. Namely, suppose that A|U corresponds to the graded R-algebra A, that M|U corresponds to the graded A-module M , and that L|V corresponds to the invertible C-module L. The map ψ gives rise to a graded R-algebra map γ : A(d) → L ⊗n fT )|V is the quasi-coherent sheaf . (Tensor powers of L over C.) Then (M n≥0 L associated to the C-module  M  NR,C,A,M,γ = M (d) ⊗A(d) ,γ L⊗n n∈Z

0

By assumption we may even cover T by affine opens V such that there exists some a ∈ Ad such that γ(a) ∈ L is a C-basis P for the module L. In that case any element of NR,C,A,M,γ is a sum of pure tensors mi ⊗ γ(a)−ni with m ∈ Mni d . In fact we may multiply each mi with a suitable positive power of a and collect terms to see that each element of NR,C,A,M,γ can be written as m ⊗ γ(a)−n with m ∈ Mnd and n  0. In other words we see that in this case NR,C,A,M,γ = M(a) ⊗A(a) C where the map A(a) → C is the map x/an 7→ γ(x)/γ(a)n . In other words, this f on D+ (a) ⊂ Proj(A) pulled back to Spec(C) via the morphism is the value of M Spec(C) → D+ (a) coming from γ.

1324

22. CONSTRUCTIONS OF SCHEMES

Lemma 22.17.1. In Situation 22.15.1. For any quasi-coherent sheaf of graded A-modules M on S, there exists a canonical associated sheaf of OProj (A) -modules S f with the following properties: M (1) Given a scheme T and a quadruple (T → S, d, L, ψ) over T corresponding fT = to a morphism h : T → ProjS (A) there is a canonical isomorphism M f where M fT is defined by (22.17.0.1). h∗ M (2) The isomorphisms of (1) are compatible with pullbacks. (3) There is a canonical map f π ∗ M0 −→ M. f is functorial in M. (4) The construction M 7→ M f is exact. (5) The construction M 7→ M (6) There are canonical maps f ⊗O M Proj

S

(A)

e −→ M^ N ⊗A N

as in Lemma 22.9.1. (7) There exist canonical maps π ∗ M −→

M n∈Z

^ M(n)

generalizing (22.10.1.6). f commutes with base change. (8) The formation of M Proof. Omitted. We should split this lemma into parts and prove the parts separately.  22.18. Functoriality of relative Proj This section is the analogue of Section 22.11 for the relative Proj. Let S be a scheme. A graded OS -algebra map ψ : A → B does not always give rise to a morphism of associated relative Proj. The correct result is stated as follows. Lemma 22.18.1. Let S be a scheme. Let A, B be two graded quasi-coherent OS algebras. Set p : X = ProjS (A) → S and q : Y = ProjS (B) → S. Let ψ : A → B be a homomorphism of graded OS -algebras. There is a canonical open U (ψ) ⊂ Y and a canonical morphism of schemes rψ : U (ψ) −→ X over S and a map of Z-graded OU (ψ) -algebras M  M ∗ θ = θψ : rψ OX (d) −→ d∈Z

d∈Z

OU (ψ) (d).

The triple (U (ψ), rψ , θ) is characterized by the property that for any affine open W ⊂ S the triple (U (ψ) ∩ p−1 W,

rψ |U (ψ)∩p−1 W : U (ψ) ∩ p−1 W → q −1 W,

θ|U (ψ)∩p−1 W )

is equal to the triple associated to ψ : A(W ) → B(W ) in Lemma 22.11.1 via the identifications p−1 W = Proj(A(W )) and q −1 W = Proj(B(W )) of Section 22.15. Proof. This lemma proves itself by glueing the local triples.



22.19. INVERTIBLE SHEAVES AND MORPHISMS INTO RELATIVE PROJ

1325

Lemma 22.18.2. Let S be a scheme. Let A, B, and C be quasi-coherent graded OS -algebras. Set X = ProjS (A), Y = ProjS (B) and Z = ProjS (C). Let ϕ : A → B, ψ : B → C be graded OS -algebra maps. Then we have U (ψ ◦ ϕ) = rϕ−1 (U (ψ))

and

rψ◦ϕ = rϕ ◦ rψ |U (ψ◦ϕ) .

In addition we have ∗ θψ ◦ rψ θϕ = θψ◦ϕ

with obvious notation. Proof. Omitted.



Lemma 22.18.3. With hypotheses and notation as in Lemma 22.18.1 above. Assume Ad → Bd is surjective for d  0. Then (1) U (ψ) = Y , (2) rψ : Y → X is a closed immersion, and ∗ (3) the maps θ : rψ OX (n) → OY (n) are surjective but not isomorphisms in general (even if A → B is surjective). Proof. Follows on combining Lemma 22.18.1 with Lemma 22.11.3.



Lemma 22.18.4. With hypotheses and notation as in Lemma 22.18.1 above. Assume Ad → Bd is an isomorphism for all d  0. Then (1) U (ψ) = Y , (2) rψ : Y → X is an isomorphism, and ∗ (3) the maps θ : rψ OX (n) → OY (n) are isomorphisms. Proof. Follows on combining Lemma 22.18.1 with Lemma 22.11.4.



Lemma 22.18.5. With hypotheses and notation as in Lemma 22.18.1 above. Assume Ad → Bd is surjective for d  0 and that A is generated by A1 over A0 . Then (1) U (ψ) = Y , (2) rψ : Y → X is a closed immersion, and ∗ (3) the maps θ : rψ OX (n) → OY (n) are isomorphisms. Proof. Follows on combining Lemma 22.18.1 with Lemma 22.11.5.



22.19. Invertible sheaves and morphisms into relative Proj It seems that we may need the following lemma somewhere. The situation is the following: (1) Let S be a scheme. (2) Let A be a quasi-coherent graded OS -algebra. (3) Denote π : ProjS (A) → S the relative homogeneous spectrum over S. (4) Let f : X → S be a morphism of schemes. (5) Let L be an invertible OX -module. L (6) Let ψ : f ∗ A → d≥0 L⊗d be a homomorphism of graded OX -algebras. Given this data set [ U (ψ) = Uψ(a) (U,V,a)

where (U, V, a) satisfies: (1) V ⊂ S affine open,

1326

22. CONSTRUCTIONS OF SCHEMES

(2) U = f −1 (V ), and (3) a ∈ A(V )+ is homogeneous. Namely, then ψ(a) ∈ Γ(U, L⊗ deg(a) ) and Uψ(a) is the corresponding open (see Modules, Lemma 15.21.7). Lemma 22.19.1. With assumptions and notation as above. The morphism ψ induces a canonical morphism of schemes over S rL,ψ : U (ψ) −→ ProjS (A) together with a map of graded OU (ψ) -algebras M  M ∗ θ : rL,ψ OProj (A) (d) −→ d≥0

d≥0

S

L⊗d |U (ψ)

characterized by the following properties: (1) For every open V ⊂ S and every d ≥ 0 the diagram Ad (V )

/ Γ(f −1 (V ), L⊗d )

ψ

ψ

 Γ(π −1 (V ), OProj

restrict

S

θ

 / Γ(f −1 (V ) ∩ U (ψ), L⊗d )

(A) (d))

is commutative. (2) For any d ≥ 1 and any open subscheme W ⊂ X such that ψ|W : f ∗ Ad |W → L⊗d |W is surjective the restriction of the morphism rL,ψ agrees with the morphism W → ProjS (A) which exists by the construction of the relative homogeneous spectrum, see Definition 22.16.7. (3) For any affine open V ⊂ S, the restriction (U (ψ) ∩ f −1 (V ), rL,ψ |U (ψ)∩f −1 (V ) , θ|U (ψ)∩f −1 (V ) ) agrees via iV (see Lemma 22.15.4) with the triple (U (ψ 0 ), rL,ψ0 , θ0 ) of Lemma 22.14.1 associated to the map ψ 0 : A = A(V ) → Γ∗ (f −1 (V ), L|f −1 (V ) ) induced by ψ. Proof. Use characterization (3) to construct the morphism rL,ψ and θ locally over S. Use the uniqueness of Lemma 22.14.1 to show that the construction glues. Details omitted.  22.20. Twisting by invertible sheaves and relative Proj L Let S be a scheme. Let A = d≥0 Ad be a quasi-coherent graded OS -algebra. Let L be an invertible sheaf on S. In this situation we obtain another quasi-coherent graded OS -algebra, namely M B= Ad ⊗OS L⊗d d≥0

It turns out that A and B have isomorphic relative homogeneous spectra. Lemma 22.20.1. With notation S, A, L and B as above. There is a canonical isomorphism / Proj (B) = P 0 P = ProjS (A) g S π

%

S

y

π0

22.21. PROJECTIVE BUNDLES

1327

with the following properties (1) There are isomorphisms θn : g ∗ OP 0 (n) → OP (n) ⊗ π ∗ L⊗n which fit together to give an isomorphism of Z-graded algebras M  M θ : g∗ OP 0 (n) −→ OP (n) ⊗ π ∗ L⊗n n∈Z

n∈Z

(2) For every open V ⊂ S the diagrams An (V ) ⊗ L⊗n (V )

/ Bn (V )

multiply

ψ⊗π ∗

 Γ(π −1 V, OP (n)) ⊗ Γ(π −1 V, π ∗ L⊗n )

ψ

multiply

 Γ(π −1 V, OP (n) ⊗ π ∗ L⊗n ) o

θn

 Γ(π 0−1 V, OP 0 (n))

are commutative. (3) Add more here as necessary. Proof. This is the identity map when L ∼ = OS . In general choose an open covering of S such that L is trivialized over the pieces and glue the corresponding maps. Details omitted.  22.21. Projective bundles Let S be a scheme. Let E be a quasi-coherent sheaf of OS -modules. By Modules, Lemma 15.18.6 the symmetric algebra Sym(E) of E over OS is a quasi-coherent sheaf of OS -algebras. Note that it is generated in degree 1 over OS . Hence it makes sense to apply the construction of the previous section to it, specifically Lemmas 22.16.5 and 22.16.11. Definition 22.21.1. Let S be a scheme. Let E be a quasi-coherent OS -module3. We denote π : P(E) = ProjS (Sym(E)) −→ S and we call it the projective bundle associated to E. The symbol OP(E) (n) indicates the invertible OP(E) -modules introduced in Lemma 22.16.5 and is called the nth twist of the structure sheaf. Note that according to Lemma 22.16.5 there are canonical OS -module homomorphisms Symn (E) −→ π∗ (OP(E) (n)) for all n ≥ 0. This, combined with the fact that OP(E) (1) is the canonical relatively ample invertible sheaf on P(E), is a good way to remember how we have normalized our construction of P(E). Namely, in some references the space P(E) is only defined for E finite locally free on S, and sometimes P(E) is actually defined as our P(E ∧ ) where E ∧ is the dual of the sheaf E. 3The reader may expect here the condition that E is finite locally free. We do not do so in order to be consistent with [DG67, II, Definition 4.1.1].

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Example 22.21.2. The map Symn (E) → π∗ (OP(E) (n)) is an isomorphism if E is locally free, but in general need not be an isomorphism. In fact we will give an example where this map is not injective for n = 1. Set S = Spec(A) with A = k[u, v, s1 , s2 , t1 , t2 ]/I where k is a field and I = (−us1 + vt1 + ut2 , vs1 + us2 − vt2 , vs2 , ut1 ). Denote u the class of u in A and similarly for the other variables. Let M = (Ax ⊕ Ay)/A(ux + vy) so that Sym(M ) = A[x, y]/(ux + vy) = k[x, y, u, v, s1 , s2 , t1 , t2 ]/J where J = (−us1 + vt1 + ut2 , vs1 + us2 − vt2 , vs2 , ut1 , ux + vy). f on In this case the projective bundle associated to the quasi-coherent sheaf E = M S = Spec(A) is the scheme P = Proj(Sym(M )). Note that this scheme as an affine open covering P = D+ (x) ∪ D+ (y). Consider the element m ∈ M which is the image of the element us1 x + vt2 y. Note that x(us1 x + vt2 y) = (s1 x + s2 y)(ux + vy) mod I and y(us1 x + vt2 y) = (t1 x + t2 y)(ux + vy) mod I. The first equation implies that m maps to zero as a section of OP (1) on D+ (x) and the second that it maps to zero as a section of OP (1) on D+ (y). This shows that m maps to zero in Γ(P, OP (1)). On the other hand we claim that m 6= 0, so that m gives an example of a nonzero global section of E mapping to zero in Γ(P, OP (1)). Assume m = 0 to get a contradiction. In this case there exists an element f ∈ k[u, v, s1 , s2 , t1 , t2 ] such that us1 x + vt2 y = f (ux + vy) mod I Since I is generated by homogeneous polynomials of degree 2 we may decompose f into its homogeneous components and take the degree 1 component. In other words we may assume that f = au + bv + α1 s1 + α2 s2 + β1 t1 + β2 t2 for some a, b, α1 , α2 , β1 , β2 ∈ k. The resulting conditions are that us1 − u(au + bv + α1 s1 + α2 s2 + β1 t1 + β2 t2 ) ∈ I vt2 − v(au + bv + α1 s1 + α2 s2 + β1 t1 + β2 t2 ) ∈ I There are no terms u2 , uv, v 2 in the generators of I and hence we see a = b = 0. Thus we get the relations us1 − u(α1 s1 + α2 s2 + β1 t1 + β2 t2 ) ∈ I vt2 − v(α1 s1 + α2 s2 + β1 t1 + β2 t2 ) ∈ I We may use the first generator of I to replace any occurence of us1 by vt1 + ut2 , the second generator of I to replace any occurence of vs1 by −us2 + vt2 , the third

22.22. OTHER CHAPTERS

1329

generator to remove occurences of vs2 and the third to remove occurences of ut1 . Then we get the relations (1 − α1 )vt1 + (1 − α1 )ut2 − α2 us2 − β2 ut2 = 0 (1 − α1 )vt2 + α1 us2 − β1 vt1 − β2 vt2 = 0 This implies that α1 should be both 0 and 1 which is a contradiction as desired. Lemma 22.21.3. Let S be a scheme. The structure morphism P(E) → S of a projective bundle over S is separated. Proof. Immediate from Lemma 22.16.9.



Lemma 22.21.4. Let S be a scheme. Let n ≥ 0. Then over S.

PnS

is a projective bundle

Proof. Note that   , . . . , Tn ] PnZ = Proj(Z[T0 , . . . , Tn ]) = ProjSpec(Z) Z[T0^ where the grading on the ring Z[T0 , . . . , Tn ] is given by deg(Ti ) = 1 and the elements of Z are in degree 0. Recall that PnS is defined as PnZ ×Spec(Z) S. Moreover, forming the relative homogeneous spectrum commutes with base change, see Lemma 22.16.10. For any scheme g : S → Spec(Z) we have g ∗ OSpec(Z) [T0 , . . . , Tn ] = OS [T0 , . . . , Tn ]. Combining the above we see that PnS = ProjS (OS [T0 , . . . , Tn ]). Finally, note that OS [T0 , . . . , Tn ] = Sym(OS⊕n+1 ). Hence we see that PnS is a projective bundle over S.  22.22. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes

(22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces

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(43) Decent Algebraic Spaces (44) Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces (48) More on Morphisms of Spaces (49) Quot and Hilbert Spaces (50) Spaces over Fields (51) Stacks (52) Formal Deformation Theory (53) Groupoids in Algebraic Spaces (54) More on Groupoids in Spaces (55) Bootstrap (56) Examples of Stacks (57) Quotients of Groupoids (58) Algebraic Stacks

(59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 23

Properties of Schemes 23.1. Introduction In this chapter we introduce some absolute properties of schemes. A foundational reference is [DG67]. 23.2. Constructible sets Constructible and locally construcible sets are introduced in Topology, Section 5.10. We may characterize locally constructible subsets of schemes as follows. Lemma 23.2.1. Let X be a scheme. A subset E of X is locally constructible in X if and only if E ∩ U is constructible in U for every affine open U of X. Proof. S Assume E is locally constructible. Then there exists an open covering X = Ui such that E ∩ Ui is constructible in Ui for each i. Let V ⊂ X be any affine open. We can find a finite open affine covering V = V1 ∪. . .∪Vm such that for each j we have Vj ⊂ Ui for some i = i(j). By Topology, Lemma 5.10.4 we see that each E ∩ Vj is construcible in Vj . Since the inclusions Vj → V are quasi-compact (see Schemes, Lemma 21.19.2) we conclude that E ∩ V is constructible in V by Topology, Lemma 5.10.5. The converse implication is immediate.  Lemma 23.2.2. Let X be a quasi-separated scheme. The intersection of any two quasi-compact opens of X is a quasi-compact open of X. Every quasi-compact open of X is retrocompact in X. Proof. If U and V are quasi-compact open then U ∩ V = ∆−1 (U × V ), where ∆ : X → X × X is the diagonal. As X is quasi-separated we see that ∆ is quasicompact. Hence we see that U ∩ V is quasi-compact as U × V is quasi-compact (details omitted; use Schemes, Lemma 21.17.4 to see U × V is a finite union of affines). The other assertions follow from the first and Topology, Lemma 5.18.2.  Lemma 23.2.3. Let X be a quasi-compact and quasi-separated scheme. Any locally constructible subset of X is constructible. Proof. As X is quasi-compact we can choose a finite affine open covering X = V1 ∪ . . . ∪ Vm . As X is quasi-separated each Vi is retrocompact in X by Lemma 23.2.2. Hence by Topology, Lemma 5.10.5 we see that E ⊂ X is constructible in X if and only if E ∩ Vj is constructible in Vj . Thus we win by Lemma 23.2.1.  Lemma 23.2.4. Let X be a scheme. A subset Z of X is retrocompact in X if and only if E ∩ U is quasi-compact for every affine open U of X. Proof. Immediate from the fact that every quasi-compact open of X is a finite union of affine opens.  1331

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23.3. Integral, irreducible, and reduced schemes Definition 23.3.1. Let X be a scheme. We say X is integral if it is nonempty and for every nonempty affine open Spec(R) = U ⊂ X the ring R is an integral domain. Lemma 23.3.2. Let X be a scheme. The following are equivalent. (1) The scheme X is reduced, see Schemes, Definition 21.12.1. S (2) There exists an affine open covering X = Ui such that each Γ(Ui , OX ) is reduced. (3) For every affine open U ⊂ X the ring OX (U ) is reduced. (4) For every open U ⊂ X the ring OX (U ) is reduced. Proof. See Schemes, Lemmas 21.12.2 and 21.12.3.



Lemma 23.3.3. Let X be a scheme. The following are equivalent. (1) The scheme X is irreducible. S (2) There exists an affine open covering X = i∈I Ui such that I is not empty, Ui is irreducible for all i ∈ I, and Ui ∩ Uj 6= ∅ for all i, j ∈ I. (3) The scheme X is nonempty and every nonempty affine open U ⊂ X is irreducible. Proof. Assume (1). By Schemes, Lemma 21.11.1 we see that X has a unique generic point η. Then X = {η}. Hence η is an element of every nonempty affine open U ⊂ X. This implies that U = {η} and that any two nonempty affines meet. Thus (1) implies both (2) and (3). Assume (2). Suppose X = Z1 ∪ Z2 is a union of two closed subsets. For every i we see that either Ui ⊂ Z1 or Ui ⊂ Z2 . Pick some i ∈ I and assume Ui ⊂ Z1 (possibly after renumbering Z1 , Z2 ). For any j ∈ I the open subset Ui ∩ Uj is dense in Uj and contained in the closed subset Z1 ∩ Uj . We conclude that also Uj ⊂ Z1 . Thus X = Z1 as desired. S Assume (3). Choose an affine open covering X = i∈I Ui . We may assume that each Ui is nonempty. Since X is nonempty we see that I is not empty. By assumption each`Ui is irreducible. Suppose Ui ∩ Uj = ∅ for some pair i, j ∈ I. Then the open Ui Uj = Ui ∪Uj is affine, see Schemes, Lemma 21.6.8. Hence it is irreducible by assumption which is absurd. We conclude that (3) implies (2). The lemma is proved.  Lemma 23.3.4. A scheme X is integral if and only if it is reduced and irreducible. Proof. If X is irreducible, then every affine open Spec(R) = U ⊂ X is irreducible. If X is reduced, then R is reduced, by Lemma 23.3.2 above. Hence R is reduced and (0) is a prime ideal, i.e., R is an integral domain. If X is integral, then for every nonempty affine open Spec(R) = U ⊂ X the ring R is reduced and hence X is reduced by Lemma 23.3.2. Moreover, every nonempty affine open is irreducible. Hence X is irreducible, see Lemma 23.3.3.  Example 23.3.5. We give an example of an affine scheme X = Spec(A) which is connected, all of whose local rings are domains, but which is not integral. Connectedness for A means A has no nontrivial idempotents, see Algebra, Lemma 7.19.3. Integrality means A is a domain (see above). Local rings being domains means that

23.4. TYPES OF SCHEMES DEFINED BY PROPERTIES OF RINGS

1333

whenever f g = 0 in A, every point of X has a neighborhood where either f or g vanishes. Roughly speaking, the construction is as follows: let X0 be the cross (the union of coordinate axes) on the affine plane. Then let X1 be the (reduced) full preimage of X0 on the blow-up of the plane (X1 has three rational components forming a chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational components), etc. Take X to be the inverse limit. The only problem with this construction is that blow-ups glue in a projective line, so X1 is not affine. Let us correct this by glueing in an affine line instead (so our scheme will be an open subset in what was described above). Here is a completely algebraic construction: For every k ≥ 0, let Ak be the following ring: its elements are collections of polynomials pi ∈ C[x] where i = 0, . . . , 2k such that pi (1) = pi+1 (0). Set Xk = Spec(Ak ). Observe that Xk is a union of 2k +1 affine lines that meet transversally in a chain. Define a ring homomorphism Ak → Ak+1 by (p0 , . . . , p2k ) 7−→ (p0 , p0 (1), p1 , p1 (1), . . . , p2k ), in other words, every other polynomial is constant. This identifies Ak with a subring of Ak+1 . Let A be the direct limit of Ak (basically, their union). Set X = Spec(A). For every k, we have a natural embedding Ak → A, that is, a map X → Xk . Each Ak is connected but not integral; this implies that A is connected but not integral. It remains to show that the local rings of A are domains. Take f, g ∈ A with f g = 0 and x ∈ X. Let us construct a neighborhood of x on which one of f and g vanishes. Choose k such that f, g ∈ Ak−1 (note the k − 1 index). Let y be the image of x in Xk . It suffices to prove that y has a neighborhood on which either f or g viewed as sections of OXk vanishes. If y is a smooth point of Xk , that is, it lies on only one of the 2k + 1 lines, this is obvious. We can therefore assume that y is one of the 2k singular points, so two components of Xk pass through y. However, on one of these two components (the one with odd index), both f and g are constant, since they are pullbacks of functions on Xk−1 . Since f g = 0 everywhere, either f or g (say, f ) vanishes on the other component. This implies that f vanishes on both components, as required. 23.4. Types of schemes defined by properties of rings In this section we study what properties of rings allow one to define local properties of schemes. Definition 23.4.1. Let P be a property of rings. We say that P is local if the following hold: (1) For any ring R, and any f ∈ R we have P (R) ⇒ P (Rf ). (2) For any ring R, and fi ∈ R such that (f1 , . . . , fn ) = R then ∀i, P (Rfi ) ⇒ P (R). Definition 23.4.2. Let P be a property of rings. Let X be a scheme. We say X is locally P if for any x ∈ X there exists an affine open neighbourhood U of x in X such that OX (U ) has property P .

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This is only a good notion if the property is local. Even if P is a local property we will not automatically use this definition to say that a scheme is “locally P ” unless we also explicitly state the definition elsewhere. Lemma 23.4.3. Let X be a scheme. Let P be a local property of rings. The following are equivalent: (1) The scheme X is locally P . (2) For every affine open U ⊂ X the property PS (OX (U )) holds. (3) There exists an affine open covering X = Ui such that each OX (Ui ) satisfies P . S (4) There exists an open covering X = Xj such that each open subscheme Xj is locally P . Moreover, if X is locally P then every open subscheme is locally P . Proof. Of course (1) ⇔ (3) and (2) ⇒ (1). If (3) ⇒ (2), then the final statement of the lemma holds and it follows easily that (4) is also equivalent to (1). Thus we show (3) ⇒ (2). S Let X = Ui be an affine open covering, say Ui = Spec(Ri ). Assume P (Ri ). Let Spec(R) = U ⊂ X be an arbitrary affine open. By Schemes, Lemma 21.11.6 there exists a standard covering of U = Spec(R) by standard opens D(fj ) such that each ring Rfj is a principal localization of one of the rings Ri . By Definition 23.4.1 (1) we get P (Rfj ). Whereupon P (R) by Definition 23.4.1 (2).  Here is a sample application. Lemma 23.4.4. Let X be a scheme. Then X is reduced if and only if X is “locally reduced” in the sense of Definition 23.4.2. Proof. This is clear from Lemma 23.3.2.



Lemma 23.4.5. The following properties of a ring R are local. (1) (Cohen-Macauley.) The ring R is Noetherian and CM, see Algebra, Definition 7.97.6. (2) (Regular.) The ring R is Noetherian and regular, see Algebra, Definition 7.103.6. (3) (Absolutely Noetherian.) The ring R is of finite type over Z. (4) Add more here as needed.1 Proof. Omitted.

 23.5. Noetherian schemes

Recall that a ring R is Noetherian if it satsifies the ascending chain condition of ideals. Equivalently every ideal of R is finitely generated. Definition 23.5.1. Let X be a scheme. (1) We say X is locally Noetherian if every x ∈ X has an affine open neighbourhood Spec(R) = U ⊂ X such that the ring R is Noetherian. (2) We say X is Noetherian if X is Noetherian and quasi-compact. Here is the standard result characterizing locally Noetherian schemes. 1But we only list those properties here which we have not already dealt with separately somewhere else.

23.5. NOETHERIAN SCHEMES

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Lemma (1) (2) (3)

23.5.2. Let X be a scheme. The following are equivalent: The scheme X is locally Noetherian. For every affine open U ⊂ X the ring OX (U S ) is Noetherian. There exists an affine open covering X = Ui such that each OX (Ui ) is Noetherian. S (4) There exists an open covering X = Xj such that each open subscheme Xj is locally Noetherian. Moreover, if X is locally Noetherian then every open subscheme is locally Noetherian. Proof. To show this it suffices to show that being Noetherian is a local property of rings, see Lemma 23.4.3. Any localization of a Noetherian ring is Noetherian, see Algebra, Lemma 7.29.1. By Algebra, Lemma 7.22.2 we see the second property to Definition 23.4.1. 

Lemma 23.5.3. Any immersion Z → X with X locally Noetherian is quasicompact. Proof. A closed immersion is clearly quasi-compact. A composition of quasicompact morphisms is quasi-compact, see Topology, Lemma 5.9.2. Hence it suffices to show that an open immersion into a locally Noetherian scheme is quasi-compact. Using Schemes, Lemma 21.19.2 we reduce to the case where X is affine. Any open subset of the spectrum of a Noetherian ring is quasi-compact (for example combine Algebra, Lemma 7.29.5 and Topology, Lemmas 5.6.2 and 5.9.9).  Lemma 23.5.4. A locally Noetherian scheme is quasi-separated. Proof. By Schemes, Lemma 21.21.7 we have to show that the intersection U ∩ V of two affine opens of X is quasi-compact. This follows from Lemma 23.5.3 above on considering the open immersion U ∩ V → U for example. (But really it is just because any open of the spectrum of a Noetherian ring is quasi-compact.)  Lemma 23.5.5. A (locally) Noetherian scheme has a (locally) Noetherian underlying topological space, see Topology, Definition 5.6.1. Proof. This is because a Noetherian scheme is a finite union of spectra of Noetherian rings and Algebra, Lemma 7.29.5 and Topology, Lemma 5.6.4.  Lemma 23.5.6. Any morphism of schemes f : X → Y with X Noetherian is quasi-compact. Proof. Use Lemma 23.5.5 and use that any subset of a Noetherian topological space is quasi-compact (see Topology, Lemmas Lemmas 5.6.2 and 5.9.9).  Lemma 23.5.7. Any locally closed subscheme of a (locally) Noetherian scheme is (locally) Noetherian. Proof. Omitted. Hint: Any quotient, and any localization of a Noetherian ring is Noetherian. For the Noetherian case use again that any subset of a Noetherian space is a Noetherian space (with induced topology).  Here is a fun lemma. It says that every locally Noetherian scheme has plenty of closed points (at least one in every closed subset).

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Lemma 23.5.8. Any locally Noetherian scheme has a closed point. Any closed subset of a locally Noetherian scheme has a closed point. Equivalently, any point of a locally Noetherian scheme specializes to a closed point. Proof. The second assertion follows from the first (using Schemes, Lemma 21.12.4 and Lemma 23.5.7). Consider any nonempty affine open U ⊂ X. Let x ∈ U be a closed point. If x is a closed point of X then we are done. If not, let y ∈ {x} be a specialization of x. Note that y ∈ X \ U . Consider the local ring R = OX,y . This is a Noetherian local ring. Denote V ⊂ Spec(R) the inverse image of U in Spec(R) by the canonical morphism Spec(R) → X (see Schemes, Section 21.13.) By construction V is a singleton with unique point corresponding to x (use Schemes, Lemma 21.13.2). Say V = {q}. Consider the Noetherian local domain R/q. By Algebra, Lemma 7.59.1 we see that dim(R/q) = 1. In other words, we see that y is an immediate specialization of x (see Topology, Definition 5.16.1). In other words, any point y 6= x such that x y is an immediate specialization of x. Clearly each of these points is a closed point, and we win.  Lemma 23.5.9. Let X be a locally Noetherian scheme. Let x0 x be a specialization of points of X. Then (1) there exists a discrete valuation ring R and a morphism f : Spec(R) → X such that the generic point η of Spec(R) maps to x0 and the special point maps to x, and (2) given a finitely generated field extension κ(x0 ) ⊂ K we may arrange it so that the extension κ(x0 ) ⊂ κ(η) induced by f is isomorphic to the given one. Proof. Let x0 x be a specialization in X, and let κ(x0 ) ⊂ K be a finitely generated extension of fields. By Schemes, Lemma 21.13.2 and the discussion following Schemes, Lemma 21.13.3 this leads to ring maps OX,x → κ(x0 ) → K. Let R ⊂ K be any discrete valuation ring whose field of fractions is K and which dominates the image of OX,x → K, see Algebra, Lemma 7.111.11. The ring map OX,x → R induces the morphism f : Spec(R) → X, see Schemes, Lemma 21.13.1. This morphism has all the desired properties by construction.  23.6. Jacobson schemes Recall that a space is said to be Jacobson if the closed points are dense in every closed subset, see Topology, Section 5.13. Definition 23.6.1. A scheme S is said to be Jacobson if its underlying topological space is Jacobson. Recall that a ring R is Jacobson if every radical ideal of R is the intersection of maximal ideals, see Algebra, Definition 7.32.1. Lemma 23.6.2. An affine scheme Spec(R) is Jacobson if and only if the ring R is Jacobson. Proof. This is Algebra, Lemma 7.32.4.



Here is the standard result characterizing Jacobson schemes. Intuitively it claims that Jacobson ⇔ locally Jacobson. Lemma 23.6.3. Let X be a scheme. The following are equivalent:

23.7. NORMAL SCHEMES

1337

(1) (2) (3) (4)

The scheme X is Jacobson. The scheme X is “locally Jacobson” in the sense of Definition 23.4.2. For every affine open U ⊂ X the ring OX (U S ) is Jacobson. There exists an affine open covering X = Ui such that each OX (Ui ) is Jacobson. S (5) There exists an open covering X = Xj such that each open subscheme Xj is Jacobson. Moreover, if X is Jacobson then every open subscheme is Jacobson. Proof. The final assertion of the lemma holds by Topology, Lemma 5.13.5. The equivalence of (5) and (1) is Topology, Lemma 5.13.4. Hence, using Lemma 23.6.2, we see that (1) ⇔ (2). To finish proving the lemma it suffices to show that “Jacobson” is a local property of rings, see Lemma 23.4.3. Any localization of a Jacobson ring at an element is Jacobson, see Algebra, Lemma 7.32.14. Suppose R is a ring, f1 , . . . , fn ∈SR generate the unit ideal and each Rfi is Jacobson. Then we see that Spec(R) = D(fi ) is a union of open subsets which are all Jacobson, and hence Spec(R) is Jacobson by Topology, Lemma 5.13.4 again. This proves the second property of Definition 23.4.1.  Many schemes used commonly in algebraic geometry are Jacobson, see Morphisms, Lemma 24.17.10. We mention here the following interesting case. Lemma 23.6.4. Let R be a Noetherian local ring with maximal ideal m. In this case the scheme S = Spec(R) \ {m} is Jacobson. Proof. Since Spec(R) is a Noetherian scheme, hence S is a Noetherian scheme (Lemma 23.5.7). Hence S is a sober, Noetherian topological space (use Schemes, Lemma 21.11.1). Assume S is not Jacobson to get a contradiction. By Topology, Lemma 5.13.3 there exists some non-closed point ξ ∈ S such that {ξ} is locally closed. This corresponds to a prime p ⊂ R such that (1) there exists a prime q, p ⊂ q ⊂ m with both inclusions strict, and (2) {p} is open in Spec(R/p). This is impossible by Algebra, Lemma 7.59.1.  23.7. Normal schemes Recall that a ring R is said to be normal if all its local rings are normal domains, see Algebra, Definition 7.34.10. A normal domain is a domain which is integrally closed in its field of fractions, see Algebra, Definition 7.34.1. Thus it makes sense to define a normal scheme as follows. Definition 23.7.1. A scheme X is normal if and only if for all x ∈ X the local ring OX,x is a normal domain. This seems to be the definition used in EGA, see [DG67, 0, 4.1.4]. Suppose X = Spec(A), and A is reduced. Then saying that X is normal is not equivalent to saying that A is integrally closed in its total ring of fractions. However, if A is Noetherian then this is the case (see Algebra, Lemma 7.34.14). Lemma (1) (2) (3)

23.7.2. Let X be a scheme. The following are equivalent: The scheme X is normal. For every affine open U ⊂ X the ring OX (U S ) is normal. There exists an affine open covering X = Ui such that each OX (Ui ) is normal.

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S (4) There exists an open covering X = Xj such that each open subscheme Xj is normal. Moreover, if X is normal then every open subscheme is normal. Proof. This is clear from the definitions.



Lemma 23.7.3. A normal scheme is reduced. Proof. Immediate from the defintions.



Lemma 23.7.4. Let X be an integral scheme. Then X is normal if and only if for every affine open U ⊂ X the ring OX (U ) is a normal domain. Proof. This follows from Algebra, Lemma 7.34.9.



Lemma 23.7.5. Let X be a scheme with a finite number of irreducible components. The following are equivalent: (1) X is normal, and (2) X is a finite disjoint union of normal integral schemes. Proof. It is immediate from the definitions that (2) implies (1). Let X be a normal scheme with a finite number of irreducible components. If X isSaffine then X satisfies (2) by Algebra, Lemma 7.34.14. For a general X, let X = Xi be an affine open covering. Note that also each Xi has but a finite number of irreducible components, and the lemma holds for each Xi . Let T ⊂ X be an irreducible component. By the affine case each intersection T ∩ Xi is open in Xi and an integral normal scheme. Hence T ⊂ X is open, and an integral normal scheme. This proves that X is the disjoint union of its irreducible components, which are integral normal schemes. There are only finitely many by assumption.  Lemma 23.7.6. Let X be a Noetherian scheme. The following are equivalent: (1) X is normal, and (2) X is a finite disjoint union of normal integral schemes. Proof. This is a special case of Lemma 23.7.5 because a Noetherian scheme has a Noetherian underlying topological space (Lemma 23.5.5 and Topology, Lemma 5.6.2.  Lemma 23.7.7. Let X be a locally Noetherian normal scheme. The following are equivalent: (1) X is normal, and (2) X is a disjoint union of integral normal schemes. Proof. Omitted. Hint: This is purely topological from Lemma 23.7.6.



Remark 23.7.8. Let X be a normal scheme. If X is locally Noetherian then we see that X is integral if and only if X is connected, see Lemma 23.7.7. But there exists a connected affine scheme X such that OX,x is a domain for all x ∈ X, but X is not irreducible, see Example 23.3.5. This example is even a normal scheme (proof omitted), so beware! Lemma 23.7.9. Let X be an integral normal scheme. Then Γ(X, OX ) is a normal domain.

23.9. REGULAR SCHEMES

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Proof. Set R = Γ(X, OX ). It is clear that R is a domain. Suppose f = a/b d is P an elementi of its fraction field which is integral over R. Say we have f + i=1,...,d ai f = 0 with ai ∈ R. Let U ⊂ X be affine open. Since b ∈ R is not zero and since X is integral we see that also b|U ∈ OX (U ) is not zero. Hence a/b is an element of the fraction field of OXP(U ) which is integral over OX (U ) (because we can use the same polynomial f d + i=1,...,d ai |U f i = 0 on U ). Since OX (U ) is a normal domain (Lemma 23.7.2), we see that fU = (a|U )/(b|U ) ∈ OX (U ). It is easy to see that fU |V = fV whenever V ⊂ U ⊂ X are affine open. Hence the local sections fU glue to a global section f as desired.  23.8. Cohen-Macaulay schemes Recall, see Algebra, Definition 7.97.1, that a local Noetherian ring (R, m) is said to be Cohen-Macaulay if depthm (R) = dim(R). Recall that a Noetherian ring R is said to be Cohen-Macaulay if every local ring Rp of R is Cohen-Macaulay, see Algebra, Definition 7.97.6. Definition 23.8.1. Let X be a scheme. We say X is Cohen-Macaulay if for every x ∈ X there exists an affine open neighbourhood U ⊂ X of x such that the ring OX (U ) is Noetherian and Cohen-Macaulay. Lemma (1) (2) (3)

23.8.2. Let X be a scheme. The following are equivalent: X is Cohen-Macaulay, X is locally Noetherian and all of its local rings are Cohen-Macaulay, and X is locally Noetherian and for any closed point x ∈ X the local ring OX,x is Cohen-Macaulay.

Proof. Algebra, Lemma 7.97.5 says that the localization of a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows by combining this with Lemma 23.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 23.5.8), and the definitions.  Lemma 23.8.3. Let X be a scheme. The following are equivalent: (1) The scheme X is Cohen-Macaulay. (2) For every affine open U ⊂ X the ring OX (U ) is Noetherian and CohenMacaulay. S (3) There exists an affine open covering X = Ui such that each OX (Ui ) is Noetherian and Cohen-Macaulay. S (4) There exists an open covering X = Xj such that each open subscheme Xj is Cohen-Macaulay. Moreover, if X is Cohen-Macaulay then every open subscheme is Cohen-Macaulay. Proof. Combine Lemmas 23.5.2 and 23.8.2.



More information on Cohen-Macaulay schemes and depth can be found in Cohomology of Schemes, Section 25.13. 23.9. Regular schemes Recall, see Algebra, Definition 7.58.9, that a local Noetherian ring (R, m) is said to be regular if m can be generated by dim(R) elements. Recall that a Noetherian ring R is said to be regular if every local ring Rp of R is regular, see Algebra, Definition 7.103.6.

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Definition 23.9.1. Let X be a scheme. We say X is regular, or nonsingular if for every x ∈ X there exists an affine open neighbourhood U ⊂ X of x such that the ring OX (U ) is Noetherian and regular. Lemma (1) (2) (3)

23.9.2. Let X be a scheme. The following are equivalent: X is regular, X is locally Noetherian and all of its local rings are regular, and X is locally Noetherian and for any closed point x ∈ X the local ring OX,x is regular.

Proof. By the discussion in Algebra preceding Algebra, Definition 7.103.6 we know that the localization of a regular local ring is regular. The lemma follows by combining this with Lemma 23.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 23.5.8), and the definitions.  Lemma (1) (2) (3)

23.9.3. Let X be a scheme. The following are equivalent: The scheme X is regular. For every affine open U ⊂ X the ring OX (U S ) is Noetherian and regular. There exists an affine open covering X = Ui such that each OX (Ui ) is Noetherian and regular. S (4) There exists an open covering X = Xj such that each open subscheme Xj is regular. Moreover, if X is regular then every open subscheme is regular. Proof. Combine Lemmas 23.5.2 and 23.9.2.



Lemma 23.9.4. A regular scheme is normal. Proof. See Algebra, Lemma 7.141.5.



23.10. Dimension The dimension of a scheme is just the dimension of its underlying topological space. Definition 23.10.1. Let X be a scheme. (1) The dimension of X is just the dimension of X as a topological spaces, see Topology, Definition 5.7.1. (2) For x ∈ X we denote dimx (X) the dimension of the underlying topological space of X at x as in Topology, Definition 5.7.1. We say dimx (X) is the dimension of X at x. As a scheme has a sober underlying topological space (Schemes, Lemma 21.11.1) we may compute the dimension of X as the supremum of the lengths n of chains T0 ⊂ T1 ⊂ . . . ⊂ Tn of irreducible closed subsets of X, or as the supremum of the lengths n of chains of specializations ξn ξn−1 ... ξ0 of points of X. Lemma 23.10.2. Let X be a scheme. The following are equal (1) The dimension of X. (2) The supremum of the dimensions of the local rings of X.

23.11. CATENARY SCHEMES

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(3) The supremum of dimx (X) for x ∈ X. Proof. Note that given a chain of specializations ξn

ξn−1

...

ξ0

of points of X all of the points ξi correspond to prime ideals of the local ring of X at ξ0 by Schemes, Lemma 21.13.2. Hence we see that the dimension of X is the supremum of the dimensions of its local rings. In particular dimx (X) ≥ dim(OX,x ) as dimx (X) is the minimum of the dimensions of open neighbourhoods of x. Thus supx∈X dimx (X) ≥ dim(X). On the other hand, it is clear that supx∈X dimx (X) ≤ dim(X) as dim(U ) ≤ dim(X) for any open subset of X.  23.11. Catenary schemes Recall that a topological space X is called catenary if for every pair of irreducible closed subsets T ⊂ T 0 there exist a maximal chain of irreducible closed subsets T = T0 ⊂ T1 ⊂ . . . ⊂ Te = T 0 and every such chain has the same length. See Topology, Definition 5.8.1. Definition 23.11.1. Let S be a scheme. We say S is catenary if the underlying topological space of S is catenary. Recall that a ring A is called catenary if for any pair of prime ideals p ⊂ q there exists a maximal chain of primes p = p0 ⊂ . . . ⊂ pe = q and all of these have the same length. See Algebra, Definition 7.98.1. Lemma 23.11.2. Let S be a scheme. The following are equivalent (1) S is catenary, (2) there exists an open covering of S all of whose members are catenary schemes, (3) for every affine open Spec(R) = U ⊂ S theSring R is catenary, and (4) there exists an affine open covering S = Ui such that each Ui is the spectrum of a catenary ring. Moreover, in this case any locally closed subscheme of S is catenary as well. Proof. Combine Topology, Lemma 5.8.2, and Algebra, Lemma 7.98.2.



Lemma 23.11.3. Let S be a locally Noetherian scheme. The following are equivalent: (1) S is catenary, and (2) locally in the Zariski topology there exists a dimension function on S (see Topology, Definition 5.16.1). Proof. This follows from Topology, Lemmas 5.8.2, 5.16.2, and 5.16.4, Schemes, Lemma 21.11.1 and finally Lemma 23.5.5.  Lemma 23.11.4. Let X be a scheme. Let Y ⊂ X be an irreducible closed subset. Let ξ ∈ Y be the generic point. Then codim(Y, X) = dim(OX,ξ ) where the codimension is as defined in Topology, Definition 5.8.3.

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Proof. By Topology, Lemma 5.8.4 we may replace X by an affine open neighbourhood of ξ. In this case the result follows easily from Algebra, Lemma 7.24.2.  In particular the dimension of a scheme is the supremum of the dimensions of all of its local rings. It turns out that we can use this lemma to characterize a catenary scheme as a scheme all of whose local rings are catenary. Lemma 23.11.5. Let X be a scheme. The following are equivalent (1) X is catenary, and (2) for any x ∈ X the local ring OX,x is catenary. Proof. Assume X is catenary. Let x ∈ X. By Lemma 23.11.2 we may replace X by an affine open neighbourhood of x, and then Γ(X, OX ) is a catenary ring. By Algebra, Lemma 7.98.3 any localization of a catenary ring is catenary. Whence OX,x is catenary. Conversely assume all local rings of X are catenary. Let Y ⊂ Y 0 be an inclusion of irreducible closed subsets of X. Let ξ ∈ Y be the generic point. Let p ⊂ OX,ξ be the prime corresponding to the generic point of Y 0 , see Schemes, Lemma 21.13.2. By that same lemma the irreducible closed subsets of X in between Y and Y 0 correspond to primes q ⊂ OX,ξ with p ⊂ q ⊂ mξ . Hence we see all maximal chains of these are finite and have the same length as OX,ξ is a catenary ring.  23.12. Serre’s conditions Here are two technical notions that are often useful. See also Cohomology of Schemes, Section 25.13. Definition 23.12.1. Let X be a locally Noetherian scheme. Let k ≥ 0. (1) We say X is regular in codimension k, or we say X has property (Rk ) if for every x ∈ X we have dim(OX,x ) ≤ k ⇒ OX,x is regular (2) We say X has property (Sk ) if for every x ∈ X we have depth(OX,x ) ≥ min(k, dim(OX,x )). The phrase “regular in codimension k” makes sense since we have seen in Section 23.11 that if Y ⊂ X is irreducible closed with generic point x, then dim(OX,x ) = codim(Y, X). For example condition (R0 ) means that for every generic point η ∈ X of an irreducible component of X the local ring OX,η is a field. But for general Noetherian schemes it can happen that the regular locus of X is badly behaved, so care has to be taken. Lemma 23.12.2. Let X be a locally Noetherian scheme. Then X is CohenMacaulay if and only if X has (Sk ) for all k ≥ 0. Proof. By Lemma 23.8.2 we reduce to looking at local rings. Hence the lemma is true because a Noetherian local ring is Cohen-Macaulay if and only if it has depth equal to its dimension.  Lemma 23.12.3. Let X be a locally Noetherian scheme. Then X is reduced if and only if X has properties (S1 ) and (R0 ). Proof. This is Algebra, Lemma 7.141.3.



23.13. JAPANESE AND NAGATA SCHEMES

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Lemma 23.12.4. Let X be a locally Noetherian scheme. Then X is normal if and only if X has properties (S2 ) and (R1 ). Proof. This is Algebra, Lemma 7.141.4.



23.13. Japanese and Nagata schemes The notions considered in this section are not prominently defined in EGA. A “universally Japanese scheme” is mentioned and defined in [DG67, IV Corollary 5.11.4]. A “Japanese scheme” is mentioned in [DG67, IV Remark 10.4.14 (ii)] but no definition is given. A Nagata scheme (as given below) occurs in a few places in the literature (see for example [Liu02, Definition 8.2.30] and [Gre76, Page 142]). We briefly recall that a domain R is called Japanese if the integral closure of R in any finite extension of its fraction field is finite over R. A ring R is called universally Japanese if for any finite type ring map R → S with S a domain S is Japanese. A ring R is called Nagata if it is Noetherian and R/p is Japanese for every prime p of R. Definition 23.13.1. Let X be a scheme. (1) Assume X integral. We say X is Japanese if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring OX (U ) is Japanese (see Algebra, Definition 7.145.1). (2) We say X is universally Japanese if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring OX (U ) is universally Japanese (see Algebra, Definition 7.145.15). (3) We say X is Nagata if for every x ∈ X there exists an affine open neighbourhood x ∈ U ⊂ X such that the ring OX (U ) is Nagata (see Algebra, Definition 7.145.15). Being Nagata is the same thing as being locally Noetherian and universally Japanese, see Lemma 23.13.8. Remark 23.13.2. In [Hoo72] a (locally Noetherian) scheme X is called Japanese if for every x ∈ X and every associated prime p of OX,x the ring OX,x /p is Japanese. We do not use this definition since it is not clear that this gives the same notion as above for Noetherian integral schemes. In other words, we do not know whether a Noetherian domain all of whose local rings are Japanese is Japanese. If you do please email [email protected]. On the other hand, we could circumvent this problem by calling a scheme X Japanese if for every affine open Spec(A) ⊂ X the ring A/p is Japanese for every associated prime p of A. Lemma 23.13.3. A Nagata scheme is locally Noetherian. Proof. This is true because a Nagata ring is Noetherian by definition. Lemma (1) (2) (3)



23.13.4. Let X be an integral scheme. The following are equivalent: The scheme X is Japanese. For every affine open U ⊂ X the domain O SX (U ) is Japanese. There exists an affine open covering X = Ui such that each OX (Ui ) is Japanese. S (4) There exists an open covering X = Xj such that each open subscheme Xj is Japanese.

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Moreover, if X is Japanese then every open subscheme is Japanese. Proof. This follows from Lemma 23.4.3 and Algebra, Lemmas 7.145.3 and 7.145.4.  Lemma (1) (2) (3)

23.13.5. Let X be a scheme. The following are equivalent: The scheme X is universally Japanese. For every affine open U ⊂ X the ring OX (U S ) is universally Japanese. There exists an affine open covering X = Ui such that each OX (Ui ) is universally Japanese. S (4) There exists an open covering X = Xj such that each open subscheme Xj is universally Japanese. Moreover, if X is universally Japanese then every open subscheme is universally Japanese. Proof. This follows from Lemma 23.4.3 and Algebra, Lemmas 7.145.18 and 7.145.21.  Lemma (1) (2) (3)

23.13.6. Let X be a scheme. The following are equivalent: The scheme X is Nagata. For every affine open U ⊂ X the ring OX (U S ) is Nagata. There exists an affine open covering X = Ui such that each OX (Ui ) is Nagata. S (4) There exists an open covering X = Xj such that each open subscheme Xj is Nagata. Moreover, if X is Nagata then every open subscheme is Nagata. Proof. This follows from Lemma 23.4.3 and Algebra, Lemmas 7.145.20 and 7.145.21.  Lemma 23.13.7. Let X be a locally Noetherian scheme. Then X is Nagata if and only if every integral closed subscheme Z ⊂ X is Japanese. Proof. Assume X is Nagata. Let Z ⊂ X be an integral closed subscheme. Let z ∈ Z. Let Spec(A) = U ⊂ X be an affine open containing z such that A is Nagata. Then Z ∩ U ∼ = Spec(A/p) for some prime p, see Schemes, Lemma 21.10.1 (and Definition 23.3.1). By Algebra, Definition 7.145.15 we see that A/p is Japanese. Hence Z is Japanese by definition.

Assume every integral closed subscheme of X is Japanese. Let Spec(A) = U ⊂ X be any affine open. As X is locally Noetherian we see that A is Noetherian (Lemma 23.5.2). Let p ⊂ A be a prime ideal. We have to show that A/p is Japanese. Let T ⊂ U be the closed subset V (p) ⊂ Spec(A). Let T ⊂ X be the closure. Then T is irreducible as the closure of an irreducible subset. Hence the reduced closed subscheme defined by T is an integral closed subscheme (called T again), see Schemes, Lemma 21.12.4. In other words, Spec(A/p) is an affine open of an integral closed subscheme of X. This subscheme is Japanese by assumption and by Lemma 23.13.4 we see that A/p is Japanese.  Lemma 23.13.8. Let X be a scheme. The following are equivalent: (1) X is Nagata, and (2) X is locally Noetherian and universally Japanese.

23.15. QUASI-AFFINE SCHEMES

Proof. This is Algebra, Proposition 7.145.30.

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This discussion will be continued in Morphisms, Section 24.19. 23.14. The singular locus Here is the definition. Definition 23.14.1. Let X be a locally Noetherian scheme. The regular locus Reg(X) of X is the set of x ∈ X such that OX,x is a regular local ring. The singular locus Sing(X) is the complement X \ Reg(X), i.e., the set of points x ∈ X such that OX,x is not a regular local ring. The regular locus of a locally Noetherian scheme is stable under generalizations, see the discussion preceding Algebra, Definition 7.103.6. However, for general locally Noetherian schemes the regular locus need not be open. In More on Algebra, Section 12.38 the reader can find some criteria for when this is the case. We will discuss this further in Morphisms, Section 24.20. 23.15. Quasi-affine schemes Definition 23.15.1. A scheme X is called quasi-affine if it is quasi-compact and isomorphic to an open subscheme of an affine scheme. Lemma 23.15.2. Let X be a scheme. Let f ∈ Γ(X, OX ). Denote Xf the maximal open subscheme of X where f is invertible, see Schemes, Lemma 21.6.2 or Modules, Lemma 15.21.7. If X is quasi-compact and quasi-separated, the canonical map Γ(X, OX )f −→ Γ(Xf , OX ) is an isomorphism. Moreover, if F is a quasi-coherent sheaf of OX -modules the map Γ(X, F)f −→ Γ(Xf , F) is an isomorphism. Proof. Write R = Γ(X, OX ). Consider the canonical morphism ϕ : X −→ Spec(R) of schemes, see Schemes, Lemma 21.6.4. Then the inverse image of the standard open D(f ) on the right hand side is Xf on the left hand side. Moreover, since X is assumed quasi-compact and quasi-separated the morphism ϕ is quasi-compact and quasi-separated, see Schemes, Lemma 21.19.2 and 21.21.14. Hence by Schemes, f Lemma 21.24.1 we see that ϕ∗ F is quasi-coherent. Hence we see that ϕ∗ F = M with M = Γ(X, F) as an R-module. Thus we see that f) = Mf Γ(Xf , F) = Γ(D(f ), ϕ∗ F) = Γ(D(f ), M which is exactly the content of the lemma. The case of F = OX will given the first displayed isomorphism of the lemma.  Lemma 23.15.3. Let X be a scheme. Let f ∈ Γ(X, OX ). Assume X is quasicompact and quasi-separated and assume that Xf is affine. Then the canonical morphism j : X −→ Spec(Γ(X, OX )) from Schemes, Lemma 21.6.4 induces an isomorphism of Xf = j −1 (D(f )) onto the standard affine open D(f ) ⊂ Spec(Γ(X, OX )).

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Proof. This is clear as j induces an isomorphism of rings Γ(X, OX )f → OX (Xf ) by Lemma 23.15.2 above.  Lemma 23.15.4. Let X be a scheme. Then X is quasi-affine if and only if the canonical morphism X −→ Spec(Γ(X, OX )) from Schemes, Lemma 21.6.4 is a quasi-compact open immersion. Proof. If the displayed morphism is a quasi-compact open immersion then X is ismorphic to a quasi-compact open subscheme of Spec(Γ(X, OX )) and clearly X is quasi-affine. Assume X is quasi-affine, say X ⊂ Spec(R) is quasi-compact open. This in particular implies that X is separated, see Schemes, Lemma 21.23.8. Let A = Γ(X, OX ). Consider the ring map R → A coming from R = Γ(Spec(R), OSpec(R) ) and the restriction mapping of the sheaf OSpec(R) . By Schemes, Lemma 21.6.4 we obtain a factorization: X −→ Spec(A) −→ Spec(R) of the inclusion morphism. Let x ∈ X. Choose r ∈ R such that x ∈ D(r) and D(r) ⊂ X. Denote f ∈ A the image of r in A. The open Xf of Lemma 23.15.2 above is equal to D(r) ⊂ X and hence Af ∼ = Rr by the conclusion of that lemma. Hence D(r) → Spec(A) is an isomorphism onto the standard affine open D(f ) of Spec(A). Since X can be covered by such affine opens D(f ) we win.  23.16. Characterizing modules of finite type and finite presentation Let X be a scheme. Let F be a quasi-coherent OX -module. The following lemma implies that F is of finite type (see Modules, Definition 15.9.1) if and only if F is f for some finite type A-module on each open affine Spec(A) = U ⊂ X of the form M M . Similarly, F is of finite presentation (see Modules, Definition 15.11.1) if and f for some finitely only if F is on each open affine Spec(A) = U ⊂ X of the form M presented A-module M . Lemma 23.16.1. Let X = Spec(R) be an affine scheme. The quasi-coherent sheaf f is a finite type OX -module if and only if M is a finite R-module. of OX -modules M f is a finite type OX -module. This means there exists an open Proof. Assume M f restricted to the members of this covering is globally covering of X such that M generated by finitely many sections. Thus there also exists a standard open covering S f|D(f ) is generated by finitely many sections. X = i=1,...,n D(fi ) such that M i Thus Mfi is finitely generated for each i. Hence we conclude by Algebra, Lemma 7.22.2.  Lemma 23.16.2. Let X = Spec(R) be an affine scheme. The quasi-coherent sheaf f is an OX -module of finite presentation if and only if M is an of OX -modules M R-module of finite presentation. f is an OX -module of finite presentation. By Lemma 23.16.1 we Proof. Assume M see that M is a finite R-module. Choose a surjection Rn → M with kernel K. By Schemes, Lemma 21.5.4 there is a short exact sequence M ⊕n e → f→0 0→K OX →M

23.18. LOCALLY FREE MODULES

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e is a finite type OX -module. Hence by By Modules, Lemma 15.11.3 we see that K Lemma 23.16.1 again we see that K is a finite R-module. Hence M is an R-module of finite presentation. 

23.17. Flat modules On any ringed space (X, OX ) we know what it means for an OX -module to be flat (at a point), see Modules, Definition 15.16.1 (Definition 15.16.3). On an affine scheme this matches the notion defined in the algebra chapter. f for some Lemma 23.17.1. Let X = Spec(R) be an affine scheme. Let F = M R-module M . The quasi-coherent sheaf F is a flat OX -module of if and only if M is a flat R-module. Proof. Flatness of F may be checked on the stalks, see Modules, Lemma 15.16.2. The same is true in the case of modules over a ring, see Algebra, Lemma 7.36.19. And since Fx = Mp if x corresponds to p the lemma is true. 

23.18. Locally free modules On any ringed space we know what it means for an OX -module to be (finite) locally free. On an affine scheme this matches the notion defined in the algebra chapter. f for some Lemma 23.18.1. Let X = Spec(R) be an affine scheme. Let F = M R-module M . The quasi-coherent sheaf F is a (finite) locally free OX -module of if and only if M is a (finite) locally free R-module. Proof. Follows from the definitions, see Modules, Definition 15.14.1 and Algebra, Definition 7.73.1.  We can characterize finite locally free modules in many different ways. Lemma 23.18.2. Let X be a scheme. Let F be a quasi-coherent OX -module. The following are equivalent: (1) F is a flat OX -module of finite presentation, (2) F is OX -module of finite presentation and for all x ∈ X the stalk Fx is a free OX,x -module, (3) F is a locally free, finite type OX -module, (4) F is a finite locally free OX -module, and (5) F is an OX -module of finite type, for every x ∈ X the the stalk Fx is a free OX,x -module, and the function ρF : X → Z,

x 7−→ dimκ(x) Fx ⊗OX,x κ(x)

is locally constant in the Zariski topology on X. Proof. This lemma immediately reduces to the affine case. In this case the lemma is a reformulation of Algebra, Lemma 7.73.2. The translation uses Lemmas 23.16.1, 23.16.2, 23.17.1, and 23.18.1. 

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23.19. Locally projective modules A consequence of the work done in the algebra chapter is that it makes sense to define a locally projective module as follows. Definition 23.19.1. Let X be a scheme. Let F be a quasi-coherent OX -module. We say F is locally projective if for every affine open U ⊂ X the OX (U )-module F(U ) is projective. Lemma 23.19.2. Let X be a scheme. Let F be a quasi-coherent OX -module. The following are equivalent (1) F is locally projective, and S (2) there exists an affine open covering X = Ui such that the OX (Ui )module F(Ui ) is projective for every i. f then F is locally projective if and only In particular, if X = Spec(A) and F = M if M is a projective A-module. Proof. First, note that if M is a projective A-module and A → B is a ring map, then M ⊗A B is a projective B-module, see Algebra, Lemma 7.89.1. Hence if U is an affine open such that F(U ) is a projective OX (U )-module, then the standard open D(f ) is an affine open such that F(D(f )) is a projective OX (D(f ))-module for all f ∈ OX (U ). Assume (2) holds. Let U ⊂ X be an arbitrary affine open. We can S find an open covering U = j=1,...,m D(fj ) by finitely many standard opens D(fj ) such that for each j the open D(fj ) is a standard open of some Ui , see Schemes, Lemma 21.11.5. Hence, if we set A = OX (U ) and if M is an A-module such that F|U corresponds Qto M , then we see that Mfj is a projective Afj -module. It follows that A → B = Afj is a faithfully flat ring map such that M ×A B is a projective B-module. Hence M is projective by Algebra, Theorem 7.90.5.  Lemma 23.19.3. Let f : X → Y be a morphism of schemes. Let G be a quasicoherent OY -module. If G is locally projective on Y , then f ∗ G is locally projective on X. Proof. See Algebra, Lemma 7.89.1.



23.20. Extending quasi-coherent sheaves It is sometimes useful to be able to show that a given quasi-coherent sheaf on an open subscheme extends to the whole scheme. Lemma 23.20.1. Let j : U → X be a quasi-compact open immersion of schemes. (1) Any quasi-coherent sheaf on U extends to a quasi-coherent sheaf on X. (2) Let F be a quasi-coherent sheaf on X. Let G ⊂ F |U be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf H of F such that H|U = G as subsheaves of F|U . (3) Let F be a quasi-coherent sheaf on X. Let G be a quasi-coherent sheaf on U . Let ϕ : G → F |U be a morphism of OU -modules. There exists a quasi-coherent sheaf H of OX -modules and a map ψ : H → F such that H|U = G and that ψ|U = ϕ. Proof. An immersion is separated (see Schemes, Lemma 21.23.7) and j is quasicompact by assumption. Hence for any quasi-coherent sheaf G on U the sheaf j∗ G is an extension to X. See Schemes, Lemma 21.24.1 and Sheaves, Section 6.31.

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Assume F, G are as in (2). Then j∗ G is a quasi-coherent sheaf on X (see above). It is a subsheaf of j∗ j ∗ F. Hence the kernel H = ker(F ⊕ j∗ G −→ j∗ j ∗ F) is quasi-coherent as well, see Schemes, Section 21.24. It is formal to check that H ⊂ F and that H|U = G (using the material in Sheaves, Section 6.31 again). The same proof as above works. Just take H = ker(F ⊕ j∗ G → j∗ j ∗ F) with its obvious map to F and its obvious identification with G over U .  Lemma 23.20.2. Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open. Let F be a quasi-coherent OX -module. Let G ⊂ F|U be a quasi-coherent OU -submodule which is of finite type. Then there exists a quasicoherent submodule G 0 ⊂ F which is of finite type such that G 0 |U = G. Proof. Let n S be the minimal number of affine opens Ui ⊂ X, i = 1, . . . , n such that X = U ∪ Ui . (Here we use that X is quasi-compact.) Suppose we can prove the lemma for the case n = 1. Then we can successively extend G to a G1 over U ∪ U1 to a G2 over U ∪ U1 ∪ U2 to a G3 over U ∪ U1 ∪ U2 ∪ U3 , and so on. Thus we reduce to the case n = 1. Thus we may assume that X = U ∪ V with V affine. Since X is quasi-separated and U , V are quasi-compact open, we see that U ∩ V is a quasi-compact open. It suffices to prove the lemma for the system (V, U ∩ V, F|V , G|U ∩V ) since we can glue the resulting sheaf G 0 over V to the given sheaf G over U along the common value over U ∩ V . Thus we reduce to the case where X is affine. f for some R-module M . By Lemma 23.20.1 Assume X = Spec(R). Write F = M above we may find a quasi-coherent subsheaf H ⊂ F which restricts to G over U . e for some R-module N . For every u ∈ U there exists an f ∈ R such Write H = N that u ∈ D(f ) ⊂ U and such that Nf is finitely generated, see Lemma 23.16.1. Since U is quasi-compact we can cover it by finitely many D(fi ) such that Nfi is generated by finitely many elements, say xi,1 /fiN , . . . , xi,ri /fiN . Let N 0 ⊂ N be the f0 ⊂ H ⊂ F submodule generated by the elements xi,j . Then the subsheaf G := N works.  Lemma 23.20.3. Let X be a quasi-compact and quasi-separated scheme. Any quasi-coherent sheaf of OX -modules is the directed colimit of its quasi-coherent OX submodules which are of finite type. Proof. The colimit is direct because if G1 , G2 are quasi-coherent subsheaves of finite type, then G1 + G2 ⊂ F is a quasi-coherent subsheaf of finite type. Let U ⊂ X be any affine open, and let s ∈ Γ(U, F) be any section. Let G ⊂ F|U be the subsheaf generated by s. Then clearly G is quasi-coherent and has finite type as an OU -module. By Lemma 23.20.2 we see that G is the restriction of a quasi-coherent subsheaf G 0 ⊂ F which has finite type. Since X has a basis for the topology consisting of affine opens we conclude that every local section of F is locally contained in a quasi-coherent submodule of finite type. Thus we win.  Lemma 23.20.4. (Variant of Lemma 23.20.2 dealing with modules of finite presentation.) Let X be a quasi-compact and quasi-separated scheme. Let F be a quasi-coherent OX -module. Let U ⊂ X be a quasi-compact open. Let G be an

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OU -module which of finite presentation. Let ϕ : G → F|U be a morphism of OU modules. Then there exists an OX -module G 0 of finite presentation, and a morphism of OX -modules ϕ0 : G 0 → F such that G 0 |U = G and such that ϕ0 |U = ϕ. Proof. The beginning of the proof is a repeat of the beginning of the proof of Lemma 23.20.2. We write it out carefully anyway. Let n be Sthe minimal number of affine opens Ui ⊂ X, i = 1, . . . , n such that X = U ∪ Ui . (Here we use that X is quasi-compact.) Suppose we can prove the lemma for the case n = 1. Then we can successively extend the pair (G, ϕ) to a pair (G1 , ϕ1 ) over U ∪ U1 to a pair (G2 , ϕ2 ) over U ∪ U1 ∪ U2 to a pair (G3 , ϕ3 ) over U ∪ U1 ∪ U2 ∪ U3 , and so on. Thus we reduce to the case n = 1. Thus we may assume that X = U ∪ V with V affine. Since X is quasi-separated and U quasi-compact, we see that U ∩ V ⊂ V is quasi-compact. Suppose we prove the lemma for the system (V, U ∩ V, F|V , G|U ∩V , ϕ|U ∩V ) thereby producing (G 0 , ϕ0 ) over V . Then we can glue G 0 over V to the given sheaf G over U along the common value over U ∩ V , and similarly we can glue the map ϕ0 to the map ϕ along the common value over U ∩ V . Thus we reduce to the case where X is affine. Assume X = Spec(R). By Lemma 23.20.1 above we may find a quasi-coherent sheaf H with a map ψ : H → F over X which restricts to G and ϕ over U . By Lemma 23.20.2 we can find a finite type quasi-coherent OX -submodule H0 ⊂ H such that H0 |U = G. Thus after replacing H by H0 and ψ by the restriction of ψ to H0 we may assume that H is of finite type. By Lemma 23.16.2 we conclude that e with N a finitely generated R-module. Hence there exists a surjection as H=N in the following short exact sequence of quasi-coherent OX -modules ⊕n 0 → K → OX →H→0

where K is defined as the kernel. Since G is of finite presentation and H|U = G by Modules, Lemma 15.11.3 the restriction K|U is an OU -module of finite type. Hence by Lemma 23.20.2 again we see that there exists a finite type quasi-coherent OX -submodule K0 ⊂ K such that K0 |U = K|U . The solution to the problem posed in the lemma is to set ⊕n G 0 = OX /K0 which is clearly of finite presentation and restricts to give G on U with ϕ0 equal to the composition ψ

⊕n ⊕n G 0 = OX /K0 → OX /K = H − → F.

This finishes the proof of the lemma.



The following lemma says that every quasi-coherent sheaf on a quasi-compact and quasi-separated scheme is a filtered colimit of O-modules of finite presentation. Actually, we reformulate this in (perhaps more familiar) terms of directed colimits over posets in the next lemma. Lemma 23.20.5. Let X be a scheme. Assume X is quasi-compact and quasiseparated. Let F be a quasi-coherent OX -module. There exist (1) a filtered index category I (see Categories, Definition 4.17.1), (2) a diagram I → Mod(OX ) (see Categories, Section 4.13), i 7→ Fi , (3) morphisms of OX -modules ϕi : Fi → F

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such that each Fi is of finite presentation and such that the morphisms ϕi induce an isomorphism colimi Fi = F. Proof. Choose a set I and for each i ∈ I an OX -module of finite presentation and a homomorphism of OX -modules ϕi : Fi → F with the following property: For any ψ : G → F with G of finite presentation there is an i ∈ I such that there exists an isomorphism α : Fi → G with ϕi = ψ ◦ α. It is clear from Modules, Lemma 15.9.8 that such a set exists (see also its proof). We denote I the category with Ob(I) = I and given i, i0 ∈ I we set MorI (i, i0 ) = {α : Fi → Fi0 | α ◦ ϕi0 = ϕi }. We claim that I is a filtered category and that F = colimi Fi . Let i, i0 ∈ I. Then we can consider the morphism Fi ⊕ Fi0 −→ F which is the direct sum of ϕi and ϕi0 . Since a direct sum of finitely presented OX -modules is finitely presented we see that there exists some i00 ∈ I such that ϕi00 : Fi00 → F is isomorphic to the displayed arrow towards F above. Since there are commutative diagrams Fi

/F

 Fi ⊕ Fi0

/F

Fi0

/F

 F i ⊕ F i0

/F

and

we see that there are morphisms i → i00 and i0 → i00 in I. Next, suppose that we have i, i0 ∈ I and morphisms α, β : i → i0 (corresponding to OX -module maps α, β : Fi → Fi0 ). In this case consider the coequalizer α−β

G = Coker(Fi −−−→ Fi0 ) Note that G is an OX -module of finite presentation. Since by definition of morphisms in the category I we have ϕi0 ◦ α = ϕi0 ◦ β we see that we get an induced map ψ : G → F. Hence again the pair (G, ψ) is isomorphic to the pair (Fi00 , ϕi00 ) for some i00 . Hence we see that there exists a morphism i0 → i00 in I which equalizes α and β. Thus we have shown that the category I is filtered. We still have to show that the colimit of the diagram is F. By definition of the colimit, and by our definition of the category I there is a canonical map ϕ : colimi Fi −→ F. Pick x ∈ X. Let us show that ϕx is an isomorphism. Recall that (colimi Fi )x = colimi Fi,x , see Sheaves, Section 6.29. First we show that the map ϕx is injective. Suppose that s ∈ Fi,x is an element such that s maps to zero in Fx . Then there exists a quasi-compact open U such that s comes from s ∈ Fi (U ) and such that ϕi (s) = 0 in F(U ). By Lemma 23.20.2 we can find a finite type quasi-coherent subsheaf K ⊂ Ker(ϕi ) which restricts to the quasi-coherent OU -submodule of Fi generated by s: K|U = OU · s ⊂ Fi |U . Clearly, Fi /K is of finite presentation and the map ϕi factors through the quotient map Fi → Fi /K. Hence we can find an i0 ∈ I

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and a morphism α : Fi → Fi0 in I which can be identified with the quotient map Fi → Fi /K. Then it follows that the section s maps to zero in Fi0 (U ) and in particular in (colimi Fi )x = colimi Fi,x . The injectivity follows. Finally, we show that the map ϕx is surjective. Pick s ∈ Fx . Choose a quasi-compact open neighbourhood U ⊂ X of x such that s corresponds to a section s ∈ F(U ). Consider the map s : OU → F (multiplication by s). By Lemma 23.20.4 there exists an OX -module G of finite presentation and an OX -module map G → F such that G|U → F |U is identified with s : OU → F. Again by definition of I there exists an i ∈ I such that G → F is isomorphic to ϕi : Fi → F. Clearly there exists a section s0 ∈ Fi (U ) mapping to s ∈ F(U ). This proves surjectivity and the proof of the lemma is complete.  Lemma 23.20.6. Let X be a scheme. Assume X is quasi-compact and quasiseparated. Let F be a quasi-coherent OX -module. There exist (1) a directed partially ordered set I (see Categories, Definition 4.19.2), (2) a system (Fi , ϕii0 ) over I in Mod(OX ) (see Categories, Definition 4.19.1) (3) morphisms of OX -modules ϕi : Fi → F such that each Fi is of finite presentation and such that the morphisms ϕi induce an isomorphism colimi Fi = F. Proof. This is a direct consequence of Lemma 23.20.5 and Categories, Lemma 4.19.3 (combined with the fact that colimits exist in the category of sheaves of OX -modules, see Sheaves, Section 6.29).  Lemma 23.20.7. Let X be a scheme. Assume X is quasi-compact and quasiseparated. Let F be a quasi-coherent OX -module. Then F is the directed colimit of its finite type quasi-coherent submodules. Proof. If G, H ⊂ F are finite type quasi-coherent OX -submodules then the image of G ⊕ H → F is another finite type quasi-coherent OX -submodule which contains both of them. In this way we see that the system is directed. To show that F is the colimit of this system, write F = colimi Fi as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 23.20.6. Then the images Gi = Im(Fi → F) are finite type quasi-coherent subsheaves of F. Since F is the colimit of these the result follows.  Let X be a scheme. In the following lemma we use the notion of a quasi-coherent OX -algebra A of finite presentation. This means that for every affine open Spec(R) ⊂ e where A is a (commutative) R-algebra which is of finite presenX we have A = A tation as an R-algebra. Lemma 23.20.8. Let X be a scheme. Assume X is quasi-compact and quasiseparated. Let A be a quasi-coherent OX -algebra. There exist (1) a directed partially ordered set I (see Categories, Definition 4.19.2), (2) a system (Ai , ϕii0 ) over I in the category of OX -algebras, (3) morphisms of OX -algebras ϕi : Ai → A such that each Ai is a quasi-coherent OX -algebra of finite presentation and such that the morphisms ϕi induce an isomorphism colimi Ai = A.

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Proof. First we write A = colimi Fi as a directed colimit of finitely presented quasi-coherent sheaves as in Lemma 23.20.6. For each i let Bi = Sym(Fi ) be the symmetric algebra on Fi over OX . Write Ii = ker(Bi → A). Write Ii = colimj Fi,j where Fi,j is a finite type quasi-coherent submodule of Ii , see Lemma 23.20.7. Set Ii,j ⊂ Ii equal to the Bi -ideal generated by Fi,j . Set Ai,j = Bi /Ii,j . Then Ai,j is a quasi-coherent finitely presented OX -algebra. Define (i, j) ≤ (i0 , j 0 ) if i ≤ i0 and the map Bi → Bi0 maps the ideal Ii,j into the ideal Ii0 ,j 0 . Then it is clear that A = colimi,j Ai,j .  Let X be a scheme. In the following lemma we use the notion of a quasi-coherent OX -algebra A of finite type. This means that for every affine open Spec(R) ⊂ X e where A is a (commutative) R-algebra which is of finite type as an we have A = A R-algebra. Lemma 23.20.9. Let X be a scheme. Assume X is quasi-compact and quasiseparated. Let A be a quasi-coherent OX -algebra. Then A is the directed colimit of its finite type quasi-coherent OX -subalgebras. Proof. Omitted. Hint: Compare with the proof of Lemma 23.20.7.



23.21. Gabber’s result In this section we prove a result of Gabber which guarantees that on every scheme there exists a cardinal κ such that every quasi-coherent module F is the union of its quasi-coherent κ-generated subsheaves. It follows that the category of quasicoherent sheaves on a scheme is a Grothendieck abelian category having limits and enough injectives2. Definition 23.21.1. Let (X, OX ) be a ringed space. Let κ be an infinite cardinal. We say S a sheaf of OX -modules F is κ-generated if there exists an open covering X = Ui such that F|Ui is generated by a subset Ri ⊂ F(Ui ) whose cardinality is at most κ. Note that a direct sum of at most κ κ-generated modules is again κ-generated because κ ⊗ κ = κ, see Sets, Section 3.6. In particular this holds for the direct sum of two κ-generated modules. Moreover, a quotient of a κ-generated sheaf is κ-generated. (But the same needn’t be true for submodules.) Lemma 23.21.2. Let (X, OX ) be a ringed space. Let κ be a cardinal. There exists a set T and a family (Ft )t∈T of κ-generated OX -modules such that every κ-generated OX -module is isomorphic to one of the Ft . Proof. S There is a set of coverings of X (provided we disallow repeats). Suppose X = Ui is a covering and suppose Fi is an OUi -module. Then there is a set of isomorphism classes of OX -modules F with the property that F|Ui ∼ = Fi since there is a set of glueing maps. This reduces us to proving there is a set of (isomorphism classes of) quotients ⊕k∈κ OX → F for any ringed space X. This is clear.  Here is the result the title of this section refers to. Lemma 23.21.3. Let X be a scheme. There exists a cardinal κ such that every quasi-coherent module F is the directed colimit of its quasi-coherent κ-generated quasi-coherent subsheaves. 2Nicely explained in a blog post by Akhil Mathew.

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S Proof. Choose an affine open covering X = i∈I Ui . For each pair i, j choose S an affine open covering Ui ∩ Uj = k∈Iij Uijk . Write Ui = Spec(Ai ) and Uijk = Spec(Aijk ). Let κ be any infinite cardinal ≥ than the cardinality of any of the sets I, Iij . Let F be a quasi-coherent sheaf. Set Mi = F(Ui ) and Mijk = F(Uijk ). Note that Mi ⊗Ai Aijk = Mijk = Mj ⊗Aj Aijk . see Schemes, Lemma 21.7.3. Using the axiom of choice we choose a map (i, j, k, m) 7→ S(i, j, k, m) which associates to every i, j ∈ I, k ∈ Iij and m ∈ Mi a finite subset S(i, j, k, m) ⊂ Mj such that we have X m⊗1= m0 ⊗ am0 0 m ∈S(i,j,k,m)

in Mijk for some am0 ∈ Aijk . Moreover, let’s agree that S(i, i, k, m) = {m} for all i, j = i, k, m as above. Fix such a map. Given a family S = (Si )i∈I of subsets Si ⊂ Mi of cardinality at most κ we set S 0 = (Si0 ) where [ Sj0 = S(i, j, k, m) Si0 .

(i,j,k,m) such that m∈Si that Si0 has cardinality

Note that Si ⊂ Note at most κ because it is a union over a set of cardinality at most κ of finite sets. Set S (0) = S, S (1) = S 0 and by S (∞) induction S (n+1) = (S (n) )0 . Then set S (∞) = n≥0 S (n) . Writing S (∞) = (Si ) (∞)

we see that for any element m ∈ Si the image of m in Mijk can be written as a P 0 (∞) 0 finite sum m ⊗ am0 with m ∈ Sj . In this way we see that setting (∞)

Ni = Ai -submodule of Mi generated by Si we have Ni ⊗Ai Aijk = Nj ⊗Aj Aijk .

as submodules of Mijk . Thus there exists a quasi-coherent subsheaf G ⊂ F with G(Ui ) = Ni . Moreover, by construction the sheaf G is κ-generated. Let {Gt }t∈T be the set of κ-generated quasi-coherent subsheaves. If t, t0 ∈ T then Gt + Gt0 is also a κ-generated quasi-coherent subsheaf as it is the image of the map Gt ⊕ Gt0 → F. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of F over Ui is in one of the Gt (because we can start with S such that the given section is an element of Si ). Hence colimt Gt → F is both injective and surjective as desired.  Proposition 23.21.4. Let X be a scheme. The inclusion functor QCoh(OX ) → Mod(OX ) has a right adjoint Q3 : Mod(OX ) −→ QCoh(OX ) such that for every quasi-coherent sheaf F the adjunction mapping Q(F) → F is an isomorphism. Moreover, the category QCoh(OX ) has limits and enough injectives. 3This functor is sometimes called the coherator.

23.22. SECTIONS WITH SUPPORT IN A CLOSED

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Proof. The two assertions about Q(F) → F and limits in QCoh(OX ) are formal consequences of the existence of Q, the fact that the inclusion is fully faithful, and the fact that Mod(OX ) has limits (see Modules, Section 15.3). The existence of injectives follows from the existence of injectives in Mod(OX ) (see Injectives, Lemma 17.9.1) and Homology, Lemma 10.22.3. Thus it suffices to construct Q. Pick a cardinal κ as in Lemma 23.21.3. Pick a collection (Ft )t∈T of κ-generated quasi-coherent sheaves as in Lemma 23.21.2. Given an object G of QCoh(OX ) we set Q(G) = colim(t,α) Ft The colimit is over the category of pairs (t, α) where t ∈ T and α : Ft → G is a morphism of OX -modules. A morphism (t, α) → (t0 , α0 ) is given by a morphism β : Ft → Ft0 such that α0 ◦ β = α. By Schemes, Section 21.24 the colimit is quasi-coherent. Note that there is a canonical map Q(G) → G by definition of the colimit. The formula Hom(H, Q(G)) = Hom(H, G) holds for κ-generated quasi-coherent modules H by choice of the system (Ft )t∈T . It follows formally from Lemma 23.21.3 that this equality continuous to hold for any quasi-coherent module H on X. This finishes the proof.  23.22. Sections with support in a closed Given any topological space X, a closed subset Z ⊂ X, and an abelian sheaf F you can take the subsheaf of sections whose support is contained in Z. If X is a scheme, Z a closed subscheme, and F a quasi-coherent module there is a variant where you take sections which are scheme theoretically supported on Z. However, in the scheme setting you have to be careful because the resulting OX -module may not be quasi-coherent. Lemma 23.22.1. Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be an open subscheme. The following are equivalent: (1) U is retrocompact in X, (2) U is quasi-compact, (3) U is a finite union of affine opens, and (4) there exists a finite type quasi-coherent sheaf of ideals I ⊂ OX such that X \ U = V (I) (set theoretically). Proof. The equivalence of (1), (2), and (3) follows from Lemma 23.2.2. Assume (1), (2), (3). Let T = X \ U . By Schemes, Lemma 21.12.4 there exists a unique quasi-coherent sheaf of ideals J cutting out the reduced induced closed subscheme structure on T . Note that J |U = OU which is an OU -modules of finite type. By Lemma 23.20.2 there exists a quasi-coherent subsheaf I ⊂ J which is of finite type and has the property that I|U = J |U . Then X \ U = V (I) and we obtain (4). Conversely, if I is as in (4) hold and W = Spec(R) ⊂ X is an affine open, then I|W = Ie for some finitely generated ideal I ⊂ R, see Lemma 23.16.1. It follows that U ∩ W = Spec(R) \ V (I) is quasi-compact, see Algebra, Lemma 7.27.1. Hence U ⊂ X is restrocompact by Lemma 23.2.4.  Lemma 23.22.2. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals. Let F be a quasi-coherent OX -module. Consider the sheaf of OX -modules

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F 0 which associates to every open U ⊂ X F 0 (U ) = {s ∈ F(U ) | Is = 0} Assume I is of finite type. Then (1) F 0 is a quasi-coherent sheaf of OX -modules, (2) on any affine open U ⊂ X we have F 0 (U ) = {s ∈ F(U ) | I(U )s = 0}, and (3) Fx0 = {s ∈ Fx | Ix s = 0}. Proof. It is clear that the rule defining F 0 gives a subsheaf of F (the sheaf condition is easy to verify). Hence we may work locally on X to verify the other statements. f and I = I. e It is clear In other words we may assume that X = Spec(A), F = M 0 0 e that in this case F (U ) = {x ∈ M | Ix = 0} =: M because I is generated by its global sections I which proves (2). To show F 0 is quasi-coherent it suffices to show that for every f ∈ A we have {x ∈ Mf | If x = 0} = (M 0 )f . Write I = (g1 , . . . , gt ), which is possible because I is of finite type, see Lemma 23.16.1. If x = y/f n and If x = 0, then that means that for every i there exists an m ≥ 0 such that f m gi x = 0. We may choose one m which works for all i (and this is where we use that I is finitely generated). Then we see that f m x ∈ M 0 and x/f n = f m x/f n+m in (M 0 )f as desired. The proof of (3) is similar and omitted.  Definition 23.22.3. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals of finite type. Let F be a quasi-coherent OX -module. The subsheaf F 0 ⊂ F defined in Lemma 23.22.2 above is called the subsheaf of sections annihilated by I. Lemma 23.22.4. Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. Let I ⊂ OY be a quasi-coherent sheaf of ideals of finite type. Let F be a quasi-coherent OX -module. Let F 0 ⊂ F be the subsheaf of sections annihilated by f −1 IOX . Then f∗ F 0 ⊂ f∗ F is the subsheaf of sections annihilated by I. Proof. Omitted. (Hint: The assumption that f is quasi-compact and quasiseparated implies that f∗ F is quasi-coherent so that Lemma 23.22.2 applies to I and f∗ F.)  Lemma 23.22.5. Let X be a scheme. Let Z ⊂ X be a closed subset. Let F be a quasi-coherent OX -module. Consider the sheaf of OX -modules F 0 which associates to every open U ⊂ X F 0 (U ) = {s ∈ F(U ) | the support of s is contained in Z ∩ U } If X \ Z is a retrocompact open in X, then (1) for an affine open U ⊂ X there exist a finitely generated ideal I ⊂ OX (U ) such that Z ∩ U = V (I), (2) for U and I as in (1) we have F 0 (U ) = {x ∈ F(U ) | I n x = 0 for some n}, (3) F 0 is a quasi-coherent sheaf of OX -modules. Proof. Part (1) is Algebra, Lemma 7.27.1. Let U = Spec(A) and I be as in (1). Then F|U is the quasi-coherent sheaf associated to some A-module M . We have F 0 (U ) = {x ∈ M | x = 0 in Mp for all p 6∈ Z}. by Modules, Definition 15.5.1. Thus x ∈ F 0 (U ) if and only if V (Ann(x)) ⊂ V (I), see Algebra, Lemma 7.60.7. Since I is finitely generated this is equivalent to I n x = 0 for some n. This proves (2).

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The rule for F 0 indeed defines a submodule of F. Hence we may work locally on X to verify (3). Let U , I and M be as above. Let I ⊂ OX be the quasi-coherent sheaf of ideals corresponding to I. Part (2) implies sections of F 0 over any affine open of U are the sections of F which are annihilated by some power of I. Hence we see that F 0 |U = colim Fn , where Fn ⊂ F|U is the subsheaf of sections annihilated by I n , see Definition 23.22.3. Thus (3) follows from Lemma 23.22.2 and that colimits of quasi-coherent modules are quasi-coherent, see Schemes, Section 21.24.  Lemma 23.22.6. Let f : X → Y be a quasi-compact and quasi-separated morphism of schemes. Let Z ⊂ Y be a closed subset such that Y \ Z is retrocompact in Y . Let F be a quasi-coherent OX -module. Let F 0 ⊂ F be the subsheaf of sections supported in f −1 Z. Then f∗ F 0 ⊂ f∗ F is the subsheaf of sections supported in Z. Proof. Omitted. (Hint: The assumption that f is quasi-compact and quasiseparated implies that X \ f −1 Z is retrocompact in X so that Lemma 23.22.5 applies to f −1 Z and F.)  23.23. Sections of quasi-coherent sheaves Here is a computation of sections of a quasi-coherent sheaf on a quasi-compact open of an affine spectrum. Lemma 23.23.1. Let A be a ring. Let I ⊂ A be a finitely generated ideal. Let M be an A-module. Then there is a canonical map f). colimn HomA (I n , M ) −→ Γ(Spec(A) \ V (I), M This map is always injective. If for all x ∈ M we have Ix = 0 ⇒ x = 0 then this map is an isomorphism. In general, set Mn = {x ∈ M | I n x = 0}, then there is an isomorphism f). colimn HomA (I n , M/Mn ) −→ Γ(Spec(A) \ V (I), M Proof. Since I n ⊂ I n+1 and Mn ⊂ Mn+1 we can use composition via these maps to get canonical maps of A-modules HomA (I n , M ) −→ HomA (I n+1 , M ) and HomA (I n , M/Mn ) −→ HomA (I n+1 , M/Mn+1 ) which we will use as the transition maps in the systems. Given an A-module map f which we can restrict to ϕ : I n → M , then we get a map of sheaves ϕ e : Ie → M e the open Spec(A) \ V (I). Since I restricted to this open gives the structure sheaf f). We omit the verification that this is we get an element of Γ(Spec(A) \ V (I), M compatible with the transition maps in the system HomA (I n , M ). This gives the ^n agree over the f and M/M first arrow. To get the second arrow we note that M g open Spec(A) \ V (I) since the sheaf M n is clearly supported on V (I). Hence we can use the same mechanism as before. Next, we work out how toSdefine this arrow in terms of algebra. Say I = (f1 , . . . , ft ). Then Spec(A) \ V (I) = i=1,...,t D(fi ). Hence M M f) → 0 → Γ(Spec(A) \ V (I), M Mfi → Mfi fj i

i,j

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is exact. Suppose that ϕ : I n → M is an A-module map. Consider the vector of elements ϕ(fin )/fin ∈ Mfi . It is easy to see that this vector maps to zero in the second direct sum of the exact sequence above. Whence an element of f). We omit the verification that this description agrees with Γ(Spec(A) \ V (I), M the one given above. Let us show that the first arrow is injective using this description. Namely, if ϕ maps to zero, then for each i the element ϕ(fin )/fin is zero in Mfi . In other words we see that for each i we have fim ϕ(fin ) = 0 for some m ≥ 0. We may choose a single m which works for all i. Then we see that ϕ(fin+m ) = 0 for all i. It is easy to see that this means that ϕ|I t(n+m−1)+1 = 0 in other words that ϕ maps to zero in the t(n + m − 1) + 1st term of the colimit. Hence injectivity follows. Note that each Mn = 0 in case we have Ix = 0 ⇒ x = 0 for x ∈ M . Thus to finish the proof of the lemma it suffices to show that the second arrow is an isomorphism. Let us attempt to construct an inverse of the second map of the lemma. Let f). This corresponds to a vector xi /f n with xi ∈ M of the s ∈ Γ(Spec(A) \ V (I), M i first direct sum of the exact sequence above. Hence for each i, j there exists m ≥ 0 such that fim fjm (fjn xi − fin xj ) = 0 in M . We may choose a single m which works for all pairs i, j. After replacing xi by fim xi and n by n + m we see that we get fjn xi = fin xj in M for all i, j. Let us introduce Kn = {x ∈ M | f1n x = . . . = ftn x = 0} We claim there is an A-module map ϕ : I t(n−1)+1 −→ M/Kn P which maps the monomial f1e1 . . . ftet with ei = t(n − 1) + 1 to the class modulo Kn of the expression f1e1 . . . fiei −n . . . ftet xi where i is chosen such that ei ≥ n (note that there is at least one such i). To see that this is indeed the case suppose that X aE f1e1 . . . ftet = 0 E=(e1 ,...,et ),|E|=t(n−1)+1

is a relation between the monomials with coefficients aE in A. Then we would map this to X ei(E) −n z= aE f1e1 . . . fi(E) . . . ftet xi E=(e1 ,...,et ),|E|=t(n−1)+1

where for each multiindex E we have chosen a particular i(E) such that ei(E) ≥ n. Note that if we multiply this by fjn for any j, then we get zero, since by the relations fjn xi = fin xj above we get X ei(E) −n e +n fjn z = aE f1e1 . . . fj j . . . fi(E) . . . ftet xi E=(e1 ,...,et ),|E|=t(n−1)+1 X = aE f1e1 . . . ftet xj = 0. E=(e1 ,...,et ),|E|=t(n−1)+1

Hence z ∈ Kn and we see that every relation gets mapped to zero in M/Kn . This proves the claim. Note that Kn ⊂ Mt(n−1)+1 . Hence the map ϕ in particular gives rise to a A-module map I t(n−1)+1 → M/Mt(n−1)+1 . This proves the second arrow of the lemma is surjective. We omit the proof of injectivity. 

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Example 23.23.2. Let k be a field. Consider the ring A = k[f, g, x, y, {an , bn }n≥1 ]/(f y − gx, {an f n + bn g n }n≥1 ). Then x/f ∈ Af and y/g ∈ Ag map to the same element of Af g . Hence these define a section s of the structure sheaf of Spec(A) over D(f ) ∪ D(g) = Spec(A) \ V (I). Here I = (f, g) ⊂ A. However, there is no n ≥ 0 such that s comes from an A-module map ϕ : I n → A as in the source of the first displayed arrow of Lemma 23.23.1. Namely, given such a module map set xn = ϕ(f n ) and yn = ϕ(g n ). Then f m xn = f n+m−1 x and g m yn = g n+m−1 y for some m ≥ 0 (see proof of the lemma). But then we would have 0 = ϕ(0) = ϕ(an+m f n+m + bn+m g n+m ) = an+m f n+m−1 x + bn+m g n+m−1 y which is not the case in the ring A. Lemma 23.23.3. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals. Let Z ⊂ X be the closed subscheme defined by I and set U = X \Z. Let F be a quasi-coherent OX -module. Assume that X is quasi-compact and quasi-separated and that I is of finite type. Let Fn ⊂ F be subsheaf of sections annihilated by I n . The canonical map colimn HomOX (I n , F) −→ Γ(U, F) is injective and the canonical map colimn HomOX (I n , F/Fn ) −→ Γ(U, F) is an isomorphism. f for some Proof. Let Spec(A) = W ⊂ X be an affine open. Write F|W = M e A-module M and I|W = I for some ideal I ⊂ A. We omit the verification that g Fn = M n where Mn ⊂ M is defined as in Lemma 23.23.1. This proves (1). It also follows from Lemma 23.23.1 that we have an injection colimn HomOW (I n |W , F|W ) −→ Γ(U ∩ W, F) and a bijection colimn HomOW (I n |W , (F/Fn )|W ) −→ Γ(U ∩ W, F) for any such affine open W . S To see (2) we choose a finite affine open covering X = j=1,...,m Wj . The injectivity of the first arrow of (2) follows immediately from the above and the finiteness of the covering. Moreover for each pair j, j 0 we choose a finite affine open covering [ Wj ∩ Wj 0 = Wjj 0 k . k=1,...,mjj 0

Let s ∈ Γ(U, F). As seen above for each j there exists an nj and a map ϕj : I nj |Wj → (F/Fnj )|Wj which corresponds to s|Wj . By the same token for each triple (j, j 0 , k) there exists an integer njj 0 k such that the restriction of ϕj and ϕj 0 as maps I njj0 k → F/Fnjj0 k agree over Wjj 0 l . Let n = max{nj , njj 0 k } and we see that the ϕj glue as maps I n → F/Fn over X. This proves surjectivity of the map. We omit the proof of injectivity. 

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23.24. Ample invertible sheaves Recall from Modules, Lemma 15.21.7 that given an invertible sheaf L on a locally ringed space X, and given a global section s of L the set Xs = {x ∈ X | s 6∈ mx Lx } is open. A general remark is that Xs ∩ Xs0 = Xss0 , where ss0 denote the section s ⊗ s0 ∈ Γ(X, L ⊗ L0 ). Definition 23.24.1. Let X be a scheme. Let L be an invertible OX -module. We say L is ample if (1) X is quasi-compact, and (2) for every x ∈ X there exists an n ≥ 1 and s ∈ Γ(X, L⊗n ) such that x ∈ Xs and Xs is affine. Lemma 23.24.2. Let X be a scheme. Let L be an invertible OX -module. Let n ≥ 1. Then L is ample if and only if L⊗n is ample. Proof. This follows from the fact that Xsn = Xs .



Lemma 23.24.3. Let X be a scheme. Let L be an ample invertible OX -module. For any closed subscheme Z ⊂ X the restriction of L to Z is ample. Proof. This is clear since a closed subset of a quasi-compact space is quasi-compact and a closed subscheme of an affine scheme is affine (see Schemes, Lemma 21.8.2).  Lemma 23.24.4. Let X be a scheme. Let L be an invertible OX -module. Let s ∈ Γ(X, L). For any affine U ⊂ X the intersection U ∩ Xs is affine. Proof. This translates into the following algebra problem. Let R be a ring. Let N be an invertible R-module (i.e., locally free of rank 1). Let s ∈ N be an element. Then U = {p | s 6∈ pN } is an affine open subset of Spec(R). This you can see as follows. Think of s as an R-module map R → N . This gives rise to R-module maps N ⊗k → N ⊗k+1 . Consider R0 = colimn N ⊗n with transition maps as above. Define an R-algebra structure on R0 by the rule x · y = x ⊗ y ∈ N ⊗n+m if x ∈ N ⊗n and y ∈ N ⊗m . We claim that Spec(R0 ) → Spec(R) is an open immersion with image U . To prove this is a local question on Spec(R). Let p ∈ Spec(R). Pick f ∈ f 6∈ p such that Nf ∼ = Rf as a module. Replacing R by Rf , N by Nf and R0 ⊗n Rf0 = colim Nf we may assume that N ∼ = R. Say N = R. In this case s is 0 ∼ element of R and it is easy to see that R = Rs . Thus the lemma follows.

R, by an 

Recall that given L a scheme X and an invertible sheaf L on X we get a graded ring Γ∗ (X, L) = n≥0 Γ(X, L⊗n ), see Modules, Definition 15.21.4. Also, given a sheaf of OX -modules we have the graded Γ∗ (X, L)-module Γ∗ (X, F) = Γ∗ (X, L, F). Lemma 23.24.5. Let X be a scheme. Let L be an invertible sheaf on X. Let s ∈ Γ(X, L). If X is quasi-compact and quasi-separated, the canonical map Γ∗ (X, L)(s) −→ Γ(Xs , O) n

−n

which maps a/s to a ⊗ s OX -module then the map

is an isomorphism. Moreover, if F is a quasi-coherent

Γ∗ (X, L, F)(s) −→ Γ(Xs , F)

23.24. AMPLE INVERTIBLE SHEAVES

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is an isomorphism. Proof. Consider the scheme π : L∗ = SpecX

M n∈Z

 L⊗n −→ X

see Constructions, Section 22.4. Since the inverse image π −1 (U ) of every affine open U ⊂ X is affine (see Constructions, Lemma 22.4.6), it follows that L∗ quasi-compact and separated, since X is assumed quasi-compact and separated (use Schemes, Lemma 21.21.7). Note that s gives rise to an element f ∈ Γ(L∗ , O), via π∗ OL∗ = L ⊗n . Note that (L∗ )f = π −1 (Xs ). Hence we have n∈Z L M  Γ(X, L⊗n ) = Γ(L∗ , OL∗ )f n∈Z

s

Γ((L∗ )f , OL∗ ) M = Γ(Xs , L⊗n ) =

n∈Z

where the middle “=” is Lemma 23.15.2. The first statement of the lemma follows from this equality by looking at degree zero terms. The second statement also follows from Lemma 23.15.2 applied to the quasi-coherent sheaf of OL∗ -modules π ∗ F using that M  M π∗ π ∗ F = F ⊗OX L⊗n = F ⊗OX L⊗n n∈Z

n∈Z

which is proved by computing both sides on affine opens of X.



Lemma 23.24.6. Let X be a scheme. Let L be an invertible OX -module. Assume the open sets Xs , where s ∈ Γ(X, L⊗n ) and n ≥ 1, form a basis for the topology on X. Then among those opens, the open sets Xs which are affine form a basis for the topology on X. Proof. Let x ∈ X. Choose an affine open neighbourhood Spec(R) = U ⊂ X of x. By assumption, there exists a n ≥ 1 and a s ∈ Γ(X, L⊗n ) such that Xs ⊂ U . By Lemma 23.24.4 above the intersection Xs = U ∩ Xs is affine. Since U can be chosen arbitrarily small we win.  Lemma 23.24.7. Let X be a scheme. Let L be an invertible OX -module. Assume for every point x of X there exists n ≥ 1 and s ∈ Γ(X, L⊗n ) such that x ∈ Xs and Xs is affine. Then X is quasi-separated. Proof. By assumption we can find a covering of X by affine opens of the form Xs . By Schemes, Lemma 21.21.7 it suffices to show that Xs ∩ Xs0 is quasi-compact whenever Xs is affine. This is true by Lemma 23.24.4.  Lemma 23.24.8. Let X be a scheme. Let L be an invertible OX -module. Set S = Γ∗ (X, L) as a graded ring. If every point of X is contained in one of the open subschemes Xs , for some s ∈ S+ homogeneous, then there is a canonical morphism of schemes f : X −→ Y = Proj(S), to the homogeneous spectrum of S (see Constructions, Section 22.8). This morphism has the following properties (1) f −1 (D+ (s)) = Xs for any s ∈ S+ homogeneous, (2) there are OY -module maps f ∗ OY (n) → L⊗n compatible with multiplication maps, see Constructions, Equation (22.10.1.1),

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(3) the compositions Sn → Γ(Y, OY (n)) → Γ(X, L⊗n ) are equal to the identity maps, and (4) for every x ∈ X there is an integer d ≥ 1 and an open neighbourhood U ⊂ X of x such that f ∗ OY (dn)|U → L⊗dn |U is an isomorphism for all n ∈ Z. Proof. Denote ψ : S → Γ∗ (X, L) the identity map. We are going to use the triple (U (ψ), rL,ψ , θ) of Constructions, Lemma 22.14.1. By assumption the open subscheme U (ψ) of equals X. Hence rL,ψ : U (ψ) → Y is defined on all of X. We set f = rL,ψ . The maps in part (2) are the components of θ. Part (3) follows from condition (2) in the lemma cited above. Part (1) follows from (3) combined with condition (1) in the lemma cited above. Part (4) follows from the last statement in Constructions, Lemma 22.14.1 since the map α mentioned there is an isomorphism.  Lemma 23.24.9. Let X be a scheme. Let L be an invertible OX -module. Set S = Γ∗ (X, L). Assume (a) every point of X is contained in one of the open subschemes Xs , for some s ∈ S+ homogeneous, and (b) X is quasi-compact. Then the canonical morphism of schemes f : X −→ Proj(S) of Lemma 23.24.8 above is quasi-compact. −1 Proof. It suffices to show S that f (D+ (s)) is quasi-compact for any s ∈ S+ homogeneous. Write X = i=1,...,n Xi as a finite union of affine opens. By Lemma S 23.24.4 each intersection Xs ∩ Xi is affine. Hence Xs = i=1,...,n Xs ∩ Xi is quasicompact. 

Lemma 23.24.10. Let X be a scheme. Let L be an invertible OX -module. Set S = Γ∗ (X, L). Assume L is ample. Then the canonical morphism of schemes f : X −→ Proj(S) of Lemma 23.24.8 is an open immersion. Proof. By Lemma 23.24.7 we see that X is quasi-separated. Choose S finitely many s1 , . . . , sn ∈ S+ homogeneous such that Xsi are affine, and X = Xsi . Say si has degree di . The inverse image of D+ (si ) under f is Xsi , see Lemma 23.24.8. By Lemma 23.24.5 the ring map (S (di ) )(si ) = Γ(D+ (si ), OProj(S) ) −→ Γ(Xsi , OX ) is an isomorphism. Hence f induces an isomorphism Xsi → D+ (si ). Thus f is an S isomorphism of X onto the open subscheme i=1,...,n D+ (si ) of Proj(S).  Lemma 23.24.11. Let X be a scheme. Let S be a graded ring. Assume X is quasi-compact, and assume there exists an open immersion j : X −→ Y = Proj(S). ∗

Then j OY (d) is an invertible ample sheaf for some d > 0. Proof. This is Constructions, Lemma 22.10.6.



Proposition 23.24.12. Let X be a quasi-compact scheme. Let L be an invertible sheaf on X. Set S = Γ∗ (X, L). The following are equivalent: (1) L is ample, (2) the open sets Xs , with s ∈ S+ homogeneous, cover X and the associated morphism X → Proj(S) is an open immersion,

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(3) the open sets Xs , with s ∈ S+ homogeneous, form a basis for the topology of X, (4) the open sets Xs , with s ∈ S+ homogeneous, which are affine form a basis for the topology of X, (5) for every quasi-coherent sheaf F on X the sum of the images of the canonical maps Γ(X, F ⊗OX L⊗n ) ⊗Z L⊗−n −→ F with n ≥ 1 equals F, (6) same property as (5) with F ranging over all quasi-coherent sheaves of ideals, (7) X is quasi-separated and for every quasi-coherent sheaf F of finite type on X there exists an integer n0 such that F ⊗OX L⊗n is globally generated for all n ≥ n0 , (8) X is quasi-separated and for every quasi-coherent sheaf F of finite type on X there exist integers n > 0, k ≥ 0 such that F is a quotient of a direct sum of k copies of L⊗−n , and (9) same as in (8) with F ranging over all sheaves of ideals of finite type on X. Proof. Lemma 23.24.10 is (1) ⇒ (2). Lemmas 23.24.2 and 23.24.11 provide the implication (1) ⇐ (2). The implications (2) ⇒ (4) ⇒ (3) are clear from Constructions, Section 22.8. Lemma 23.24.6 is (3) ⇒ (1). Thus we see that the first 4 conditions are all equivalent. Assume the equivalent conditions (1) – (4). Note that in particular X is separated (as an open subscheme of the separated scheme Proj(S)). Let F be a quasi-coherent sheaf on X. Choose s ∈ S+ homogeneous such that Xs is affine. We claim that any section m ∈ Γ(Xs , F) is in the image of one of the maps displayed in (5) above. This will imply (5) since these affines Xs cover X. Namely, by Lemma 23.24.5 we may write m as the image of m0 ⊗ s−n for some n ≥ 1, some m0 ∈ Γ(X, F ⊗ L⊗n ). This proves the claim. Clearly (5) ⇒ (6). Let us assume (6) and prove L is ample. Pick x ∈ X. Let U ⊂ X be an affine open which contains x. Set Z = X \U . We may think of Z as a reduced closed subscheme, see Schemes, Section 21.12. Let I ⊂ OX be the quasi-coherent sheaf of ideals corresponding to the closed subscheme Z. By assumption (6), there exists an n ≥ 1 and a section s ∈ Γ(X, I ⊗ L⊗n ) such that s does not vanish at x (more precisely such that s 6∈ mx Ix ⊗ L⊗n x ). We may think of s as a section of L⊗n . Since it clearly vanishes along Z we see that Xs ⊂ U . Hence Xs is affine, see Lemma 23.24.4. This proves that L is ample. At this point we have proved that (1) – (6) are equivalent. Assume the equivalent conditions (1) – (6). In the following we will use the fact that the tensor product of two sheaves of modules which are globally generated is globally generated without further mention (see Modules, Lemma 15.4.3). By S (1) we can find elements si ∈ Sdi with di ≥ 1 such that X = i=1,...,n Xsi . Set d = d1 . . . dn . It follows that L⊗d is globally generated by d/d1

s1

n , . . . , sd/d . n

This means that if L⊗j is globally generated then so is L⊗j+dn for all n ≥ 0. Fix a j ∈ {0, . . . , d − 1}. For any point x ∈ X there exists an n ≥ 1 and a global section

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s of Lj+dn which does not vanish at x, as follows from (5) applied to F = L⊗j and ample invertible sheaf L⊗d . Since X is quasi-compact there we may find a finite list of integers ni and global sections si of L⊗j+dni which do not vanish at any point of X. Since L⊗d is globally generated this means that L⊗j+dn is globally generated where n = max{ni }. Since we proved this for every congruence class mod d we conclude that there exists an n0 = n0 (L) such that L⊗n is globally generated for all n ≥ n0 . At this point we see that if F is globally generated then so is F ⊗ L⊗n for all n ≥ n0 . We continue to assume the equivalent conditions (1) – (6). Let F be a quasicoherent sheaf of OX -modules of finite type. Denote Fn ⊂ F the image of the canonical map of (5). By construction Fn ⊗ L⊗n is globally generated. By (5) we see F is the sum P of the subsheaves Fn , n ≥ 1. By Modules, Lemma 15.9.7 we see that F = n=1,...,N Fn for some N ≥ 1. It follows that F ⊗ L⊗n is globally generated whenever n ≥ N + n0 (L) with n0 (L) as above. We conclude that (1) – (6) implies (7). Assume (7). Let F be a quasi-coherent sheaf of OX -modules of finite type. By (7) there exists an integer n ≥ 1 such that the canonical map Γ(X, F ⊗OX L⊗n ) ⊗Z L⊗−n −→ F is surjective. Let I be the set of finite subsets of Γ(X, F ⊗OX L⊗n ) partially ordered by inclusion. Then I is a directed partially ordered set. For i = {s1 , . . . , sr(i) } let Fi ⊂ F be the image of the map M L⊗−n −→ F j=1,...,r(i)

which is multiplication by sj on the jth factor. The surjectivity above implies that F = colimi∈I Fi . Hence Modules, Lemma 15.9.7 applies and we conclude that F = Fi for some i. Hence we have proved (8). In other words, (7) ⇒ (8). The implication (8) ⇒ (9) is trivial. Finally, assume (9). Let I ⊂ OX be a quasi-coherent sheaf of ideals. By Lemma 23.20.3 (this is where we use the condition that X be quasi-separated) we see that I = colimα Iα with each Iα quasi-coherent of finite type. Since by assumption each of the Iα is a quotient of negative tensor powers of L we conclude the same for I (but of course without the finiteness or boundedness of the powers). Hence we conclude that (9) implies (6). This ends the proof of the proposition.  23.25. Affine and quasi-affine schemes Lemma 23.25.1. Let X be a scheme. Then X is quasi-affine if and only if OX is ample. Proof. Suppose that X is quasi-affine. Consider the open immersion j : X −→ Spec(Γ(X, OX )) from Lemma 23.15.4. Note that Spec(A) = Proj(A[T ]), see Constructions, Example 22.8.14. Hence we can apply Lemma 23.24.11 to deduce that OX is ample. ∼ Γ(X, OX )[T ] as graded rings. Suppose that OX is ample. Note that Γ∗ (X, OX ) = Hence the result follows from Lemmas 23.24.10 and 23.15.4 taking into account that Spec(A) = Proj(A[T ]) for any ring A as seen above. 

23.26. QUASI-COHERENT SHEAVES AND AMPLE INVERTIBLE SHEAVES

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Lemma 23.25.2. Let X be a scheme. Suppose that there exist finitely many elements f1 , . . . , fn ∈ Γ(X, OX ) such that (1) each Xfi is an affine open of X, and (2) the ideal generated by f1 , . . . , fn in Γ(X, OX ) is equal to the unit ideal. Then X is affine. P Proof. Assume we have f1 , . . . , fn as in the lemma. We S may write 1 = gi fi for some gj ∈ Γ(X, OX ) and hence it is clear that X = Xfi . (The fi ’s cannot all vanish at a point.) Since each Xfi is quasi-compact (being affine) it follows that X is quasi-compact. Hence we see that X is quasi-affine by Lemma 23.25.1 above. Consider the open immersion j : X → Spec(Γ(X, OX )), see Lemma 23.15.4. The inverse image of the standard open D(fi ) on the right hand side is equal to Xfi on the left hand side and the morphism j induces an isomorphism Xfi ∼ 23.15.3. Since the fi generate the unit ideal = D(fi ), see Lemma S we see that Spec(Γ(X, OX )) = i=1,...,n D(fi ). Thus j is an isomorphism.  23.26. Quasi-coherent sheaves and ample invertible sheaves Situation 23.26.1. Let X be a scheme. Let L be an invertible sheaf on X. Assume L is ample. Set S = Γ∗ (X, L) as a graded ring. Set Y = Proj(S). Let f : X → Y be the canonical morphism of Lemma 23.24.8. It comes equipped with a Z-graded L ∗ L ⊗n OX -algebra map f OY (n) → L . The following lemma is really a special case of the next lemma but it seems like a good idea to point out its validity first. Lemma 23.26.2. In Situation 23.26.1. The canonical morphism f : X → Y maps X into the open subscheme W = W1 ⊂ Y where OY (1) is invertible and where all multiplication maps OY (n) ⊗OY OY (m) → OY (n + m) are isomorphisms (see Constructions, Lemma 22.10.4). Moreover, the maps f ∗ OY (n) → L⊗n are all isomorphisms. Proof. By Proposition 23.24.12 there exists an integer n0 such that L⊗n is globally generated for all n ≥ n0 . Let x ∈ X be a point. By the above we can find a ∈ Sn0 and b ∈ Sn0 +1 such that a and b do not vanish at x. Hence f (x) ∈ D+ (a)∩D+ (b) = D+ (ab). By Constructions, Lemma 22.10.4 we see that f (x) ∈ W1 as desired. By Constructions, Lemma 22.14.1 which was used in the construction of the map f the maps f ∗ OY (n0 ) → L⊗n0 and f ∗ OY (n0 + 1) → L⊗n0 +1 are isomorphisms in a neighbourhood of x. By compatibility with the algebra structure and the fact that f maps into W we conclude all the maps f ∗ OY (n) → L⊗n are isomorphisms in a neighbourhood of x. Hence we win.  Recall from Modules, Definition 15.21.4 that given a locally ringed space X, an invertible sheaf L, and a OX -module F we have the graded Γ∗ (X, L)-module M Γ(X, L, F) = Γ(X, F ⊗OX L⊗n ). n∈Z

The following lemma says that, in Situation 23.26.1, we can recover a quasi-coherent OX -module F from this graded module. Take a look also at Constructions, Lemma 22.13.7 where we prove this lemma in the special case X = PnR .

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Lemma 23.26.3. In Situation 23.26.1. Let F be a quasi-coherent sheaf on X. Set M = Γ∗ (X, L, F) as a graded S-module. There are isomorphisms f −→ F f ∗M f) → Γ(X, F) is the identity map. functorial in F such that M0 → Γ(Proj(S), M Proof. Let s ∈ S+ be homogeneous such that Xs is affine open in X. Recall f|D (s) corresponds to the S(s) -module M(s) , see Constructions, Lemma that M + 22.8.4. Recall that f −1 (D+ (s)) = Xs . As X carries an ample invertible sheaf it is quasi-compact and quasi-separated, see Section 23.24. By Lemma 23.24.5 there is a canonical isomorphism M(s) = Γ∗ (X, L, F)(s) → Γ(Xs , F). Since F is quasi-coherent this leads to a canonical isomorphism f|X → F|X f ∗M s s Since L is ample on X we know that X is covered by the affine opens of the form Xs . Hence it suffices to prove that the displayed maps glue on overlaps. Proof of this is omitted.  Remark 23.26.4. With assumptions and notation of Lemma 23.26.3. Denote the displayed map of the lemma by θF . Note that the isomorphism f ∗ OY (n) → L⊗n of Lemma 23.26.2 is just θL⊗n . Consider the multiplication maps ^ f ⊗O OY (n) −→ M M (n) Y see Constructions, Equation (22.10.1.5). Pull this back to X and consider f ⊗O f ∗ OY (n) f ∗M X θF ⊗θL⊗n



F ⊗ L⊗n

id

^ / f ∗M (n) 

θF ⊗L⊗n

/ F ⊗ L⊗n

Here we have used the obvious identification M (n) = Γ∗ (X, L, F ⊗ L⊗n ). This diagram commutes. Proof omitted. 23.27. Finding suitable affine opens In this section we collect some results on the existence of affine opens in more and less general situations. Lemma 23.27.1. Let X be a quasi-separated scheme. Let Z1 , . . . , Zn be pairwise distinct irreducible components of X, see Topology, Section 5.5. Let ηi ∈ Zi be their generic points, see Schemes, Lemma 21.11.1. There exist affine open neighbourhoods ηi ∈ Ui such that Ui ∩ Uj = ∅ for all i 6= j. In particular, U = U1 ∪ . . . ∪ Un is an affine open containing all of the points η1 , . . . , ηn . Proof. Let Vi be any affine open containing ηi and disjoint from the closed set Z1 ∪ S . . . Zˆi . . . ∪ Zn . Since X is quasi-separated for each i the union Wi = j,j6=i Vi ∩ Vj is a quasi-compact open of Vi not containing ηi . We can find open neighbourhoods Ui ⊂ Vi containing ηi and disjoint from Wi by Algebra, Lemma 7.24.4. Finally, U is affine since it is the spectrum of the ring R1 × . . . × Rn where Ri = OX (Ui ), see Schemes, Lemma 21.6.8. 

23.27. FINDING SUITABLE AFFINE OPENS

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Remark 23.27.2. Lemma 23.27.1 above is false if X is not quasi-separated. Here is an example. Take R = Q[x, y1 , y2 , . . .]/((x − i)yi ). Consider the minimal prime ideal p = (y1 , y2 , . . .) of R. Glue two copies of Spec(R) along the (not quasicompact) open Spec(R) \ V (p) to get a scheme X (glueing as in Schemes, Example 21.14.3). Then the two maximal points of X corresponding to p are not contained in a common affine open. The reason is that any open of Spec(R) containing p contains infinitely many of the “lines” x = i, yj = 0, j 6= i with parameter yi . Details omitted. Notwithstanding the example above, for “most” finite sets of irreducible closed subsets one can apply Lemma 23.27.1 above, at least if X is quasi-compact. This is true because X contains a dense open which is separated. Lemma 23.27.3. Let X be a quasi-compact scheme. There exists a dense open V ⊂ X which is separated. S Proof. Say X = i=1,...,n Ui is a union of n affine open subschemes. We will prove S the lemma by induction on n. It is trivial for n = 1. Let V 0 ⊂ i=1,...,n−1 Ui be a separated dense open subscheme, which exists by induction hypothesis. Consider a V = V 0 (Un \ V 0 ). It is clear that V is separated and a dense open subscheme of X.



Here is a slight refinement of Lemma 23.27.1 above. Lemma 23.27.4. Let X be a quasi-separated scheme. Let Z1 , . . . , Zn be pairwise distinct irreducible components of X. Let ηi ∈ Zi be their generic points. Let x ∈ X be arbitrary. There exists an affine open U ⊂ X containing x and all the ηi . Proof. Suppose that x ∈ Z1 ∩ . . . ∩ Zr and x 6∈ Zr+1 , . . . , Zn . Then we may choose an affine open W ⊂ X such that x ∈ W and W ∩ Zi = ∅ for i = r + 1, . . . , n. Note that clearly ηi ∈ W for i = 1, . . . , r. By Lemma 23.27.1 we may choose affine opens Ui ⊂ X which are pairwise disjoint such that ηi ∈ Ui for i = r + 1, . . . , n. Since X is quasi-separated the opens W ∩ Ui are quasi-compact and do not contain ηi for i = r + 1, . . . , n. Hence by Algebra, Lemma 7.24.4 we may S shrink Ui such that W ∩ Ui = ∅ for i = r + 1, . . . , n. Then the union U = W ∪ i=r+1,...,n Ui is disjoint and hence (by Schemes, Lemma 21.6.8) a suitable affine open.  Lemma 23.27.5. Let X be a scheme. Assume either (1) The scheme X is quasi-affine. (2) The scheme X is isomorphic to a locally closed subscheme of an affine scheme. (3) There exists an ample invertible sheaf on X. (4) The scheme X is isomorphic to a locally closed subscheme of Proj(S) for some graded ring S. Then for any finite subset E ⊂ X there exists an affine open U ⊂ X with E ⊂ U . Proof. By Properties, Definition 23.15.1 a quasi-affine scheme is a quasi-compact open subscheme of an affine scheme. Any affine scheme Spec(R) is isomorphic to Proj(R[X]) where R[X] is graded by setting deg(X) = 1. By Properties, Proposition 23.24.12 if X has an ample invertible sheaf then X is isomorphic to an open

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subscheme of Proj(S) for some graded ring S. Hence, it suffices to prove the lemma in case (4). (We urge the reader to prove case (2) directly for themselves.) Thus assume X ⊂ Proj(S) is a locally closed subscheme where S is some graded ring. Let T = X \ X. Recall that the standard opens D+ (f ) form a basis of the topology on Proj(S). Since E is finite we may choose finitely many homogeneous elements fi ∈ S+ such that E ⊂ D+ (f1 ) ∪ . . . ∪ D+ (fn ) ⊂ Proj(S) \ T Suppose that E = {p1 , . . . , pm } as a subset of Proj(S). Consider the ideal I = (f1 , . . . , fn ) ⊂ S. Since I 6⊂ pj for all j = 1, . . . , m we see from Algebra, Lemma 7.54.6 that there exists a homogeneous element f ∈ I, f 6∈ pj for all j = 1, . . . , m. Then E ⊂ D+ (f ) ⊂ D+ (f1 ) ∪ . . . ∪ D+ (fn ). Since D+ (f ) does not meet T we see that X ∩ D+ (f ) is a closed subscheme of the affine scheme D+ (f ), hence is an affine open of X as desired.  23.28. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules

(33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60) (61) (62)

More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks

23.28. OTHER CHAPTERS

(63) Morphisms of Algebraic Stacks (64) Cohomology of Algebraic Stacks (65) Introducing Algebraic Stacks (66) Examples (67) Exercises (68) Guide to Literature

(69) (70) (71) (72)

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Desirables Coding Style Obsolete GNU Free Documentation License (73) Auto Generated Index

CHAPTER 24

Morphisms of Schemes 24.1. Introduction In this chapter we introduce some types of morphisms of schemes. A basic reference is [DG67]. 24.2. Closed immersions In this section we elucidate some of the results obtained previously on closed immersions of schemes. Recall that a morphism of schemes i : Z → X is defined to be a closed immersion if (a) i induces a homeomorphism onto a closed subset of X, (b) i] : OX → i∗ OZ is surjective, and (c) the kernel of i] is locally generated by sections, see Schemes, Definitions 21.10.2 and 21.4.1. It turns out that, given that Z and X are schemes, there are many different ways of characterizing a closed immersion. Lemma 24.2.1. Let i : Z → X be a morphism of schemes. The following are equivalent: (1) The morphism i is a closed immersion. (2) For every affine open Spec(R) = U ⊂ X, there exists an ideal I ⊂ R such that i−1 (U ) = Spec(R/I) as schemes overSU = Spec(R). (3) There exists an affine open covering X = j∈J Uj , Uj = Spec(Rj ) and for every j ∈ J there exists an ideal Ij ⊂ Rj such that i−1 (Uj ) = Spec(Rj /Ij ) as schemes over Uj = Spec(Rj ). (4) The morphism i induces a homeomorphism of Z with a closed subset of X and i] : OX → i∗ OZ is surjective. (5) The morphism i induces a homeomorphism of Z with a closed subset of X, the map i] : OX → i∗ OZ is surjective, and the kernel Ker(i] ) ⊂ OX is a quasi-coherent sheaf of ideals. (6) The morphism i induces a homeomorphism of Z with a closed subset of X, the map i] : OX → i∗ OZ is surjective, and the kernel Ker(i] ) ⊂ OX is a sheaf of ideals which is locally generated by sections. Proof. Condition (6) is our definition of a closed immersion, see Schemes, Definitions 21.4.1 and 21.10.2. So (6) ⇔ (1). We have (1) ⇒ (2) by Schemes, Lemma 21.10.1. Trivially (2) ⇒ (3). Assume (3). Each of the morphisms Spec(Rj /Ij ) → Spec(Rj ) is a closed immersion, see Schemes, Example 21.8.1. Hence i−1 (Uj ) → Uj is a homeomorphism onto its image and i] |Uj is surjective. Hence i is a homeomorphism onto its image and i] is surjective since this may be checked locally. We conclude that (3) ⇒ (4). The implication (4) ⇒ (1) is Schemes, Lemma 21.24.2. The implication (5) ⇒ (6) is trivial. And the implication (6) ⇒ (5) follows from Schemes, Lemma 21.10.1.  1371

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Lemma 24.2.2. Let X be a scheme. Suppose i : Z → X and i0 : Z 0 → X are closed immersions corresponding to the quasi-coherent ideal sheaves I = Ker(i] ) and I 0 = Ker((i0 )] ) of OX . (1) The morphism i : Z → X factors as Z → Z 0 → X for some a : Z → Z 0 if and only if I 0 ⊂ I. If this happens, then a is a closed immersion. (2) We have Z ∼ = Z 0 as schemes over X if and only if I = I 0 . Proof. This follows from our discussion of closed subspaces in Schemes, Section 21.4 especially Schemes, Lemma 21.4.6. It also follows in a straightforward way from characterization (3) in Lemma 24.2.1 above.  Lemma 24.2.3. Let X be a scheme. Let I ⊂ OX be a sheaf of ideals. The following are equivalent: (1) The sheaf of ideals I is locally generated by sections as a sheaf of OX modules. (2) The sheaf of ideals I is quasi-coherent as a sheaf of OX -modules. (3) There exists a closed immersion i : Z → X whose corresponding sheaf of ideals Ker(i] ) is equal to I. Proof. In Schemes, Section 21.4 we constructed the closed subspace associated to a sheaf of ideals locally generated by sections. This closed subspace is a scheme by Schemes, Lemma 21.10.1. Hence we see that (1) ⇒ (3) by our definition of a closed immersion of schemes. By Lemma 24.2.1 above we see that (3) ⇒ (2). And of course (2) ⇒ (1).  Lemma 24.2.4. The base change of a closed immersion is a closed immersion. Proof. See Schemes, Lemma 21.18.2.



Lemma 24.2.5. A composition of closed immersions is a closed immersion. Proof. We have seen this in Schemes, Lemma 21.24.3, but here is another proof. Namely, it follows from the characterization (3) of closed immersions in Lemma 24.2.1. Since if I ⊂ R is an ideal, and J ⊂ R/I is an ideal, then J = J/I for some ideal J ⊂ R which contains I and (R/I)/J = R/J.  Lemma 24.2.6. A closed immersion is quasi-compact. Proof. This lemma is a duplicate of Schemes, Lemma 21.19.5.



Lemma 24.2.7. A closed immersion is separated. Proof. This lemma is a special case of Schemes, Lemma 21.23.7.



24.3. Immersions In this section we collect some facts on immersions. Lemma 24.3.1. Let Z → Y → X be morphisms of schemes. (1) If Z → X is an immersion, then Z → Y is an immersion. (2) If Z → X is a quasi-compact immersion and Z → Y is quasi-separated, then Z → Y is a quasi-compact immersion. (3) If Z → X is a closed immersion and Y → X is separated, then Z → Y is a closed immersion.

24.3. IMMERSIONS

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Proof. In each case the proof is to contemplate the commutative diagram Z

/ Y ×X Z

/Z

#  Y

 /X

where the composition of the top horizontal arrows is the identity. Let us prove (1). The first horizontal arrow is a section of Y ×X Z → Z, whence an immersion by Schemes, Lemma 21.21.12. The arrow Y ×X Z → Y is a base change of Z → X hence an immersion (Schemes, Lemma 21.18.2). Finally, a composition of immersions is an immersion (Schemes, Lemma 21.24.3). This proves (1). The other two results are proved in exactly the same manner.  Lemma 24.3.2. Let h : Z → X be an immersion. If h is quasi-compact, then we can factor h = i ◦ j with j : Z → Z an open immersion and i : Z → X a closed immersion. Proof. Note that h is quasi-compact and quasi-separated (see Schemes, Lemma 21.23.7). Hence h∗ OZ is a quasi-coherent sheaf of OX -modules by Schemes, Lemma 21.24.1. This implies that I = Ker(OX → h∗ OZ ) is a quasi-coherent sheaf of ideals, see Schemes, Section 21.24. Let Z ⊂ X be the closed subscheme corresponding to I, see Lemma 24.2.3. By Schemes, Lemma 21.4.6 the morphism h factors as h = i ◦ j where i : Z → X is the inclusion morphism. To see that j is an open immersion, choose an open subscheme U ⊂ X such that h induces a closed immersion of Z into U . Then it is clear that I|U is the sheaf of ideals corresponding to the closed immersion Z → U . Hence we see that Z = Z ∩ U .  Lemma 24.3.3. Let h : Z → X be an immersion. If Z is reduced, then we can factor h = i ◦ j with j : Z → Z an open immersion and i : Z → X a closed immersion. Proof. Let Z ⊂ X be the closure of h(Z) with the reduced induced closed subscheme structure, see Schemes, Definition 21.12.5. By Schemes, Lemma 21.12.6 the morphism h factors as h = i ◦ j with i : Z → X the inclusion morphism and j : Z → Z. From the definition of an immersion we see there exists an open subscheme U ⊂ X such that h factors through a closed immersion into U . Hence Z ∩ U and h(Z) are reduced closed subschemes of U with the same underlying closed set. Hence by the uniqueness in Schemes, Lemma 21.12.4 we see that h(Z) ∼ = Z ∩ U. So j induces an isomorphism of Z with Z ∩ U . In other words j is an open immersion.  Example 24.3.4. Here is an example of an immersion which is not a composition of an open immersion followed by a closed immersion. Let k be a field. Let X = S∞ Spec(k[x1 , x2 , x3 , . . .]). Let U = n=1 D(xn ). Then U → X is an open immersion. Consider the ideals In = (xn1 , xn2 , . . . , xnn−1 , xn − 1, xn+1 , xn+2 , . . .) ⊂ k[x1 , x2 , x3 , . . .][1/xn ]. Note that In k[x1 , x2 , x3 , . . .][1/xn xm ] = (1) for any m 6= n. Hence the quasicoherent ideals Ien on D(xn ) agree on D(xn xm ), namely Ien |D(xn xm ) = OD(xn xm ) if n 6= m. Hence these ideals glue to a quasi-coherent sheaf of ideals I ⊂ OU . Let Z ⊂ U be the closed subscheme corresponding to I. Thus Z → X is an immersion.

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We claim that we cannot factor Z → X as Z → Z → X, where Z → X is closed and Z → Z is open. Namely, Z would have to be defined by an ideal I ⊂ k[x1 , x2 , x3 , . . .] such that In = Ik[x1 , x2 , x3 , . . .][1/xn ]. But the only element f ∈ k[x1 , x2 , x3 , . . .] which ends up in all In is 0! Hence I does not exist. 24.4. Closed immersions and quasi-coherent sheaves The following lemma finally does for quasi-coherent sheaves on schemes what Modules, Lemma 15.6.1 does for abelian sheaves. See also the discussion in Modules, Section 15.13. Lemma 24.4.1. Let i : Z → X be a closed immersion of schemes. Let I ⊂ OX be the quasi-coherent sheaf of ideals cutting out Z. The functor i∗ : QCoh(OZ ) −→ QCoh(OX ) is exact, fully faithful, with essential image those quasi-coherent OX -modules G such that IG = 0. Proof. A closed immersion is quasi-compact and separated, see Lemmas 24.2.6 and 24.2.7. Hence Schemes, Lemma 21.24.1 applies and the pushforward of a quasicoherent sheaf on Z is indeed a quasi-coherent sheaf on X. By Modules, Lemma 15.6.1 the functor i∗ is faithful. We claim that for any quasicoherent sheaf F on Z the canonical map i∗ i∗ F −→ F is an isomorphism. This claim implies in particular that i∗ is fully faithful. To prove the claim let U = Spec(R) be any affine open of X, and write Z∩U = Spec(R/I), see f where M is an R/I-module (see Lemma 24.2.1 above. We may write F|U ∩Z = M Schemes, Section 21.24). By Schemes, Lemma 21.7.3 we see that i∗ F|U corresponds to MR and then i∗ i∗ F|Z∩U corresponds to MR ⊗R R/I. In other words, we have to see that for any R/I-module M the canonical map MR ⊗R R/I −→ M, m ⊗ f 7−→ f m is an isomorphism. Proof of this easy algebra fact is omitted. Now we turn to the description of the essential image of the functor i∗ . It is clear that I(i∗ F) = 0 for any quasi-coherent OZ -module, for example by our local description above. Next, suppose that G is any quasi-coherent OX -module such that IG = 0. It suffices to show that the canonical map G −→ i∗ i∗ G is an isomorphism. By exactly the same arguments as above we see that it suffices to prove the following algebraic statement: Given a ring R, an ideal I and an R-module N such that IN = 0 the canonical map N −→ N ⊗R R/I, n 7−→ n ⊗ 1 is an isomorphism of R-modules. Proof of this easy algebra fact is omitted.



Let i : Z → X be a closed immersion. Because of the lemma above we often, by abuse of notation, denote F the sheaf i∗ F on X.

24.5. SUPPORTS OF MODULES

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Lemma 24.4.2. Let X be a scheme. Let F be a quasi-coherent OX -module. Let G ⊂ F be a OX -submodule. There exists a unique quasi-coherent OX -submodule G 0 ⊂ G with the following property: For every quasi-coherent OX -module H the map HomOX (H, G 0 ) −→ HomOX (H, G) is bijective. In particular G 0 is the largest quasi-coherent OX -submodule of F contained in G. Proof. Let Ga , a ∈ A be the set of quasi-coherent OX -submodules contained in G. Then the image G 0 of M Ga −→ F a∈A

is quasi-coherent as the image of a map of quasi-coherent sheaves on X is quasicoherent and since a direct sum of quasi-coherent sheaves is quasi-coherent, see Schemes, Section 21.24. The module G 0 is contained in G. Hence this is the largest quasi-coherent OX -module contained in G. To prove the formula, let H be a quasi-coherent OX -module and let α : H → G be an OX -module map. The image of the composition H → G → F is quasi-coherent as the image of a map of quasi-coherent sheaves. Hence it is contained in G 0 . Hence α factors through G 0 as desired.  Lemma 24.4.3. Let i : Z → X be a closed immersion of schemes. There is a functor1 i! : QCoh(OX ) → QCoh(OZ ) which is a right adjoint to i∗ . (Compare Modules, Lemma 15.6.3.) Proof. Given quasi-coherent OX -module G we consider the subsheaf HZ (G) of G of local sections annihilated by I. By Lemma 24.4.2 there is a canonical largest quasi-coherent OX -submodule HZ (G)0 . By construction we have HomOX (i∗ F, HZ (G)0 ) = HomOX (i∗ F, G) for any quasi-coherent OZ -module F. Hence we can set i! G = i∗ (HZ (G)0 ). Details omitted.  24.5. Supports of modules In this section we collect some elementary results on supports of quasi-coherent modules on schemes. Recall that the support of a sheaf of modules has been defined in Modules, Section 15.5. On the other hand, the support of a module was defined in Algebra, Section 7.60. These match. Lemma 24.5.1. Let X be a scheme. Let F be a quasi-coherent sheaf on X. Let Spec(A) = U ⊂ X be an affine open, and set M = Γ(U, F). Let x ∈ U , and let p ⊂ A be the corresponding prime. The following are equivalent (1) p is in the support of M , and (2) x is in the support of F. Proof. This follows from the equality Fx = Mp , see Schemes, Lemma 21.5.4 and the definitions.  Lemma 24.5.2. Let X be a scheme. Let F be a quasi-coherent sheaf on X. The support of F is closed under specialization. 1This is likely nonstandard notation.

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Proof. If x0 x is a specialization and Fx = 0 then Fx0 is zero, as Fx0 is a localization of the module Fx . Hence the complement of Supp(F) is closed under generalization.  For finite type quasi-coherent modules the support is closed, can be checked on fibres, and commutes with base change. Lemma 24.5.3. Let F be a finite type quasi-coherent module on a scheme X. Then (1) The support of F is closed. (2) For x ∈ X we have x ∈ Supp(F) ⇔ Fx 6= 0 ⇔ Fx ⊗OX,x κ(x) 6= 0. (3) For any morphism of schemes f : Y → X the pullback f ∗ F is of finite type as well and we have Supp(f ∗ F) = f −1 (Supp(F)). Proof. Part (1) is a reformulation of Modules, Lemma 15.9.6. You can also combine Lemma 24.5.1, Properties, Lemma 23.16.1, and Algebra, Lemma 7.60.6 to see this. The first equivalence in (2) is the definition of support, and the second equivalence follows from Nakayama’s lemma, see Algebra, Lemma 7.18.1. Let f : Y → X be a morphism of schemes. Note that f ∗ F is of finite type by Modules, Lemma 15.9.2. For the final assertion, let y ∈ Y with image x ∈ X. Recall that (f ∗ F)y = Fx ⊗OX,x OY,y , see Sheaves, Lemma 6.26.4. Hence (f ∗ F)y ⊗ κ(y) is nonzero if and only if Fx ⊗ κ(x) is nonzero. By (2) this implies x ∈ Supp(F) if and only if y ∈ Supp(f ∗ F), which is the content of assertion (3).  Lemma 24.5.4. Let F be a finite type quasi-coherent module on a scheme X. There exists a smallest closed subscheme i : Z → X such that there exists a quasicoherent OZ -module G with i∗ G ∼ = F. Moreover: f then Z ∩Spec(A) = (1) If Spec(A) ⊂ X is any affine open, and F|Spec(A) = M Spec(A/I) where I = AnnA (M ). (2) The quasi-coherent sheaf G is unique up to unique isomorphism. (3) The quasi-coherent sheaf G is of finite type. (4) The support of G and of F is Z. Proof. Suppose that i0 : Z 0 → X is a closed subscheme which satisfies the description on open affines from the lemma. Then by Lemma 24.4.1 we see that F ∼ = i0∗ G 0 0 0 for some unique quasi-coherent sheaf G on Z . Furthermore, it is clear that Z 0 is the smallest closed subscheme with this property (by the same lemma). Finally, using Properties, Lemma 23.16.1 and Algebra, Lemma 7.5.6 it follows that G 0 is of finite type. We have Supp(G 0 ) = Z by Algebra, Lemma 7.60.6. Hence, in order to prove the lemma it suffices to show that the characterization in (1) actually does define a closed subscheme. And, in order to do this it suffices to prove that the given rule produces a quasi-coherent sheaf of ideals, see Lemma 24.2.3. This comes down to the following algebra fact: If A is a ring, f ∈ A, and M is a finite A-module, then AnnA (M )f = AnnAf (Mf ). We omit the proof.  Definition 24.5.5. Let X be a scheme. Let F be a quasi-coherent OX -module of finite type. The scheme theoretic support of F is the closed subscheme Z ⊂ X constructed in Lemma 24.5.4.

24.6. SCHEME THEORETIC IMAGE

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In this situation we often think of F as a quasi-coherent sheaf of finite type on Z (via the equivalence of categories of Lemma 24.4.1). Lemma 24.5.6. Let f : Y → X be a morphism of schemes. Let F be a finite type quasi-coherent OX -module with scheme theoretic support Z ⊂ X. If f is flat, then f −1 (Z) is the scheme theoretic support of f ∗ F. Proof. Using the characterization of scheme theoretic support on affines as given in Lemma 24.5.4 we reduce to Algebra, Lemma 7.60.5.  24.6. Scheme theoretic image Caution: Some of the material in this section is ultra-general and behaves differently from what you might expect. Lemma 24.6.1. Let f : X → Y be a morphism of schemes. There exists a closed subscheme Z ⊂ Y such that f factors through Z and such that for any other closed subscheme Z 0 ⊂ Y such that f factors through Z 0 we have Z ⊂ Z 0 . Proof. Let I = Ker(OY → f∗ OX ). If I is quasi-coherent then we just take Z to be the closed subscheme determined by I, see Lemma 24.2.3. This works by Schemes, Lemma 21.4.6. In general the same lemma requires us to show that there exists a largest quasi-coherent sheaf of ideals I 0 contained in I. This follows from Lemma 24.4.2.  Definition 24.6.2. Let f : X → Y be a morphism of schemes. The scheme theoretic image of f is the smallest closed subscheme Z ⊂ Y through which f factors, see Lemma 24.6.1 above. We often just denote f : X → Z the factorization of f . If the morphism f is not quasi-compact, then (in general) the construction of the scheme theoretic image does not commute with restriction to open subschemes to Y . Namely, if f is the immersion Z → X of Example 24.3.4 above then the scheme theoretic image of Z → X is X. But clearly the scheme theoretic image of Z = Z ∩ U → U is just Z. Lemma 24.6.3. Let f : X → Y be a morphism of schemes. Let Z ⊂ Y be the scheme theoretic image of f . If f is quasi-compact then (1) the sheaf of ideals I = Ker(OY → f∗ OX ) is quasi-coherent, (2) the scheme theoretic image Z is the closed subscheme determined by I, (3) for any open U ⊂ Y the scheme theoretic image of f |f −1 (U ) : f −1 (U ) → U is equal to Z ∩ U , and (4) the image f (X) ⊂ Z is a dense subset of Z, in other words the morphism X → Z is dominant (see Definition 24.8.1). Proof. Part (4) follows from part (3). To show (3) it suffices to prove (1) since the formation of I commutes with restriction to open subschemes of Y . And if (1) holds then in the proof of Lemma 24.6.1 we showed (2). Thus it suffices to prove that I is quasi-coherent. Since the property of being quasi-coherent is local we may assume S Y is affine. As f is quasi-compact, we can find a finite affine open covering X = i=1,...,n Ui . Denote f 0 the composition a X0 = Ui −→ X −→ Y. Then f∗ OX is a subsheaf of f∗0 OX 0 , and hence I = Ker(OY → OX 0 ). By Schemes, Lemma 21.24.1 the sheaf f∗0 OX 0 is quasi-coherent on Y . Hence we win. 

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Example 24.6.4. If A → B is a ring map with kernel I, then the scheme theoretic image of Spec(B) → Spec(A) is the closed subscheme Spec(A/I) of Spec(A). This follows from Lemma 24.6.3. If the morphism is quasi-compact, then the scheme theoretic image only adds points which are specializations of points in the image. Lemma 24.6.5. Let f : X → Y be a quasi-compact morphism. Let Z be the scheme theoretic image of f . Let z ∈ Z. There exists a valuation ring A with fraction field K and a commutative diagram /X

Spec(K)  Spec(A)

 /Y

/Z

such that the closed point of Spec(A) maps to z. In particular any point of Z is the specialization of a point of f (X). Proof. Let z ∈ Spec(R) = V ⊂ Y be an affine open neighbourhood of z. By Lemma 24.6.3 we have Z ∩ V is the scheme theoretic closure of f −1 (V ) → V , and hence we may replace Y by V and assume Y = Spec(R) is affine. In this case X is quasi-compact as f is quasi-compact. Say X = U1 ∪ . . . ∪ Un is a finite affine open covering. Write Ui = Spec(Ai ). Let I = Ker(R → A1 × . . . × An ). By Lemma 24.6.3 again we see that Z corresponds to the closed subscheme Spec(R/I) of Y . If p ⊂ R is the prime corresponding to z, then we see that Ip ⊂ Rp is not an equality. Hence (as localization is exact, see Algebra, Proposition 7.9.12) we see that Rp → (A1 )p × . . . × (A1 )p is not zero. Hence one of the rings (Ai )p is not zero. Hence there exists an i and a prime qi ⊂ Ai lying over a prime pi ⊂ p. By Algebra, Lemma 7.47.2 we can choose a valuation ring A ⊂ K = f.f.(Ai /qi ) dominating the local ring Rp /p1 Rp ⊂ f.f.(Ai /qi ). This gives the desired diagram. Some details omitted.  Lemma 24.6.6. Let f1 : X → Y1 and Y1 → Y2 be morphisms of schemes. Let f2 : X → Y2 be the composition. Let Zi ⊂ Yi , i = 1, 2 be the scheme theoretic image of fi . Then the morphism Y1 → Y2 induces a morphism Z1 → Z2 and a commutative diagram / Z1 / Y1 X  Z2 Proof. See Schemes, Lemma 21.4.6.

 / Y2 

Lemma 24.6.7. Let f : X → Y be a morphism of schemes. If X is reduced, then the scheme theoretic image of f is the reduced induced scheme structure on f (X). Proof. This is true because the reduced induced scheme structure on f (X) is clearly the smallest closed subscheme of Y through which f factors, see Schemes, Lemma 21.12.6. 

24.7. SCHEME THEORETIC CLOSURE AND DENSITY

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24.7. Scheme theoretic closure and density We take the following definition from [DG67, IV, Definition 11.10.2]. Definition 24.7.1. Let X be a scheme. Let U ⊂ X be an open subscheme. (1) The scheme theoretic image of the morphism U → X is called the scheme theoretic closure of U in X. (2) We say U is scheme theoretically dense in X if for every open V ⊂ X the scheme theoretic closure of U ∩ V in V is equal to V . With this definition it is not the case that U is scheme theoretically dense in X if and only if the scheme theoretic closure of U is X, see Example 24.7.2. This is somewhat inelegant; but see Lemmas 24.7.3 and 24.7.8 below. On the other hand, with this definition U is scheme theoretically dense in X if and only if for every V ⊂ X open the ring map OX (V ) → OX (U ∩ V ) is injective, see Lemma 24.7.5 below. In particular we see that scheme theoretically dense implies dense which is pleasing. Example 24.7.2. Here is an example where scheme theoretic closure being X does not imply dense for the underlying topological spaces. Let k be a field. Set A = k[x, z1 , z2 , . . .]/(xn zn ) Set I = (z1 , z2 , . . .) ⊂ A. Consider the affine Q scheme X = Spec(A) and the open subscheme U = X \ V (I). Since A → n Azn is injective we see that the scheme theoretic closure of U is X. Consider the morphism X → Spec(k[x]). This morphism is surjective (set all zn = 0 to see this). But the restriction of this morphism to U is not surjective because it maps to the point x = 0. Hence U cannot be topologically dense in X. Lemma 24.7.3. Let X be a scheme. Let U ⊂ X be an open subscheme. If the inclusion morphism U → X is quasi-compact, then U is scheme theoretically dense in X if and only if the scheme theoretic closure of U in X is X. Proof. Follows from Lemma 24.6.3 part (3).



Example 24.7.4. Let A be a ring and X = Spec(A). Let f1 , . . . , fn ∈ A and let Q U = D(f1 ) ∪ . . . ∪ D(fn ). Let I = Ker(A → Afi ). Then the scheme theoretic closure of U in X is the closed subscheme Spec(A/I) of X. Note that U → X is quasi-compact. Hence by Lemma 24.7.3 we see U is scheme theoretically dense in X if and only if I = 0. Lemma 24.7.5. Let j : U → X be an open immersion of schemes. Then U is scheme theoretically dense in X if and only if OX → j∗ OU is injective. Proof. If OX → j∗ OU is injective, then the same is true when restricted to any open V of X. Hence the scheme theoretic closure of U ∩ V in V is equal to V , see Lemma 24.6.3 for example. Conversely, suppose that the scheme theoretic closure of U ∩ V is equal to V for all opens V . Suppose that OX → j∗ OU is not injective. Then we can find an affine open, say Spec(A) = V ⊂ X and a nonzero element f ∈ A such that f maps to zero in Γ(V ∩ U, OX ). In this case the scheme theoretic closure of V ∩ U in V is clearly contained in Spec(A/(f )) a contradiction.  Lemma 24.7.6. Let X be a scheme. If U , V are scheme theoretically dense open subschemes of X, then so is U ∩ V .

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Proof. Let W ⊂ X be any open. Consider the map OX (W ) → OX (W ∩ V ) → OX (W ∩ V ∩ U ). By Lemma 24.7.5 both maps are injective. Hence the composite is injective. Hence by Lemma 24.7.5 U ∩ V is scheme theoretically dense in X.  Lemma 24.7.7. Let Z → X be an immersion. Assume either Z → X is quasicompact or Z is reduced. Let Z ⊂ X be the scheme theoretic image of h. Then the morphism Z → Z is an open immersion which identifies Z with a scheme theoretically dense open subscheme of Z. Moreover, Z is topologically dense in Z. Proof. By Lemma 24.3.2 or Lemma 24.3.3 we can factor Z → X as Z → Z 1 → X with Z → Z 1 open and Z 1 → X closed. On the other hand, let Z → Z ⊂ X be the scheme theoretic closure of Z → X. We conclude that Z ⊂ Z 1 . Since Z is an open subscheme of Z 1 it follows that Z is an open subscheme of Z as well. In the case that Z is reduced we know that Z ⊂ Z 1 is topologically dense by the construction of Z 1 in the proof of Lemma 24.3.3. Hence Z 1 and Z have the same underlying topological spaces. Thus Z ⊂ Z 1 is a closed immersion into a reduced scheme which induces a bijection on underlying topological spaces, and hence it is an isomorphism. In the case that Z → X is quasi-compact we argue as follows: The assertion that Z is scheme theoretically dense in Z follows from Lemma 24.6.3 part (3). The last assertion follows from Lemma 24.6.3 part (4).  Lemma 24.7.8. Let X be a reduced scheme and let U ⊂ X be an open subscheme. Then the following are equivalent (1) U is topologically dense in X, (2) the scheme theoretic closure of U in X is X, and (3) U is scheme theoretically dense in X. Proof. This follows from Lemma 24.7.7 and the fact that the a closed subscheme Z of X whose underlying topological space equals X must be equal to X as a scheme.  Lemma 24.7.9. Let X be a scheme and let U ⊂ X be a reduced open subscheme. Then the following are equivalent (1) the scheme theoretic closure of U in X is X, and (2) U is scheme theoretically dense in X. If this holds then X is a reduced scheme. Proof. This follows from Lemma 24.7.7 and the fact that the scheme theoretic closure of U in X is reduced by Lemma 24.6.7.  Lemma 24.7.10. Let S be a scheme. Let X, Y be schemes over S. Let f, g : X → Y be morphisms of schemes over S. Let U ⊂ X be an open subscheme such that f |U = g|U . If the scheme theoretic closure of U in X is X and Y → S is separated, then f = g. Proof. Follows from the definitions and Schemes, Lemma 21.21.5.



24.8. Dominant morphisms The definition of a morphism of schemes being dominant is a little different from what you might expect if you are used to the notion of a dominant morphism of varieties.

24.8. DOMINANT MORPHISMS

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Definition 24.8.1. A morphism f : X → S of schemes is called dominant if the image of f is a dense subset of S. So for example, if k is an infinite field and λ1 , λ2 , . . . is a countable collection of elements of k, then the morphism a Spec(k) −→ Spec(k[x]) i=1,2,...

with ith factor mapping to the point x = λi is dominant. Lemma 24.8.2. Let f : X → S be a morphism of schemes. If every generic point of every irreducible component of S is in the image of f , then f is dominant. Proof. This is a topological fact which follows directly from the fact that the topological space underlying a scheme is sober, see Schemes, Lemma 21.11.1, and that every point of S is contained in an irreducible component of S, see Topology, Lemma 5.5.3.  The expectation that morphisms are dominant only if generic points of the target are in the image does hold if the morphism is quasi-compact. Lemma 24.8.3. Let f : X → S be a quasi-compact morphism of schemes. Then f is dominant (if and) only if for every irreducible component Z ⊂ S the generic point of Z is in the image of f . Proof. Let V ⊂ S be an affine open. Because f is quasi-compact we may choose finitely many affine opens Ui ⊂ f −1 (V ), i = 1, . . . , n covering f −1 (V ). Consider the morphism of affines a f0 : Ui −→ V. i=1,...,n

A disjoint union of affines is affine, see Schemes, Lemma 21.6.8. Generic points of irreducible components of V are exactly the generic points of the irreducible components of S that meet V . Also, f is dominant if and only f 0 is dominant no matter what choices of V, n, Ui we make above. Thus we have reduced the lemma to the case of a morphism of affine schemes. The affine case is Algebra, Lemma 7.28.6.  Here is a slightly more useful variant of the lemma above. Lemma 24.8.4. Let f : X → S be a quasi-compact morphism of schemes. Let η ∈ S be a generic point of an irreducible component of S. If η 6∈ f (X) then there exists an open neighbourhood V ⊂ S of η such that f −1 (V ) = ∅. Proof. Let Z ⊂ S be the scheme theoretic image of f . We have to show that η 6∈ Z. This follows from Lemma 24.6.5 but can also be seen as follows. By Lemma 24.6.3 the morphism X → Z is dominant, which by Lemma 24.8.3 means all the generic points of all irreducible components of Z are in the image of X → Z. By assumption we see that η 6∈ Z since η would be the generic point of some irreducible component of Z if it were in Z.  There is another case where dominant is the same as having all generic points of irreducible components in the image.

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Lemma 24.8.5. Let f : X → S be a morphism of schemes. Suppose that X has finitely many irreducible components. Then f is dominant (if and) only if for every irreducible component Z ⊂ S the generic point of Z is in the image of f . If so, then S has finitely many irreducible components as well. Proof. Assume f is dominant. Say X = Z1 ∪ Z2 ∪ . . . ∪ Zn is the decomposition of X into irreducible components. Let ξi ∈ Zi be its generic point, so Zi = {ξi }. Note that f (Zi ) is an irreducible subset of S. Hence [ [ S = f (X) = f (Zi ) = {f (ξi )} is a finite union of irreducible subsets whose generic points are in the image of f . The lemma follows.  24.9. Birational morphisms You may be used to the notion of a birational map of varieties having the property that it is an isomorphism over an open subset of the target. However, in general a birational morphism may not be an isomorphism over any nonempty open, see Example 24.9.3. Here is the formal definition. Definition 24.9.1. Let X, Y be schemes. Assume X and Y have finitely many irreducible components. We say a morphism f : X → Y is birational if (1) f induces a bijection between the set of generic points of irreducible components of X and the set of generic points of the irreducible components of Y , and (2) for every generic point η ∈ X of an irreducible component of X the local ring map OY,f (η) → OX,η is an isomorphism. Lemma 24.9.2. Let f : X → Y be a morphism of schemes having finitely many irreducible components. If f is birational then f is dominant. Proof. Follows immediately from the definitions.



Example 24.9.3. Here is an example of a birational morphism which is not an isomorphism over any open of the target. Let k be an infinite field. Let A = k[x]. Let B = k[x, {yα }α∈k ]/((x − α)yα , yα yβ ). There is an inclusion A ⊂ B and a retraction B → A setting all yα equal to zero. Both the morphism Spec(A) → Spec(B) and the morphism Spec(B) → Spec(A) are birational but not an isomorphism over any open. 24.10. Rational maps Let X be a scheme. Note that if U , V are dense open in X, then so is U ∩ V . Definition 24.10.1. Let X, Y be schemes. (1) Let f : U → Y , g : V → Y be morphisms of schemes defined on dense open subsets U , V of X. We say that f is equivalent to g if f |W = g|W for some W ⊂ U ∩ V dense open in X. (2) A rational map from X to Y is an equivalence class for the equivalence relation defined in (1). (3) If X, Y are schemes over a base scheme S we say that a rational map from X to Y is an S-rational map from X to Y if there exists a representative f : U → Y of the equivalence class which is an S-morphism.

24.10. RATIONAL MAPS

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We say that two morphisms f , g as in (1) of the definition define the same rational map instead of saying that they are equivalent. Definition 24.10.2. Let X be a scheme. A rational function on X is a rational morphism from X to A1Z . See Constructions, Definition 22.5.1 for the definition of the affine line A1 . Let X be a scheme over S. For any open U ⊂ X a morphism U → A1Z is the same as a morphism U → A1S over S. Hence a rational function is also the same as a S-rational map from X into A1S . Recall that we have the canonical identification Mor(T, A1Z ) = Γ(T, OT ) for any scheme T , see Schemes, Example 21.15.2. Hence A1Z is a ring-object in the category of schemes. More precisely, the morphisms + : A1Z × A1Z

−→

(f, g) 7−→ ∗:

A1Z

×

A1Z

−→

(f, g) 7−→

A1Z f +g A1Z fg

satisfy all the axioms of the addition and multiplication in a ring (commutative with 1 as always). Hence also the set of rational maps into A1Z has a natural ring structure. Definition 24.10.3. Let X be a scheme. The ring of rational functions on X is the ring R(X) whose elements are rational functions with addition and multiplication as just described. Lemma 24.10.4. Let X be an irreducible scheme. Let η ∈ X be the generic point of X. There is a canonical identification R(X) ∼ = OX,η . If X is integral then R(X) = κ(η) = OX,η is a field. Proof. Omitted.



Definition 24.10.5. Let X be an integral scheme. The function field, or the field of rational functions of X is the field R(X). We may occasionally indicate this field k(X) instead of R(X). Remark 24.10.6. There is a variant of Definition 24.10.1 where we consider only those morphism U → Y defined on scheme theoretically dense open subschemes U ⊂ X. We use Lemma 24.7.6 to see that we obtain an equivalence relation. An equivalence class of these is called a pseudo-morphism from X to Y . If X is reduced the two notions coincide. Here is a fun application of these notions. Note that by Lemma 24.10.4 on an integral scheme every local ring OX,x may be viewed as a local subring of R(X). Lemma 24.10.7. Let X be an integral separated scheme. Let Z1 , Z2 be distinct irreducible closed subsets of X. Let ηi be the generic point of Zi . If Z1 6⊂ Z2 , then OX,η1 6⊂ OX,η2 as subrings of R(X). In particular, if Z1 = {x} consists of one closed point x, there exists a function regular in a neighborhood of x which is not in OX,η2 .

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Proof. First observe that under the assumption of X being seperated, there is a unique map of schemes Spec(OX,η2 ) → X over X such that the composition Spec(R(X)) −→ Spec(OX,η2 ) −→ X is the canonical map Spec(R(X)) → X. Namely, there is the canonical map can : Spec(OX,η2 ) → X, see Schemes, Equation (21.13.1.1). Given a second morphism a to X, we have that a agrees with can on the generic point of Spec(OX,η2 ) by assumption. Now being X being seperated guarantees that the subset in Spec(OX,η2 ) where these two maps agree is closed, see Schemes, Lemma 21.21.5. Hence a = can on all of Spec(OX,η2 ). Assume Z1 6⊂ Z2 and assume on the contrary that OX,η1 ⊂ OX,η2 as subrings of R(X). Then we would obtain a second morphism Spec(OX,η2 ) −→ Spec(OX,η1 ) −→ X. By the above this composition would have to be equal to can. This implies that η2 specializes to η1 (see Schemes, Lemma 21.13.2). But this contradicts our assumption Z1 6⊂ Z2 .  24.11. Surjective morphisms Definition 24.11.1. A morphism of schemes is said to be surjective if it is surjective on underlying topological spaces. Lemma 24.11.2. The composition of surjective morphisms is surjective. Proof. Omitted.



Lemma 24.11.3. Let X and Y be schemes over a base scheme S. Given points x ∈ X and y ∈ Y , there is a point of X ×S Y mapping to x and y under the projections if and only if x and y lie above the same point of S. Proof. The condition is obviously necessary, and the converse follows from the proof of Schemes, Lemma 21.17.5.  Lemma 24.11.4. The base change of a surjective morphism is surjective. Proof. Let f : X → Y be a morphism of schemes over a base scheme S. If S 0 → S is a morphism of schemes, let p : XS 0 → X and q : YS 0 → Y be the canonical projections. The commutative square XS 0 fS 0



YS 0

p

/X f

q

 / Y.

identifies XS 0 as a fibre product of X → Y and YS 0 → Y . Let Z be a subset of the underlying topological space of X. Then q −1 (f (Z)) = fS 0 (p−1 (Z)), because y 0 ∈ q −1 (f (Z)) if and only if q(y 0 ) = f (x) for some x ∈ Z, if and only if, by Lemma 24.11.3, there exists x0 ∈ XS 0 such that fS 0 (x0 ) = y 0 and p(x0 ) = x. In particular taking Z = X we see that if f is surjective so is the base change fS 0 : XS 0 → YS 0 . 

24.12. RADICIAL AND UNIVERSALLY INJECTIVE MORPHISMS

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Example 24.11.5. Bijectivity is not stable under base change, and so neither is injectivity. For example consider the bijection Spec(C) → Spec(R). The base change Spec(C ⊗R C) → Spec(C) is not injective, since there is an isomorphism C ⊗R C ∼ ) and = C × C (the decomposition comes from the idempotent 1⊗1+i⊗i 2 hence Spec(C ⊗R C) has two points. Lemma 24.11.6. Let X

/Y

f p



q

Z be a commutative diagram of morphisms of schemes. If f is surjective and p is quasi-compact, then q is quasi-compact. Proof. Let W ⊂ Z be a quasi-compact open. By assumption p−1 (W ) is quasicompact. Hence by Topology, Lemma 5.9.5 the inverse image q −1 (W ) = f (p−1 (W )) is quasi-compact too. This proves the lemma.  24.12. Radicial and universally injective morphisms In this section we define what it means for a morphism of schemes to be radicial and what it means for a morphism of schemes to be universally injective. We then show that these notions agree. The reason for introducing both is that in the case of algebraic spaces there are corresponding notions which may not always agree. Definition 24.12.1. Let f : X → S be a morphism. (1) We say that f is universally injective if and only if for any morphism of schemes S 0 → S the base change f 0 : XS 0 → S 0 is injective (on underlying topological spaces). (2) We say f is radicial if f is injective as a map of topological spaces, and for every x ∈ X the field extension κ(x) ⊃ κ(f (x)) is purely inseparable. Lemma 24.12.2. Let f : X → S be a morphism of schemes. The following are equivalent: (1) For every field K the induced map Mor(Spec(K), X) → Mor(Spec(K), S) is injective. (2) The morphism f is universally injective. (3) The morphism f is radicial. (4) The diagonal morphism ∆X/S : X −→ X ×S X is surjective. Proof. Let K be a field, and let s : Spec(K) → S be a morphism. Giving a morphism x : Spec(K) → X such that f ◦ x = s is the same as giving a section of the projection XK = Spec(K) ×S X → Spec(K), which in turn is the same as giving a point x ∈ XK whose residue field is K. Hence we see that (2) implies (1). Conversely, suppose that (1) holds. Assume that x, x0 ∈ XS 0 map to the same point s0 ∈ S 0 . Choose a commutative diagram KO o

κ(x) O

κ(x0 ) o

κ(s0 )

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of fields. By Schemes, Lemma 21.13.3 we get two morphisms a, a0 : Spec(K) → XS 0 . One corresponding to the point x and the embedding κ(x) ⊂ K and the other corresponding to the point x0 and the embedding κ(x0 ) ⊂ K. Also we have f 0 ◦ a = f 0 ◦ a0 . Condition (1) now implies that the compositions of a and a0 with XS 0 → X are equal. Since XS 0 is the fibre product of S 0 and X over S we see that a = a0 . Hence x = x0 . Thus (1) implies (2). If there are two different points x, x0 ∈ X mapping to the same point of s then (2) is violated. If for some s = f (x), x ∈ X the field extension κ(s) ⊂ κ(x) is not purely inseparable, then we may find a field extension κ(s) ⊂ K such that κ(x) has two κ(s)-homomorphisms into K. By Schemes, Lemma 21.13.3 this implies that the map Mor(Spec(K), X) → Mor(Spec(K), S) is not injective, and hence (1) is violated. Thus we see that the equivalent conditions (1) and (2) imply f is radicial, i.e., they imply (3). Assume (3). By Schemes, Lemma 21.13.3 a morphism Spec(K) → X is given by a pair (x, κ(x) → K). Property (3) says exactly that associating to the pair (x, κ(x) → K) the pair (s, κ(s) → κ(x) → K) is injective. In other words (1) holds. At this point we know that (1), (2) and (3) are all equivalent. Finally, we prove the equivalence of (4) with (1), (2) and (3). A point of X ×S X is given by a quadruple (x1 , x2 , s, p), where x1 , x2 ∈ X, f (x1 ) = f (x2 ) = s and p ⊂ κ(x1 ) ⊗κ(s) κ(x2 ) is a prime ideal, see Schemes, Lemma 21.17.5. If f is universally injective, then by taking S 0 = X in the definition of universally injective, ∆X/S must be surjective since it is a section of the injective morphism X ×S X −→ X. Conversely, if ∆X/S is surjective, then always x1 = x2 = x and there is exactly one such prime ideal p, which means that κ(s) ⊂ κ(x) is purely inseparable. Hence f is radicial. Alternatively, if ∆X/S is surjective, then for any S 0 → S the base change ∆XS0 /S 0 is surjective which implies that f is universally injective. This finishes the proof of the lemma.  Lemma 24.12.3. A universally injective morphism is separated. Proof. Combine Lemma 24.12.2 with the remark that X → S is separated if and only if the image of ∆X/S is closed in X ×S X, see Schemes, Definition 21.21.3 and the discussion following it.  Lemma 24.12.4. A base change of a universally injective morphism is universally injective. Proof. This is formal.



Lemma 24.12.5. A composition of radicial morphisms is radicial, and so the same holds for the equivalent condition of being universally injective. Proof. Omitted.

 24.13. Affine morphisms

Definition 24.13.1. A morphism of schemes f : X → S is called affine if the inverse image of every affine open of S is an affine open of X. Lemma 24.13.2. An affine morphism is separated and quasi-compact.

24.13. AFFINE MORPHISMS

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Proof. Let f : X → S be affine. Quasi-compactness is immediate from Schemes, Lemma 21.19.2. We will show f is separated using Schemes, Lemma 21.21.8. Let x1 , x2 ∈ X be points of X which map to the same point s ∈ S. Choose any affine open W ⊂ S containing s. By assumption f −1 (W ) is affine. Apply the lemma cited with U = V = f −1 (W ).  Lemma 24.13.3. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is affine. S (2) There exists an affine open covering S = Wj such that each f −1 (Wj ) is affine. (3) There exists a quasi-coherent sheaf of OS -algebras A and an isomorphism X ∼ = SpecS (A) of schemes over S. See Constructions, Section 22.4 for notation. Moreover, in this case X = SpecS (f∗ OX ). Proof. It is obvious that (1) implies (2). S Assume S = j∈J Wj is an affine open covering such that each f −1 (Wj ) is affine. By Schemes, Lemma 21.19.2 we see that f is quasi-compact. By Schemes, Lemma 21.21.7 we see the morphism f is quasi-separated. Hence by Schemes, Lemma 21.24.1 the sheaf A = f∗ OX is a quasi-coherent sheaf of OX -algebras. Thus we have the scheme g : Y = SpecS (A) → S over S. The identity map id : A = f∗ OX → f∗ OX provides, via the definition of the relative spectrum, a morphism can : X → Y over S, see Constructions, Lemma 22.4.7. By assumption and the lemma just cited the restriction can|f −1 (Wj ) : f −1 (Wj ) → g −1 (Wj ) is an isomorphism. Thus can is an isomorphism. We have shown that (2) implies (3). Assume (3). By Constructions, Lemma 22.4.6 we see that the inverse image of every affine open is affine, and hence the morphism is affine by definition.  Remark 24.13.4. We can also argue S directly that (2) implies (1) in Lemma 24.13.3 above as follows. Assume S = Wj is an affine open covering such that each f −1 (Wj ) is affine. First argue that A = f∗ OX is quasi-coherent as in the proof above. Let Spec(R) = V ⊂ S be affine open. We have to show that f −1 (V ) is affine. Set A = A(V ) = f∗ OX (V ) = OX (f −1 (V )). By Schemes, Lemma 21.6.4 there is a canonical morphism ψ : f −1 (V ) → Spec(A) over Spec(R) = V . By Schemes, S Lemma 21.11.6 there exists an integer n ≥ 0, a standard open covering V = i=1,...,n D(hi ), hi ∈ R, and a map a : {1, . . . , n} → J such that each D(hi ) is also a standard open of the affine scheme Wa(i) . The inverse image of a standard open under a morphism of affine schemes is standard open, see Algebra, Lemma 7.16.4. Hence we see that f −1 (D(hi )) is a standard open of f −1 (Wa(i) ), in particular that f −1 (D(hi )) is affine. Because A is quasi-coherent we have Ahi = A(D(hi )) = OX (f −1 (D(hi ))), so f −1 (D(hi )) is the spectrum of Ahi . It follows that the morphism ψ induces an isomorphism the open f −1 (D(hi )) with S of S the open −1 −1 Spec(Ahi ) of Spec(A). Since f (V ) = f (D(hi )) and Spec(A) = Spec(Ahi ) we win. Lemma 24.13.5. Let S be a scheme. There is an anti-equivalence of categories Schemes affine quasi-coherent sheaves ←→ over S of OS -algebras

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which associates to f : X → S the sheaf f∗ OX . Proof. Omitted.



Lemma 24.13.6. Let f : X → S be an affine morphism of schemes. Let A = f∗ OX . The functor F 7→ f∗ F induces an equivalence of categories     category of quasi-coherent category of quasi-coherent −→ OX -modules A-modules Moreover, an A-module is quasi-coherent as an OS -module if and only if it is quasicoherent as an A-module. Proof. Omitted.



Lemma 24.13.7. The composition of affine morphisms is affine. Proof. Let f : X → Y and g : Y → Z be affine morphisms. Let U ⊂ Z be affine open. Then g −1 (U ) is affine by assumption on g. Whereupon f −1 (g −1 (U )) is affine by assumption on f . Hence (g ◦ f )−1 (U ) is affine.  Lemma 24.13.8. The base change of an affine morphism is affine. Proof. Let f : X → S be an affine morphism. Let S 0 → S be any morphism. Denote f 0 : XS 0 = S 0 ×S X → S 0 the base change of f . For every s0 ∈ S 0 there exists an open affine neighbourhood s0 ∈ V ⊂ S 0 which maps into some open affine U ⊂ S. By assumption f −1 (U ) is affine. By the material in Schemes, Section 21.17 we see that f −1 (U )V = V ×U f −1 (U ) is affine and equal to (f 0 )−1 (V ). This proves that S 0 has an open covering by affines whose inverse image under f 0 is affine. We conclude by Lemma 24.13.3 above.  Lemma 24.13.9. A closed immersion is affine. Proof. The first indication of this is Schemes, Lemma 21.8.2. See Schemes, Lemma 21.10.1 for a complete statement.  Lemma 24.13.10. Let X be a scheme. Let L be an invertible OX -module. Let s ∈ Γ(X, L). The inclusion morphism j : Xs → X is affine. Proof. This follows from Properties, Lemma 23.24.4 and the definition.



Lemma 24.13.11. Suppose g : X → Y is a morphism of schemes over S. If X is affine over S and Y is separated over S, then g is affine. In particular, any morphism from an affine scheme to a separated scheme is affine. Proof. The base change X ×S Y → Y is affine by Lemma 24.13.8. The morphism X → X ×S Y is a closed immersion as Y → S is separated, see Schemes, Lemma 21.21.12. A closed immersion is affine (see Lemma 24.13.9) and the composition of affine morphisms is affine (see Lemma 24.13.7). Thus we win.  Lemma 24.13.12. A morphism between affine schemes is affine. Proof. Immediate from Lemma 24.13.11 with S = Spec(Z). It also follows directly from the equivalence of (1) and (2) in Lemma 24.13.3.  Lemma 24.13.13. Let S be a scheme. Let A be an Artinian ring. Any morphism Spec(A) → S is affine. Proof. Omitted.



24.14. QUASI-AFFINE MORPHISMS

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24.14. Quasi-affine morphisms Recall that a scheme X is called quasi-affine if it is quasi-compact and isomorphic to an open subscheme of an affine scheme, see Properties, Definition 23.15.1. Definition 24.14.1. A morphism of schemes f : X → S is called quasi-affine if the inverse image of every affine open of S is a quasi-affine scheme. Lemma 24.14.2. A quasi-affine morphism is separated and quasi-compact. Proof. Let f : X → S be quasi-affine. Quasi-compactness is immediate from Schemes, Lemma 21.19.2. We will show f is separated using Schemes, Lemma 21.21.8. Let x1 , x2 ∈ X be points of X which map to the same point s ∈ S. Choose any affine open W ⊂ S containing s. By assumption f −1 (W ) is isomorphic to an open subscheme of an affine scheme, say f −1 (W ) → Y is such an open immersion. Choose affine open neighbourhoods x1 ∈ U ⊂ f −1 (W ) and x2 ∈ V ⊂ f −1 (W ). We may think of U and V as open subschemes of Y and hence we see that U ∩ V is affine and that O(U ) ⊗Z O(V ) → O(U ∩ V ) is surjective (by the lemma cited above applied to U, V in Y ). Hence by the lemma cited we conclude that f is separated.  Lemma 24.14.3. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is quasi-affine. S (2) There exists an affine open covering S = Wj such that each f −1 (Wj ) is quasi-affine. (3) There exists a quasi-coherent sheaf of OS -algebras A and a quasi-compact open immersion / Spec (A) S

X 

S

{

over S. (4) Same as in (3) but with A = f∗ OX and the horizontal arrow the canonical morphism of Constructions, Lemma 22.4.7. Proof. It is obvious that (1) implies (2) and that (4) implies (3). S Assume S = j∈J Wj is an affine open covering such that each f −1 (Wj ) is quasiaffine. By Schemes, Lemma 21.19.2 we see that f is quasi-compact. By Schemes, Lemma 21.21.7 we see the morphism f is quasi-separated. Hence by Schemes, Lemma 21.24.1 the sheaf A = f∗ OX is a quasi-coherent sheaf of OX -algebras. Thus we have the scheme g : Y = SpecS (A) → S over S. The identity map id : A = f∗ OX → f∗ OX provides, via the definition of the relative spectrum, a morphism can : X → Y over S, see Constructions, Lemma 22.4.7. By assumption, the lemma just cited, and Properties, Lemma 23.15.4 the restriction can|f −1 (Wj ) : f −1 (Wj ) → g −1 (Wj ) is a quasi-compact open immersion. Thus can is a quasicompact open immersion. We have shown that (2) implies (4). Assume (3). Choose any affine open U ⊂ S. By Constructions, Lemma 22.4.6 we see that the inverse image of U in the relative spectrum is affine. Hence we conclude

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that f −1 (U ) is quasi-affine (note that quasi-compactness is encoded in (3) as well). Thus (3) implies (1).  Lemma 24.14.4. The composition of quasi-affine morphisms is quasi-affine. Proof. Let f : X → Y and g : Y → Z be quasi-affine morphisms. Let U ⊂ Z be affine open. Then g −1 (U ) is quasi-affine by assumption on g. Let j : g −1 (U ) → V be a quasi-compact open immersion into an affine scheme V . By Lemma 24.14.3 above we see that f −1 (g −1 (U )) is a quasi-compact open subscheme of the relative spectrum Specg−1 (U ) (A) for some quasi-coherent sheaf of Og−1 (U ) -algebras A. By Schemes, Lemma 21.24.1 the sheaf A0 = j∗ A is a quasi-coherent sheaf of OV algebras with the property that j ∗ A0 = A. Hence we get a commutative diagram f −1 (g −1 (U ))

/ Spec

g −1 (U )

/ Spec (A0 ) V

(A)

 g −1 (U )

j

 /V

with the square being a fibre square, see Constructions, Lemma 22.4.6. Note that the upper right corner is an affine scheme. Hence (g ◦ f )−1 (U ) is quasi-affine.  Lemma 24.14.5. The base change of a quasi-affine morphism is quasi-affine. Proof. Let f : X → S be a quasi-affine morphism. By Lemma 24.14.3 above we can find a quasi-coherent sheaf of OS -algebras A and a quasi-compact open immersion X → SpecS (A) over S. Let g : S 0 → S be any morphism. Denote f 0 : XS 0 = S 0 ×S X → S 0 the base change of f . Since the base change of a quasi-compact open immersion is a quasi-compact open immersion we see that XS 0 → SpecS 0 (g ∗ A) is a quasi-compact open immersion (we have used Schemes, Lemmas 21.19.3 and 21.18.2 and Constructions, Lemma 22.4.6). By Lemma 24.14.3 again we conclude that XS 0 → S 0 is quasi-affine.  Lemma 24.14.6. A quasi-compact immersion is quasi-affine. Proof. Let X → S be a quasi-compact immersion. We have to show the inverse image of every affine open is quasi-affine. Hence, assuming S is an affine scheme, we have to show X is quasi-affine. By Lemma 24.7.7 the morphism X → S factors as X → Z → S where Z is a closed subscheme of S and X ⊂ Z is a quasi-compact open. Since S is affine Lemma 24.2.1 implies Z is affine. Hence we win.  Lemma 24.14.7. Let S be a scheme. Let X be an affine scheme. A morphism f : X → S is quasi-affine if and only if it is quasi-compact. In particular any morphism from an affine scheme to a quasi-separated scheme is quasi-affine. Proof. Let V ⊂ S be an affine open. Then f −1 (V ) is an open subscheme of the affine scheme X, hence quasi-affine if and only if it is quasi-compact. This proves the first assertion. The quasi-compactness of any f : X → S where X is affine and S quasi-separated follows from Schemes, Lemma 21.21.15 applied to X → S → Spec(Z).  Lemma 24.14.8. Suppose g : X → Y is a morphism of schemes over S. If X is quasi-affine over S and Y is quasi-separated over S, then g is quasi-affine. In particular, any morphism from a quasi-affine scheme to a quasi-separated scheme is quasi-affine.

24.15. TYPES OF MORPHISMS DEFINED BY PROPERTIES OF RING MAPS

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Proof. The base change X ×S Y → Y is quasi-affine by Lemma 24.14.5. The morphism X → X ×S Y is a quasi-compact immersion as Y → S is quasi-separated, see Schemes, Lemma 21.21.12. A quasi-compact immersion is quasi-affine by Lemma 24.14.6 and the composition of quasi-affine morphisms is quasi-affine (see Lemma 24.14.4). Thus we win.  24.15. Types of morphisms defined by properties of ring maps In this section we study what properties of ring maps allow one to define local properties of morphisms of schemes. Definition 24.15.1. Let P be a property of ring maps. (1) We say that P is local if the following hold: (a) For any ring map R → A, and any f ∈ R we have P (R → A) ⇒ P (Rf → Af ). (b) For any rings R, A, any f ∈ R, a ∈ A, and any ring map Rf → A we have P (Rf → A) ⇒ P (R → Aa ). (c) For any ring map R → A, and ai ∈ A such that (a1 , . . . , an ) = A then ∀i, P (R → Aai ) ⇒ P (R → A). (2) We say that P is stable under base change if for any ring maps R → A, R → R0 we have P (R → A) ⇒ P (R0 → R0 ⊗R A). (3) We say that P is stable under composition if for any ring maps A → B, B → C we have P (A → B) ∧ P (B → C) ⇒ P (A → C). Definition 24.15.2. Let P be a property of ring maps. Let f : X → S be a morphisms of schemes. We say f is locally of type P if for any x ∈ X there exists an affine open neighbourhood U of x in X which maps into an affine open V ⊂ S such that the induced ring map OS (V ) → OX (U ) has property P . This is not a “good” definition unless the property P is a local property. Even if P is a local property we will not automatically use this definition to say that a morphism is “locally of type P ” unless we also explicitly state the definition elsewhere. Lemma 24.15.3. Let f : X → S be a morphism of schemes. Let P be a property of ring maps. Let U be an affine open of X, and V an affine open of S such that f (U ) ⊂ V . If f is locally of type P and P is local, then P (OS (V ) → OX (U )) holds. Proof. As f is locally of type P for every u ∈ U there exists an affine open Uu ⊂ X mapping into an affine open Vu ⊂ S such that P (OS (Vu ) → OX (Uu )) holds. Choose an open neighbourhood Uu0 ⊂ U ∩ Uu of u which is standard affine open in both U and Uu , see Schemes, Lemma 21.11.5. By Definition 24.15.1 (1)(b) we see that P (OS (Vu ) → OX (Uu0 )) holds. Hence we may assume that Uu ⊂ U is a standard affine open. Choose an open neighbourhood Vu0 ⊂ V ∩ Vu of f (u) which is standard affine open in both V and Vu , see Schemes, Lemma 21.11.5. Then Uu0 = f −1 (Vu0 ) ∩ Uu is a standard affine open of Uu (hence of U ) and we have P (OS (Vu0 ) → OX (Uu0 )) by Definition 24.15.1 (1)(a). Hence we may assume both Uu ⊂ U and Vu ⊂ V are standard affine open. Applying Definition 24.15.1 (1)(b) one more time we conclude that P (OS (V ) → OX (Uu )) holds. Because U is quasi-compact we may choose a finite number of points u1 , . . . , un ∈ U such that U = Uu1 ∪ . . . ∪ Uun . By Definition 24.15.1 (1)(c) we conclude that P (OS (V ) → OX (U )) holds.



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Lemma 24.15.4. Let P be a local property of ring maps. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is locally of type P . (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V we have P (OS (V ) → OX (U )). S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the morphisms Ui → Vj , j ∈ J, i ∈ Ij is locally of type P . S (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that P (OS (Vj ) → OX (Ui )) holds, for all j ∈ J, i ∈ Ij . Moreover, if f is locally of type P then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is locally of type P . Proof. This follows from Lemma 24.15.3 above.



Lemma 24.15.5. Let P be a property of ring maps. Assume P is local and stable under composition. The composition of morphisms locally of type P is locally of type P . Proof. Let f : X → Y and g : Y → Z be morphisms locally of type P . Let x ∈ X. Choose an affine open neighbourhood W ⊂ Z of g(f (x)). Choose an affine open neighbourhood V ⊂ g −1 (W ) of f (x). Choose an affine open neighbourhood U ⊂ f −1 (V ) of x. By Lemma 24.15.4 the ring maps OZ (W ) → OY (V ) and OY (V ) → OX (U ) satisfy P . Hence OZ (W ) → OX (U ) satisfies P as P is assumed stable under composition.  Lemma 24.15.6. Let P be a property of ring maps. Assume P is local and stable under base change. The base change of a morphism locally of type P is locally of type P . Proof. Let f : X → S be a morphism locally of type P . Let S 0 → S be any morphism. Denote f 0 : XS 0 = S 0 ×S X → S 0 the base change of f . For every s0 ∈ S 0 there exists an open affine neighbourhood s0 ∈ V 0 ⊂ S 0 which maps into some open affine V ⊂ S. By Lemma 24.15.4 the open f −1 (V ) is a union of affines Ui such that the ring maps OS (V ) → OX (Ui ) all satisfy P . By the material in Schemes, Section 21.17 we see that f −1 (U )V 0 = V 0 ×V f −1 (V ) is the union of the affine opens V 0 ×V Ui . Since OXS0 (V 0 ×V Ui ) = OS 0 (V 0 ) ⊗OS (V ) OX (Ui ) we see that the ring maps OS 0 (V 0 ) → OXS0 (V 0 ×V Ui ) satisfy P as P is assumed stable under base change.  Lemma 24.15.7. The following properties of a ring map R → A are local. (1) (Isomorphism on local rings.) For every prime q of A lying over p ⊂ R the ring map R → A induces an isomorphism Rp → Aq . (2) (Open immersion.) For every prime q of A there exists an f ∈ R, ϕ(f ) 6∈ q such that the ring map ϕ : R → A induces an isomorphism Rf → Af . (3) (Reduced fibres.) For every prime p of R the fibre ring A ⊗R κ(p) is reduced. (4) (Fibres of dimension at most n.) For every prime p of R the fibre ring A ⊗R κ(p) has Krull dimension at most n.

24.16. MORPHISMS OF FINITE TYPE

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(5) (Locally Noetherian on the target.) The ring map R → A has the property that A is Noetherian. (6) Add more here as needed2. Proof. Omitted.



Lemma 24.15.8. The following properties of ring maps are stable under base change. (1) (Isomorphism on local rings.) For every prime q of A lying over p ⊂ R the ring map R → A induces an isomorphism Rp → Aq . (2) (Open immersion.) For every prime q of A there exists an f ∈ R, ϕ(f ) 6∈ q such that the ring map ϕ : R → A induces an isomorphism Rf → Af . (3) (Reduced fibres.) For every prime p of R the fibre ring A ⊗R κ(p) is reduced. (4) (Fibres of dimension at most n.) For every prime p of R the fibre ring A ⊗R κ(p) has Krull dimension at most n. (5) Add more here as needed3. Proof. Omitted.



Lemma 24.15.9. The following properties of ring maps are stable under composition. (1) (Isomorphism on local rings.) For every prime q of A lying over p ⊂ R the ring map R → A induces an isomorphism Rp → Aq . (2) (Open immersion.) For every prime q of A there exists an f ∈ R, ϕ(f ) 6∈ q such that the ring map ϕ : R → A induces an isomorphism Rf → Af . (3) (Locally Noetherian on the target.) The ring map R → A has the property that A is Noetherian. (4) Add more here as needed4. Proof. Omitted.

 24.16. Morphisms of finite type

Recall that a ring map R → A is said to be of finite type if A is isomorphic to a quotient of R[x1 , . . . , xn ] as an R-algebra, see Algebra, Definition 7.6.1. Definition 24.16.1. Let f : X → S be a morphism of schemes. (1) We say that f is of finite type at x ∈ X if there exists an affine open neighbourhood Spec(A) = U ⊂ X of x and and an affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is of finite type. (2) We say that f is locally of finite type if it is of finite type at every point of X. (3) We say that f is of finite type if it is locally of finite type and quasicompact. Lemma 24.16.2. Let f : X → S be a morphism of schemes. The following are equivalent 2But only those properties that are not already dealt with separately elsewhere. 3But only those properties that are not already dealt with separately elsewhere. 4But only those properties that are not already dealt with separately elsewhere.

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(1) The morphism f is locally of finite type. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is of finite type. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the morphisms Ui → Vj , j ∈ J, i ∈ Ij is locally of finite type. S (4) There exists an affine open covering S = j∈J Vj and affine open coverS ings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is of finite type, for all j ∈ J, i ∈ Ij . Moreover, if f is locally of finite type then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is locally of finite type. Proof. This follows from Lemma 24.15.3 if we show that the property “R → A is of finite type” is local. We check conditions (a), (b) and (c) of Definition 24.15.1. By Algebra, Lemma 7.13.2 being of finite type is stable under base change and hence we conclude (a) holds. By the same lemma being of finite type is stable under composition and trivially for any ring R the ring map R → Rf is of finite type. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 7.22.3.  Lemma 24.16.3. The composition of two morphisms which locally of finite type is locally of finite type. The same is true for morphisms of finite type. Proof. In the proof of Lemma 24.16.2 we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being of finite type is a property of ring maps that is stable under composition, see Algebra, Lemma 7.6.2. By the above and the fact that compositions of quasi-compact morphisms are quasi-compact, see Schemes, Lemma 21.19.4 we see that the composition of morphisms of finite type is of finite type.  Lemma 24.16.4. The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type. Proof. In the proof of Lemma 24.16.2 we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being of finite type is a property of ring maps that is stable under base change, see Algebra, Lemma 7.13.2. By the above and the fact that a base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 21.19.3 we see that the base change of a morphism of finite type is a morphism of finite type.  Lemma 24.16.5. A closed immersion is of finite type. An immersion is locally of finite type. Proof. This is true because an open immersion is a local isomorphism, and a closed immersion is obviously of finite type.  Lemma 24.16.6. Let f : X → S be a morphism. If S is (locally) Noetherian and f (locally) of finite type then X is (locally) Noetherian.

24.17. POINTS OF FINITE TYPE AND JACOBSON SCHEMES

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Proof. This follows immediately from the fact that a ring of finite type over a Noetherian ring is Noetherian, see Algebra, Lemma 7.29.1. (Also: use the fact that the source of a quasi-compact morphism with quasi-compact target is quasicompact.)  Lemma 24.16.7. Let f : X → S be locally of finite type with S locally Noetherian. Then f is quasi-separated. Proof. In fact, it is true that X is quasi-separated, see Properties, Lemma 23.5.4 and Lemma 24.16.6 above. Then apply Schemes, Lemma 21.21.14 to conclude that f is quasi-separated.  Lemma 24.16.8. Let X → Y be a morphism of schemes over a base scheme S. If X is locally of finite type over S, then X → Y is locally of finite type. Proof. Via Lemma 24.16.2 this translates into the following algebra fact: Given ring maps A → B → C such that A → C is of finite type, then B → C is of finite type. (See Algebra, Lemma 7.6.2).  24.17. Points of finite type and Jacobson schemes Let S be a scheme. A finite type point s of S is a point such that the morphism Spec(κ(s)) → S is of finite type. The reason for studying this is that finite type points can replace closed points in a certain sense and in certain situations. There are always enough of them for example. Moreover, a scheme is Jacobson if and only if all finite type points are closed points. Lemma 24.17.1. Let S be a scheme. Let k be a field. Let f : Spec(k) → S be a morphism. The following are equivalent: (1) The morphism f is of finite type. (2) The morphism f is locally of finite type. (3) There exists an affine open U = Spec(R) of S such that f corresponds to a finite ring map R → k. (4) There exists an affine open U = Spec(R) of S such that the image of f consists of a closed point u in U and the field extension κ(u) ⊂ k is finite. Proof. The equivalence of (1) and (2) is obvious as Spec(k) is a singleton and hence any morphism from it is quasi-compact. Suppose f is locally of finite type. Choose any affine open Spec(R) = U ⊂ S such that the image of f is contained in U , and the ring map R → k is of finite type. Let p ⊂ R be the kernel. Then R/p ⊂ k is of finite type. By Algebra, Lemma 7.31.2 there exist a f ∈ R/p such that (R/p)f is a field and (R/p)f → k is a finite field extension. If f ∈ R is a lift of f , then we see that k is a finite Rf -module. Thus (2) ⇒ (3). Suppose that Spec(R) = U ⊂ S is an affine open such that f corresponds to a finite ring map R → k. Then f is locally of finite type by Lemma 24.16.2. Thus (3) ⇒ (2). Suppose R → k is finite. The image of R → k is a field over which k is finite by Algebra, Lemma 7.33.16. Hence the kernel of R → k is a maximal ideal. Thus (3) ⇒ (4). The implication (4) ⇒ (3) is immediate.



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Lemma 24.17.2. Let S be a scheme. Let A be an Artinian local ring with residue field κ. Let f : Spec(A) → S be a morphism of schemes. Then f is of finite type if and only if the composition Spec(κ) → Spec(A) → S is of finite type. Proof. Since the morphism Spec(κ) → Spec(A) is of finite type it is clear that if f is of finite type so is the composition Spec(κ) → S (see Lemma 24.16.3). For the converse, note that Spec(A) → S maps into some affine open U = Spec(B) of S as Spec(A) has only one point. To finish apply Algebra, Lemma 7.51.3 to B → A.  Recall that given a point s of a scheme S there is a canonical morphism Spec(κ(s)) → S, see Schemes, Section 21.13. Definition 24.17.3. Let S be a scheme. Let us say that a point s of S is a finite type point if the canonical morphism Spec(κ(s)) → S is of finite type. We denote Sft-pts the set of finite type points of S. We can describe the set of finite type points as follows. Lemma 24.17.4. Let S be a scheme. We have [ Sft-pts = U0 U ⊂S open

where U0 is the set of closed points of U . Here we may let U range over all opens or over all affine opens of S. Proof. Immediate from Lemma 24.17.1.



Lemma 24.17.5. Let f : T → S be a morphism of schemes. If f is locally of finite type, then f (Tft-pts ) ⊂ Sft-pts . Proof. If T is the spectrum of a field this is Lemma 24.17.1. In general it follows since the composition of morphisms locally of finite type is locally of finite type (Lemma 24.16.3).  Lemma 24.17.6. Let f : T → S be a morphism of schemes. If f is locally of finite type and surjective, then f (Tft-pts ) = Sft-pts . Proof. We have f (Tft-pts ) ⊂ Sft-pts by Lemma 24.17.5. Let s ∈ S be a finite type point. As f is surjective the scheme Ts = Spec(κ(s)) ×S T is nonempty, therefore has a finite type point t ∈ Ts by Lemma 24.17.4. Now Ts → T is a morphism of finite type as a base change of s → S (Lemma 24.16.4). Hence the image of t in T is a finite type point by Lemma 24.17.5 which maps to s by construction.  Lemma 24.17.7. Let S be a scheme. For any locally closed subset T ⊂ S we have T 6= ∅ ⇒ T ∩ Sft-pts 6= ∅. In particular, for any closed subset T ⊂ S we see that T ∩ Sft-pts is dense in T . Proof. Note that T carries a scheme structure (see Schemes, Lemma 21.12.4) such that T → S is a locally closed immersion. Any locally closed immersion is locally of finite type, see Lemma 24.16.5. Hence by Lemma 24.17.5 we see Tft-pts ⊂ Sft-pts . Finally, any nonempty affine open of T has at least one closed point which is a finite type point of T by Lemma 24.17.4.  It follows that most of the material from Topology, Section 5.13 goes through with the set of closed points replaced by the set of points of finite type. In fact, if S is Jacobson then we recover the closed points as the finite type points.

24.18. UNIVERSALLY CATENARY SCHEMES

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Lemma 24.17.8. Let S be a scheme. The following are equivalent: (1) For every finite type morphism f : Spec(k) → S with k a field the image consists of a closed point of S. In the terminology introduced above: finite type points of S are closed points of S. (2) For every locally finite type morphism T → S closed points map to closed points. (3) For every locally finite type morphism f : T → S any closed point t ∈ T maps to a closed point s ∈ S and κ(s) ⊂ κ(t) is finite. (4) The scheme S is Jacobson. Proof. We have trivially (3) ⇒ (2) ⇒ (1). The discussion above shows that (1) implies (4). Hence it suffices to show that (4) implies (3). Suppose that T → S is locally of finite type. Choose t ∈ T with s = f (t) as in (3). Choose affine open neighbourhoods Spec(R) = U ⊂ S of s and Spec(A) = V ⊂ T of t with f (V ) ⊂ U . The induced ring map R → A is of finite type (see Lemma 24.16.2) and R is Jacobson by Properties, Lemma 23.6.3. Thus the result follows from Algebra, Proposition 7.32.18.  Lemma 24.17.9. Let S be a Jacobson scheme. Any scheme locally of finite type over S is Jacobson. Proof. This is clear from Algebra, Proposition 7.32.18 (and Properties, Lemma 23.6.3 and Lemma 24.16.2).  Lemma (1) (2) (3)

24.17.10. The following types of schemes are Jacobson. Any scheme locally of finite type over a field. Any scheme locally of finite type over Z. Any scheme locally of finite type over a 1-dimensional Noetherian domain with infinitely many primes. (4) A scheme of the form Spec(R) \ {m} where (R, m) is a Noetherian local ring. Also any scheme locally of finite type over it.

Proof. We will use Lemma 24.17.9 without mention. The spectrum of a field is clearly Jacobson. The spectrum of Z is Jacobson, see Algebra, Lemma 7.32.6. For (3) see Algebra, Lemma 7.59.2. For (4) see Properties, Lemma 23.6.4.  24.18. Universally catenary schemes Recall that a topological space X is called catenary if for every pair of irreducible closed subsets T ⊂ T 0 there exist a maximal chain of irreducible closed subsets T = T0 ⊂ T1 ⊂ . . . ⊂ Te = T 0 and every such chain has the same length. See Topology, Definition 5.8.1. Recall that a scheme is catenary if its underlying topological space is catenary. See Properties, Definition 23.11.1. Definition 24.18.1. Let S be a scheme. Assume S is locally Noetherian. We say S is universally catenary if for every morphism X → S locally of finite type the scheme X is catenary. This is a “better” notion than catenary as there exist Noetherian schemes which are catenary but not universally catenary. See Examples, Section 66.9. Many schemes are universally catenary, see Lemma 24.18.4 below.

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Recall that a ring A is called catenary if for any pair of prime ideals p ⊂ q there exists a maximal chain of primes p = p0 ⊂ . . . ⊂ pe = q and all of these have the same length. See Algebra, Definition 7.98.1. We have seen the relationship between catenary schemes and catenary rings in Properties, Section 23.11. Recall that a ring A is called universally catenary if A is Noetherian and for every finite type ring map A → B the ring B is catenary. See Algebra, Definition 7.98.5. Many interesting rings which come up in algebraic geometry satisfy this property. Lemma 24.18.2. Let S be a locally Noetherian scheme. The following are equivalent (1) S is universally catenary, (2) there exists an open covering of S all of whose members are universally catenary schemes, (3) for every affine open Spec(R) = U ⊂ S the ring R is universally catenary, and S (4) there exists an affine open covering S = Ui such that each Ui is the spectrum of a universally catenary ring. Moreover, in this case any scheme locally of finite type over S is universally catenary as well. Proof. By Lemma 24.16.5 an open immersion is locally of finite type. A composition of morphisms locally of finite type is locally of finite type (Lemma 24.16.3). Thus it is clear that if S is universally catenary then any open and any scheme locally of finite type over S is universally catenary as well. This proves the final statement of the lemma and that (1) implies (2). If Spec(R) is a universally catenary scheme, then every scheme Spec(A) with A a finite type R-algebra is catenary. Hence all these rings A are catenary by Algebra, Lemma 7.98.2. Thus R is universally catenary. Combined with the remarks above we conclude that (1) implies (3), and (2) implies (4). Of course (3) implies (4) trivially. To finish the proof we show that (4) implies (1). Assume (4) and let X → S S be a morphism locally of finite type. We can find an affine open covering X = Vj such that each Vj → S maps into one of the Ui . By Lemma 24.16.2 the induced ring map O(Ui ) → O(Vj ) is of finite type. Hence O(Vj ) is catenary. Hence X is catenary by Properties, Lemma 23.11.2.  Lemma 24.18.3. Let S be a locally Noetherian scheme. The following are equivalent: (1) S is universally catenary, and (2) all local rings OS,s of S are universally catenary. Proof. Assume that all local rings of S are universally catenary. Let f : X → S be locally of finite type. We know that X is catenary if and only if OX,x is catenary for all x ∈ X. If f (x) = s, then OX,x is essentially of finite type over OS,s . Hence OX,x is catenary by the assumption that OS,s is universally catenary.

24.20. THE SINGULAR LOCUS, REPRISE

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Conversly, assume that S is universally catenary. Let s ∈ S. We may replace S by an affine open neighbourhood of s by Lemma 24.18.2. Say S = Spec(R) and s corresponds to the prime ideal p. Any finite type Rp -algebra A0 is of the form Ap for some finite type R-algebra A. By assumption (and Lemma 24.18.2 if you like) the ring A is catenary, and hence A0 (a localization of A) is catenary. Thus Rp is universally catenary.  Lemma (1) (2) (3) (4) (5)

24.18.4. The following types of schemes are universally catenary. Any scheme locally of finite type over a field. Any scheme locally of finite type over a Cohen-Macaulay scheme. Any scheme locally of finite type over Z. Any scheme locally of finite type over a 1-dimensional Noetherian domain. And so on.

Proof. All of these follow from the fact that a Cohen-Macaulay ring is universally catenary, see Algebra, Lemma 7.98.6. Also, use the last assertion of Lemma 24.18.2. Some details omitted.  24.19. Nagata schemes, reprise See Properties, Section 23.13 for the definitions and basic properties of Nagata and universally Japanese schemes. Lemma 24.19.1. Let f : X → S be a morphism. If S is Nagata and f locally of finite type then X is Nagata. If S is universally Japanese and f locally of finite type then X is universally Japanese. Proof. For “universally Japanese” this follows from Algebra, Lemma 7.145.18. For “Nagata” this follows from Algebra, Proposition 7.145.30.  Lemma (1) (2) (3) (4)

24.19.2. The following types of schemes are Nagata. Any scheme locally of finite type over a field. Any scheme locally of finite type over a Noetherian complete local ring. Any scheme locally of finite type over Z. Any scheme locally of finite type over a Dedeking ring of characteristic zero. (5) And so on.

Proof. By Lemma 24.19.1 we only need to show that the rings mentioned above are Nagata rings. For this see Algebra, Proposition 7.145.31.  24.20. The singular locus, reprise We look for a criterion that implies openness of the regular locus for any scheme locally of finite type over the base. Here is the definition. Definition 24.20.1. Let X be a locally Noetherian scheme. We say X is J-2 if for every morphism Y → X which is locally of finite type the regular locus Reg(Y ) is open in Y . This is the analogue of the corresponding notion for Noetherian rings, see More on Algebra, Definition 12.38.1. Lemma 24.20.2. Let X be a locally Noetherian scheme. The following are equivalent

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(1) (2) (3) (4)

X is J-2, there exists an open covering of X all of whose members are J-2 schemes, for every affine open Spec(R) = U ⊂ X S the ring R is J-2, and there exists an affine open covering S = Ui such that each O(Ui ) is J-2 for all i. Moreover, in this case any scheme locally of finite type over X is J-2 as well. Proof. By Lemma 24.16.5 an open immersion is locally of finite type. A composition of morphisms locally of finite type is locally of finite type (Lemma 24.16.3). Thus it is clear that if X is J-2 then any open and any scheme locally of finite type over X is J-2 as well. This proves the final statement of the lemma. If Spec(R) is J-2, then for every finite type R-algebra A the regular locus of the scheme Spec(A) is open. Hence R is J-2, by definition (see More on Algebra, Definition 12.38.1). Combined with the remarks above we conclude that (1) implies (3), and (2) implies (4). Of course (1) ⇒ (2) and (3) ⇒ (4) trivially. To finish the proof we show that (4) implies (1). Assume (4) and let Y → S X be a morphism locally of finite type. We can find an affine open covering Y = Vj such that each Vj → X maps into one of the Ui . By Lemma 24.16.2 the induced ring map O(Ui ) → O(Vj ) is of finite type. Hence the regular locus of Vj = Spec(O(Vj )) is open. Since Reg(Y )∩Vj = Reg(Vj ) we conclude that Reg(Y ) is open as desired.  Lemma (1) (2) (3) (4)

24.20.3. The following types of schemes are J-2. Any scheme locally of finite type over a field. Any scheme locally of finite type over a Noetherian complete local ring. Any scheme locally of finite type over Z. Any scheme locally of finite type over a Dedeking ring of characteristic zero. (5) And so on.

Proof. By Lemma 24.20.2 we only need to show that the rings mentioned above are J-2. For this see More on Algebra, Proposition 12.39.6.  24.21. Quasi-finite morphisms A solid treatment of quasi-finite morphisms is the basis of many developments further down the road. It will lead to various versions of Zariski’s Main Theorem, behaviour of dimensions of fibres, descent for ´etale morphisms, etc, etc. Before reading this section it may be a good idea to take a look at the algebra results in Algebra, Section 7.114. Recall that a finite type ring map R → A is quasi-finite at a prime q if q defines an isolated point of its fibre, see Algebra, Definition 7.114.3. Definition 24.21.1. Let f : X → S be a morphism of schemes. (1) We say that f is quasi-finite at a point x ∈ X if there exist an affine neighbourhood Spec(A) = U ⊂ X of x and an affine open Spec(R) = V ⊂ S such that f (U ) ⊂ V , the ring map R → A is of finite type, and R → A is quasi-finite at the prime of A corresponding to x (see above). (2) We say f is locally quasi-finite if f is quasi-finite at every point x of X. (3) We say that f is quasi-finite if f is of finite type and every point x is an isolated point of its fibre.

24.21. QUASI-FINITE MORPHISMS

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Trivially, a locally quasi-finite morphism is locally of finite type. We will see below that a morphism f which is locally of finite type is quasi-finite at x if and only if x is isolated in its fibre. Moreover, the set of points at which a morphism is quasi-finite is open; we will see this in Section 24.49 on Zariski’s Main Theorem. Lemma 24.21.2. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f (x). If κ(s) ⊃ κ(x) is an algebraic field extension, then (1) x is a closed point of its fibre, and (2) if in addition s is a closed point of S, then x is a closed point of X. Proof. The second statement follows from the first by elementary topology. According to Schemes, Lemma 21.18.5 to prove the first statement we may replace X by Xs and S by Spec(κ(s)). Thus we may assume that S = Spec(k) is the spectrum of a field. In this case, let Spec(A) = U ⊂ X be any affine open containing x. The point x corresponds to a prime ideal q ⊂ A such that k ⊂ κ(q) is an algebraic field extension. By Algebra, Lemma 7.32.9 we see that q is a maximal ideal, i.e., x ∈ U is a closed point. Since the affine opens form a basis of the topology of X we conclude that {x} is closed.  The following lemma is a version of the Hilbert Nullstellensatz. Lemma 24.21.3. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f (x). Assume f is locally of finite type. Then x is a closed point of its fibre if and only if κ(s) ⊂ κ(x) is a finite field extension. Proof. If the extension is finite, then x is a closed point of the fibre by Lemma 24.21.2 above. For the converse, assume that x is a closed point of its fibre. Choose affine opens Spec(A) = U ⊂ X and Spec(R) = V ⊂ S such that f (U ) ⊂ V . By Lemma 24.16.2 the ring map R → A is of finite type. Let q ⊂ A, resp. p ⊂ R be the prime ideal corresponding to x, resp. s. Consider the fibre ring A = A ⊗R κ(p). Let q be the prime of A corresponding to q. The assumption that x is a closed point of its fibre implies that q is a maximal ideal of A. Since A is an algebra of finite type over the field κ(p) we see by the Hilbert Nullstellensatz, see Algebra, Theorem 7.31.1, that κ(q) is a finite extension of κ(p). Since κ(s) = κ(p) and κ(x) = κ(q) = κ(q) we win.  Lemma 24.21.4. Let f : X → S be a morphism of schemes which is locally of finite type. Let g : S 0 → S be any morphism. Denote f 0 : X 0 → S 0 the base change. If x0 ∈ X 0 maps to a point x ∈ X which is closed in Xf (s) then x0 is closed in Xf0 0 (x0 ) . Proof. The residue field κ(x0 ) is a quotient of κ(f 0 (x0 )) ⊗κ(f (x)) κ(x), see Schemes, Lemma 21.17.5. Hence it is a finite extension of κ(f 0 (x0 )) as κ(x) is a finite extension of κ(f (x)) by Lemma 24.21.3. Thus we see that x0 is closed in its fibre by applying that lemma one more time.  Lemma 24.21.5. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f (x). If f is quasi-finite at x, then the residue field extension κ(s) ⊂ κ(x) is finite. Proof. This is clear from Algebra, Definition 7.114.3.



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Lemma 24.21.6. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f (x). Let Xs be the fibre of f at s. Assume f is locally of finite type. The following are equivalent: (1) The morphism f is quasi-finite at x. (2) The point x is isolated in Xs . (3) The point x is closed in Xs and there is no point x0 ∈ Xs , x0 6= x which specializes to x. (4) For any pair of affine opens Spec(A) = U ⊂ X, Spec(R) = V ⊂ S with f (U ) ⊂ V and x ∈ U corresponding to q ⊂ A the ring map R → A is quasi-finite at q. Proof. Assume f is quasi-finite at x. By assumption there exist opens U ⊂ X, V ⊂ S such that f (U ) ⊂ V , x ∈ U and x an isolated point of Us . Hence {x} ⊂ Us is an open subset. Since Us = U ∩ Xs ⊂ Xs is also open we conclude that {x} ⊂ Xs is an open subset also. Thus we conclude that x is an isolated point of Xs . Note that Xs is a Jacobson scheme by Lemma 24.17.10 (and Lemma 24.16.4). If x is isolated in Xs , i.e., {x} ⊂ Xs is open, then {x} contains a closed point (by the Jacobson property), hence x is closed in Xs . It is clear that there is no point x0 ∈ Xs , distinct from x, specializing to x. Assume that x is closed in Xs and that there is no point x0 ∈ Xs , distinct from x, specializing to x. Consider a pair of affine opens Spec(A) = U ⊂ X, Spec(R) = V ⊂ S with f (U ) ⊂ V and x ∈ U . Let q ⊂ A correspond to x and p ⊂ R correspond to s. By Lemma 24.16.2 the ring map R → A is of finite type. Consider the fibre ring A = A ⊗R κ(p). Let q be the prime of A corresponding to q. Since Spec(A) is an open subscheme of the fibre Xs we see that q is a maximal ideal of A and that there is no point of Spec(A) specializing to q. This implies that dim(Aq ) = 0. Hence by Algebra, Definition 7.114.3 we see that R → A is quasi-finite at q, i.e., X → S is quasi-finite at x by definition. At this point we have shown conditions (1) – (3) are all equivalent. It is clear that (4) implies (1). And it is also clear that (2) implies (4) since if x is an isolated point of Xs then it is also an isolated point of Us for any open U which contains it.  Lemma 24.21.7. Let f : X → S be a morphism of schemes. Let s ∈ S. Assume that (1) f is locally of finite type, and (2) f −1 ({s}) is a finite set. Then Xs is a finite discrete topological space, and f is quasi-finite at each point of X lying over s. Proof. Suppose T is a scheme which (a) is locally of finite type over a field k, and (b) has finitely many points. Then Lemma 24.17.10 shows T is a Jacobson scheme. A finite sober Jacobson space is discrete, see Topology, Lemma 5.13.6. Apply this remark to the fibre Xs which is locally of finite type over Spec(κ(s)) to see the first statement. Finally, apply Lemma 24.21.6 to see the second.  Lemma 24.21.8. Let f : X → S be a morphism of schemes. Assume f is locally of finite type. Then the following are equivalent (1) f is locally quasi-finite, (2) for every s ∈ S the fibre Xs is a discrete topological space, and

24.21. QUASI-FINITE MORPHISMS

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(3) for every morphism Spec(k) → S where k is a field the base change Xk has an underlying discrete topological space. Proof. It is immediate that (3) implies (2). Lemma 24.21.6 shows that (2) is equivalent to (1). Assume (2) and let Spec(k) → S be as in (3). Denote s ∈ S the image of Spec(k) → S. Then Xk is the base change of Xs via Spec(k) → Spec(κ(s)). Hence every point of Xk is closed by Lemma 24.21.4. As Xk → Spec(k) is locally of finite type (by Lemma 24.16.4), we may apply Lemma 24.21.6 to conclude that every point of Xk is isolated, i.e., Xk has a discrete underlying topological space.  Lemma 24.21.9. Let f : X → S be a morphism of schemes. Then f is quasi-finite if and only if f is locally quasi-finite and quasi-compact. Proof. Assume f is quasi-finite. It is quasi-compact by Definition 24.16.1. Let x ∈ X. We see that f is quasi-finite at x by Lemma 24.21.6. Hence f is quasicompact and locally quasi-finite. Assume f is quasi-compact and locally quasi-finite. Then f is of finite type. Let x ∈ X be a point. By Lemma 24.21.6 we see that x is an isolated point of its fibre. The lemma is proved.  Lemma 24.21.10. Let f : X → S be a morphism of schemes. The following are equivalent: (1) f is quasi-finite, and (2) f is locally of finite type, quasi-compact, and has finite fibres. Proof. Assume f is quasi-finite. In particular f is locally of finite type and quasicompact (since it is Sof finite type). Let s ∈ S. Since every x ∈ Xs is isolated in Xs we see that Xs = x∈Xs {x} is an open covering. As f is quasi-compact, the fibre Xs is quasi-compact. Hence we see that Xs is finite. Conversely, assume f is locally of finite type, quasi-compact and has finite fibres. Then it is locally quasi-finite by Lemma 24.21.7. Hence it is quasi-finite by Lemma 24.21.9.  Recall that a ring map R → A is quasi-finite if it is of finite type and quasi-finite at all primes of A, see Algebra, Definition 7.114.3. Lemma 24.21.11. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is locally quasi-finite. (2) For every pair of affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is quasi-finite. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the morphisms Ui → Vj , j ∈ J, i ∈ Ij is locally quasi-finite. S (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is quasi-finite, for all j ∈ J, i ∈ Ij . Moreover, if f is locally quasi-finite then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is locally quasi-finite.

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Proof. For a ring map R → A let us define P (R → A) to mean “R → A is quasifinite” (see remark above lemma). We claim that P is a local property of ring maps. We check conditions (a), (b) and (c) of Definition 24.15.1. In the proof of Lemma 24.16.2 we have seen that (a), (b) and (c) hold for the property of being “of finite type”. Note that, for a finite type ring map R → A, the property R → A is quasifinite at q depends only on the local ring Aq as an algebra over Rp where p = R ∩ q (usual abuse of notation). Using these remarks (a), (b) and (c) of Definition 24.15.1 follow immediately. For example, suppose R → A is a ring map such that all of the ring maps R → Aai are quasi-finite for a1 , . . . , an ∈ A generating the unit ideal. We conclude that R → A is of finite type. Also, for any prime q ⊂ A the local ring Aq is isomorphic as an R-algebra to the local ring (Aai )qi for some i and some qi ⊂ Aai . Hence we conclude that R → A is quasi-finite at q. We conclude that Lemma 24.15.3 applies with P as in the previous paragraph. Hence it suffices to prove that f is locally quasi-finite is equivalent to f is locally of type P . Since P (R → A) is “R → A is quasi-finite” which means R → A is quasi-finite at every prime of A, this follows from Lemma 24.21.6.  Lemma 24.21.12. The composition of two morphisms which are locally quasi-finite is locally quasi-finite. The same is true for quasi-finite morphisms. Proof. In the proof of Lemma 24.21.11 we saw that P =“quasi-finite” is a local property of ring maps, and that a morphism of schemes is locally quasi-finite if and only if it is locally of type P as in Definition 24.15.2. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being quasifinite is a property of ring maps that is stable under composition, see Algebra, Lemma 7.114.7. By the above, Lemma 24.21.9 and the fact that compositions of quasi-compact morphisms are quasi-compact, see Schemes, Lemma 21.19.4 we see that the composition of quasi-finite morphisms is quasi-finite.  Lemma 24.21.13. Let f : X → S be a morphism of schemes. Let g : S 0 → S be a morphism of schemes. Denote f 0 : XS 0 → S 0 the base change of f by g and denote g 0 : XS 0 → X the projection. Assume X is locally of finite type over S. (1) Let U ⊂ X (resp. U 0 ⊂ X 0 ) be the set of points where f (resp. f 0 ) is quasi-finite. Then U 0 = US 0 = (g 0 )−1 (U ). (2) The base change of a locally quasi-finite morphism is locally quasi-finite. (3) The base change of a quasi-finite morphism is quasi-finite. Proof. The first and second assertion follow from the corresponding algebra result, see Algebra, Lemma 7.114.8 (combined with the fact that f 0 is also locally of finite type by Lemma 24.16.4). By the above, Lemma 24.21.9 and the fact that a base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 21.19.3 we see that the base change of a quasi-finite morphism is quasi-finite.  Lemma 24.21.14. Any immersion is locally quasi-finite. Proof. This is true because an open immersion is a local isomorphism and a closed immersion is clearly quasi-finite.  Lemma 24.21.15. Let X → Y be a morphism of schemes over a base scheme S. Let x ∈ X. If X → S is quasi-finite at x, then X → Y is quasi-finite at x. If X is locally quasi-finite over S, then X → Y is locally quasi-finite.

24.22. MORPHISMS OF FINITE PRESENTATION

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Proof. Via Lemma 24.21.11 this translates into the following algebra fact: Given ring maps A → B → C such that A → C is quasi-finite, then B → C is quasifinite. This follows from Algebra, Lemma 7.114.6 with R = A, S = S 0 = C and R0 = B.  24.22. Morphisms of finite presentation Recall that a ring map R → A is of finite presentation if A is isomorphic to R[x1 , . . . , xn ]/(f1 , . . . , fm ) as an R-algebra for some n, m and some polynomials fj , see Algebra, Definition 7.6.1. Definition 24.22.1. Let f : X → S be a morphism of schemes. (1) We say that f is of finite presentation at x ∈ X if there exists a affine open neighbourhood Spec(A) = U ⊂ X of x and and affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is of finite presentation. (2) We say that f is locally of finite presentation if it is of finite presentation at every point of X. (3) We say that f is of finite presentation if it is locally of finite presentation, quasi-compact and quasi-separated. Note that a morphism of finite presentation is not just a quasi-compact morphism which is locally of finite presentation. Later we will characterize morphisms which are locally of finite presentation as those morphisms such that colim MorS (Ti , X) = MorS (lim Ti , X) for any directed system of affine schemes Ti over S. See Limits, Proposition 27.4.1. In Limits, Section 27.6 we show that, if S = limi Si is a limit of affine schemes, any scheme X of finite presentation over S descends to a scheme Xi over Si for some i. Lemma 24.22.2. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is locally of finite presentation. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is of finite presentation. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the morphisms Ui → Vj , j ∈ J, i ∈ Ij is locally of finite presentation. S (4) There exists an affine open covering S = j∈J Vj and affine open coverS ings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is of finite presentation, for all j ∈ J, i ∈ Ij . Moreover, if f is locally of finite presentation then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is locally of finite presentation. Proof. This follows from Lemma 24.15.3 if we show that the property “R → A is of finite presentation” is local. We check conditions (a), (b) and (c) of Definition 24.15.1. By Algebra, Lemma 7.13.2 being of finite presentation is stable under base change and hence we conclude (a) holds. By the same lemma being of finite presentation is stable under composition and trivially for any ring R the ring map R → Rf is of finite presentation. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 7.22.3. 

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Lemma 24.22.3. The composition of two morphisms which locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation. Proof. In the proof of Lemma 24.22.2 we saw that being of finite presentation is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being of finite presentation is a property of ring maps that is stable under composition, see Algebra, Lemma 7.6.2. By the above and the fact that compositions of quasi-compact, quasi-separated morphisms are quasi-compact and quasi-separated, see Schemes, Lemmas 21.19.4 and 21.21.13 we see that the composition of morphisms of finite presentation is of finite presentation.  Lemma 24.22.4. The base change of a morphism which is locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation. Proof. In the proof of Lemma 24.22.2 we saw that being of finite presentation is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being of finite presentation is a property of ring maps that is stable under base change, see Algebra, Lemma 7.13.2. By the above and the fact that a base change of a quasi-compact, quasi-separated morphism is quasi-compact and quasi-separated, see Schemes, Lemmas 21.19.3 and 21.21.13 we see that the base change of a morphism of finite presentation is a morphism of finite presentation.  Lemma 24.22.5. Any open immersion is locally of finite presentation. Proof. This is true because an open immersion is a local isomorphism.



Lemma 24.22.6. Any open immersion is of finite presentation if and only if it is quasi-compact. Proof. We have seen (Lemma 24.22.5) that an open immersion is locally of finite presentation. We have see (Schemes, Lemma 21.23.7) that an immersion is separated and hence quasi-separated. From this and Definition 24.22.1 the lemma follows.  Lemma 24.22.7. Any closed immersion i : Z → X is of finite presentation if and only if the associated quasi-coherent sheaf of ideals I = Ker(OX → i∗ OZ ) is of finite type (as an OX -module). Proof. On any affine open Spec(R) ⊂ X we have i−1 (Spec(R)) = Spec(R/I) and e Moreover, I is of finite type if and only if I is a finite R-module for every I = I. such affine open (see Properties, Lemma 23.16.1). And R/I is of finite presentation over R if and only if I is a finite R-module. Hence we win.  Lemma 24.22.8. A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type. Proof. Omitted.



Lemma 24.22.9. Let f : X → S be a morphism. (1) If S is locally Noetherian and f locally of finite type then f is locally of finite presentation.

24.23. CONSTRUCTIBLE SETS

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(2) If S is locally Noetherian and f of finite type then f is of finite presentation. Proof. The first statement follows from the fact that a ring of finite type over a Noetherian ring is of finite presentation, see Algebra, Lemma 7.29.4. Suppose that f is of finite type and S is locally Noetherian. Then f is quasi-compact and locally of finite presentation by (1). Hence it suffices to prove that f is quasi-separated. This follows from Lemma 24.16.7 (and Lemma 24.22.8).  Lemma 24.22.10. Let S be a scheme which is quasi-compact and quasi-separated. If X is of finite presentation over S, then X is quasi-compact and quasi-separated. Proof. Omitted.



Lemma 24.22.11. Let f : X → Y be a morphism of schemes over S. (1) If X is locally of finite presentation over S and Y is locally of finite type over S, then f is locally of finite presentation. (2) If X is of finite presentation over S and Y is quasi-separated and locally of finite type over S, then f is of finite presentation. Proof. Proof of (1). Via Lemma 24.22.2 this translates into the following algebra fact: Given ring maps A → B → C such that A → C is of finite presentation and A → B is of finite type, then B → C is of finite type. See Algebra, Lemma 7.6.2. Part (2) follows from (1) and Schemes, Lemmas 21.21.14 and 21.21.15.



24.23. Constructible sets Constructible and locally construcible sets of schemes have been discussed in Properties, Section 23.2. In this section we prove some results concerning images and inverse images of (locally) constructible sets. The main result is Chevalley’s theorem which states that the image of a locally constructible set under a morphism of finite presentation is locally constructible. Lemma 24.23.1. Let f : X → Y be a morphism of schemes. Let E ⊂ Y be a subset. If E is (locally) construcible in Y , then f −1 (E) is (locally) constructible in X. Proof. To show that the inverse image of every construcible subset is constructible it suffices to show that the inverse image of every retrocompact open V of Y is retrocompact in X, see Topology, Lemma 5.10.3. The significance of V being retrocompact in Y is just that the open immersion V → Y is quasi-compact. Hence the base change f −1 (V ) = X ×Y V → X is quasi-compact too, see Schemes, Lemma 21.19.3. Hence we see f −1 (V ) is retrocompact in X. Suppose E is locally constructible in Y . Choose x ∈ X. Choose an affine neighbourhood V of f (x) and an affine neighbourhood U ⊂ X of x such that f (U ) ⊂ V . Thus we think of f |U : U → V as a morphism into V . By Properties, Lemma 23.2.1 we see that E ∩ V is constructible in V . By the constructible case we see that (f |U )−1 (E ∩ V ) is constructible in U . Since (f |U )−1 (E ∩ V ) = f −1 (E) ∩ U we win.  Lemma 24.23.2. Let f : X → Y be a morphism of schemes. Assume (1) f is quasi-compact and locally of finite presentation, and (2) Y is quasi-compact and quasi-separated. Then the image of every constructible subset of X is constructible in Y .

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Proof. By Properties, Lemma 23.2.3 it suffices to prove this lemma in case Y is affine. In this case X is quasi-compact. Hence we can write X = U1 ∪ . . . ∪ Un with each Ui affine open in X. If E ⊂ X is constructible, then each S E ∩ Ui is constructible too, see Topology, Lemma 5.10.4. Hence, since f (E) = f (E ∩ Ui ) and since finite unions of constructible sets are constructible, this reduces us to the case where X is affine. In this case the result is Algebra, Theorem 7.27.9.  Theorem 24.23.3 (Chevalley’s Theorem). Let f : X → Y be a morphism of schemes. Assume f is quasi-compact and locally of finite presentation. Then the image of every locally constructible subset is locally constructible. Proof. Let E ⊂ X be locally constructible. We have to show that f (E) is locally constructible too. We will show that f (E) ∩ V is constructible for any affine open V ⊂ Y . Thus we reduce to the case where Y is affine. In this case X is quasicompact. Hence we can write X = U1 ∪. . .∪Un with each Ui affine open in X. If E ⊂ X is locally constructible, thenSeach E ∩ Ui is constructible, see Properties, Lemma 23.2.1. Hence, since f (E) = f (E ∩ Ui ) and since finite unions of constructible sets are constructible, this reduces us to the case where X is affine. In this case the result is Algebra, Theorem 7.27.9.  Lemma 24.23.4. Let X be a scheme. Let x ∈ X. Let E ⊂ X be a locally constructible subset. If {x0 | x0 x} ⊂ E, then E contains an open neighbourhood of x. Proof. Assume {x0 | x0 x} ⊂ E. We may assume X is affine. In this case E is constructible, see Properties, Lemma 23.2.1. In particular, also the complement E c is constructible. By Algebra, Lemma 7.27.3 we can find a morphism of affine schemes f : Y → X such that E c = f (Y ). Let Z ⊂ X be the scheme theoretic image of f . By Lemma 24.6.5 and the assumption {x0 | x0 x} ⊂ E we see that x 6∈ Z. Hence X \ Z ⊂ E is an open neighbourhood of x contained in E.  24.24. Open morphisms Definition 24.24.1. Let f : X → S be a morphism. (1) We say f is open if the map on underlying topological spaces is open. (2) We say f is universally open if for any morphism of schemes S 0 → S the base change f 0 : XS 0 → S 0 is open. According to Topology, Lemma 5.14.6 generalizations lift along certain types of open maps of topological spaces. In fact generalizations lift along any open morphism of schemes (see Lemma 24.24.5). Also, we will see that generalizations lift along flat morphisms of schemes (Lemma 24.26.8). This sometimes in turn implies that the morphism is open. Lemma 24.24.2. Let f : X → S be a morphism. (1) If f is locally of finite presentation and generalizations lift along f , then f is open. (2) If f is locally of finite presentation and generalizations lift along every base change of f , then f is universally open. Proof. It suffices to prove the first assertion. This reduces to the case where both X and S are affine. In this case the result follows from Algebra, Lemma 7.37.3 and Proposition 7.37.8. 

24.26. FLAT MORPHISMS

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See also Lemma 24.26.9 for the case of a morphism flat of finite presentation. Lemma 24.24.3. A composition of (universally) open morphisms is (universally) open. Proof. Omitted.



Lemma 24.24.4. Let k be a field. Let X be a scheme over k. The structure morphism X → Spec(k) is universally open. Proof. Let S → Spec(k) be a morphism. We have to show that the base change XS → S is open. The question is local on S and X, hence we may assume that S and X are affine. In this case the result is Algebra, Lemma 7.37.10.  Lemma 24.24.5. Let ϕ : X → Y be a morphism of schemes. If ϕ is open, then ϕ is generizing (i.e., generalizations lift along ϕ). If ϕ is universally open, then ϕ is universally generizing. Proof. Assume ϕ is open. Let y 0 y be a specialization of points of Y . Let x ∈ X with ϕ(x) = y. Choose affine opens U ⊂ X and V ⊂ Y such that ϕ(U ) ⊂ V and x ∈ U . Then also y 0 ∈ V . Hence we may replace X by U and Y by V and assume X, Y affine. The affine case is Algebra, Lemma 7.37.2 (combined with Algebra, Lemma 7.37.3).  Lemma 24.24.6. Let f : X → Y be a morphism of schemes. Let g : Y 0 → Y be open and surjective such that the base change f 0 : X 0 → Y 0 is quasi-compact. Then f is quasi-compact. Proof. Let V ⊂ Y be a quasi-compact open. As g is open and surjective we can find a quasi-compact open W 0 ⊂ W such that g(W 0 ) = V . By assumption (f 0 )−1 (W 0 ) is quasi-compact. The image of (f 0 )−1 (W 0 ) in X is equal to f −1 (V ), see Lemma 24.11.3. Hence f −1 (V ) is quasi-compact as the image of a quasi-compact space, see Topology, Lemma 5.9.5. Thus f is quasi-compact.  24.25. Submersive morphisms Definition 24.25.1. Let f : X → Y be a morphism of schemes. (1) We say f is submersive5 if the continuous map of underlying topological spaces is submersive, see Topology, Definition 5.15.1. (2) We say f is universally submersive if for every morphism of schemes Y 0 → Y the base change Y 0 ×Y X → Y 0 is submersive. We note that a submersive morphism is in particular surjective. 24.26. Flat morphisms Flatness is one of the most important technical tools in algebraic geometry. In this section we introduce this notion. We intentionally limit the discussion to straightforward observations, apart from Lemma 24.26.9. A very important class of results, namely criteria for flatness will be discussed (insert future reference here). Recall that a module M over a ring R is flat if the functor −⊗R M : ModR → ModR is exact. A ring map R → A is said to be flat if A is flat as an R-module. See Algebra, Definition 7.36.1. 5This is very different from the notion of a submersion of differential manifolds.

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Definition 24.26.1. Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of OX -modules. (1) We say f is flat at a point x ∈ X if the local ring OX,x is flat over the local ring OS,f (x) . (2) We say that F is flat over S at a point x ∈ X if the stalk Fx is a flat OS,f (x) -module. (3) We say f is flat if f is flat at every point of X. (4) We say that F is flat over S if F is flat over S at every point x of X. Thus we see that f is flat if and only if the structure sheaf OX is flat over S. Lemma 24.26.2. Let f : X → S be a morphism of schemes. Let F be a quasicoherent sheaf of OX -modules. The following are equivalent (1) The sheaf F is flat over S. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the OS (V )-module F(U ) is flat. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the modules F|Ui is flat over Vj , for all j ∈ J, i ∈ Ij . S (4) There exists an affine open covering S = j∈J Vj and affine open coverS ings f −1 (Vj ) = i∈Ij Ui such that F(Ui ) is a flat OS (Vj )-module, for all j ∈ J, i ∈ Ij . Moreover, if F is flat over S then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction F|U is flat over V . Proof. Let R → A be a ring map. Let M be an A-module. If M is R-flat, then for all primes q the module Mq is flat over Rp with p the prime of R lying under q. Conversely, if Mq is flat over Rp for all primes q of A, then M is flat over R. See Algebra, Lemma 7.36.19. This equivalence easily implies the statements of the lemma.  Lemma 24.26.3. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is flat. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is flat. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i → Vj , j ∈ J, i ∈ Ij is flat. i∈Ij Ui such that each of the morphisms US (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that OS (Vj ) → OX (Ui ) is flat, for all j ∈ J, i ∈ Ij . Moreover, if f is flat then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is flat. Proof. This is a special case of Lemma 24.26.2 above.



Lemma 24.26.4. Let X → Y → Z be morphisms of schemes. Let F be a quasicoherent OX -module. If F is flat over Y , and Y is flat over Z, then F is flat over Z. Proof. See Algebra, Lemma 7.36.3.



24.26. FLAT MORPHISMS

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Lemma 24.26.5. The composition of flat morphisms is flat. Proof. This is a special case of Lemma 24.26.4.



Lemma 24.26.6. Let f : X → S be a morphism of schemes. Let F be a quasicoherent sheaf of OX -modules. Let g : S 0 → S be a morphism of schemes. Denote g 0 : X 0 = XS 0 → X the projection. Let x0 ∈ X 0 be a point with image x = g(x0 ) ∈ X. If F is flat over S at x, then (g 0 )∗ F is flat over S 0 at x0 . In particular, if F is flat over S, then (g 0 )∗ F is flat over S 0 . Proof. See Algebra, Lemma 7.36.6.



Lemma 24.26.7. The base change of a flat morphism is flat. Proof. This is a special case of Lemma 24.26.6.



Lemma 24.26.8. Let f : X → S be a flat morphism of schemes. Then generalizations lift along f , see Topology, Definition 5.14.3. Proof. See Algebra, Section 7.37.



Lemma 24.26.9. A flat morphism locally of finite presentation is universally open. Proof. This follows from Lemmas 24.26.8 and Lemma 24.24.2 above. We can also argue directly as follows. Let f : X → S be flat locally of finite presentation. S To show f is open it suffices to show that we may cover X by open affines X = Ui such that Ui → S is open. By definition we may cover X by affine opens Ui ⊂ X such that each Ui maps into an affine open Vi ⊂ S and such that the induced ring map OS (Vi ) → OX (Ui ) is of finite presentation. Thus Ui → Vi is open by Algebra, Proposition 7.37.8. The lemma follows.  Lemma 24.26.10. Let f : X → Y be a quasi-compact, surjective, flat morphism. A subset T ⊂ Y is open (resp. closed) if and only f −1 (T ) is open (resp. closed). In other words, f is a submersive morphism. Proof. The question is local on Y , hence we may assume that Y is affine. In this case X is quasi-compact as f is quasi-compact. ` Write ` X = X1 ∪ . . . ∪ Xn as a finite union of affine opens. Then f 0 : X 0 = X1 . . . Xn → Y is a surjective flat morphism affine schemes. Note that for T ⊂ Y we have (f 0 )−1 (T ) = f −1 (T ) ∩ ` `of −1 X1 . . . f (T ) ∩ Xn . Hence, f −1 (T ) is open if and only if (f 0 )−1 (T ) is open. Thus we may assume both X and Y are affine. Let f : Spec(B) → Spec(A) be a surjective morphism of affine schemes corresponding to a flat ring map A → B. Suppose that f −1 (T ) is closed, say f −1 (T ) = V (I) for I ⊂ A an ideal. Then T = f (f −1 (T )) = f (V (I)) is the image of Spec(A/I) → Spec(B) (here we use that f is surjective). On the other hand, generalizations lift along f (Lemma 24.26.8). Hence by Topology, Lemma 5.14.5 we see that Y \ T = f (X \ f −1 (T )) is stable under generalization. Hence T is stable under specialization (Topology, Lemma 5.14.2). Thus T is closed by Algebra, Lemma 7.37.5.  Lemma 24.26.11. Let h : X → Y be a morphism of schemes over S. Let G be a quasi-coherent sheaf on Y . Let x ∈ X with y = h(x) ∈ Y . If h is flat at x, then G flat over S at y ⇔ h∗ G flat over S at x.

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In particular: If h is surjective and flat, then G is flat over S, if and only if h∗ G is flat over S. If h is surjective and flat, and X is flat over S, then Y is flat over S. Proof. You can prove this by applying Algebra, Lemma 7.36.8. Here is a direct proof. Let s ∈ S be the image of y. Consider the local ring maps OS,s → OY,y → OX,x . By assumption the ring map OY,y → OX,x is faithfully flat, see Algebra, Lemma 7.36.16. Let N = Gy . Note that h∗ Gx = N ⊗OY,y OX,x , see Sheaves, Lemma 6.26.4. Let M 0 → M be an injection of OS,s -modules. By the faithful flatness mentioned above we have Ker(M 0 ⊗OS,s N → M ⊗OS,s N ) ⊗OY,y OX,x = Ker(M 0 ⊗OS,s N ⊗OY,y OX,x → M ⊗OS,s N ⊗OY,y OX,x ) Hence the equivalence of the lemma follows from the second characterization of flatness in Algebra, Lemma 7.36.4.  24.27. Flat closed immersions Connected components of schemes are not always open. But they do always have a canonical scheme structure. We explain this in this section. Lemma 24.27.1. Let X be a scheme. The rule which associates to a closed subscheme of X its underlying closed subset defines a bijection     closed subschemes Z ⊂ X closed subsets Z ⊂ X ↔ such that Z → X is flat closed under generalizations Proof. The affine case is Algebra, Lemma 7.101.4. In general the lemma follows by covering X by affines and glueing. Details omitted.  Note that a connected component T of a scheme X is a closed subset stable under generalization. Hence the following definition makes sense. Definition 24.27.2. Let X be a scheme. Let T ⊂ X be a connected component. The canonical scheme structure on T is the unique scheme structure on T such that the closed immersion T → X is flat, see Lemma 24.27.1. It turns out that we can determine when every finite flat OX -module is finite locally free using the previous lemma. Lemma 24.27.3. Let X be a scheme. The following are equivalent (1) every finite flat quasi-coherent OX -module is finite locally free, and (2) every closed subset Z ⊂ X which is closed under generalizations is open. Proof. In the affine case this is Algebra, Lemma 7.101.6. The scheme case does not follow directly from the affine case, so we simply repeat the arguments. Assume (1). Consider a closed immersion i : Z → X such that i is flat. Then i∗ OZ is quasi-coherent and flat, hence finite locally free by (1). Thus Z = Supp(i∗ OZ ) is also open and we see that (2) holds. Hence the implication (1) ⇒ (2) follows from the characterization of flat closed immersions in Lemma 24.27.1. For the converse assume that X satisfies (2). Let F be a finite flat quasi-coherent OX -module. The support Z = Supp(F) of F is closed, see Modules, Lemma 15.9.6.

24.28. GENERIC FLATNESS

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On the other hand, if x x0 is a specialization, then by Algebra, Lemma 7.73.4 the module Fx0 is free over OX,x0 , and Fx = Fx0 ⊗OX,x0 OX,x . 0

Hence x ∈ Supp(F) ⇒ x ∈ Supp(F), in other words, the support is closed under generalization. As X satisfies (2) we see that the support of F is open and closed. The modules ∧i (F), i = 1, 2, 3, . . . are finite flat quasi-coherent OX -modules also, see Modules, Section 15.18. Note that Supp(∧i+1 (F)) ⊂ Supp(∧i (F)). Thus we see that there exists a decomposition a a a X = U0 U1 U2 ... by open and closed subsets such that the support of ∧i (F) is Ui ∪ Ui+1 ∪ . . . for all i. Let x be a point of X, and say x ∈ Ur . Note that ∧i (F)x ⊗ κ(x) = ∧i (Fx ⊗ κ(x)). Hence, x ∈ Ur implies that Fx ⊗ κ(x) is a vector space of dimension r. By Nakayama’s lemma, see Algebra, Lemma 7.18.1 we can choose an affine open neighbourhood U ⊂ Ur ⊂ X of x and sections s1 , . . . , sr ∈ F(U ) such that the induced map X ⊕r OU −→ F|U , (f1 , . . . , fr ) 7−→ fi si is surjective. This means that ∧r (F|U ) is a finite flat quasi-coherent OU -module whose support is all of U . By the above it is generated by a single element, namely s1 ∧ . . . ∧ sr . Hence ∧r (F|U ) ∼ = OU /I for some quasi-coherent sheaf of ideals I such that OU /I is flat over OU and such that V (I) = U . It follows that I = 0 by applying Lemma 24.27.1. Thus s1 ∧ . . . ∧ sr is a basis for ∧r (F|U ) and it follows that the displayed map is injective as well as surjective. This proves that F is finite locally free as desired.  24.28. Generic flatness A scheme of finite type over an integral base is flat over a dense open of the base. In Algebra, Section 24.28 we proved a Noetherian version, a version for morphisms of finite presentation, and a general version. We only state and prove the general version here. However, it turns out that this will be superseded by Proposition 24.28.2 which shows the result holds if we only assume the base is reduced. Proposition 24.28.1 (Generic flatness). Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of OX -modules. Assume (1) S is integral, (2) f is of finite type, and (3) F is a finite type OX -module. Then there exists an open dense subscheme U ⊂ S such that XU → U is flat and of finite presentation and such that F|XU is flat over U and of finite presentation over OXU . Proof. As S is integral it is irreducible (see Properties, Lemma 23.3.4) and any nonempty open is dense. Hence we may replace S by an affine open of S and assume that S = Spec(A) is affine. As S is integral we see that A is a domain. As f is of finite type, it is quasi-compact, so X is quasi-compact. Hence we can find a S finite affine open cover X = i=1,...,n Xi . Write Xi = Spec(Bi ). Then Bi is a finite type A-algebra, see Lemma 24.16.2. Moreover there are finite type Bi -modules Mi such that F|Xi is the quasi-coherent sheaf associated to the Bi -module Mi ,

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see Properties, Lemma 23.16.1. Next, for each pair of indices i, j choose an ideal Iij ⊂ Bi such that Xi \ Xi ∩ Xj = V (Iij ) inside Xi = Spec(Bi ). Set Mij = Bi /Iij and think of it as a Bi -module. Then V (Iij ) = Supp(Mij ) and Mij is a finite Bi -module. At this point we apply Algebra, Lemma 7.110.3 the pairs (A → Bi , Mij ) and to the pairs (A → Bi , Mi ). Thus we obtain nonzero fij , fi ∈ A such that (a) Afij → Bi,fij is flat and of finite presentation and Mij,fij is flat over Afij and of finite presentation over Bi,fij , and (b) Bi,fi is flat and of finite presentation Q Q over Af and Mi,fi is flat and of finite presentation over Bi,fi . Set f = ( fi )( fij ). We claim that taking U = D(f ) works. To prove our claim we may replace A by Af , i.e., perform the base change by U = Spec(Af ) → S. After this base change we see that each of A → Bi is flat and of finite presentation and that Mi , Mij are flat over A and of finite presentation over Bi . This already proves that X → S is quasi-compact, locally of finite presentation, flat, and that F is flat over S and of finite presentation over OX , see Lemma 24.22.2 and Properties, Lemma 23.16.2. Since Mij is of finite presentation over Bi we see that Xi ∩ Xj = Xi \ Supp(Mij ) is a quasi-compact open of Xi , see Algebra, Lemma 7.60.8. Hence we see that X → S is quasi-separated by Schemes, Lemma 21.21.7. This proves the proposition.  It actually turns out that there is also a version of generic flatness over an arbitrary reduced base. Here it is. Proposition 24.28.2 (Generic flatness, reduced case). Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf of OX -modules. Assume (1) S is reduced, (2) f is of finite type, and (3) F is a finite type OX -module. Then there exists an open dense subscheme U ⊂ S such that XU → U is flat and of finite presentation and such that F|XU is flat over U and of finite presentation over OXU . Proof. For the impatient reader: This proof is a repeat of the proof of Proposition 24.28.1 using Algebra, Lemma 7.110.7 instead of Algebra, Lemma 7.110.3. Since being flat and being of finite presentation is local on the base, see Lemmas 24.26.2 and 24.22.2, we may work affine locally on S. Thus we may assume that S = Spec(A), where A is a reduced ring (see Properties, Lemma 23.3.2). As f is of finite type, it is quasi-compact, so X is quasi-compact. Hence we can find a S finite affine open cover X = i=1,...,n Xi . Write Xi = Spec(Bi ). Then Bi is a finite type A-algebra, see Lemma 24.16.2. Moreover there are finite type Bi -modules Mi such that F|Xi is the quasi-coherent sheaf associated to the Bi -module Mi , see Properties, Lemma 23.16.1. Next, for each pair of indices i, j choose an ideal Iij ⊂ Bi such that Xi \ Xi ∩ Xj = V (Iij ) inside Xi = Spec(Bi ). Set Mij = Bi /Iij and think of it as a Bi -module. Then V (Iij ) = Supp(Mij ) and Mij is a finite Bi -module. At this point we apply Algebra, Lemma 7.110.7 the pairs (A → Bi , Mij ) and to the pairs (A → Bi , Mi ). Thus we obtain dense opens U (A → Bi , Mij ) ⊂ S and dense

24.29. MORPHISMS AND DIMENSIONS OF FIBRES

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opens U (A → Bi , Mi ) ⊂ S with notation as in Algebra, Equation (7.110.3.2). Since a finite intersection of dense opens is dense open, we see that \ \ U= U (A → Bi , Mij ) ∩ U (A → Bi , Mi ) i,j

i

is open and dense in S. We claim that U is the desired open. Pick u ∈ U . By definition of the loci U (A → Bi , Mij ) and U (A → B, Mi ) there exist fij , fi ∈ A such that (a) u ∈ D(fi ) and u ∈ D(fij ), (b) Afij → Bi,fij is flat and of finite presentation and Mij,fij is flat over Afij and of finite presentation over Bi,fij , and (c) Bi,fi is flat and of finite presentation over Af and Mi,fi is flat and of Q Q finite presentation over Bi,fi . Set f = ( fi )( fij ). Now it suffices to prove that X → S is flat and of finite presentation over D(f ) and that F restricted to XD(f ) is flat over D(f ) and of finite presentation over the structure sheaf of XD(f ) . Hence we may replace A by Af , i.e., perform the base change by Spec(Af ) → S. After this base change we see that each of A → Bi is flat and of finite presentation and that Mi , Mij are flat over A and of finite presentation over Bi . This already proves that X → S is quasi-compact, locally of finite presentation, flat, and that F is flat over S and of finite presentation over OX , see Lemma 24.22.2 and Properties, Lemma 23.16.2. Since Mij is of finite presentation over Bi we see that Xi ∩ Xj = Xi \ Supp(Mij ) is a quasi-compact open of Xi , see Algebra, Lemma 7.60.8. Hence we see that X → S is quasi-separated by Schemes, Lemma 21.21.7. This proves the proposition.  Remark 24.28.3. The results above are a first step towards more refined flattening techniques for morphisms of schemes. The article [GR71] by Raynaud and Gruson contains many wonderful results in this direction. 24.29. Morphisms and dimensions of fibres Let X be a topological space, and x ∈ X. Recall that we have defined dimx (X) as the minimum of the dimensions of the open neighbourhoods of x in X. See Topology, Definition 5.7.1. Lemma 24.29.1. Let f : X → S be a morphism of schemes. Let x ∈ X and set s = f (x). Assume f is locally of finite type. Then dimx (Xs ) = dim(OXs ,x ) + trdegκ(s) (κ(x)). Proof. This immediately reduces to the case S = s, and X affine. In this case the result follows from Algebra, Lemma 7.108.3.  Lemma 24.29.2. Let f : X → Y and g : Y → S be morphisms of schemes. Let x ∈ X and set y = f (x), s = g(y). Assume f and g locally of finite type. Then dimx (Xs ) ≤ dimx (Xy ) + dimy (Ys ). Moreover, equality holds if OXs ,x is flat over OYs ,y , which holds for example if OX,x is flat over OY,y . Proof. Note that trdegκ(s) (κ(x)) = trdegκ(y) (κ(x)) + trdegκ(s) (κ(y)). Thus by Lemma 24.29.1 the statement is equivalent to dim(OXs ,x ) ≤ dim(OXy ,x ) + dim(OYs ,y ). For this see Algebra, Lemma 7.104.6. For the flat case see Algebra, Lemma 7.104.7. 

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Lemma 24.29.3. Let X0 f0

g0

/X f

  g /S S0 be a fibre product diagram of schemes. Assume f locally of finite type. Suppose that x0 ∈ X 0 , x = g 0 (x0 ), s0 = f 0 (x0 ) and s = g(s0 ) = f (x). Then dimx (Xs ) = dimx0 (Xs0 0 ). Proof. Follows immediately from Algebra, Lemma 7.108.6.



The following lemma follows from a nontrivial algebraic result. Namely, the algebraic version of Zariski’s main theorem. Lemma 24.29.4. Let f : X → S be a morphism of schemes. Let n ≥ 0. Assume f is locally of finite type. The set Un = {x ∈ X | dimx Xf (x) ≤ n} is open in X. Proof. This is immediate from Algebra, Lemma 7.117.6



Lemma 24.29.5. Let f : X → S be a morphism of schemes. Let n ≥ 0. Assume f is locally of finite presentation. The open Un = {x ∈ X | dimx Xf (x) ≤ n} of Lemma 24.29.4 is retrocompact in X. (See Topology, Definition 5.9.1.) Proof. The topological space X has a basis for its topology consisting of affine opens U ⊂ X such that the infuced morphism f |U : U → S factors through an affine open V ⊂ S. Hence it is enough to show that U ∩ Un is quasi-compact for such a U . Note that Un ∩ U is the same as the open {x ∈ U | dimx Uf (x) ≤ n}. This reduces us to the case where X and S are affine. In this case the lemma follows from Algebra, Lemma 7.117.8 (and Lemma 24.22.2).  Lemma 24.29.6. Let f : X → S be a morphism of schemes. Let x x0 be a nontrivial specialization of points in X lying over the same point s ∈ S. Assume f is locally of finite type. Then (1) dimx (Xs ) ≤ dimx0 (Xs ), (2) trdegκ(s) (κ(x)) > trdegκ(s) (κ(x0 )), and (3) dim(OXs ,x ) < dim(OXs ,x0 ). Proof. The first inequality follows from Lemma 24.29.4. The third inequality follows since OXs ,x is a localization of OXs ,x in a prime ideal, hence any chain of prime ideals in OXs ,x is part of a strictly longer chain of primes in OXs ,x0 . The second inequality follows from Algebra, Lemma 7.108.2.  24.30. Morphisms of given relative dimension In order to be able to speak comfortably about morphisms of a given relative dimension we introduce the following notion. Definition 24.30.1. Let f : X → S be a morphism of schemes. Assume f is locally of finite type.

24.30. MORPHISMS OF GIVEN RELATIVE DIMENSION

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(1) We say f is of relative dimension ≤ d at x if dimx (Xf (x) ) ≤ d. (2) We say f is of relative dimension ≤ d if dimx (Xf (x) ) ≤ d for all x ∈ X. (3) We say f is of relative dimension d if all nonempty fibres Xs are equidimensional of dimension d. This is not a particularly well behaved notion, but it works well in a number of situations. Lemma 24.30.2. Let f : X → S be a morphism of schemes which is locally of finite type. If f has relative dimension d, then so does any base change of f . Same for relative dimension ≤ d. Proof. This is immediate from Lemma 24.29.3.



Lemma 24.30.3. Let f : X → Y , g : Y → Z be locally of finite type. If f has relative dimension ≤ d and g has relative dimension ≤ e then g ◦ f has relative dimension ≤ d + e. If (1) f has relative dimension d, (2) g has relative dimension e, and (3) f is flat, then g ◦ f has relative dimension d + e. Proof. This is immediate from Lemma 24.29.2.



In general it is not possible to decompose a morphism into its pieces where the relative dimension is a given one. However, it is possible if the morphism has Cohen-Macaulay fibres and is flat of finite presentation. Lemma 24.30.4. Let f : X → S be a morphism of schemes. Assume that (1) f is flat, (2) f is locally of finite presentation, and (3) for all s ∈ S the fibre Xs is Cohen-Macaulay (Properties, Definition 23.8.1) ` Then there exist open and closed subschemes Xd ⊂ X such that X = d≥0 Xd and f |Xd : Xd → S has relative dimension d. Proof. This is immediate from Algebra, Lemma 7.122.8.



Lemma 24.30.5. Let f : X → S be a morphism of schemes. Assume f is locally of finite type. Let x ∈ X with s = f (x). Then f is quasi-finite at x if and only if dimx (Xs ) = 0. In particular, f is locally quasi-finite if and only if f has relative dimension 0. Proof. If f is quasi-finite at x then κ(x) is a finite extension of κ(s) (by Lemma 24.21.5) and x is isolated in Xs (by Lemma 24.21.6), hence dimx (Xs ) = 0 by Lemma 24.29.1. Conversely, if dimx (Xs ) = 0 then by Lemma 24.29.1 we see κ(s) ⊂ κ(x) is algebraic and there are no other points of Xs specializing to x. Hence x is closed in its fibre by Lemma 24.21.2 and by Lemma 24.21.6 (3) we conclude that f is quasi-finite at x. 

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24.31. The dimension formula For morphisms between Noetherian schemes we can say a little more about dimensions of local rings. Here is an important (and not so hard to prove) result. Recall that R(X) denotes the function field of an integral scheme X. Lemma 24.31.1. Let S be a scheme. Let f : X → S be a morphism of schemes. Let x ∈ X, and set s = f (x). Assume (1) (2) (3) (4)

S is locally Noetherian, f is locally of finite type, X and S integral, and f dominant.

We have (24.31.1.1)

dim(OX,x ) ≤ dim(OS,s ) + trdegR(S) R(X) − trdegκ(s) κ(x).

Moreover, equality holds if S is universally catenary. Proof. The corresponding algebra statement is Algebra, Lemma 7.105.1.



An application is the construction of a dimension function on any scheme of finite type over a universally catenary scheme endowed with a dimension function. For the definition of dimension functions, see Topology, Definition 5.16.1. Lemma 24.31.2. Let S be a universally catenary scheme. Let δ : S → Z be a dimension function. Let f : X → S be a morphism of schemes. Assume f locally of finite type. Then the map δ = δX/S : X −→ Z x 7−→ δ(f (x)) + trdegκ(f (x)) κ(x) is a dimension function on X. Proof. Let f : X → S be locally of finite type. Let x y, x 6= y be a specialization in X. We have to show that δX/S (x) > δX/S (y) and that δX/S (x) = δX/S (y) + 1 if y is an immediate specialization of x. Choose an affine open V ⊂ S containing the image of y and choose an affine open U ⊂ X mapping into V and containing y. We may clearly replace X by U and S by V . Thus we may assume that X = Spec(A) and S = Spec(R) and that f is given by a ring map R → A. The ring R is universally catenary (Lemma 24.18.2) and the map R → A is of finite type (Lemma 24.16.2). Let q ⊂ A be the prime ideal corresponding to the point x and let p ⊂ R be the prime ideal corresponding to f (x). The restriction δ 0 of δ to S 0 = Spec(R/p) ⊂ S is a dimension function. The ring R/p is universally catenary. The restriction of δX/S to X 0 = Spec(A/q) is clearly equal to the function δX 0 /S 0 constructed using the dimension function δ 0 . Hence we may assume in addition to the above that R ⊂ A are domains, in other words that X and S are integral schemes. Note that OX,x is a localization of OX,y at a non-maximal prime (Schemes, Lemma 21.13.2). Hence dim(OX,x ) < dim(OX,y ) and dim(OX,x ) = dim(OX,y ) − 1 if y is an immediate specialization of x.

24.31. THE DIMENSION FORMULA

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Write s = f (x) 6= f (y) = s0 . We see, using equality in (24.31.1.1), that δX/S (x) − δX/S (y) = δ(s) − δ(s0 ) + dim(OS,s ) − dim(OS,s0 ) − dim(OX,x ) + dim(OX,y ). Since δ is a dimension function on the scheme S the difference δ(s) − δ(s0 ) is equal to codim({s0 }, {s}) by Topology, Lemma 5.16.2. As S is integral, catenary this is equal to codim({s0 }, S) − codim({s}, S) (Topology, Lemma 5.8.6). And this in turn is equal to dim(OS,s0 ) − dim(OS,s ) by Properties, Lemma 23.11.4. Hence we conclude that δX/S (x) − δX/S (y) = − dim(OX,x ) + dim(OX,y ) and hence the lemma follows from our remarks about the dimensions of these local rings above.  Another application of the dimension formula is that the dimension does not change under “alterations” (to be defined later). Lemma (1) (2) (3) (4) Then we

24.31.3. Let f : X → Y be a morphism of schemes. Assume that Y is locally Noetherian, X and Y are integral schemes, f is dominant, and f is locally of finite type. have dim(X) ≤ dim(Y ) + trdegR(Y ) R(X).

If f is closed6 then equality holds. Proof. Let f : X → Y be as in the lemma. Let ξ0 ξ1 ... ξe be a sequence of specializations in X. We may assume that x = ξe is a closed point of X, see Properties, Lemma 23.5.8. In particular, setting y = f (x), we see x is a closed point of its fibre Xy . By the Hilbert Nullstellensatz we see that κ(x) is a finite extension of κ(y), see Lemma 24.21.3. By the dimension formula, Lemma 24.31.1, we see that dim(OX,x ) ≤ dim(OY,y ) + trdegR(Y ) R(X) Hence we conclude that e ≤ dim(Y ) + trdegR(Y ) R(X) as desired. Next, assume f is also closed. Say ξ 0 ξ1 ... ξ d is a sequence of specializations in Y . We want to show that dim(X) ≥ d + r. We may assume that ξ 0 = η is the generic point of Y . The generic fibre Xη is a scheme locally of finite type over κ(η) = R(Y ). It is nonempty as f is dominant. Hence by Lemma 24.17.10 it is a Jacobson scheme. Thus by Lemma 24.17.8 we can find a closed point ξ0 ∈ Xη and the extension κ(η) ⊂ κ(ξ0 ) is a finite extension. Note that OX,ξ0 = OXη ,ξ0 because η is the generic point of Y . Hence we see that dim(OX,ξ0 ) = r by Lemma 24.31.1 applied to the scheme Xη over the universally catenary scheme Spec(κ(η)) (see Lemma 24.18.4) and the point ξ0 . This means that we can find ξ−r ... ξ−1 ξ0 in X. On the other hand, as f is closed specializations lift along f , see Topology, Lemma 6For example if f is proper, see Definition 24.42.1.

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5.14.6. Thus, as ξ0 lies over η = ξ 0 we can find specializations ξ0 lying over ξ 0 ξ1 ... ξ d . In other words we have ξ−r

...

ξ−1

ξ0

ξ1

...

ξ1

...

ξd

ξd

which means that dim(X) ≥ d + r as desired.



24.32. Syntomic morphisms ∼ An algebra A over a field k is called a global complete intersection over k if A = k[x1 , . . . , xn ]/(f1 , . . . , fc ) and dim(A) = n − c. An algebra A over a field k is called a local complete intersection if Spec(A) can be covered by standard opens each of which are global complete intersections over k. See Algebra, Section 7.125. Recall that a ring map R → A is syntomic if it is of finite presentation, flat with local complete intersection rings as fibres, see Algebra, Definition 7.126.1. Definition 24.32.1. Let f : X → S be a morphism of schemes. (1) We say that f is syntomic at x ∈ X if there exists a affine open neighbourhood Spec(A) = U ⊂ X of x and and affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is syntomic. (2) We say that f is syntomic if it is syntomic at every point of X. (3) If S = Spec(k) and f is syntomic, then we say that X is a local complete intersection over k. (4) A morphism of affine schemes f : X → S is called standard syntomic if there exists a global relative complete intersection R → R[x1 , . . . , xn ]/(f1 , . . . , fc ) (see Algebra, Definition 7.126.5) such that X → S is isomorphic to Spec(R[x1 , . . . , xn ]/(f1 , . . . , fc )) → Spec(R). In the literature a syntomic morphism is sometimes referred to as a flat local complete intersection morphism. It turns out this is a convenient class of morphisms. For example one can define a syntomic topology using these, which is finer than the smooth and ´etale topologies, but has many of the same formal properties. A global relative complete intersection (which we used to define standard syntomic ring maps) is in particular flat. In More on Morphisms, Section 33.39 we will consider morphisms X → S which locally are of the form Spec(R[x1 , . . . , xn ]/(f1 , . . . , fc )) → Spec(R). for some Koszul-regular sequence f1 , . . . , fr in R[x1 , . . . , xn ]. Such a morphism will be called a local complete intersection morphism. One we have this definition in place it will be the case that a morphism is syntomic if and only if it is a flat, local complete intersection morphism. Note that there is no separation or quasi-compactness hypotheses in the definition of a syntomic morphism. Hence the question of being syntomic is local in nature on the source. Here is the precise result. Lemma 24.32.2. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is syntomic. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is syntomic.

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S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the morphisms Ui → Vj , j ∈ J, i ∈ Ij is syntomic. S (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is syntomic, for all j ∈ J, i ∈ Ij . Moreover, if f is syntomic then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is syntomic. Proof. This follows from Lemma 24.15.3 if we show that the property “R → A is syntomic” is local. We check conditions (a), (b) and (c) of Definition 24.15.1. By Algebra, Lemma 7.126.3 being syntomic is stable under base change and hence we conclude (a) holds. By Algebra, Lemma 7.126.18 being syntomic is stable under composition and trivially for any ring R the ring map R → Rf is syntomic. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 7.126.4.  Lemma 24.32.3. The composition of two morphisms which are syntomic is syntomic. Proof. In the proof of Lemma 24.32.2 we saw that being syntomic is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being syntomic is a property of ring maps that is stable under composition, see Algebra, Lemma 7.126.18.  Lemma 24.32.4. The base change of a morphism which is syntomic is syntomic. Proof. In the proof of Lemma 24.32.2 we saw that being syntomic is a local property of ring maps. Hence the lemma follows from Lemma 24.15.5 combined with the fact that being syntomic is a property of ring maps that is stable under base change, see Algebra, Lemma 7.126.3.  Lemma 24.32.5. Any open immersion is syntomic. Proof. This is true because an open immersion is a local isomorphism.



Lemma 24.32.6. A syntomic morphism is locally of finite presentation. Proof. True because a syntomic ring map is of finite presentation by definition.



Lemma 24.32.7. A syntomic morphism is flat. Proof. True because a syntomic ring map is flat by definition.



Lemma 24.32.8. A syntomic morphism is universally open. Proof. Combine Lemmas 24.32.6, 24.32.7, and 24.26.9.



Let k be a field. Let A be a local k-algebra essentially of finite type over k. Recall that A is called a complete intersection over k if we can write A ∼ = R/(f1 , . . . , fc ) where R is a regular local ring essentially of finite type over k, and f1 , . . . , fc is a regular sequence in R, see Algebra, Definition 7.125.5. Lemma 24.32.9. Let k be a field. Let X be a scheme locally of finite type over k. The following are equivalent:

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(1) X is a local complete intersection over k, (2) for every x ∈ X there exists an affine open U = Spec(R) ⊂ X neighbourhood of x such that R ∼ = k[x1 , . . . , xn ]/(f1 , . . . , fc ) is a global complete intersection over k, and (3) for every x ∈ X the local ring OX,x is a complete intersection over k. Proof. The corresponding algebra results can be found in Algebra, Lemmas 7.125.8 and 7.125.9.  The following lemma says locally any syntomic morphism is standard syntomic. Hence we can use standard syntomic morphisms as a local model for a syntomic morphism. Moreover, it says that a flat morphism of finite presentation is syntomic if and only if the fibres are local complete intersection schemes. Lemma 24.32.10. Let f : X → S be a morphism of schemes. Assume f locally of finite presentation. Let x ∈ X be a point. Set s = f (x). The following are equivalent (1) The morphism f is syntomic at x. (2) There exist affine opens U ⊂ X, and V ⊂ S such that x ∈ U , f (U ) ⊂ V and the induced morphism f |U : U → V is standard syntomic. (3) The local ring map OS,s → OX,x is flat and OX,x /ms OX,x is a complete intersection over κ(s) (see Algebra, Definition 7.125.5). Proof. Follows from the definitions and Algebra, Lemma 7.126.16.



Lemma 24.32.11. Let f : X → S be a morphism of schemes. If f is flat, locally of finite presentation, and all fibres Xs are local complete intersections, then f is syntomic. Proof. Clear from Lemmas 24.32.9 and 24.32.10 and the isomorphisms of local  rings OX,x /ms OX,x ∼ = OXs ,x . Lemma 24.32.12. Let f : X → S be a morphism of schemes. Assume f locally of finite type. Formation of the set T = {x ∈ X | OXf (x) ,x is a complete intersection over κ(f (x))} commutes with arbitrary base change: For any morphism g : S 0 → S, consider the base change f 0 : X 0 → S 0 of f and the projection g 0 : X 0 → X. Then the corresponding set T 0 for the morphism f 0 is equal to T 0 = (g 0 )−1 (T ). In particular, if f is assumed flat, and locally of finite presentation then the same holds for the open set of points where f is syntomic. Proof. Let s0 ∈ S 0 be a point, and let s = g(s0 ). Then we have Xs0 0 = Spec(κ(s0 )) ×Spec(κ(s)) Xs In other words the fibres of the base change are the base changes of the fibres. Hence the first part is equivalent to Algebra, Lemma 7.125.10. The second part follows from the first because in that case T is the set of points where f is syntomic according to Lemma 24.32.10.  Lemma 24.32.13. Let R be a ring. Let R → A = R[x1 , . . . , xn ]/(f1 , . . . , fc ) be a relative global complete intersection. Set S = Spec(R) and X = Spec(A). Consider the morphism f : X → S associated to the ring map R → A. The function x 7→ dimx (Xf (x) ) is constant with value n − c.

24.32. SYNTOMIC MORPHISMS

1423

Proof. By Algebra, Definition 7.126.5 R → A being a relative global complete intersection means all nonzero fibre rings have dimension n − c. Thus for a prime p of R the fibre ring κ(p)[x1 , . . . , xn ]/(f 1 , . . . , f c ) is either zero or a global complete intersection ring of dimension n−c. By the discussion following Algebra, Definition 7.125.1 this implies it is equidimensional of dimension n−c. Whence the lemma.  Lemma 24.32.14. Let f : X → S be a syntomic morphism. The function x 7→ dimx (Xf (x) ) is locally constant on X. Proof. By Lemma 24.32.10 the morphism f locally looks like a standard syntomic morphism of affines. Hence the result follows from Lemma 24.32.13.  Lemma 24.32.14 says that the following definition makes sense. Definition 24.32.15. Let d ≥ 0 be an integer. We say a morphism of schemes f : X → S is syntomic of relative dimension d if f is syntomic and the function dimx (Xf (x) ) = d for all x ∈ X. In other words, f is syntomic and the nonempty fibres are equidimensional of dimension d. Lemma 24.32.16. Let X

/Y

f p



S



q

be a commutative diagram of morphisms of schemes. Assume that (1) f is surjective and syntomic, (2) p is syntomic, and (3) q is locally of finite presentation7. Then q is syntomic. Proof. By Lemma 24.26.11 we see that q is flat. Hence it suffices to show that the fibres of Y → S are local complete intersections, see Lemma 24.32.11. Let s ∈ S. Consider the morphism Xs → Ys . This is a base change of the morphism X → Y and hence surjective, and syntomic (Lemma 24.32.4). For the same reason Xs is syntomic over κ(s). Moreover, Ys is locally of finite type over κ(s) (Lemma 24.16.4). In this way we reduce to the case where S is the spectrum of a field. Assume S = Spec(k). Let y ∈ Y . Choose an affine open Spec(A) ⊂ Y neighbourhood of y. Let Spec(B) ⊂ X be an affine open such that f (Spec(B)) ⊂ Spec(A), containing a point x ∈ X such that f (x) = y. Choose a surjection k[x1 , . . . , xn ] → A with kernel I. Choose a surjection A[y1 , . . . , ym ] → B, which gives rise in turn to a surjection k[xi , yj ] → B with kernel J. Let q ⊂ k[xi , yj ] be the prime corresponding to y ∈ Spec(B) and let p ⊂ k[xi ] the prime corresponding to x ∈ Spec(A). Since x maps to y we have p = q ∩ k[xi ]. Consider the following commutative diagram of 7In fact this is implied by (1) and (2), see Descent, Lemma 31.10.3. See also Descent, Remark 31.10.7 for further discussion.

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local rings: OX,x O

Bq o O

k[x1 , . . . , xn , y1 , . . . , ym ]q O

OY,y

Ap o

k[x1 , . . . , xn ]p

We claim that the hypotheses of Algebra, Lemma 7.125.12 are satisfied. Conditions (1) and (2) are trivial. Condition (4) follows as X → Y is flat. Condition (3) follows as the rings OY,y and OXy ,x = OX,x /my OX,x are complete intersection rings by our assumptions that f and p are syntomic, see Lemma 24.32.10. The output of Algebra, Lemma 7.125.12 is exactly that OY,y is a complete intersection ring! Hence by Lemma 24.32.10 again we see that Y is syntomic over k at y as desired.  24.33. Conormal sheaf of an immersion Let i : Z → X be a closed immersion. Let I ⊂ OX be the corresponding quasicoherent sheaf of ideals. Consider the short exact sequence 0 → I 2 → I → I/I 2 → 0 of quasi-coherent sheaves on X. Since the sheaf I/I 2 is annihilated by I it corresponds to a sheaf on Z by Lemma 24.4.1. This quasi-coherent OZ -module is called the conormal sheaf of Z in X and is often simply denoted I/I 2 by the abuse of notation mentioned in Section 24.4. In case i : Z → X is a (locally closed) immersion we define the conormal sheaf of i as the conormal sheaf of the closed immersion i : Z → X \ ∂Z, where ∂Z = Z \ Z. It is often denoted I/I 2 where I is the ideal sheaf of the closed immersion i : Z → X \ ∂Z. Definition 24.33.1. Let i : Z → X be an immersion. The conormal sheaf CZ/X of Z in X or the conormal sheaf of i is the quasi-coherent OZ -module I/I 2 described above. In [DG67, IV Definition 16.1.2] this sheaf is denoted NZ/X . We will not follow this convention since we would like to reserve the notation NZ/X for the normal sheaf of the immersion. It is defined as NZ/X = Hom OZ (CZ/X , OZ ) = Hom OZ (I/I 2 , OZ ) provided the conormal sheaf is of finite presentation (otherwise the normal sheaf may not even be quasi-coherent). We will come back to the normal sheaf later (insert future reference here). Lemma 24.33.2. Let i : Z → X be an immersion. The conormal sheaf of i has the following properties: (1) Let U ⊂ X be any open such that i(Z) is a closed subset of U . Let I ⊂ OU be the sheaf of ideals corresponding to the closed subscheme i(Z) ⊂ U . Then CZ/X = i∗ I = i−1 (I/I 2 ) (2) For any affine open Spec(R) = U ⊂ X such that Z ∩ U = Spec(R/I) there is a canonical isomorphism Γ(Z ∩ U, CZ/X ) = I/I 2 .

24.33. CONORMAL SHEAF OF AN IMMERSION

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Proof. Mostly clear from the definitions. Note that given a ring R and an ideal I of R we have I/I 2 = I ⊗R R/I. Details omitted.  Lemma 24.33.3. Let Z

i

g

f

 Z0

/X

i0

 / X0

be a commutative diagram in the category of schemes. Assume i, i0 immersions. There is a canonical map of OZ -modules f ∗ CZ 0 /X 0 −→ CZ/X characterized by the following property: For every pair of affine opens (Spec(R) = U ⊂ X, Spec(R0 ) = U 0 ⊂ X 0 ) with f (U ) ⊂ U 0 such that Z ∩ U = Spec(R/I) and Z 0 ∩ U 0 = Spec(R0 /I 0 ) the induced map Γ(Z 0 ∩ U 0 , CZ 0 /X 0 ) = I 0 /I 02 −→ I/I 2 = Γ(Z ∩ U, CZ/X ) is the one induced by the ring map f ] : R0 → R which has the property f ] (I 0 ) ⊂ I. Proof. Let ∂Z 0 = Z 0 \ Z 0 and ∂Z = Z \ Z. These are closed
subsets of X 0 and of X. Replacing X 0 by X 0 \ ∂Z 0 and X by X \ g −1 (∂Z 0 ) ∪ ∂Z we see that we may assume that i and i0 are closed immersions. The fact that g ◦ i factors through i0 implies that g ∗ I 0 maps into I under the canonical map g ∗ I 0 → OX , see Schemes, Lemmas 21.4.6 and 21.4.7. Hence we get an induced map of quasi-coherent sheaves g ∗ (I 0 /(I 0 )2 ) → I/I 2 . Pulling back by i gives i∗ g ∗ (I 0 /(I 0 )2 ) → i∗ (I/I 2 ). Note that i∗ (I/I 2 ) = CZ/X . On the other hand, i∗ g ∗ (I 0 /(I 0 )2 ) = f ∗ (i0 )∗ (I 0 /(I 0 )2 ) = f ∗ CZ 0 /X 0 . This gives the desired map. Checking that the map is locally described as the given map I 0 /(I 0 )2 → I/I 2 is a matter of unwinding the definitions and is omitted. Another observation is that given any x ∈ i(Z) there do exist affine open neighbourhoods U , U 0 with f (U ) ⊂ U 0 and Z ∩ U as well as U 0 ∩ Z 0 closed such that x ∈ U . Proof omitted. Hence the requirement of the lemma indeed characterizes the map (and could have been used to define it).  Lemma 24.33.4. Let Z

i

g

f

 Z0

/X

i

0

 / X0

be a fibre product diagram in the category of schemes with i, i0 immersions. Then the canonical map f ∗ CZ 0 /X 0 → CZ/X of Lemma 24.33.3 is surjective. If g is flat, then it is an isomorphism. Proof. Let R0 → R be a ring map, and I 0 ⊂ R0 an ideal. Set I = I 0 R. Then I 0 /(I 0 )2 ⊗R0 R → I/I 2 is surjective. If R0 → R is flat, then I = I 0 ⊗R0 R and I 2 = (I 0 )2 ⊗R0 R and we see the map is an isomorphism. 

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Lemma 24.33.5. Let Z → Y → X be immersions of schemes. Then there is a canonical exact sequence i∗ CY /X → CZ/X → CZ/Y → 0 where the maps come from Lemma 24.33.3 and i : Z → Y is the first morphism. Proof. Via Lemma 24.33.3 this translates into the following algebra fact. Suppose that C → B → A are surjective ring maps. Let I = Ker(B → A), J = Ker(C → A) and K = Ker(C → B). Then there is an exact sequence K/K 2 ⊗B A → J/J 2 → I/I 2 → 0. This follows immediately from the observation that I = J/K.



24.34. Sheaf of differentials of a morphism We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 7.123). Definition 24.34.1. Let f : X → S be a morphism of schemes. Let F be an OX -module. A derivation or more precisely an S-derivation into F is a map D : OX → F which is additive, annihilates the image of f −1 OS → OX , and satisfies the Leibniz rule D(ab) = aD(b) + D(a)b for all a, b local sections of OX (wherever they are both defined). We denote DerS (OX , F) the set of S-derivations into F. This is the sheaf theoretic analogue of Algebra, Definition 24.34.1. Given a derivation D : OX → F as in the definition the map on global sections D : Γ(X, OX ) −→ Γ(X, F) clearly is a Γ(S, OS )-derivation as in the algebra definition. Lemma 24.34.2. Let R → A be a ring map. Let F be a sheaf of OX -modules on X = Spec(A). Set S = Spec(R). The rule which associates to an S-derivation on F its action on global sections defines a bijection between the set of S-derivations of F and the set of R-derivations on M = Γ(X, F). Proof. Let D : A → M be an R-derivation. We have to show there exists a unique S-derivation on F which gives rise to D on global sections. Let U = D(f ) ⊂ X be a standard affine open. Any element of Γ(U, OX ) is of the form a/f n for some a ∈ A and n ≥ 0. By the Leibniz rule we have D(a)|U = a/f n D(f n )|U + f n D(a/f n ) in Γ(U, F). Since f acts invertibly on Γ(U, F) this completely determines the value of D(a/f n ) ∈ Γ(U, F). This proves uniqueness. Existence follows by simply defining D(a/f n ) := (1/f n )D(a)|U − a/f 2n D(f n )|U and proving this has all the desired properties (on the basis of standard opens of X). Details omitted.  Here is a particular situation where derivations come up naturally.

24.34. SHEAF OF DIFFERENTIALS OF A MORPHISM

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Lemma 24.34.3. Let f : X → S be a morphism of schemes. Consider a short exact sequence 0 → I → A → OX → 0 Here A is a sheaf of f −1 OS -algebras, π : A → OX is a surjection of sheaves of f −1 OS -algebras, and I = Ker(π) is its kernel. Assume I an ideal sheaf with square zero in A. So I has a natural structure of an OX -module. A section s : OX → A of π is a f −1 OS -algebra map such that π ◦ s = id. Given any section s : OX → I of π and any S-derivation D : OX → I the map s + D : OX → A is a section of π and every section s0 is of the form s + D for a unique S-derivation D. ˜ (multiplicaProof. Recall that the OX -module structure on I is given by hτ = hτ ˜ is a local lift of h to a local section tion in A) where h is a local section of OX , and h ˜ = s(h). To of A, and τ is a local section of I. In particular, given s, we may use h verify that s + D is a homomorphism of sheaves of rings we compute (s + D)(ab)

= s(ab) + D(ab) = s(a)s(b) + aD(b) + D(a)b = s(a)s(b) + s(a)D(b) + D(a)s(b) =

(s(a) + D(a))(s(b) + D(b))

by the Leibniz rule. In the same manner one shows s + D is a f −1 OS -algebra map because D is an S-derivation. Conversely, given s0 we set D = s0 − s. Details omitted.  Let f : X → S be a morphism of schemes. We now esthablish the existence of a couple of “global” sheaves and maps of sheaves, and in the next paragraph we describe the constructions over some affine opens. Recall that ∆ = ∆X/S : X → X×S X is an immersion, see Schemes, Lemma 21.21.2. Let J be the ideal sheaf of the immersion. It lives over any open subscheme U of X ×S X such that ∆(X) ⊂ U is closed. For example the one from the proof of the lemma just cited; if f is separated then we can take U = X ×S X. Note that the sheaf of rings OU /J 2 is supported on ∆(X). Moreover it sits in a short exact sequence of sheaves 0 → J /J 2 → OU /J 2 → ∆∗ OX → 0. Using ∆−1 we can think of this as a surjection of sheaves of f −1 OS -algebras with kernel the conormal sheaf of ∆ (see Definition 24.33.1 and Lemma 24.33.2). 0 → CX/X×S X → ∆−1 (OU /J 2 ) → OX → 0 This places us in the sitation of Lemma 24.34.3. The projection morphisms pi : X ×S X → X, i = 1, 2 induce maps of sheaves of rings p]i : p−1 i OX → OX×S X . We 2 may restrict to U and divide by J 2 to get p−1 O → O /J . Since ∆−1 p−1 X U i i OX = OX we get maps si : OX → ∆−1 (OU /J 2 ). Both s1 and s2 are sections to the map ∆−1 (OU /J 2 ) → OX , as in Lemma 24.34.3. Thus we get an S-derivation d = s2 − s1 : OX → CX/X×S X .

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Let us work this out on a suitable affine open. We can cover X by affine opens Spec(A) = W ⊂ X whose image is contained in an affine open Spec(R) = V ⊂ S. According to the proof of Schemes, Lemma 21.21.2 W ×V W ⊂ X ×S X is an affine open contained in the open U mentioned above. Also W ×V W = Spec(A ⊗R A). The sheaf J corresponds to the ideal J = Ker(A ⊗R A → A). The short exact sequence to the short exact sequence of A ⊗R A-modules 0 → J/J 2 → (A ⊗R A)/J 2 → A → 0 The sections si correspond to the ring maps A −→ (A ⊗R A)/J 2 , s1 : a 7→ a ⊗ 1, s2 : a 7→ 1 ⊗ a. By Lemma 24.33.2 the conormal sheaf of ∆X/S restricted to U ×V U is the quasicoherent sheaf associated to the A-module J/J 2 . Comparing with Algebra, Lemma 7.123.13 (or by a direct computation) we see that the induced map d : A → J/J 2 is isomorphic to the universal R-derivation on A. Thus the following definition makes sense. Definition 24.34.4. Let f : X → S be a morphism of schemes. (1) The sheaf of differentials ΩX/S of X over S is the conormal sheaf of the immersion ∆X/S : X → X ×S X, see Definition 24.33.1. (2) The universal S-derivation is the S-derivation dX/S : OX −→ ΩX/S which maps a local section f of OX to the class of the local section d(f ) = dX/S (f ) = s2 (f ) − s1 (f ) with s2 and s1 as described above. Here is the universal property of the universal derivation. If you have any other construction of the sheaf of relative differentials which satisfies this universal property then, by the Yoneda lemma, it will be canonically isomorphic to the one defined above. Lemma 24.34.5. Let f : X → S be a morphism of schemes. The map HomOX (ΩX/S , F) −→ DerS (OX , F), α 7−→ α ◦ dX/S is an isomorphism of functors from the category of OX -modules to the category of sets. Proof. Let F be an OX -module. Let D ∈ DerS (OX , F). We have to show there exists a unique OX -linear map α : ΩX/S → F such that D = α ◦ dX/S . We claim that the image of dX/S : OX → ΩX/S generates ΩX/S as an OX -module. To see this it suffices to prove this is true on suitable affine opens. We can cover X by affine opens Spec(A) = W ⊂ X whose image is contained in an affine open Spec(R) = V ⊂ S. As seen in the discussion leading up to Definition 24.34.4 we have ]2 ΩX/S |W = J/J with J = Ker(A⊗R A → A). Now clearly J is generated by the elements 1⊗f −f ⊗1. Hence the claim follows. This claim implies immediately that α, if it exists, is unique. Next, we come to existence of α. Note that the construction of the pair (ΩX/S , dX/S ) commutes with restriction to open subschemes (in both X and S). Proof omitted. By the uniqueness just shown, it therefore suffices to prove existence in case both X

24.34. SHEAF OF DIFFERENTIALS OF A MORPHISM

1429

and S are affine. Thus we may write X = Spec(A), S = Spec(S) and M = Γ(X, F). ]2 . According to Algebra, Set as usual J = Ker(A ⊗R A → A) so that ΩX/S = J/J Lemmas 7.123.3 and 7.123.13 there exists a unique A-linear map α0 : J/J 2 → M such that the composition d ◦ α0 : A → J/J 2 → M is equal to the action of D on global sections over X. By Schemes, Lemma 21.7.1 the A-linear map α0 corresponds ]2 → F. Then the derivations α ◦ dX/S and D have the to a map α : ΩX/S = J/J same effect on global sections and hence agree by Lemma 24.34.2. This proves existence and we win.  Lemma 24.34.6. Let f : X → S be a morphism of schemes. Let U ⊂ X, V ⊂ S be open subschemes such that f (U ) ⊂ V . Then there is a unique isomorphism ΩX/S |U = ΩU/V of OU -modules such that dX/S |U = dU/V . Proof. The existence of the isomorphism is clear from the construction of ΩX/S . Uniqueness comes from the fact, seen in the proof of Lemma 24.34.5, that the image of dX/S : OX → ΩX/S generates ΩX/S as an OX -module.  From now on we will use these canonical identifications and simply write ΩU/S or ΩU/V for the restriction of ΩX/S to U . Lemma 24.34.7. Let f : X → S be a morphism of schemes. For any pair of affine opens Spec(A) = U ⊂ X, Spec(R) = V ⊂ S with f (U ) ⊂ V there is a unique isomorphism Γ(U, OX/S ) = ΩA/R . compatible with dX/S and d : A → ΩA/R . Proof. During the construction of ΩX/S we have seen that the restriction of ΩX/S to U is isomorphic to the quasi-coherent sheaf associated to the A-module J/J 2 where J = Ker(A ⊗R A → A). Hence the result follows from Algebra, Lemma 7.123.13. An alternative proof is to show that the A-module M = Γ(U, ΩX/S ) = Γ(U, ΩU/V ) together with dX/S = dU/V : A → M is a universal R-derivation of A. This follows by combining Lemmas 24.34.2 and 24.34.5 above. The universal property of d : A → ΩA/R (see Algebra, Lemma 7.123.3) and the Yoneda lemma (Categories, Lemma 4.3.5) imply there is a unique isomorphism of A-modules M ∼ = ΩA/R compatible with derivations. This gives the second proof.  Remark 24.34.8. The lemma above gives a second way of constructing the module of differentials. Namely, let f : X → S be a morphism of schemes. Consider the collection of all affine opens U ⊂ X which map into an affine open of S. These form a basis for the topology on X. Thus it suffices to define Γ(U, ΩX/S ) for such U . We simply set Γ(U, ΩX/S ) = ΩA/R if A, R are as in Lemma 24.34.7 above. This works, but it takes somewhat more algebraic preliminaries to construct the restriction mappings and to verify the sheaf condition with this ansatz. Lemma 24.34.9. Let X0  S0

f

/X  /S

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24. MORPHISMS OF SCHEMES

be a commutative diagram of schemes. The canonical map OX → f∗ OX 0 composed with the map f∗ dX 0 /S 0 : f∗ OX 0 → f∗ ΩX 0 /S 0 is a S-derivation. Hence we obtain a canonical map of OX -modules ΩX/S → f∗ ΩX 0 /S 0 , and by adjointness of f∗ and f ∗ a canonical OX 0 -module homomorphism cf : f ∗ ΩX/S −→ ΩX 0 /S 0 . It is uniquely characterized by the property that f ∗ dX/S (h) mapsto dX 0 /S 0 (f ∗ h) for any local section h of OX . Proof. Everything but the last assertion of the lemma is proven in the lemma; the universal property of ΩX/S is Lemma 24.34.5. The last assertion means that cf is the unique OX 0 -linear map such that whenever U ⊂ X is open and h ∈ OX (U ), then the pullback f ∗ h ∈ OX 0 (f −1 U ) of h satisfies dX 0 /S 0 (f ∗ h) = cf (f ∗ dX/S (h)). We omit the proof. We can also use the functoriality of the conormal sheaves (see Lemma 24.33.3) to define cf . Or we can use the characterization in the last line of the lemma to glue maps defined on affine patches (see Algebra, Equation (7.123.5.1)).  Lemma 24.34.10. Let X 00

g

/ X0

f

/X

   / S0 /S S 00 be a commutative diagram of schemes. Then we have cf ◦g = cg ◦ g ∗ cf as maps (f ◦ g)∗ ΩX/S → ΩX 00 /S 00 . Proof. Omitted. One way to see this is to restrict to affine opens.



Lemma 24.34.11. Let f : X → Y , g : Y → S be morphisms of schemes. Then there is a canonical exact sequence f ∗ ΩY /S → ΩX/S → ΩX/Y → 0 where the maps come from applications of Lemma 24.34.9. Proof. This is the sheafified version of Algebra, Lemma 7.123.7.



Lemma 24.34.12. Let X → S be a morphism of schemes. Let g : S 0 → S be a morphism of schemes. Let X 0 = XS 0 be the base change of X. Denote g 0 : X 0 → X the projection. Then the map (g 0 )∗ ΩX/S → ΩX 0 /S 0 of Lemma 24.34.9 is an isomorphism. Proof. This is the sheafified version of Algebra, Lemma 7.123.12.



Lemma 24.34.13. Let f : X → S and g : Y → S be morphisms of schemes with the same target. Let p : X ×S Y → X and q : X ×S Y → Y be the projection morphisms. The maps from Lemma 24.34.9 p∗ ΩX/S ⊕ q ∗ ΩY /S −→ ΩX×S Y /S give an isomorphism.

24.34. SHEAF OF DIFFERENTIALS OF A MORPHISM

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Proof. By Lemma 24.34.12 the composition p∗ ΩX/S → ΩX×S Y /S → ΩX×S Y /Y is an isomorphism, and similarly for q. Moreover, the cokernel of p∗ ΩX/S → ΩX×S Y /S is ΩX×S Y /X by Lemma 24.34.11. The result follows.  Lemma 24.34.14. Let f : X → S be a morphism of schemes. If f is locally of finite type, then ΩX/S is a finite type OX -module. Proof. Immediate from Algebra, Lemma 7.123.16, Lemma 24.34.7, Lemma 24.16.2, and Properties, Lemma 23.16.1.  Lemma 24.34.15. Let f : X → S be a morphism of schemes. If f is locally of finite type, then ΩX/S is an OX -module of finite presentation. Proof. Immediate from Algebra, Lemma 7.123.15, Lemma 24.34.7, Lemma 24.22.2, and Properties, Lemma 23.16.2.  Lemma 24.34.16. If X → S is an immersion, or more generally a monomorphism, then ΩX/S is zero. Proof. This is true because ∆X/S is an isomorphism in this case and hence has trivial conormal sheaf. The algebraic version is Algebra, Lemma 7.123.5.  Lemma 24.34.17. Let i : Z → X be an immersion of schemes over S. There is a canonical exact sequence CZ/X → i∗ ΩX/S → ΩZ/S → 0 where the first arrow is induced by dX/S and the second arrow comes from Lemma 24.34.9. Proof. This is the sheafified version of Algebra, Lemma 7.123.9. However we should make sure we can define the first arrow globally. Hence we explain the meaning of “induced by dX/S ” here. Namely, we may assume that i is a closed immersion by shrinking X. Let I ⊂ OX be the sheaf of ideals corresponding to Z ⊂ X. Then dX/S : I → ΩX/S maps the subsheaf I 2 ⊂ I to IΩX/S . Hence it induces a map I/I 2 → ΩX/S /IΩX/S which is OX /I-linear. By Lemma 24.4.1 this corresponds to a map CZ/X → i∗ ΩX/S as desired.  Lemma 24.34.18. Let i : Z → X be an immersion of schemes over S, and assume i (locally) has a left inverse. Then the canonical sequence 0 → CZ/X → i∗ ΩX/S → ΩZ/S → 0 of Lemma 24.34.17 is (locally) split exact. In particular, if s : S → X is a section of the structure morphism X → S then the map CS/X → s∗ ΩX/S induced by dX/S is an isomorphism. Proof. Follows from Algebra, Lemma 7.123.10. Clarification: if g : X → Z is a left inverse of i, then i∗ cg is a right inverse of the map i∗ ΩX/S → ΩZ/S . Also, if s is a section, then it is an immersion s : Z = S → X over S (see Schemes, Lemma 21.21.12) and in that case ΩZ/S = 0.  Remark 24.34.19. Let X → S be a morphism of schemes. According to Lemma 24.34.13 we have ΩX×S X/S = pr∗1 ΩX/S ⊕ pr∗2 ΩX/S

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On the other hand, the diagonal morphism ∆ : X → X ×S X is an immersion, which locally has a left inverse. Hence by Lemma 24.34.18 we obtain a canonical short exact sequence 0 → CX/X×S X → ΩX/S ⊕ ΩX/S → ΩX/S → 0 Note that the right arrow is (1, 1) which is indeed a split surjection. On the other hand, by our very definition we have ΩX/S = CX/X×S X . Because we chose dX/S (f ) = s2 (f ) − s1 (f ) in Definition 24.34.4 it turns out that the left arrow is the map (−1, 1)8. Lemma 24.34.20. Let Z

i

/X

 Y be a commutative diagram of schemes where i and j are immersions. Then there is a canonical exact sequence j

CZ/Y → CZ/X → i∗ ΩX/Y → 0 where the first arrow comes from Lemma 24.33.3 and the second from Lemma 24.34.17. Proof. The algebraic version of this is Algebra, Lemma 7.124.6.



24.35. Smooth morphisms Let f : X → Y be a map of topological spaces. Consider the following condition: (*) For every x ∈ X there exist open neighbourhoods x ∈ U ⊂ X and f (x) ∈ V ⊂ Y , and an integer d such that f (U ) = V and such that there is an isomorphism V × Bd (0, 1)  V

∼ =

/U

/X

 V

 /Y

where Bd (0, 1) ⊂ Rd is a ball of radius 1 around 0. Smooth morphisms are the analogue of such morphisms in the category of schemes. See Lemma 24.35.11 and Lemma 24.37.20. Contrary to expectations (perhaps) the notion of a smooth ring map is not defined solely in terms of the module of differentials. Namely, recall that R → A is a smooth ring map if A is of finite presentation over R and if the naive cotangent complex of A over R is quasi-isomorphic to a projective module placed in degree 0, see Algebra, Definition 7.127.1. Definition 24.35.1. Let f : X → S be a morphism of schemes. (1) We say that f is smooth at x ∈ X if there exists a affine open neighbourhood Spec(A) = U ⊂ X of x and and affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is smooth. 8Namely, the local section d X/S (f ) = 1⊗f −f ⊗1 of the ideal sheaf of ∆ maps via dX×S X/X

to the local section 1⊗1⊗1⊗f −1⊗f ⊗1⊗1−1⊗1⊗f ⊗1+f ⊗1⊗1⊗1 = pr∗2 dX/S (f )−pr∗1 dX/S (f ).

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(2) We say that f is smooth if it is smooth at every point of X. (3) A morphism of affine schemes f : X → S is called standard smooth there exists a standard smooth ring map R → R[x1 , . . . , xn ]/(f1 , . . . , fc ) (see Algebra, Definition 7.127.6) such that X → S is isomorphic to Spec(R[x1 , . . . , xn ]/(f1 , . . . , fc )) → Spec(R). A pleasing feature of this definition is that the set of points where a morphism is smooth is automatically open. Note that there is no separation or quasi-compactness hypotheses in the definition. Hence the question of being smooth is local in nature on the source. Here is the precise result. Lemma 24.35.2. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is smooth. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is smooth. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S → Vj , j ∈ J, i ∈ Ij is smooth. i∈Ij Ui such that each of the morphisms UiS (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is smooth, for all j ∈ J, i ∈ Ij . Moreover, if f is smooth then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is smooth. Proof. This follows from Lemma 24.15.3 if we show that the property “R → A is smooth” is local. We check conditions (a), (b) and (c) of Definition 24.15.1. By Algebra, Lemma 7.127.4 being smooth is stable under base change and hence we conclude (a) holds. By Algebra, Lemma 7.127.14 being smooth is stable under composition and for any ring R the ring map R → Rf is (standard) smooth. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma 7.127.13.  The following lemma characterizes a smooth morphism as a flat, finitely presented morphism with smooth fibres. Note that schemes smooth over a field are discussed in more detail in Varieties, Section 28.15. Lemma 24.35.3. Let f : X → S be a morphism of schemes. If f is flat, locally of finite presentation, and all fibres Xs are smooth, then f is smooth. Proof. Follows from Algebra, Lemma 7.127.16.



Lemma 24.35.4. The composition of two morphisms which are smooth is smooth. Proof. In the proof of Lemma 24.35.2 we saw that being smooth is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being smooth is a property of ring maps that is stable under composition, see Algebra, Lemma 7.127.14.  Lemma 24.35.5. The base change of a morphism which is smooth is smooth.

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Proof. In the proof of Lemma 24.35.2 we saw that being smooth is a local property of ring maps. Hence the lemma follows from Lemma 24.15.5 combined with the fact that being smooth is a property of ring maps that is stable under base change, see Algebra, Lemma 7.127.4.  Lemma 24.35.6. Any open immersion is smooth. Proof. This is true because an open immersion is a local isomorphism.



Lemma 24.35.7. A smooth morphism is syntomic. Proof. See Algebra, Lemma 7.127.10.



Lemma 24.35.8. A smooth morphism is locally of finite presentation. Proof. True because a smooth ring map is of finite presentation by definition.



Lemma 24.35.9. A smooth morphism is flat. Proof. Combine Lemmas 24.32.7 and 24.35.7.



Lemma 24.35.10. A smooth morphism is universally open. Proof. Combine Lemmas 24.35.9, 24.35.8, and 24.26.9. Or alternatively, combine Lemmas 24.35.7, 24.32.8.  The following lemma says locally any smooth morphism is standard smooth. Hence we can use standard smooth morphisms as a local model for a smooth morphism. Lemma 24.35.11. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f (x). The following are equivalent (1) The morphism f is smooth at x. (2) There exist affine opens U ⊂ X, and V ⊂ S such that x ∈ U , f (U ) ⊂ V and the induced morphism f |U : U → V is standard smooth. Proof. Follows from the definitions and Algebra, Lemmas 7.127.7 and 7.127.10.  Lemma 24.35.12. Let f : X → S be a morphism of schemes. Assume f is smooth. Then the module of differentials ΩX/S of X over S is finite locally free and rankx (ΩX/S ) = dimx (Xf (x) ) for every x ∈ X. Proof. The statement is local on X and S. By Lemma 24.35.11 above we may assume that f is a standard smooth morphism of affines. In this case the result follows from Algebra, Lemma 7.127.7 (and the definition of a relative global complete intersection, see Algebra, Definition 7.126.5).  Lemma 24.35.12 says that the following definition makes sense. Definition 24.35.13. Let d ≥ 0 be an integer. We say a morphism of schemes f : X → S is smooth of relative dimension d if f is smooth and ΩX/S is finite locally free of constant rank d.

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In other words, f is smooth and the nonempty fibres are equidimensional of dimension d. By Lemma 24.35.14 below this is also the same as requiring: (a) f is locally of finite presentation, (b) f is flat, (c) all nonempty fibres equidimensional of dimension d, and (d) ΩX/S finite locally free of rank d. It is not enough to simply assume that f is flat, of finite presentation, and ΩX/S is finite locally free of rank d. A counter example is given by Spec(Fp [t]) → Spec(Fp [tp ]). Here is a differential criterion of smoothness at a point. There are many variants of this result all of which may be useful at some point. We will just add them here as needed. Lemma 24.35.14. Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f (x). Assume f is locally of finite presentation. The following are equivalent: (1) The morphism f is smooth at x. (2) The local ring map OS,s → OX,x is flat and the OX,x -module ΩX/S,x can be generated by at most dimx (Xf (x) ) elements. (3) The local ring map OS,s → OX,x is flat and the κ(x)-vector space ΩXs /s,x ⊗OXs ,x κ(x) = ΩX/S,x ⊗OX,x κ(x) can be generated by at most dimx (Xf (x) ) elements. (4) There exist affine opens U ⊂ X, and V ⊂ S such that x ∈ U , f (U ) ⊂ V and the induced morphism f |U : U → V is standard smooth. (5) There exist affine opens Spec(A) = U ⊂ X and Spec(R) = V ⊂ S with x ∈ U corresponding to q ⊂ A, and f (U ) ⊂ V such that there exists a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fc ) with 

∂f1 /∂x1 ∂f1 /∂x2 g = det   ... ∂f1 /∂xc

∂f2 /∂x1 ∂f2 /∂x2 ... ∂f2 /∂xc

 . . . ∂fc /∂x1 . . . ∂fc /∂x2   ... ...  . . . ∂fc /∂xc

mapping to an element of A not in q. Proof. Note that if f is smooth at x, then we see from Lemma 24.35.11 that (4) holds, and (5) is a slightly weakened version of (4). Moreover, this implies that the ring map OS,s → OX,x is flat (see Lemma 24.35.9) and that ΩX/S is finite locally free of rank equal to dimx (Xs ) (see Lemma 24.35.12). This implies (2) and (3). By Lemma 24.34.12 the module of differentials ΩXs /s of the fibre Xs over κ(s) is the pullback of the module of differentials ΩX/S of X over S. Hence the displayed equality in part (3) of the lemma. By Lemma 24.34.14 these modules are of finite type. Hence the mimimal number of generators of the modules ΩX/S,x and ΩXs /s,x is the same and equal to the dimension of this κ(x)-vector space by Nakayama’s Lemma (Algebra, Lemma 7.18.1). This in particular shows that (2) and (3) are equivalent. Combining Algebra, Lemmas 7.127.16 and 7.130.3 shows that (2) and (3) imply (1). Finally, (5) implies (4) see for example Algebra, Example 7.127.8. 

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Lemma 24.35.15. Let f : X → S be a morphism of schemes. Assume f locally of finite type. Formation of the set T = {x ∈ X | Xf (x) is smooth over κ(f (x)) at x} commutes with arbitrary base change: For any morphism g : S 0 → S, consider the base change f 0 : X 0 → S 0 of f and the projection g 0 : X 0 → X. Then the corresponding set T 0 for the morphism f 0 is equal to T 0 = (g 0 )−1 (T ). In particular, if f is assumed flat, and locally of finite presentation then the same holds for the open set of points where f is smooth. Proof. Let s0 ∈ S 0 be a point, and let s = g(s0 ). Then we have Xs0 0 = Spec(κ(s0 )) ×Spec(κ(s)) Xs In other words the fibres of the base change are the base changes of the fibres. Hence the first part is equivalent to Algebra, Lemma 7.127.18. The second part follows from the first because in that case T is the (open) set of points where f is smooth according to Lemma 24.35.3.  Here is a lemma that actually uses the vanishing of H −1 of the naive cotangent complex for a smooth ring map. Lemma 24.35.16. Let f : X → Y , g : Y → S be morphisms of schemes. Assume f is smooth. Then 0 → f ∗ ΩY /S → ΩX/S → ΩX/Y → 0 (see Lemma 24.34.11) is short exact. Proof. The algebraic version of this lemma is the following: Given ring maps A → B → C with B → C smooth, then the sequence 0 → C ⊗B ΩB/A → ΩC/A → ΩC/B → 0 of Algebra, Lemma 7.123.7 is exact. This is Algebra, Lemma 7.129.1.



Lemma 24.35.17. Let i : Z → X be an immersion of schemes over S. Assume that Z is smooth over S. Then the canonical exact sequence 0 → CZ/X → i∗ ΩX/S → ΩZ/S → 0 of Lemma 24.34.17 is short exact. Proof. The algebraic version of this lemma is the following: Given ring maps A → B → C with A → C smooth and B → C surjective with kernel J, then the sequence 0 → J/J 2 → C ⊗B ΩB/A → ΩC/A → 0 of Algebra, Lemma 7.123.9 is exact. This is Algebra, Lemma 7.129.2.  Lemma 24.35.18. Let Z

i

/X

 Y be a commutative diagram of schemes where i and j are immersions and X → Y is smooth. Then the canonical exact sequence j

0 → CZ/Y → CZ/X → i∗ ΩX/Y → 0

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of Lemma 24.34.20 is exact. Proof. The algebraic version of this lemma is the following: Given ring maps A → B → C with A → C surjective and A → B smooth, then the sequence 0 → I/I 2 → J/J 2 → C ⊗B ΩB/A → 0 of Algebra, Lemma 7.124.6 is exact. This is Algebra, Lemma 7.129.3. Lemma 24.35.19. Let X

/Y

f p







q

S be a commutative diagram of morphisms of schemes. Assume that (1) f is surjective, and smooth, (2) p is smooth, and (3) q is locally of finite presentation9. Then q is smooth. Proof. By Lemma 24.26.11 we see that q is flat. Pick a point y ∈ Y . Pick a point x ∈ X mapping to y. Suppose f has relative dimension a at x and p has relative dimension b at x. By Lemma 24.35.12 this means that ΩX/S,x is free of rank b and ΩX/Y,x is free of rank a. By the short exact sequence of Lemma 24.35.16 this means that (f ∗ ΩY /S )x is free of rank b − a. By Nakayama’s Lemma this implies that ΩY /S,y can be generated by b − a elements. Also, by Lemma 24.29.2 we see that dimy (Ys ) = b − a. Hence we conclude that Y → S is smooth at y by Lemma 24.35.14 part (2).  In the situation of the following lemma the image of σ is locally on X cut out by a regular sequence, see Divisors, Lemma 26.14.7. Lemma 24.35.20. Let f : X → S be a morphism of schemes. Let σ : S → X be a section of f . Let s ∈ S be a point such that f is smooth at x = σ(s). Then there exist affine open neighbourhoods Spec(A) = U ⊂ S of s and Spec(B) = V ⊂ X of x such that (1) f (V ) ⊂ U and σ(U ) ⊂ V , (2) with I = Ker(σ # : B → A) the module I/I 2 is a free A-module, and (3) B ∧ ∼ = A[[x1 , . . . , xd ]] as A-algebras where B ∧ denotes the completion of B with respect to I. Proof. Pick an affine open U ⊂ S containing s Pick an affine open V ⊂ f −1 (U ) containing x. Pick an affine open U 0 ⊂ σ −1 (V ) containing s. Note that V 0 = f −1 (U 0 ) ∩ V is affine as it is equal to the fibre product V 0 = U 0 ×U V . Then U 0 and V 0 satisfy (1). Write U 0 = Spec(A0 ) and V 0 = Spec(B 0 ). By Algebra, Lemma 7.129.4 the module I 0 /(I 0 )2 is finite locally free as a A0 -module. Hence after replacing U 0 by a smaller affine open U 00 ⊂ U 0 and V 0 by V 00 = V 0 ∩ f −1 (U 00 ) we obtain the situation where I 00 /(I 00 )2 is free, i.e., (2) holds. In this case (3) holds also by Algebra, Lemma 7.129.4.  9In fact this is implied by (1) and (2), see Descent, Lemma 31.10.3. Moreover, it suffices to assume f is surjective, flat and locally of finite presentation, see Descent, Lemma 31.10.5.

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24.36. Unramified morphisms We briefly discuss unramified morphisms before the (perhaps) more interesting class of ´etale morphisms. Recall that a ring map R → A is unramified if it is of finite type and ΩA/R = 0 (this is the definition of [Ray70]). A ring map R → A is called G-unramified if it is of finite presentation and ΩA/R = 0 (this is the definition of [DG67]). See Algebra, Definition 7.139.1. Definition 24.36.1. Let f : X → S be a morphism of schemes. (1) We say that f is unramified at x ∈ X if there exists a affine open neighbourhood Spec(A) = U ⊂ X of x and and affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is unramified. (2) We say that f is G-unramified at x ∈ X if there exists a affine open neighbourhood Spec(A) = U ⊂ X of x and and affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is Gunramified. (3) We say that f is unramified if it is unramified at every point of X. (4) We say that f is G-unramified if it is G-unramified at every point of X. Note that a G-unramified morphism is unramified. Hence any result for unramified morphisms implies the corresponding result for G-unramified morphisms. Moreover, if S is locally Noetherian then there is no difference between G-unramified and unramified morphisms, see Lemma 24.36.6. A pleasing feature of this definition is that the set of points where a morphism is unramified (resp. G-unramified) is automatically open. Lemma 24.36.2. Let f : X → S be a morphism of schemes. Then (1) f is unramified if and only if f is locally of finite type and ΩX/S = 0, and (2) f is G-unramified if and only if f is locally of finite presentation and ΩX/S = 0. Proof. By definition a ring map R → A is unramified (resp. G-unramified) if and only if it is of finite type (resp. finite presentation) and ΩA/R = 0. Hence the lemma follows directly from the definitions and Lemma 24.34.7.  Note that there is no separation or quasi-compactness hypotheses in the definition. Hence the question of being unramified is local in nature on the source. Here is the precise result. Lemma 24.36.3. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is unramified (resp. G-unramified). (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is unramified (resp. S G-unramified). (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S i∈Ij Ui such that each of the morphisms Ui → Vj , j ∈ J, i ∈ Ij is unramified (resp. G-unramified). S (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is unramified (resp. G-unramified), for all j ∈ J, i ∈ Ij .

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Moreover, if f is unramified (resp. G-unramified) then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is unramified (resp. G-unramified). Proof. This follows from Lemma 24.15.3 if we show that the property “R → A is unramified” is local. We check conditions (a), (b) and (c) of Definition 24.15.1. These properties are proved in Algebra, Lemma 7.139.3.  Lemma 24.36.4. The composition of two morphisms which are unramified is unramified. The same holds for G-unramified morphisms. Proof. The proof of Lemma 24.36.3 shows that being unramified (resp. G-unramified) is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being unramified (resp. G-unramified) is a property of ring maps that is stable under composition, see Algebra, Lemma 7.139.3.  Lemma 24.36.5. The base change of a morphism which is unramified is unramified. The same holds for G-unramified morphisms. Proof. The proof of Lemma 24.36.3 shows that being unramified (resp. G-unramified) is a local property of ring maps. Hence the lemma follows from Lemma 24.15.5 combined with the fact that being unramified (resp. G-unramified) is a property of ring maps that is stable under base change, see Algebra, Lemma 7.139.3.  Lemma 24.36.6. Let f : X → S be a morphism of schemes. Assume S is locally Noetherian. Then f is unramified if and only if f is G-unramified. Proof. Follows from the definitions and Lemma 24.22.9.



Lemma 24.36.7. Any open immersion is G-unramified. Proof. This is true because an open immersion is a local isomorphism.



Lemma 24.36.8. A closed immersion i : Z → X is unramified. It is G-unramified if and only if the associated quasi-coherent sheaf of ideals I = Ker(OX → i∗ OZ ) is of finite type (as an OX -module). Proof. Follows from Lemma 24.22.7 and Algebra, Lemma 7.139.3.



Lemma 24.36.9. An unramified morphism is locally of finite type. A G-unramified morphism is locally of finite presentation. Proof. An unramified ring map is of finite type by definition. A G-unramified ring map is of finite presentation by definition.  Lemma 24.36.10. Let f : X → S be a morphism of schemes. If f is unramified at x then f is quasi-finite at x. In particular, an unramified morphism is locally quasi-finite. Proof. See Algebra, Lemma 7.139.6.



Lemma 24.36.11. Fibres of unramified morphisms. (1) Let X be a scheme over a field k. The structure morphism X → Spec(k) is unramified if and only if X is a disjoint union of spectra of finite separable field extensions of k.

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(2) If f : X → S is an unramified morphism then for every s ∈ S the fibre Xs is a disjoint union of spectra of finite separable field extensions of κ(s). Proof. Part (2) follows from part (1) and Lemma 24.36.5. Let us prove part (1). We first use Algebra, Lemma 7.139.7. This lemma implies that if X is a disjoint union of spectra of finite separable field extensions of k then X → Spec(k) is unramified. Conversely, suppose that X → Spec(k) is unramified. By Algebra, Lemma 7.139.5 for every x ∈ X the residue field extension k ⊂ κ(x) is finite separable. Hence all points of X are closed points (see Lemma 24.21.2 for example). Thus X is a discrete space, in particular the disjoint union of the spectra of its local rings. By Algebra, Lemma 7.139.5 again these local rings are fields, and we win.  The following lemma characterizes an unramified morphisms as morphisms locally of finite type with unramified fibres. Lemma 24.36.12. Let f : X → S be a morphism of schemes. (1) If f is unramified then for any x ∈ X the field extension κ(f (x)) ⊂ κ(x) is finite separable. (2) If f is locally of finite type, and for every s ∈ S the fibre Xs is a disjoint union of spectra of finite separable field extensions of κ(s) then f is unramified. (3) If f is locally of finite presentation, and for every s ∈ S the fibre Xs is a disjoint union of spectra of finite separable field extensions of κ(s) then f is G-unramified. Proof. Follows from Algebra, Lemmas 7.139.5 and 7.139.7.



Here is a characterization of unramified morphisms in terms of the diagonal morphism. Lemma 24.36.13. Let f : X → S be a morphism. (1) If f is unramified, then the diagonal morphism ∆ : X → X ×S X is an open immersion. (2) If f is locally of finite type and ∆ is an open immersion, then f is unramified. (3) If f is locally of finite presentation and ∆ is an open immersion, then f is G-unramified. Proof. The first statement follows from Algebra, Lemma 7.139.4. The second statement from the fact that ΩX/S (see Definition 24.34.4) is the conormal sheaf of the diagonal morphism and hence clearly zero if ∆ is an open immersion.  Lemma 24.36.14. Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f (x). Assume f is locally of finite type (resp. locally of finite presentation). The following are equivalent: (1) The morphism f is unramified (resp. G-unramified) at x. (2) The fibre Xs is unramified over κ(s) at x. (3) The OX,x -module ΩX/S,x is zero. (4) The OXs ,x -module ΩXs /s,x is zero. (5) The κ(x)-vector space ΩXs /s,x ⊗OXs ,x κ(x) = ΩX/S,x ⊗OX,x κ(x) is zero.

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(6) We have ms OX,x = mx and the field extension κ(s) ⊂ κ(x) is finite separable. Proof. Note that if f is unramified at x, then we see that ΩX/S = 0 in a neighbourhood of x by the definitions and the results on modules of differentials in Section 24.34. Hence (1) implies (3) and the vanishing of the right hand vector space in (5). It also implies (2) because by Lemma 24.34.12 the module of differentials ΩXs /s of the fibre Xs over κ(s) is the pullback of the module of differentials ΩX/S of X over S. This fact on modules of differentials also implies the displayed equality of vector spaces in part (4). By Lemma 24.34.14 the modules ΩX/S,x and ΩXs /s,x are of finite type. Hence he modules ΩX/S,x and ΩXs /s,x are zero if and only if the corresponding κ(x)-vector space in (4) is zero by Nakayama’s Lemma (Algebra, Lemma 7.18.1). This in particular shows that (3), (4) and (5) are equivalent. The support of ΩX/S is closed in X, see Modules, Lemma 15.9.6. Assumption (3) implies that x is not in the support. Hence ΩX/S is zero in a neighbourhood of x, which implies (1). The equivalence of (1) and (3) applied to Xs → s implies the equivalence of (2) and (4). At this point we have seen that (1) – (5) are equivalent. Alternatively you can use Algebra, Lemma 7.139.3 to see the equivalence of (1) – (5) more directly. The equivalence of (1) and (6) follows from Lemma 24.36.12. It also follows more directly from Algebra, Lemmas 7.139.5 and 7.139.7.  Lemma 24.36.15. Let f : X → S be a morphism of schemes. Assume f locally of finite type. Formation of the open set T = {x ∈ X | Xf (x) is unramified over κ(f (x)) at x} = {x ∈ X | X is unramified over S at x} commutes with arbitrary base change: For any morphism g : S 0 → S, consider the base change f 0 : X 0 → S 0 of f and the projection g 0 : X 0 → X. Then the corresponding set T 0 for the morphism f 0 is equal to T 0 = (g 0 )−1 (T ). If f is assumed locally of finite presentation then the same holds for the open set of points where f is G-unramified. Proof. Let s0 ∈ S 0 be a point, and let s = g(s0 ). Then we have Xs0 0 = Spec(κ(s0 )) ×Spec(κ(s)) Xs In other words the fibres of the base change are the base changes of the fibres. In particular ΩXs /s,x ⊗OXs ,x κ(x0 ) = ΩX 0 0 /s0 ,x0 ⊗OX 0 ,x0 κ(x0 ) s

s0

see Lemma 24.34.12. Whence x0 ∈ T 0 if and only if x ∈ T by Lemma 24.36.14. The second part follows from the first because in that case T is the (open) set of points where f is G-unramified according to Lemma 24.36.14.  Lemma 24.36.16. Let f : X → Y be a morphism of schemes over S. (1) If X is unramified over S, then f is unramified. (2) If X is G-unramified over S and Y of finite type over S, then f is Gunramified.

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Proof. Assume that X is unramified over S. By Lemma 24.16.8 we see that f is locally of finite type. By assumption we have ΩX/S = 0. Hence ΩX/Y = 0 by Lemma 24.34.11. Thus f is unramified. If X is G-unramified over S and Y of finite type over S, then by Lemma 24.22.11 we see that f is locally of finite presentation and we conclude that f is G-unramified.  Lemma 24.36.17. Let S be a scheme. Let X, Y be schemes over S. Let f, g : X → Y be morphisms over S. Let x ∈ X. Assume that (1) the structure morphism Y → S is unramified, (2) f (x) = g(x) in Y , say y = f (x) = g(x), and (3) the induced maps f ] , g ] : κ(y) → κ(x) are equal. Then there exists an open neighbourhood of x in X on which f and g are equal. Proof. Consider the morphism (f, g) : X → Y ×S Y . By assumption (1) and Lemma 24.36.13 the inverse image of ∆Y /S (Y ) is open in X. And assumptions (2) and (3) imply that x is in this open subset.  ´ 24.37. Etale morphisms The Zariski topology of a scheme is a very coarse topology. This is particularly clear when looking at varieties over C. It turns out that declaring an ´etale morphism to be the analogue of a local isomorphism in topology introduces a much finer topology. On varieties over C this topology gives rise to the “correct” betti numbers when computing cohomology with finite coefficients. Another observable is that if f : X → Y is an ´etale morphism of varieties over C, and if x is a closed point of ∧ ∧ X, then f induces an isomorphism OY,f (x) → OX,x of complete local rings. In this section we start our study of these matters. In fact we deliberately restrict our discussion to a minimum since we will discuss more interesting results elsewhere. Recall that a ring map R → A is said to be ´etale if it is smooth and ΩA/R = 0, see Algebra, Definition 7.133.1. Definition 24.37.1. Let f : X → S be a morphism of schemes. (1) We say that f is ´etale at x ∈ X if there exists a affine open neighbourhood Spec(A) = U ⊂ X of x and and affine open Spec(R) = V ⊂ S with f (U ) ⊂ V such that the induced ring map R → A is ´etale. (2) We say that f is ´etale if it is ´etale at every point of X. (3) A morphism of affine schemes f : X → S is called standard ´etale if X → S is isomorphic to Spec(R[x]g /(f )) → Spec(R) where R → R[x]g /(f ) is a standard ´etale ring map, see Algebra, Definition 7.133.13, i.e., f is monic and f 0 invertible in R[x]g . A morphism is ´etale if and only if it is smooth of relative dimension 0 (see Definition 24.35.13). A pleasing feature of the definition is that the set of points where a morphism is ´etale is automatically open. Note that there is no separation or quasi-compactness hypotheses in the definition. Hence the question of being ´etale is local in nature on the source. Here is the precise result.

´ 24.37. ETALE MORPHISMS

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Lemma 24.37.2. Let f : X → S be a morphism of schemes. The following are equivalent (1) The morphism f is ´etale. (2) For every affine opens U ⊂ X, V ⊂ S with f (U ) ⊂ V the ring map OS (V ) → OX (U ) is ´etale. S (3) There exists an open covering S = j∈J Vj and open coverings f −1 (Vj ) = S etale. i → Vj , j ∈ J, i ∈ Ij is ´ i∈Ij Ui such that each of the morphisms US (4) There exists an affine open covering S = j∈J Vj and affine open covS erings f −1 (Vj ) = i∈Ij Ui such that the ring map OS (Vj ) → OX (Ui ) is ´etale, for all j ∈ J, i ∈ Ij . Moreover, if f is ´etale then for any open subschemes U ⊂ X, V ⊂ S with f (U ) ⊂ V the restriction f |U : U → V is ´etale. Proof. This follows from Lemma 24.15.3 if we show that the property “R → A is ´etale” is local. We check conditions (a), (b) and (c) of Definition 24.15.1. These all follow from Algebra, Lemma 7.133.3.  Lemma 24.37.3. The composition of two morphisms which are ´etale is ´etale. Proof. In the proof of Lemma 24.37.2 we saw that being ´etale is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 24.15.5 combined with the fact that being ´etale is a property of ring maps that is stable under composition, see Algebra, Lemma 7.133.3.  Lemma 24.37.4. The base change of a morphism which is ´etale is ´etale. Proof. In the proof of Lemma 24.37.2 we saw that being ´etale is a local property of ring maps. Hence the lemma follows from Lemma 24.15.5 combined with the fact that being ´etale is a property of ring maps that is stable under base change, see Algebra, Lemma 7.133.3.  Lemma 24.37.5. Let f : X → S be a morphism of schemes. Let x ∈ X. Then f is ´etale at x if and only if f is smooth and unramified at x. Proof. This follows immediately from the definitions.



Lemma 24.37.6. An ´etale morphism is locally quasi-finite. Proof. By Lemma 24.37.5 an ´etale morphism is unramified. By Lemma 24.36.10 an unramified morphism is locally quasi-finite.  Lemma 24.37.7. Fibres of ´etale morphisms. (1) Let X be a scheme over a field k. The structure morphism X → Spec(k) is ´etale if and only if X is a disjoint union of spectra of finite separable field extensions of k. (2) If f : X → S is an ´etale morphism, then for every s ∈ S the fibre Xs is a disjoint union of spectra of finite separable field extensions of κ(s). Proof. You can deduce this from Lemma 24.36.11 via Lemma 24.37.5 above. Here is a direct proof. We will use Algebra, Lemma 7.133.4. Hence it is clear that if X is a disjoint union of spectra of finite separable field extensions of k then X → Spec(k) is ´etale. Conversely, suppose that X → Spec(k) is ´etale. Then for any affine open U ⊂ X

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we see that U is a finite disjoint union of spectra of finite separable field extensions of k. Hence all points of X are closed points (see Lemma 24.21.2 for example). Thus X is a discrete space and we win.  The following lemma characterizes an ´etale morphism as a flat, finitely presented morphism with “´etale fibres”. Lemma 24.37.8. Let f : X → S be a morphism of schemes. If f is flat, locally of finite presentation, and for every s ∈ S the fibre Xs is a disjoint union of spectra of finite separable field extensions of κ(s), then f is ´etale. Proof. You can deduce this from Algebra, Lemma 7.133.7. Here is another proof. By Lemma 24.37.7 a fibre Xs is ´etale and hence smooth over s. By Lemma 24.35.3 we see that X → S is smooth. By Lemma 24.36.12 we see that f is unramified. We conclude by Lemma 24.37.5.  Lemma 24.37.9. Any open immersion is ´etale. Proof. This is true because an open immersion is a local isomorphism.



Lemma 24.37.10. An ´etale morphism is syntomic. Proof. See Algebra, Lemma 7.127.10 and use that an ´etale morphism is the same as a smooth morphism of relative dimension 0.  Lemma 24.37.11. An ´etale morphism is locally of finite presentation. Proof. True because an ´etale ring map is of finite presentation by definition.



Lemma 24.37.12. An ´etale morphism is flat. Proof. Combine Lemmas 24.32.7 and 24.37.10.



Lemma 24.37.13. An ´etale morphism is open. Proof. Combine Lemmas 24.37.12, 24.37.11, and 24.26.9.



The following lemma says locally any ´etale morphism is standard ´etale. This is actually kind of a tricky result to prove in complete generality. The tricky parts are hidden in the chapter on commutative algebra. Hence a standard ´etale morphism is a local model for a general ´etale morphism. Lemma 24.37.14. Let f : X → S be a morphism of schemes. Let x ∈ X be a point. Set s = f (x). The following are equivalent (1) The morphism f is ´etale at x. (2) There exist affine opens U ⊂ X, and V ⊂ S such that x ∈ U , f (U ) ⊂ V and the induced morphism f |U : U → V is standard ´etale (see Definition 24.37.1). Proof. Follows from the definitions and Algebra, Proposition 7.133.16.



Here is a differential criterion of ´etaleness at a point. There are many variants of this result all of which may be useful at some point. We will just add them here as needed. Lemma 24.37.15. Let f : X → S be a morphism of schemes. Let x ∈ X. Set s = f (x). Assume f is locally of finite presentation. The following are equivalent:

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(1) The morphism f is ´etale at x. (2) The local ring map OS,s → OX,x is flat and the OX,x -module ΩX/S,x is zero. (3) The local ring map OS,s → OX,x is flat and the κ(x)-vector space ΩXs /s,x ⊗OXs ,x κ(x) = ΩX/S,x ⊗OX,x κ(x) is zero. (4) The local ring map OS,s → OX,x is flat, we have ms OX,x = mx and the field extension κ(s) ⊂ κ(x) is finite separable. (5) There exist affine opens U ⊂ X, and V ⊂ S such that x ∈ U , f (U ) ⊂ V and the induced morphism f |U : U → V is standard smooth of relative dimension 0. (6) There exist affine opens Spec(A) = U ⊂ X and Spec(R) = V ⊂ S with x ∈ U corresponding to q ⊂ A, and f (U ) ⊂ V such that there exists a presentation A = R[x1 , . . . , xn ]/(f1 , . . . , fn ) with 

∂f1 /∂x1  ∂f1 /∂x2 g = det   ... ∂f1 /∂xn

∂f2 /∂x1 ∂f2 /∂x2 ... ∂f2 /∂xn

 . . . ∂fn /∂x1 . . . ∂fn /∂x2   ... ...  . . . ∂fn /∂xn

mapping to an element of A not in q. (7) There exist affine opens U ⊂ X, and V ⊂ S such that x ∈ U , f (U ) ⊂ V and the induced morphism f |U : U → V is standard ´etale. (8) There exist affine opens Spec(A) = U ⊂ X and Spec(R) = V ⊂ S with x ∈ U corresponding to q ⊂ A, and f (U ) ⊂ V such that there exists a presentation A = R[x]Q /(P ) = R[x, 1/Q]/(P ) with P, Q ∈ R[x], P monic and P 0 = dP/dx mapping to an element of A not in q. Proof. Use Lemma 24.37.14 and the definitions to see that (1) implies all of the other conditions. For each of the conditions (2) – (7) combine Lemmas 24.35.14 and 24.36.14 to see that (1) holds by showing f is both smooth and unramified at x and applying Lemma 24.37.5. Some details omitted.  Lemma 24.37.16. A morphism is ´etale at a point if and only if it is flat and G-unramified at that point. A morphism is ´etale if and only if it is flat and Gunramified. Proof. This is clear from Lemmas 24.37.15 and 24.36.14.



Lemma 24.37.17. Let f : X → S be a morphism of schemes. Assume f locally of finite type. Formation of the set T = {x ∈ X | Xf (x) is ´etale over κ(f (x)) at x} commutes with arbitrary base change: For any morphism g : S 0 → S, consider the base change f 0 : X 0 → S 0 of f and the projection g 0 : X 0 → X. Then the corresponding set T 0 for the morphism f 0 is equal to T 0 = (g 0 )−1 (T ). In particular,

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if f is assumed locally of finite presentation and flat then the same holds for the open set of points where f is ´etale. Proof. Combine Lemmas 24.37.16 and 24.36.15.



Our proof of the following lemma is somewhat complicated. It uses the “Crit`ere de platitude par fibres” to see that a morphism X → Y over S between schemes ´etale over S is automatically flat. The details are in the chapter on commutative algebra. Lemma 24.37.18. Let f : X → Y be a morphism of schemes over S. If X and Y are ´etale over S, then f is ´etale. Proof. See Algebra, Lemma 7.133.8.



Lemma 24.37.19. Let X

/Y

f p



S

q



be a commutative diagram of morphisms of schemes. Assume that (1) f is surjective, and ´etale, (2) p is ´etale, and (3) q is locally of finite presentation10. Then q is ´etale. Proof. By Lemma 24.35.19 we see that q is smooth. Thus we only need to see that q has relative dimension 0. This follows from Lemma 24.29.2 and the fact that f and p have relative dimension 0.  A final characterization of smooth morphisms is that a smooth morphism f : X → S is locally the composition of an ´etale morphism by a projection AdS → S. Lemma 24.37.20. Let ϕ : X → Y be a morphism of schemes. Let x ∈ X. If ϕ is smooth at x, then there exist exist an integer d ≥ 0 and affine opens V ⊂ Y and U ⊂ X with x ∈ U and ϕ(U ) ⊂ V such that there exists a commutative diagram Xo

U

 Y o

 ~ V

π

/ Ad V

where π is ´etale. Proof. By Lemma 24.35.11 we can find affine opens U and V as in the lemma such that ϕ|U : U → V is standard smooth. Write U = Spec(A) and V = Spec(R) so that we can write A = R[x1 , . . . , xn ]/(f1 , . . . , fc ) 10In fact this is implied by (1) and (2), see Descent, Lemma 31.10.3. Moreover, it suffices to assume that f is surjective, flat and locally of finite presentation, see Descent, Lemma 31.10.5.

24.38. RELATIVELY AMPLE SHEAVES

with

1447



 ∂f1 /∂x1 ∂f2 /∂x1 . . . ∂fc /∂x1 ∂f1 /∂x2 ∂f2 /∂x2 . . . ∂fc /∂x2   g = det   ... ... ... ...  ∂f1 /∂xc ∂f2 /∂xc . . . ∂fc /∂xc mapping to an invertible element of A. Then it is clear that R[xc+1 , . . . , xn ] → A is standard smooth of relative dimension 0. Hence it is smooth of relative dimension 0. In other words the ring map R[xc+1 , . . . , xn ] → A is ´etale. As An−c = V Spec(R[xc+1 , . . . , xn ]) the lemma with d = n − c.  24.38. Relatively ample sheaves Let X be a scheme and L an invertible sheaf on X. Then L is ample on X if X is quasi-compact and every point of X is contained in an affine open of the form Xs , where s ∈ Γ(X, L⊗n ) and n ≥ 1, see Properties, Definition 23.24.1. We relativize this as follows. Definition 24.38.1. Let f : X → S be a morphism of schemes. Let L be an invertible OX -module. We say L is relatively ample, or f -relatively ample, or ample on X/S, or f -ample if f : X → S is quasi-compact, and if for every affine open V ⊂ S the restriction of L to the open subscheme f −1 (V ) of X is ample. We note that the existence of a relatively ample sheaf on X does not force the morphism X → S to be of finite type. Lemma 24.38.2. Let X → S be a morphism of schemes. Let L be an invertible OX -module. Let n ≥ 1. Then L is f -ample if and only if L⊗n is f -ample. Proof. This follows from Properties, Lemma 23.24.2.



Lemma 24.38.3. Let f : X → S be a morphism of schemes. If there exists an f -ample invertible sheaf, then f is separated. Proof. Being separated is local on the base (see Schemes, Lemma 21.21.8 for example; it also follows easily from the definition). Hence we may assume S is affine and X has an ample invertible sheaf. In this case the result follows from Properties, Lemma 23.24.10 and Constructions, Lemma 22.8.8.  There are many ways to charactarize relatively ample invertible sheaves, by relativizing any of the list of equivalent conditions in Properties, Proposition 23.24.12. We will add these here as needed. Lemma 24.38.4. Let f : X → S be a quasi-compact morphism of schemes. Let L be an invertible sheaf on X. The following are equivalent: (1) The invertible sheaf L is f -ample. S (2) There exists an open covering S = Vi such that each L|f −1 (Vi ) is ample relative to f −1 (Vi ) → Vi . S (3) There exists an affine open covering S = Vi such that each L|f −1 (Vi ) is ample. (4) There exists a quasi-coherent L graded OS -algebra A and a map of graded OX -algebras ψ : f ∗ A → d≥0 L⊗d such that U (ψ) = X and rL,ψ : X −→ ProjS (A) is an open immersion (see Constructions, Lemma 22.19.1 for notation).

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(5) TheLmorphism f is quasi-separated and part (4) above holds with A = f∗ ( d≥0 L⊗d ) and ψ the adjunction mapping. (6) Same as (4) but just requiring rL,ψ to be an immersion. Proof. It is immediate from the definition that (1) implies (2) and (2) implies (3). It is clear that (5) implies (4). S Assume (3) holds for the affine open covering S = Vi . We are going to show (5) holds. Since each f −1 (Vi ) has an ample invertible sheaf we see that f −1 (Vi ) is separated (see Properties, Lemma 23.24.10 and Constructions, LemmaL 22.8.8). Hence f is separated. By Schemes, Lemma 21.24.1 we see that A = f∗ ( d≥0 L⊗d ) is a L quasi-coherent graded OS -algebra. Denote ψ : f ∗ A → d≥0 L⊗d the adjunction mapping. The description of the open U (ψ) in Constructions, Section 22.19 and the definition of ampleness of L|f −1 (Vi ) show that U (ψ) = X. Moreover, Constructions, Lemma 22.19.1 part (3) shows that the restriction of rL,ψ to f −1 (Vi ) is the same as the morphism from Properties, Lemma 23.24.8 which is an open immersion according to Properties, Lemma 23.24.10. Hence (5) holds. Let us show that (4) implies (1). Assume (4). Denote π : ProjS (A) → S the structure morphism. Choose V ⊂ S affine open. By Constructions, Definition 22.16.7 we see that π −1 (V ) ⊂ ProjS (A) is equal to Proj(A) where A = A(V ) as a graded ring. Hence rL,ψ maps f −1 (V ) isomorphically onto a quasi-compact open of Proj(A). Moreover, L⊗d is isomorphic to the pullback of OProj(A) (d) for some d ≥ 1. (See part (3) of Constructions, Lemma 22.19.1 and the final statement of Constructions, Lemma 22.14.1.) This implies that L|f −1 (V ) is ample by Properties, Lemmas 23.24.11 and 23.24.2. Assume (6). By the equivalence of (1) - (5) above we see that the property of being relatively ample on X/S is local on S. Hence we may assume that S is affine, and we have to show that L is ample on X. In this case the morphism rL,ψ is identified with the morphism, also denoted rL,ψ : X → Proj(A) associated to the map ψ : A = A(V ) → Γ∗ (X, L). (See references above.) As above we also see that L⊗d is the pullback of the sheaf OProj(A) (d) for some d ≥ 1. Moreover, since X is quasi-compact we see that X gets identified with a closed subscheme of a quasicompact open subscheme Y ⊂ Proj(A). By Constructions, Lemma 22.10.6 (see also Properties, Lemma 23.24.11) we see that OY (d0 ) is an ample invertible sheaf on Y for some d0 ≥ 1. Since the restriction of an ample sheaf to a closed subscheme 0 is ample, see Properties, Lemma 23.24.3 we conclude that the pullback of OYd is ample. Combining these results with Properties, Lemma 23.24.2 we conclude that L is ample as desired.  Lemma 24.38.5. Let f : X → S be a morphism of schemes. Let L be an invertible OX -module. Assume S affine. Then L is f -relatively ample if and only if L is ample on X. Proof. Immediate from Lemma 24.38.4 and the definitions.



24.39. Very ample sheaves Recall that given a quasi-coherent sheaf E on a scheme S the projective bundle associated to E is the morphism P(E) → S, where P(E) = ProjS (Sym(E)), see Constructions, Definition 22.21.1.

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Definition 24.39.1. Let f : X → S be a morphism of schemes. Let L be an invertible OX -module. We say L is relatively very ample or more precisely f relatively very ample, or very ample on X/S, or f -very ample if there exist a quasicoherent OS -module E and an immersion i : X → P(E) over S such that L ∼ = i∗ OP(E) (1). Since there is no assumption of quasi-compactness in this definition it is not true in general that a relatively very ample invertible sheaf is a relatively ample invertible sheaf. Lemma 24.39.2. Let f : X → S be a morphism of schemes. Let L be an invertible OX -module. If f is quasi-compact and L is a relatively very ample invertible sheaf, then L is a relatively ample invertible sheaf. Proof. By definition there exists quasi-coherent OS -module E and an immersion i : X → P(E) over S such that L ∼ = i∗ OP(E) (1). Set A = Sym(E), so P(E) = ProjS (A) by definition. The graded OS -algebra A comes equipped with a map M M ψ:A→ π∗ OP(E) (n) → f∗ L⊗n n≥0

n≥0

where the second arrow uses the identification L ∼ = i∗ OP(E) (1). By adjointness of L ∗ ∗ ⊗n f∗ and f we get a morphism ψ : f A → n≥0 L . We omit the verification that the morphism rL,ψ associated to this map is exactly the immersion i. Hence the result follows from part (6) of Lemma 24.38.4.  To arrive at the correct converse of this lemma we ask whether given a relatively ample invertible sheaf L there exists an integer n ≥ 1 such that L⊗n is relatively very ample? In general this is false. There are several things that prevent this from being true: (1) Even if S is affine, it can happen that no finite integer n works because X → S is not of finite type, see Example 24.39.4. (2) The base not being quasi-compact means the result can be prevented from being true even with f finite type. Namely, given a field k there exists a scheme Xd of finite type over k with an ample invertible sheaf OXd (1) so that the smallest tensor power of OXd (1) which is very ample is the dth power. See Example 24.39.5. Taking f to be the disjoint union of the schemes Xd mapping to the disjoint union of copies of Spec(k) gives an example. To see our version of the converse take a look at Lemma 24.40.5 below. We will do some preliminary work before proving it. Example 24.39.3. Let S be a scheme. Let A be a quasi-coherent graded OS algebra generated by A1 over A0 . Set X = ProjS (A). In this case OX (1) is a very ample invertible sheaf on X. Namely, the morphism associated to the graded OS -algebra map Sym∗OX (A1 ) −→ A is a closed immersion X → P(A1 ) which pulls back OP(A1 ) (1) to OX (1), see Constructions, Lemma 22.18.5. Example 24.39.4. Let k be a field. Consider the graded k-algebra A = k[U, V, Z1 , Z2 , Z3 , . . .]/I

with I = (U 2 − Z12 , U 4 − Z22 , U 6 − Z32 , . . .)

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with grading given by deg(U ) = deg(V ) = deg(Z1 ) = 1 and deg(Zd ) = d. Note that X = Proj(A) is covered by D+ (U ) and D+ (V ). Hence the sheaves OX (n) are all invertible and isomorphic to OX (1)⊗n . In particular OX (1) is ample and f -ample for the morphism f : X → Spec(k). We claim that no power of OX (1) is f -relatively very ample. Namely, it is easy to see that Γ(X, OX (n)) is the degree n summand of the algebra A. Hence if OX (n) were very ample, then X would be a closed subscheme of a projective space over k and hence of finite type over k. On the other hand D+ (V ) is the spectrum of k[t, t1 , t2 , . . .]/(t2 − t21 , t4 − t22 , t6 − t23 , . . .) which is not of finite type over k. Example 24.39.5. Let k be an infinite field. Let λ1 , λ2 , λ3 , . . . be pairwise distinct elements of k ∗ . (This is not strictly necessary, and in fact the example works perfectly well even if all λi are equal to 1.) Consider the graded k-algebra Y2d Ad = k[U, V, Z]/Id with Id = (Z 2 − (U − λi V )). i=1

with grading given by deg(U ) = deg(V ) = 1 and deg(Z) = d. Then Xd = Proj(Ad ) has ample invertible sheaf OXd (1). We claim that if OXd (n) is very ample, then n ≥ d. The reason for this is that Z has degree d, and hence Γ(Xd , OXd (n)) = k[U, V ]n for n < d. Details omitted. Lemma 24.39.6. Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. If L is relatively very ample on X/S then f is separated. Proof. Being separated is local on the base (see Schemes, Section 21.21). An immersion is separated (see Schemes, Lemma 21.23.7). Hence the lemma follows since locally X has an immersion into the homogeneous spectrum of a graded ring which is separated, see Constructions, Lemma 22.8.8.  Lemma 24.39.7. Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume f is quasi-compact. The following are equivalent (1) L is relatively very ample on X/S, S (2) there exists an open covering S = Vj such that L|f −1 (Vj ) is relatively very ample on f −1 (Vj )/Vj for all j, (3) there exists a quasi-coherent sheaf of graded OS -algebras A generated in L degree 1 over OS and a map of graded OX -algebras ψ : f ∗ A → n≥0 L⊗n such that f ∗ A1 → L is surjective and the associated morphism rL,ψ : X → ProjS (A) is an immersion, and (4) f is quasi-separated, the canonical map ψ : f ∗ f∗ L → L is surjective, and the associated map rL,ψ : X → P(f∗ L) is an immersion. Proof. It is clear that (1) implies (2). It is also clear that (4) implies (1); the hypothesis of quasi-spearation in (4) is used to garantee that f∗ L is quasi-coherent via Schemes, Lemma 21.24.1. S Assume (2). We will prove (4). Let S = Vj be an open covering as in (2). Set Xj = f −1 (Vj ) and fj : Xj → Vj the restriction of f . We see that f is separated by Lemma 24.39.6 (as being separated is local on the base). Consider the map ψ : f ∗ f∗ L → L. On each Vj there exists a quasi-coherent sheaf Ej and an embedding i : Xj → P(Ej ) with LXj ∼ = i∗ OP(Ej ) (1). In other words there is a map Ej → (f∗ L)|Xj such that the composition fj∗ Ej → (f ∗ f∗ L)|Xj → L|Xj

24.40. AMPLE AND VERY AMPLE SHEAVES RELATIVE TO FINITE TYPE MORPHISMS 1451

is surjective. Hence we conclude that ψ is surjective. Let rL,ψ : X → P(f∗ L) be the associated morphism. Using the maps Ej → (f∗ L)|Xj we see that there is a factorization rL,ψ / P(f∗ L)|Vj / P(Ej ) Xj which shows that rL,ψ is an immersion. At this point we see that (1), (2) and (4) are equivalent. Clearly (4) implies (3). Assume (3). We will prove (1). Let A be a quasi-coherent sheaf of graded OS algebras generated in degree 1 over OS . Consider the map of graded OS -algebras Sym(A1 ) → A. This is surjective by hypothesis and hence induces a closed immersion ProjS (A) −→ P(A1 ) which pulls back O(1) to O(1), see Constructions, Lemma 22.18.5. Hence it is clear that (3) implies (1).  24.40. Ample and very ample sheaves relative to finite type morphisms In fact most of the material in this section is about the notion of a (quasi-)projective morphism which we have not defined yet. Lemma 24.40.1. Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume that (1) the invertible sheaf L is very ample on X/S, (2) the morphism X → S is of finite type, and (3) S is affine. Then there exists an n ≥ 0 and an immersion i : X → PnS over S such that L∼ = i∗ OPnS (1). Proof. Assume (1), (2) and (3). Condition (3) means S = Spec(R) for some ring R. Condition (1) means by definition there exists a quasi-coherent OS -module E f for and an immersion α : X → P(E) such that L = α∗ OP(E) (1). Write E = M some R-module M . Thus we have P(E) = Proj(SymR (M )). Since α is an immersion, and since the topology of Proj(SymR (M )) is generated by the standard opens D+ (f ), f ∈ SymdR (M ), d ≥ 1, we can find for each x ∈ X an f ∈ SymdR (M ), d ≥ 1, with α(x) ∈ D+ (f ) such that α|α−1 (D+ (f )) : α−1 (D+ (f )) → D+ (f ) is a closed immersion. Condition (2) implies X is quasi-compact. Hence we can d find a finite collection of elements fj ∈ SymRj (M ), dj ≥ 1 such that for each S f = fj the displayed map above is a closed immersion and such that α(X) ⊂ D+ (fj ). Write Uj = α−1 (D+ (fj )). Note that Uj is affine as a closed subscheme of the affine scheme D+ (fj ). Write Uj = Spec(Aj ). Condition (2) also implies that Aj is of finite type over R, see Lemma 24.16.2. Choose finitely many xj,k ∈ Aj which generate Aj as a R-algebra. Since α|Uj is a closed immersion we see that xj,k is the image of an element e

fj,k /fj j,k ∈ SymR (M )(fj ) = Γ(D+ (fj ), OProj(SymR (M )) ).

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Finally, choose n ≥ 1 and elements y0 , . . . , yn ∈ M such that each of the polynomials fj , fj,k ∈ SymR (M ) is a polynomial in the elements yt with coefficients in R. Consider the graded ring map ψ : R[Y0 , . . . , Yn ] −→ SymR (M ),

Yi 7−→ yi .

Denote Fj , Fj,k the elements of R[Y0 , . . . , Yn ] such that ψ(Fj ) = fj and ψ(Fj,k ) = fj,k . By Constructions, Lemma 22.11.1 we obtain an open subscheme U (ψ) ⊂ Proj(SymR (M )) −1 and a morphism rψ : U (ψ) → PnR . This morphism satisfies rψ (D+ (Fj )) = D+ (fj ), and hence we see that α(X) ⊂ U (ψ). Moreover, it is clear that

i = rψ ◦ α : X −→ PnR e

is still an immersion since i] (Fj,k /Fj j,k ) = xj,k ∈ Aj = Γ(Uj , OX ) by construc∗ tion. Moreover, the morphism rψ comes equipped with a map θ : rψ OPnR (1) → OProj(SymR (M )) (1)|U (ψ) which is an isomorphism in this case (for construction θ see lemma cited above; some details omitted). Since the original map α was assumed to have the property that L = α∗ OProj(SymR (M )) (1) we win.  Lemma 24.40.2. Let π : X → S be a morphism of schemes. Assume that X is quasi-affine and that π is locally of finite type. Then there exist n ≥ 0 and an immersion i : X → AnS over S. Proof. Let A = Γ(X, OX ). By assumption X is quasi-compact and is identified with an open subscheme of Spec(A), see Properties, Lemma 23.15.4. Moreover, the set of opens Xf , for those f ∈ A such that Xf is affine, forms a basis for the topology of X, see the proof of Properties, Lemma 23.15.4.S Hence we can find a finite number of fj ∈ A, j = 1, . . . , m such that X = Xfj , and such that π(Xfj ) ⊂ Vj for some affine open Vj ⊂ S. By Lemma 24.16.2 the ring maps O(Vj ) → O(Xfj ) = Afj are of finite type. Thus we may choose a1 , . . . , aN ∈ A such that the elements a1 , . . . , aN , f1 , . . . , fm , 1/fj generate Afj over O(Vj ) for each j. Take n = N + m and let i : X −→ AnS be the morphism given by the global sections a1 , . . . , an , f1 , . . . , fn of the structure sheaf of X. Let D(xj ) ⊂ AnS be the open subscheme where the jth coordinate function is nonzero. Then it is clear that i−1 (D(xj )) is Xfj and that the induced morphism Xfj → D(xj ) factors through the affine open Spec(O(Vj )[x1 , . . . , xn , 1/xj ]) of D(xj ). Since the ring map O(Vj )[x1 , . . . , xn , 1/xj ] → Afj is surjective by construction we conclude that the restriction of i to Xfj is an immersion as desired.  Lemma 24.40.3. Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume that (1) the invertible sheaf L is ample on X, and (2) the morphism X → S is locally of finite type. Then there exists a d0 ≥ 1 such that for every d ≥ d0 there exists an n ≥ 0 and an immersion i : X → PnS over S such that L⊗d ∼ = i∗ OPnS (1). L Proof. Let A = Γ∗ (X, L) = d≥0 Γ(X, L⊗d ). By Properties, Proposition 23.24.12 the set of affine opens Xa with a ∈ A+ homogeneous forms a basis for the topology of X. Hence we can find finitely many such elements a0 , . . . , an ∈ A+ such that

24.40. AMPLE AND VERY AMPLE SHEAVES RELATIVE TO FINITE TYPE MORPHISMS 1453

S (1) we have X = i=0,...,n Xai , (2) each Xai is affine, and (3) each Xai maps into an affine open Vi ⊂ S. By Lemma 24.16.2 we see that the ring maps OS (Vi ) → OX (Xai ) are of finite type. Hence we can find finitely many elements fij ∈ OX (Xai ), j = 1, . . . , ni which generate OX (Xai ) as an OS (Vi )-algebra. By Properties, Lemma 23.24.5 e we may write each fij as aij /ai ij for some aij ∈ A+ homogeneous. Let N be a positive integer which is a common multiple of all the degrees of the elements ai , aij . Consider the elements N/ deg(ai )

ai

(N/ deg(ai ))−eij

, aij ai

∈ AN .

By construction these generate the invertible sheaf L⊗N over X. Hence they give rise to a morphism X j : X −→ Pm with m = n + ni S over S, see Constructions, Lemma 22.13.1 and Definition 22.13.2. Moreover, j ∗ OPS (1) = L⊗N . We name the homogeneous coordinates T0 , . . . , Tn , Tij instead of T0 , . . . , Tm . For i = 0, . . . , n we have i−1 (D+ (Ti )) = Xai . Moreover, pulling back the element Tij /Ti via j ] we get the element fij ∈ OX (Xai ). Hence the morphism j restricted N to Xai gives a closed immersion of Xai into the affine open D+ (Ti ) ∩ Pm Vi of PS . Hence we conclude that the morphism j is an immersion. This implies the lemma holds for some d and n which is enough in virtually all applications. This proves that for one d2 ≥ 1 (namely d2 = N above), some m ≥ 0 there exists ⊗d2 0 0 ). some immersion j : X → Pm S given by global sections s0 , . . . , sm ∈ Γ(X, L By Properties, Proposition 23.24.12 we know there exists an integer d1 such that L⊗d is globally generated for all d ≥ d1 . Set d0 = d1 + d2 . We claim that the lemma holds with this value of d0 . Namely, given an integer d ≥ d0 we may choose s001 , . . . , s00t ∈ Γ(X, L⊗d−d2 ) which generate L⊗d−d2 over X. Set n = (m + 1)t and denote s0 , . . . , sn the collection of sections s0α s00β , α = 0, . . . , m, β = 1, . . . , t. These generate L⊗d over X and therefore define a morphism i : X −→ PnS such that i∗ OPnS (1) ∼ = L⊗d . We omit the verification that since j was an immersion also the morphism i so obtained is an immersion also. (Hint: Segre embedding.)  Lemma 24.40.4. Let f : X → S be a morphism of schemes. Let L be an invertible OX -module. Assume S affine and f of finite type. The following are equivalent (1) L is ample on X, (2) L is f -ample, (3) L⊗d is f -very ample for some d ≥ 1, (4) L⊗d is f -very ample for all d  1, (5) for some d ≥ 1 there exist n ≥ 1 and an immersion i : X → PnS such that L⊗d ∼ = i∗ OPnS (1), and (6) for all d  1 there exist n ≥ 1 and an immersion i : X → PnS such that L⊗d ∼ = i∗ OPnS (1). Proof. The equivalence of (1) and (2) is Lemma 24.38.5. The implication (2) ⇒ (6) is Lemma 24.40.3. Trivially (6) implies (5). As PnS is a projective bundle over S (see Constructions, Lemma 22.21.4) we see that (5) implies (3) and (6) implies

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(4) from the definition of a relatively very ample sheaf. Trivially (4) implies (3). To finish we have to show that (3) implies (2) which follows from Lemma 24.39.2 and Lemma 24.38.2.  Lemma 24.40.5. Let f : X → S be a morphism of schemes. Let L be an invertible OX -module. Assume S quasi-compact and f of finite type. The following are equivalent (1) L is f -ample, (2) L⊗d is f -very ample for some d ≥ 1, (3) L⊗d is f -very ample for all d  1. Proof. Trivially (3) implies (2). Lemma 24.39.2 garantees that (2) implies (1) since a morphism of finite type is quasi-compact by definition. Assume that L is f -ample. Choose a finite affine open covering S = V1 ∪ . . . ∪ Vm . Write Xi = f −1 (Vi ). By Lemma 24.40.4 above we see there exists a d0 such that L⊗d is relatively very ample on Xi /Vi for all d ≥ d0 . Hence we conclude (1) implies (3) by Lemma 24.39.7.  The following two lemmas provide the most used and most useful characterizations of relatively very ample and relatively ample invertible sheaves when the morphism is of finite type. Lemma 24.40.6. Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume f is of finite type. The following are equivalent: (1) L is f -relatively very ample, and S (2) there exist an open covering S = Vj , for each j an integer nj , and immersions n

ij : Xj = f −1 (Vj ) = Vj ×S X −→ PVjj over Vj such that L|Xj ∼ = i∗j OPnj (1). Vj

Proof. We see that (1) implies (2) by taking an affine open covering of S and applying Lemma 24.40.1 to each of the restrictions of f and L. We see that (2) implies (1) by Lemma 24.39.7.  Lemma 24.40.7. Let f : X → S be a morphism of schemes. Let L be an invertible sheaf on X. Assume f is of finite type. The following are equivalent: (1) L is f -relatively ample, and S (2) there exist an open covering S = Vj , for each j an integers dj ≥ 1, nj ≥ 0, and immersions n

ij : Xj = f −1 (Vj ) = Vj ×S X −→ PVjj over Vj such that L⊗dj |Xj ∼ = i∗j OPnj (1). Vj

Proof. We see that (1) implies (2) by taking an affine open covering of S and applying Lemma 24.40.4 to each of the restrictions of f and L. We see that (2) implies (1) by Lemma 24.38.4. 

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24.41. Quasi-projective morphisms The discussion in the previous section suggests the following definitions. We take our definition of quasi-projective from [DG67]. The version with the letter “H” is the definition in [Har77]. Definition 24.41.1. Let f : X → S be a morphism of schemes. (1) We say f is quasi-projective if f is of finite type and there exists an f relatively ample invertible OX -module. (2) We say f is H-quasi-projective if f if there exists a quasi-compact immersion X → PnS over S for some n.11 (3) S We say f is locally quasi-projective if there exists an open covering S = Vj such that each f −1 (Vj ) → Vj is quasi-projective. As this definition suggests the property of being quasi-projective is not local on S. Lemma 24.41.2. Let f : X → S be a morphism of schemes. If f is quasiprojective, or H-quasi-projective or locally quasi-projective, then f is separated of finite type. Proof. Omitted.



Lemma 24.41.3. A H-quasi-projective morphism is quasi-projective. Proof. Omitted.



Lemma 24.41.4. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is locally quasi-projective. S (2) There exists an open covering S = Vj such that each f −1 (Vj ) → Vj is H-quasi-projective. Proof. By Lemma 24.41.3 we see that (2) implies (1). Assume (1). The question is local on S and hence we may assume S is affine, X of finite type over S and L is a relatively ample invertible sheaf on X/S. By Lemma 24.40.4 we may assume L is ample on X. By Lemma 24.40.3 we see that there exists an immersion of X into a projective space over S, i.e., X is H-quasi-projective over S as desired.  24.42. Proper morphisms The notion of a proper morphism plays an important role in algebraic geometry. An important example of a proper morphism will be the structure morphism PnS → S of projective n-space, and this is in fact the motivating example leading to the definition. Definition 24.42.1. Let f : X → S be a morphism of schemes. We say f is proper if f is separated, finite type, and universally closed. 11This is not exactly the same as the definition in Hartshorne. Namely, the definition in Hartshorne (8th corrected printing, 1997) is that f should be the composition of an open immersion followed by a H-projective morphism (see Definition 24.43.1), which does not imply f is quasicompact. See Lemma 24.43.11 for the implication in the other direction.

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The morphism from the affine line with zero doubled to the affine line is of finite type and universally closed, so the separation condition is necessary in the definition above. In the rest of this section we prove some of the basic properties of proper morphisms and of universally closed morphisms. Lemma 24.42.2. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is universally closed. S (2) There exists an open covering S = Vj such that f −1 (Vj ) → Vj is universally closed for all indices j. Proof. This is clear from the definition.



Lemma 24.42.3. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is proper. S (2) There exists an open covering S = Vj such that f −1 (Vj ) → Vj is proper for all indices j. Proof. Omitted.



Lemma 24.42.4. The composition of proper morphisms is proper. The same is true for universally closed morphisms. Proof. A composition of closed morphisms is closed. If X → Y → Z are universally closed morphisms and Z 0 → Z is any morphism, then we see that Z 0 ×Z X = (Z 0 ×Z Y ) ×Y X → Z 0 ×Z Y is closed and Z 0 ×Z Y → Z 0 is closed. Hence the result for universally closed morphisms. We have seen that “separated” and “finite type” are preserved under compositions (Schemes, Lemma 21.21.13 and Lemma 24.16.3). Hence the result for proper morphisms.  Lemma 24.42.5. The base change of a proper morphism is proper. The same is true for universally closed morphisms. Proof. This is true by definition for universally closed morphisms. It is true for separated morphisms (Schemes, Lemma 21.21.13). It is true for morphisms of finite type (Lemma 24.16.4). Hence it is true for proper morphisms.  Lemma 24.42.6. A closed immersion is proper, hence a fortiori universally closed. Proof. The base change of a closed immersion is a closed immersion (Schemes, Lemma 21.18.2). Hence it is universally closed. A closed immersion is separated (Schemes, Lemma 21.23.7). A closed immersion is of finite type (Lemma 24.16.5). Hence a closed immersion is proper.  Lemma 24.42.7. Suppose given a commutative diagram of schemes /Y X 

S



with Y separated over S. (1) If X → S is universally closed, then the morphism X → Y is universally closed.

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(2) If X proper over S, then the morphism X → Y is proper. In particular, in both cases the image of X in Y is closed. Proof. Assume that X → S is universally closed (resp. proper). We factor the morphism as X → X ×S Y → Y . The first morphism is a closed immersion, see Schemes, Lemma 21.21.11. Hence the first morphism is proper (Lemma 24.42.6). The projection X ×S Y → Y is the base change of a unviversally closed (resp. proper) morphism and hence universally closed (resp. proper), see Lemma 24.42.5. Thus X → Y is universally closed (resp. proper) as the composition of universally closed (resp. proper) morphisms (Lemma 24.42.4).  The following lemma says that the image of a proper scheme (in a separated scheme of finite type over the base) is proper. Lemma 24.42.8. Let S be a scheme. Let f : X → Y be a morphism of schemes over S. If X is universally closed over S and f is surjective then Y is universally closed over S. In particular, if also Y is separated and of finite type over S, then Y is proper over S. Proof. Assume X is universally closed and f surjective. Denote p : X → S, q : Y → S the structure morphisms. Let S 0 → S be a morphism of schemes. The base change f 0 : XS 0 → YS 0 is surjective (Lemma 24.11.4), and the base change p0 : XS 0 → S 0 is closed. If T ⊂ YS 0 is closed, then (f 0 )−1 (T ) ⊂ XS 0 is closed, hence p0 ((f 0 )−1 (T )) = q 0 (T ) is closed. So q 0 is closed.  The proof of the following lemma is due to Bjorn Poonen, see this location. Lemma 24.42.9. A universally closed morphism of schemes is quasi-compact. Proof. Let f : X → S be a morphism. Assume that f is not quasi-compact. Our goal is to show that f is not universally closed. By Schemes, Lemma 21.19.2 there exists an affine open V ⊂ S such that f −1 (V ) is not quasi-compact. To achieve our goal it suffices to show that f −1 (V ) → V is not universally closed, hence we may assume that S = Spec(A) for some ring A. S Write X = i∈I Xi where the Xi are affine open subschemes of X. Let T = Spec(A[y i ; i ∈ I]). Let Ti = D(yi ) ⊂ T . Let Z be the closed set (X ×S T ) − S (X × i S Ti ). It suffices to prove that the image fT (Z) of Z under fT : X ×S T → i∈I T is not closed. There exists a point s ∈ S such that there is no neighborhood U of s in S such that XU is quasi-compact. Otherwise we could cover S with finitely many such U and Schemes, Lemma 21.19.2 would imply f quasi-compact. Fix such an s ∈ S. First we check that fT (Zs ) 6= Ts . Let t ∈ T be the point lying over s with κ(t) = κ(s) such that yi = 1 in κ(t) forSall i. Then t ∈ Ti for all i, and the fiber of Zs → Ts above t is isomorphic to (X − i∈I Xi )s , which is empty. Thus t ∈ Ts − fT (Zs ). Assume fT (Z) is closed in T . Then there exists an element g ∈ A[yi ; i ∈ I] with fT (Z) ⊂ V (g) but t 6∈ V (g). Hence the image of g in κ(t) is nonzero. In particular some coefficient of g has nonzero image in κ(s). Hence this coefficient is invertible on some neighborhood U of s. Let J be the finite set of j ∈ I such that S yj appears in g. Since XU is not quasi-compact, we may choose a point x ∈ X − j∈J Xj lying above some u ∈ U . Since g has a coefficient that is invertible on U , we can find a

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point t0 ∈ T lying above u such that t0 6∈ V (g) and t0 ∈ V (yi ) for all i ∈ / J. This is true because V (yi ; i ∈ I, i 6∈ J) = Spec(A[tj ; j ∈ J]) and the set of points of this scheme lying over u is bijective with Spec(κ(u)[tj ; j ∈ J]). In other words t0 ∈ / Ti for each i ∈ / J. By Schemes, Lemma 21.17.5 we can find a point z of X ×S T mapping to x ∈ X and to t0 ∈ T . Since x 6∈ Xj for j ∈ J and t0 6∈ Ti for i ∈ I \ J we see that z ∈ Z. On the other hand fT (z) = t0 6∈ V (g) which contradicts fT (Z) ⊂ V (g). Thus the assumption “fT (Z) closed” is wrong and we conclude indeed that fT is not closed, as desired.  24.43. Projective morphisms We will use the definition of a projective morphism from [DG67]. The version of the definition with the “H” is the one from [Har77]. The resulting definitions are different. Both are useful. Definition 24.43.1. Let f : X → S be a morphism of schemes. (1) We say f is projective if X is isomorphic as an S-scheme to a closed subscheme of a projective bundle P(E) for some quasi-coherent, finite type OS -module E. (2) We say f is H-projective if there exists and integer n and a closed immersion X → PnS over S. S (3) We say f is locally projective if there exists an open covering S = Ui such that each f −1 (Ui ) → Ui is projective. As expected, a projective morphism is quasi-projective, see Lemma 24.43.10. Conversely, quasi-projective morphisms are often compositions of open immersions and projective morphisms, see Lemma 24.43.12. Example 24.43.2. Let S be a scheme. Let A be a quasi-coherent graded OS algebra generated by A1 over A0 . Assume furthermore that A1 is of finite type over OS . Set X = ProjS (A). In this case X → S is projective. Namely, the morphism associated to the graded OS -algebra map Sym∗OX (A1 ) −→ A is a closed immersion X → P(A1 ) which pulls back OP(A1 ) (1) to OX (1), see Constructions, Lemma 22.18.5. Lemma 24.43.3. An H-projective morphism is H-quasi-projective. An H-projective morphism is projective. Proof. The first statement is immediate from the definitions. The second holds as PnS is a projective bundle over S, see Constructions, Lemma 22.21.4.  Lemma 24.43.4. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is locally projective. S (2) There exists an open covering S = Ui such that each f −1 (Ui ) → Ui is H-projective. Proof. By Lemma 24.43.3 we see that (2) implies (1). Assume (1). For every point s ∈ S we can find Spec(R) = U ⊂ S an affine open neighbourhood of s such that XU is isomorphic to a closed subscheme of P(E) for some finite type, quasif for some finite type R-module M coherent sheaf of OU -modules E. Write E = M

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(see Properties, Lemma 23.16.1). Choose generators x0 , . . . , xn ∈ M of M as an R-module. Consider the surjective graded R-algebra map R[X0 , . . . , Xn ] −→ SymR (M ). According to Constructions, Lemma 22.11.3 the corresponding morphism P(E) → PnR is a closed immersion. Hence we conclude that f −1 (U ) is isomorphic to a closed subscheme of PnU (as a scheme over U ). In other words: (2) holds.  Lemma 24.43.5. A locally projective morphism is proper. Proof. Let f : X → S be locally projective. In order to show that f is proper we may work locally on the base, see Lemma 24.42.3. Hence, by Lemma 24.43.4 above we may assume there exists a closed immersion X → PnS . By Lemmas 24.42.4 and 24.42.6 it suffices to prove that PnS → S is proper. Since PnS → S is the base change of PnZ → Spec(Z) it suffices to show that PnZ → Spec(Z) is proper, see Lemma 24.42.5. By Constructions, Lemma 22.8.8 the scheme PnZ is separated. By Constructions, Lemma 22.8.9 the scheme PnZ is quasi-compact. It is clear that PnZ → Spec(Z) is locally of finite type since PnZ is covered by the affine opens D+ (Xi ) each of which is the spectrum of the finite type Z-algebra Z[X0 /Xi , . . . , Xn /Xi ]. Finally, we have to show that PnZ → Spec(Z) is universally closed. This follows from Constructions, Lemma 22.8.11 and the valuative criterion (see Schemes, Proposition 21.20.6).  Lemma 24.43.6. Let S be a scheme. There exists a closed immersion nm+n+m PnS ×S Pm S −→ PS

called the Segre embedding. Proof. It suffices to prove this when S = Spec(Z). Hence we will drop the index S and work in the absolute setting. Write Pn = Proj(Z[X0 , . . . , Xn ]), Pm = Proj(Z[Y0 , . . . , Ym ]), and Pnm+n+m = Proj(Z[Z0 , . . . , Znm+n+m ]). In order to map into Pnm+n+m we have to write down an invertible sheaf L on the left hand side and (n+1)(m+1) sections si which generate it. See Constructions, Lemma 22.13.1. The invertible sheaf we take is L = pr∗1 OPn (1) ⊗ pr∗2 OPm (1) The sections we take are s0 = X0 Y0 , s1 = X1 Y0 , . . . , sn = Xn Y0 , sn+1 = X0 Y1 , . . . , snm+n+m = Xn Ym . These generate L since the sections Xi generate OPn (1) and the sections Yj generate OPm (1). The induced morphism ϕ has the property that ϕ−1 (D+ (Zi+(n+1)j )) = D+ (Xi ) × D+ (Yj ). Hence it is an affine morphism. The corresponding ring map in case (i, j) = (0, 0) is the map Z[Z1 /Z0 , . . . , Znm+n+m /Z0 ] −→ Z[X1 /X0 , . . . , Xn /X0 , Y1 /Y0 , . . . , Yn /Y0 ]

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which maps Zi /Z0 to the element Xi /X0 for i ≤ n and the element Z(n+1)j /Z0 to the element Yj /Y0 . Hence it is surjective. A similar argument works for the other affine open subsets. Hence the morphism ϕ is a closed immersion.  Lemma 24.43.7. A composition of H-projective morphisms is H-projective. Proof. Suppose X → Y and Y → Z are H-projective. Then there exist closed immersions X → PnY over Y , and Y → Pm Z over Z. Consider the following diagram X

/ Pn

  Y

/ Pm Z

/ Pn m

Y

PZ

PnZ ×Z Pm Z

/ Pnm+n+m Z

}

 } Zu Here the rightmost top horizontal arrow is the Segre embedding, see Lemma 24.43.6. The diagram identifies X as a closed subscheme of Pnm+n+m as desired.  Z Lemma 24.43.8. A base change of a H-projective morphism is H-projective. Proof. This is true because the base change of projective space over a scheme is projective space, and the fact that the base change of a closed immersion is a closed immersion, see Schemes, Lemma 21.18.2.  Lemma 24.43.9. A base change of a (locally) projective morphism is (locally) projective. Proof. This is true because the base change of a projective bundle over a scheme is a projective bundle, the pullback of a finite type O-module is of finite type (Modules, Lemma 15.9.2) and the fact that the base change of a closed immersion is a closed immersion, see Schemes, Lemma 21.18.2. Some details omitted.  Lemma 24.43.10. A projective morphism is quasi-projective. Proof. Let f : X → S be a projective morphism. Choose a closed immersion i : X → P(E) where E is a quasi-coherent, finite type OS -module. Then L = i∗ OP(E) (1) is f -very ample. Since f is proper (Lemma 24.43.5) it is quasi-compact. Hence Lemma 24.39.2 implies that L is f -ample. Since f is proper it is of finite type. Thus we’ve checked all the defining properties of quasi-projective holds and we win.  Lemma 24.43.11. Let f : X → S be a H-quasi-projective morphism. Then f factors as X → X 0 → S where X → X 0 is an open immersion and X 0 → S is H-projective. Proof. By definition we can factor f as a quasi-compact immersion i : X → PnS followed by the projection PnS → S. By Lemma 24.7.7 there exists a closed subscheme X 0 ⊂ PnS such that i factors through an open immersion X → X 0 . The lemma follows.  Lemma 24.43.12. Let f : X → S be a quasi-projective morphism with S quasicompact and quasi-separated. Then f factors as X → X 0 → S where X → X 0 is an open immersion and X 0 → S is projective.

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Proof. Let L be f -ample. Since f is of finite type and S is quasi-compact L⊗n is f -very ample for some n > 0, see Lemma 24.40.5. Replace L by L⊗n . Write F = f∗ L. This is a quasi-coherent OS -module by Schemes, Lemma 21.24.1 (quasiprojective morphisms are quasi-compact and separated, see Lemma 24.41.2). By Properties, Lemma 23.20.6 we can find a directed partially ordered set I and a system of finite type quasi-coherent OS -modules Ei over I such that F = colim Ei . Consider the S compositions ψi : f ∗ Ei → f ∗ F → L. Choose a finite affine open covering S = j=1,...,m Vj . For each j we can choose sections sj,0 , . . . , sj,nj ∈ Γ(f −1 (Vj ), L) = f∗ L(Vj ) = F(Vj ) which generate L over f −1 Vj and define an immersion n

f −1 Vj −→ PVjj , see Lemma 24.40.1. Choose i such that there exist sections ej,t ∈ Ei (Vj ) mapping to sj,t in F for all j = 1, . . . , m and t = 1, . . . , nj . Then the map ψi is surjective as the sections f ∗ ej,t have the same image as the sections sj,t which generate L|f −1 Vj . Whence we obtian a morphism rL,ψi : X −→ P(Ei ) over S such that over Vj we have a factorization n

f −1 Vj → P(Ei )|Vj → PVjj of the immersion given S above. It follows that rL,ψi |Vj is an immersion, see Lemma 24.3.1. Since S = Vj we conclude that rL,ψi is an immersion. Note that rL,ψi is quasi-compact as X → S is quasi-compact and P(Ei ) → S is separated (see Schemes, Lemma 21.21.15). By Lemma 24.7.7 there exists a closed subscheme X 0 ⊂ P(Ei ) such that i factors through an open immersion X → X 0 . Then X 0 → S is projective by definition and we win.  24.44. Integral and finite morphisms Recall that a ring map R → A is said to be integral if every element of A satisfies a monic equation with coefficients in R. Recall that a ring map R → A is said to be finite if A is finite as an R-module. See Algebra, Definition 7.33.1. Definition 24.44.1. Let f : X → S be a morphism of schemes. (1) We say that f is integral if f is affine and if for every affine open Spec(R) = V ⊂ S with inverse image Spec(A) = f −1 (V ) ⊂ X the associated ring map R → A is integral. (2) We say that f is finite if f is affine and if for every affine open Spec(R) = V ⊂ S with inverse image Spec(A) = f −1 (V ) ⊂ X the associated ring map R → A is finite. It is clear that integral/finite morphisms are separated and quasi-compact. It is also clear that a finite morphism is a morphism of finite type. Most of the lemmas in this section are completely standard. But note the fun Lemma 24.44.7 at the end of the section. Lemma 24.44.2. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is integral.

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S (2) There exists an affine open covering S = Ui such that each f −1 (Ui ) is affine and OS (Ui ) → OX (f −1 (Ui )) is S integral. (3) There exists an open covering S = Sj such that each f −1 (Ui ) → Ui is integral. Moreover, if f is integral then for every open subscheme U ⊂ S the morphism f : f −1 (U ) → U is integral. Proof. See Algebra, Lemma 7.33.12. Some details omitted.



Lemma 24.44.3. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is finite. S (2) There exists an affine open covering S = Ui such that each f −1 (Ui ) is affine and OS (Ui ) → OX (f −1 (Ui )) is S finite. (3) There exists an open covering S = Sj such that each f −1 (Ui ) → Ui is finite. Moreover, if f is finite then for every open subscheme U ⊂ S the morphim f : f −1 (U ) → U is finite. Proof. See Algebra, Lemma 7.33.12. Some details omitted.



Lemma 24.44.4. A finite morphism is integral. An integral morphism which is locally of finite type is finite. Proof. See Algebra, Lemma 7.33.3 and Lemma 7.33.5.



Lemma 24.44.5. A composition of finite morphisms is finite. Same is true for integral morphisms. Proof. See Algebra, Lemmas 7.7.3 and 7.33.6.



Lemma 24.44.6. A base change of a finite morphism is finite. Same is true for integral morphisms. Proof. See Algebra, Lemma 7.33.11.



Lemma 24.44.7. Let f : X → S be a morphism of schemes. The following are equivalent (1) f is integral, and (2) f is affine and universally closed. Proof. Assume (1). An integral morphism is affine by definition. A base change of an integral morphism is integral so in order to prove (2) it suffices to show that an integral morphism is closed. This follows from Algebra, Lemmas 7.33.20 and 7.37.6. Assume (2). We may assume f is the morphism f : Spec(A) → Spec(R) coming from a ring map R → A. Let a be an element of A. We have to show that a is integral over R, i.e. that in the kernel I of the map R[x] → A sending x to a there is a monic polynomial. Consider the ring B = A[x]/(ax − 1) and let J be the kernel of the composition R[x] → A[x] → B. If f ∈ J there exists q ∈ A[x] such that

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P P f = (ax − 1)q in A[x] so if f = i fi xi and q = i qi xi , for all i ≥ 0 we have fi = aqi−1 − qi . For n ≥ deg q + 1 the polynomial X X X fi xn−i = (aqi−1 − qi )xn−i = (a − x) qi xn−i−1 i≥0

i≥0

i≥0

is clearly in I; if f0 = 1 this polynomial is also monic, so we are reduced to prove that J contains a polynomial with constant term 1. We do it by proving Spec(R[x]/(J + (x)) is empty. Since f is universally closed the base change Spec(A[x]) → Spec(R[x]) is closed. Hence the image of the closed subset Spec(B) ⊂ Spec(A[x]) is the closed subset Spec(R[x]/J) ⊂ Spec(R[x]), see Example 24.6.4 and Lemma 24.6.3. In particular Spec(B) → Spec(R[x]/J) is surjective. Consider the following diagram where every square is a pullback: Spec(B) O

g

/ / Spec(R[x]/J) O

/ Spec(R[x]) O 0

∅

/ Spec(R[x]/(J + (x)))

/ Spec(R)

The bottom left corner is empty because it is the spectrum of R ⊗R[x] B where the map R[x] → B sends x to an invertible element and R[x] → R sends x to 0. Since g is surjective this implies Spec(R[x]/(J + (x))) is empty, as we wanted to show.  Lemma 24.44.8. Let f : X → S be an integral morphism. Then every point of X is closed in its fibre. Proof. See Algebra, Lemma 7.33.18.



Lemma 24.44.9. A finite morphism is quasi-finite. Proof. This is implied by Algebra, Lemma 7.114.4 and Lemma 24.21.9. Alternatively, all points in fibres are closed points by Lemma 24.44.8 (and the fact that a finite morphism is integral) and use Lemma 24.21.6 (3) to see that f is quasi-finite at x for all x ∈ X.  Lemma 24.44.10. A finite morphism is proper. Proof. A finite morphism is integral and hence universally closed by Lemma 24.44.7. It is also clearly separated and of finite type. Hence it is proper by definition.  Lemma 24.44.11. A closed immersion is finite (and a fortiori integral). Proof. True because a closed immersion is affine (Lemma 24.13.9) and a surjective ring map is finite and integral.  Lemma 24.44.12. Let f : X → Y and g : Y → Z be morphisms. (1) If g ◦ f is finite and g separated then f is finite. (2) If g ◦ f is integral and g separated then f is integral. Proof. Assume g ◦ f is finite (resp. integral) and g separated. The base change X ×Z Y → Y is finite (resp. integral) by Lemma 24.44.6. The morphism X → X ×Z Y is a closed immersion as Y → Z is separated, see Schemes, Lemma 21.21.12. A closed immersion is finite (resp. integral), see Lemma 24.44.11. The composition

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of finite (resp. integral) morphisms is finite (resp. integral), see Lemma 24.44.5. Thus we win.  Lemma 24.44.13. Let f : X → Y be a morphism of schemes. If f is finite and a monomorphism, then f is a closed immersion. Proof. This reduces to Algebra, Lemma 7.100.6.



24.45. Universal homeomorphisms The following definition is really superfluous since a universal homeomorphism is really just an integral, universally injective and surjective morphism, see Lemma 24.45.3. Definition 24.45.1. A morphisms f : X → Y of schemes is called a universal homeomorphism if the base change f 0 : Y 0 ×Y X → Y 0 is a homeomorphism for every morphism Y 0 → Y . Lemma 24.45.2. Let f : X → Y be a morphism of schemes. If f is a homeomorphism then f is affine. Proof. Let y ∈ Y be a point. Let y ∈ V be an affine open neighbourhood. let x ∈ X be the unique point of X mapping to y. Let U ⊂ X be an affine open neighbourhood of x which maps into V . Since f (U ) ⊂ V is open we may choose a h ∈ Γ(V, OY ) such that y ∈ D(h) ⊂ f (U ). Denote h0 ∈ Γ(U, OX ) the restriction of f ] (h) to U . Then we see that D(h0 ) ⊂ U is equal to f −1 (D(h)). In other words, every point of Y has an open neighbourhood whose inverse image is affine. Thus f is affine, see Lemma 24.13.3.  Lemma 24.45.3. Let f : X → Y be a morphism of schemes. The following are equivalent: (1) f is a universal homeomorphism, and (2) f is integral, universally injective and surjective. Proof. Assume f is a universal homeomorphism. By Lemma 24.45.2 we see that f is affine. Since f is clearly universally closed we see that f is integral by Lemma 24.44.7. It is also clear that f is universally injective and surjective. Assume f is integral, universally injective and surjective. By Lemma 24.44.7 f is universally closed. Since it is also universally bijective (see Lemma 24.11.4) we see that it is a universal homeomorphism.  Lemma 24.45.4. Let X be a scheme. The canonical closed immersion Xred → X (see Schemes, Definition 21.12.5) is a universal homeomorphism. Proof. Omitted.

 24.46. Finite locally free morphisms

In many papers the authors use finite flat morphisms when they really mean finite locally free morphisms. The reason is that if the base is locally Noetherian then this is the same thing. But in general it is not, see Exercises, Exercise 67.4.3. Definition 24.46.1. Let f : X → S be a morphism of schemes. We say f is finite locally free if f is affine and f∗ OX is a finite locally free OS -module. In this case we say f is has rank or degree d if the sheaf f∗ OX is finite locally free of degree d.

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Note that if f : X → S is finite locally free then S is the disjoint union of open and closed subschemes Sd such that f −1 (Sd ) → Sd is finite locally free of degree d. Lemma 24.46.2. Let f : X → S be a morphism of schemes. The following are equivalent: (1) f is finite locally free, (2) f is finite, flat, and locally of finite presentation. If S is locally Noetherian these are also equivalent to (3) f is finite and flat. Proof. See Algebra, Lemma 7.73.2. The Noetherian case follows as a finite module over a Noetherian ring is a finitely presented module, see Algebra, Lemma 7.29.4.  Lemma 24.46.3. A composition of finite locally free morphisms is finite locally free. Proof. Omitted.



Lemma 24.46.4. A base change of a finite locally free morphism is finite locally free. Proof. Omitted.



Lemma 24.46.5. Let f : X → S be a finite locally free morphism of schemes. ` There exists a disjoint union decomposition S = d≥0 Sd by open and closed subschemes such that setting Xd = f −1 (Sd ) the restrictions f |Xd are finite locally free morphisms Xd → Sd of degree d. Proof. This is true because a finite locally free sheaf locally has a well defined rank. Details omittted.  Lemma 24.46.6. Let f : Y → X be a finite morphism with X affine. There exists a diagram /Y Y0 Z0 o i

 X0 where (1) (2) (3) (4) (5)

 /X

Y 0 → Y and X 0 → X are surjective finite locally free, Y 0 = X 0 ×X Y , i : Y 0 → Z 0 is a closed immersion, Z0 → S X 0 is finite locally free, and 0 Z = j=1,...,m Zj0 is a (set theoretic) finite union of closed subschemes, each of which maps isomorphically to X 0 .

Proof. Write X = Spec(A) and Y = Spec(B). See also More on Algebra, Section 12.18. Let x1 , . . . , xn ∈ B be generators of B over A. For each i we can choose a monic polynomial Pi (T ) ∈ A[T ] such that P (xi ) = 0 in B. By Algebra, Lemma 7.126.9 (applied n times) there exists a finite locally free ring extension A ⊂ A0 such that each Pi splits completely: Y Pi (T ) = (T − αik ) k=1,...,di

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for certain αik ∈ A0 . Set C = A0 [T1 , . . . , Tn ]/(P1 (T1 ), . . . , Pn (Tn )) and B 0 = A0 ⊗A B. The map C → B 0 , Ti 7→ 1 ⊗ xi is an A0 -algebra surjection. Setting X 0 = Spec(A0 ), Y 0 = Spec(B 0 ) and Z 0 = Spec(C) we see that (1) – (4) hold. Part (5) holds because set theoretically Spec(C) is the union of the closed subschemes cut out by the ideals (T1 − α1k1 , T2 − α2k2 , . . . , Tn − αnkn ) for any 1 ≤ ki ≤ di .



The following lemma is stated in the correct generality in Lemma 24.49.4 below. Lemma 24.46.7. Let f : Y → X be a finite morphism of schemes. Let T ⊂ Y be a closed nowhere dense subset of Y . Then f (T ) ⊂ X is a closed nowhere dense subset of X. S Proof. By Lemma 24.44.10 we know that f (T ) ⊂ X is closed. Let X = Xi be an affine covering. Since T is nowhere dense in Y , we see that also T ∩ f −1 (Xi ) is nowhere dense in f −1 (Xi ). Hence if we can prove the theorem in the affine case, then we see that f (T ) ∩ Xi is nowhere dense. This then implies that T is nowhere dense in X by Topology, Lemma 5.17.4. Assume X is affine. Choose a diagram Z0 o

i

Y0

a

f0

 X0

/Y f

b

 /X

as in Lemma 24.46.6. The morphisms a, b are open since they are finite locally free (Lemmas 24.46.2 and 24.26.9). Hence T 0 = a−1 (T ) is nowhere dense, see Topology, Lemma 5.17.6. The morphism b is surjective and open. Hence, if we can prove f 0 (T 0 ) = b−1 (f (T )) is nowhere dense, then f (T ) is nowhere dense, see Topology, Lemma 5.17.6. As i is a closed immersion, by Topology, Lemma 5.17.5 we see that i(T 0 ) ⊂ Z 0 is closed and nowhere dense. Thus we have reduced the problem to the case discussed in the following paragraph. S Assume that Y = i=1,...,n Yi is a finite union of closed subsets, each mapping isomorphically to X. Consider Ti = Yi ∩ T . If each of the Ti is nowhere dense in Yi , then each f (Ti ) is nowhere dense in X as Yi → X is an isomorphism. Hence f (T ) = f (Ti ) is a finite union of nowhere dense closed subsets of X and we win, see Topology, Lemma 5.17.2. Suppose not, say T1 contains a nonempty open V ⊂ Y1 . We are going to show this leads to a contradiction. Consider Y2 ∩ V ⊂ V . This is either a proper closed subset, or equal to V . In the first case we replace V by V \ V ∩ Y2 , so V ⊂ T1 is open in Y1 and does not meet Y2 . In the second case we have V ⊂ Y1 ∩ Y2 is open in both Y1 and Y2 . Repeat sequentially with i = 3, . . . , n. The result is a disjoint union decomposition a {1, . . . , n} = I1 I2 , 1 ∈ I1 and an open V of Y1 contained in T1 such that V ⊂ Yi for i ∈ I1 and V ∩ Yi = ∅ for i ∈ I2 . Set U = f (V ). This is an open of X since f |Y1 : Y1 → X is an isomorphism.

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Then f −1 (U ) = V

a [ i∈I2

(Yi ∩ f −1 (U ))

S As i∈I2 Yi is closed, this implies that V ⊂ f −1 (U ) is open, hence V ⊂ Y is open. This contradicts the assumption that T is nowhere dense in Y , as desired.  24.47. Generically finite morphisms In this section we characterize maps between schemes which are locally of finite type and which are “generically finite” in some sense. Lemma 24.47.1. Let X, Y be schemes. Let f : X → Y be locally of finite type. Let η ∈ Y be a generic point of an irreducible component of Y . The following are equivalent: (1) the set f −1 ({η}) is finite, (2) there exist affine opens S Ui ⊂ X, i = 1, . . . , n and V ⊂ Y with f (Ui ) ⊂ V , η ∈ V and f −1 ({η}) ⊂ Ui such that each f |Ui : Ui → V is finite. If f is quasi-separated, then these are also equivalent to (3) there exist affine opens V ⊂ Y , and U ⊂ X with f (U ) ⊂ V , η ∈ V and f −1 ({η}) ⊂ U such that f |U : U → V is finite. If f is quasi-compact and quasi-separated, then these are also equivalent to (4) there exists an affine open V ⊂ Y , η ∈ V such that f −1 (V ) → V is finite. Proof. The question is local on the base. Hence we may replace Y by an affine neighbourhood of η, and we may and do assume throughout the proof below that Y is affine, say Y = Spec(R). It is clear that (2) implies (1). Assume that f −1 ({η}) = {ξ1 , . . . , ξn } is finite. Choose affine opens Ui ⊂ X with ξi ∈ Ui . By Algebra, Lemma 7.114.9 we see that after replacing Y by a standard open in Y each of the morphisms Ui → Y is finite. In other words (2) holds. It is clear that (3) implies (1). Assume f −1 ({η}) = {ξ1 , . . . , ξn } and assume that f is quasi-separated. Since Y is affine this implies that X is quasi-separated. Since each ξi maps to a generic point of an irreducible component of Y , we see that each ξi is a generic point of an irreducible component of X. By Properties, Lemma 23.27.1 we can find an affine open U ⊂ X containing each ξi . By Algebra, Lemma 7.114.9 we see that after replacing Y by a standard open in Y the morphisms U → Y is finite. In other words (3) holds. It is clear that (4) implies all of (1) – (3) with no further assumptions on f . Suppose that f is quasi-compact and quasi-separated. We have to show that the equivalent conditions (1) – (3) imply (4). Let U , V be as in (3). Replace Y by V . Since f is quasi-compact and Y is quasi-compact (being affine) we see that X is quasicompact. Hence Z = X \ U is quasi-compact, hence the morphism f |Z : Z → Y is quasi-compact. By construction of Z we see that η 6∈ f (Z). Hence by Lemma 24.8.4 we see that there exists an affine open neighbourhood V 0 of η in Y such that f −1 (V 0 ) ∩ Z = ∅. Then we have f −1 (V 0 ) ⊂ U and this means that f −1 (V 0 ) → V 0 is finite. 

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Q Example 24.47.2. Let A = n∈N F2 . Every element of A is an idempotent. Hence every prime ideal is maximal with residue field F2 . Thus the topology on X = Spec(A) is totally disconnected and quasi-compact. The projection maps A → F2 define open points of Spec(A). It cannot be the case that all the points of X are open since X is quasi-compact. Let x ∈ X be a closed point which is not open. Then we can form a scheme Y which is two copies of X glued along X \ {x}. In other words, this is X with x doubled, compare Schemes, Example 21.14.3. The morphism f : Y → X is quasi-compact, finite type and has finite fibres but is not quasi-separated. The point x ∈ X is a generic point of an irreducible component of X (since X is totally disconnected). But properties (3) and (4) of Lemma 24.47.1 do not hold. The reason is that for any open neighbourhood x ∈ U ⊂ X the inverse image f −1 (U ) is not affine because functions on f −1 (U ) cannot separated the two points lying over x (proof omitted; this is a nice exercise). Hence the condition that f is quasi-separated is necessary in parts (3) and (4) of the lemma. Remark 24.47.3. An alternative to Lemma 24.47.1 is the statement that a quasifinite morphism is finite over a dense open of the target. This will be shown in More on Morphisms, Section 33.30. Lemma 24.47.4. Let X, Y be integral schemes. Let f : X → Y be locally of finite type. Assume f is dominant. The following are equivalent: (1) the extension R(Y ) ⊂ R(X) has transcendence degree 0, (2) the extension R(Y ) ⊂ R(X) is finite, (3) there exist nonempty affine opens U ⊂ X and V ⊂ Y such that f (U ) ⊂ V and f |U : U → V is finite, and (4) the generic point of X is the only point of X mapping to the generic point of Y . If f is separated, or if f is quasi-compact, then these are also equivalent to (5) there exists a nonempty affine open V ⊂ Y such that f −1 (V ) → V is finite. Proof. Choose any affine opens Spec(A) = U ⊂ X and Spec(R) = V ⊂ Y such that f (U ) ⊂ V . Then R and A are domains by definition. The ring map R → A is of finite type Lemma 24.16.2). Let K = f.f.(R) = R(Y ) and L = f.f.(A) = R(X). Then K ⊂ L is a finitely generated field extension. Hence we see that (1) is equivalent to (2). Suppose (2) holds. Let x1 , . . . , xn ∈ A be generators of A over R. By assumption there exist nonzero polynomials Pi (X) ∈ R[X] such that Pi (xi ) = 0. Let fi ∈ R be the leading coefficient of Pi . Then we conclude that Rf1 ...fn → Af1 ...fn is finite, i.e., (3) holds. Note that (3) implies (2). So now we see that (1), (2) and (3) are all equivalent. Let η be the generic point of X, and let η 0 ∈ Y be the generic point of Y . Assume (4). Then dimη (Xη0 ) = 0 and we see that R(X) = κ(η) has transcendence degree 0 over R(Y ) = κ(η 0 ) by Lemma 24.29.1. In other words (1) holds. Assume the equivalent conditions (1), (2) and (3). Suppose that x ∈ X is a point mapping to η 0 . As x is a specialization of η, this gives inclusions R(Y ) ⊂ OX,x ⊂ R(X), which implies OX,x is a field, see Algebra, Lemma 7.33.17. Hence x = η. Thus we see that (1) – (4) are all equivalent.

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It is clear that (5) implies (3) with no additional assumptions on f . What remains is to prove that if f is either separated or quasi-compact, then the equivalent conditions (1) – (4) imply (5). Assume U, V as in (3) and assume f is separated. Then U → f −1 (V ) is a morphism from a scheme proper over V Lemma 24.44.10) into a scheme separated over V . Hence U ⊂ f −1 (V ) is closed Lemma 24.42.7. Since X is irreducible we conclude U = f −1 (V ). This proves (5). Assume f is quasi-compact. Let U, V be as in (3). Then f −1 (V ) is quasi-compact. Consider the closed subset Z = f −1 (V ) \ U . Since Z does not contain the generic point of X we see that the quasi-compact morphism f : Z → V is not dominant by Lemma 24.8.3. Hence after shrinking V we may assume that f −1 (V ) = U which implies that (5) holds.  Definition 24.47.5. Let X and Y be integral schemes. Let f : X → Y be locally of finite type and dominant. Assume [R(X) : R(Y )] < ∞, or any other of the equivalent conditions (1) – (4) of Lemma 24.47.4. Then the positive integer deg(X/Y ) = [R(X) : R(Y )] is called the degree of X over Y . It is possible to extend this notion to a morphism f : X → Y if (a) Y is integral with generic point η, (b) f is locally of finite type, and (c) f −1 ({η}) is finite. Namely, in this case we can define X deg(X/Y ) = dimR(Y ) (OX,ξ ). ξ∈X, f (ξ)=η

Namely, given that R(Y ) = κ(η) = OY,η (Lemma 24.10.4) the dimensions above are finite by Lemma 24.47.1 above. However, for most applications the definition given above is the right one. Lemma 24.47.6. Let X, Y , Z be integral schemes. Let f : X → Y and g : Y → Z be dominant morphisms locally of finite type. Assume that [R(X) : R(Y )] < ∞ and [R(Y ) : R(Z)] < ∞. Then deg(X/Z) = deg(X/Y ) deg(Y /Z). Proof. This comes from the multiplicativity of degrees in towers of finite extensions of fields.  Remark 24.47.7. Let f : X → Y be a morphism of schemes which is locally of finite type. There are (at least) two properties that we could use to define generically finite morphisms. These correspond to whether you want the property to be local on the source or local on the target: (1) (Local on the target; suggested by Ravi Vakil.) Assume every quasicompact open of Y has finitely many irreducible components (for example if Y is locally Noetherian). The requirement is that the inverse image of each generic point is finite, see Lemma 24.47.1. (2) (Local on the source.) The requirement is that there exists a dense open U ⊂ X such that U → Y is locally quasi-finite.

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In case (1) the requirement can be formulated without the auxiliary condition on Y , but probably doesn’t give the right notion for general schemes. Property (2) as formulated doesn’t imply that the fibres over generic points are finite; however, if f is quasi-compact and Y is as in (1) then it does. 24.48. Normalization In this section we construct the normalization, and the normalization of one scheme in another. Lemma 24.48.1. Let X be a scheme. Let A be a quasi-coherent sheaf of OX algebras. The subsheaf A0 ⊂ A defined by the rule U 7−→ {f ∈ A(U ) | fx ∈ Ax integral over OX,x for all x ∈ U } is a quasi-coherent OX -algebra, and for any affine open U ⊂ X the ring A0 (U ) ⊂ A(U ) is the integral closure of OX (U ) in A(U ). Proof. This is a subsheaf by the local nature of the conditions. It is an OX -algebra by Algebra, Lemma 7.33.7. Let U ⊂ X be an affine open. Say U = Spec(R) and say A is the quasi-coherent sheaf associated to the R-algebra A. Then according to Algebra, Lemma 7.33.10 the value of A0 over U is given by the integral closure A0 of R in A. This proves the last assertion of the lemma. To prove that A0 is quasi-coherent, it suffices to show that A0 (D(f )) = A0f . This follows from the fact that integral closure and localization commute, see Algebra, Lemma 7.33.9.  Definition 24.48.2. Let X be a scheme. Let A be a quasi-coherent sheaf of OX algebras. The integral closure of OX in A is the quasi-coherent OX -subalgebra A0 ⊂ A constructed in Lemma 24.48.1 above. In the setting of the definition above we can consider the morphism of relative spectra / X 0 = Spec (A0 ) Y = SpecX (A) X &

x X see Lemma 24.13.5. The scheme X 0 → X will be the normalization of X in the scheme Y . Here is a slightly more general setting. Suppose we have a quasi-compact and quasi-separated morphism f : Y → X of schemes. In this case the sheaf of OX algebras f∗ OY is quasi-coherent, see Schemes, Lemma 21.24.1. Taking the integral closure O0 ⊂ f∗ OY we obtain a quasi-coherent sheaf of OX -algebras whose relative spectrum is the normalization of X in Y . Here is the formal definition. Definition 24.48.3. Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let O0 be the integral closure of OX in f∗ OY . The normalization of X in Y is the scheme12 ν : X 0 = SpecX (O0 ) → X over X. It comes equipped with a natural factorization f0

ν

Y −→ X 0 − →X of the initial morphism f . 12The scheme X 0 need not be normal, for example if Y = X and f = id , then X 0 = X. X

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The factorization is the composition of the canonical morphism Y → Spec(f∗ OY ) (see Constructions, Lemma 22.4.7) and the morphism of relative spectra coming from the inclusion map O0 → f∗ OY . We can characterize the normalization as follows. Lemma 24.48.4. Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. The factorization f = ν ◦ f 0 , where ν : X 0 → X is the normalization of X in Y is characterized by the following two properties: (1) the morphism ν is integral, and (2) for any factorization f = π ◦ g, with π : Z → X integral, there exists a commutative diagram Y f0

 X0

g h

/Z > π

ν

 /X

for some unique morphism h : X 0 → Z. Moreover, in (2) the morphism h : X 0 → Z is the normalization of Z in Y . Proof. Let O0 ⊂ f∗ OY be the integral closure of OX as in Definition 24.48.3. The morphism ν is integral by construction, which proves (1). Assume given a factorization f = π ◦ g with π : Z → X integral as in (2). By Definition 24.44.1 Then π is affine, and hence Z is the relative spectrum of a quasi-coherent sheaf of OX -algebras B. The morphism g : X → Z corresponds to a map of OX -algebras χ : B → f∗ OY . Since B(U ) is integral over OX (U ) for every affine open U ⊂ X (by Definition 24.44.1) we see from Lemma 24.48.1 that χ(B) ⊂ O0 . By the functoriality of the relative spectrum Lemma 24.13.5 this provides us with a unique morphism h : X 0 → Z. We omit the verification that the diagram commutes. It is clear that (1) and (2) characterize the factorization f = ν ◦ f 0 since it characterizes it as an initial object in a category. The morphism h in (2) is integral by Lemma 24.44.12. Given a factorization g = π 0 ◦ g 0 with π 0 : Z 0 → Z integral, we get a factorization f = (π ◦ π 0 ) ◦ g 0 and we get a morphism h0 : X 0 → Z 0 . Uniqueness implies that π 0 ◦h0 = h. Hence the characterization (1), (2) applies to the morphism h : X 0 → Z which gives the last statement of the lemma.  Lemma 24.48.5. Let Y2 f2

 X2

/ Y1 f1

 / X1

be a commutative diagram of morphisms of schemes. Assume f1 , f2 quasi-compact and quasi-separated. Let fi = νi ◦ fi0 , i = 1, 2 be the canonical factorizations, where νi : Xi0 → Xi is the normalization of Xi in Yi . Then there exists a canonical

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commutative diagram

/ Y1

Y2 f20

f10

 X20

 / X10

ν2

ν1

 X2

 / X1

Proof. By Lemmas 24.48.4 (1) and 24.44.6 the base change X2 ×X1 X10 → X2 is integral. Note that f2 factors through this morphism. Hence we get a canonical morphism X20 → X2 ×X1 X10 from Lemma 24.48.4 (2). This gives the middle horizontal arrow in the last diagram.  Lemma 24.48.6. Let f : Y → X be a quasi-compact and quasi-separated morphism of schemes. Let U ⊂ X be an open subscheme and set V = f −1 (U ). Then the normalization of U in V is the inverse image of U in the normalization of X in Y . Proof. Clear from the construction.



Lemma 24.48.7. Let f : Y → X be `a quasi-compact and quasi-separated morphism of schemes. Suppose that Y = Y1 Y2 is a disjoint union ` of two schemes. Write fi = fYi . Let Xi0 be the normalization of X in Yi . Then Y10 Y20 is the normalization of X in Y . Proof. In terms of integral closures this corresponds to the following fact: Let A → B be a ring map. Suppose that B = B1 × B2 . Let A0i be the integral closure of A in Bi . Then A01 × A02 is the integral closure of A in B. The reason this works is that the elements (1, 0) and (0, 1) of B are idempotents and hence integral over A. Thus the integral closure A0 of A in B is a product and it is not hard to see that the factors are the integral closures A0i as described above (some details omitted).  Lemma 24.48.8. Let f : Y → X be an integral morphism. Then the integral closure of X in Y is equal to Y . Proof. Omitted.



The following lemma is a generalization of the preceding one. Lemma 24.48.9. Let f : X → S be a quasi-compact, quasi-separated and universally closed morphisms of schemes. Then f∗ OX is integral over OS . In other words, the normalization of S in X is equal to the factorization X −→ SpecS (f∗ OX ) −→ S of Constructions, Lemma 22.4.7. Proof. The question is local on S, hence we may assume S = Spec(R) is affine. Let h ∈ Γ(X, OX ). We have to show that h satisfies a monic equation over R. Think of h as a morphism as in the following commutative diagram X

/ A1 S

h f



S

~

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Let Z ⊂ A1S be the scheme theoretic image of h, see Definition 24.6.2. The morphism h is quasi-compact as f is quasi-compact and A1S → S is separated, see Schemes, Lemma 21.21.15. By Lemma 24.6.3 the morphism X → Z is dominant. By Lemma 24.42.7 the morphism X → Z is closed. Hence h(X) = Z (set theoretically). Thus we can use Lemma 24.42.8 to conclude that Z → S is universally closed (and even proper). Since Z ⊂ A1S , we see that Z → S is affine and proper, hence integral by Lemma 24.44.7. Writing A1S = Spec(R[T ]) we conclude that the ideal I ⊂ R[T ] of Z contains a monic polynomial P (T ) ∈ R[T ]. Hence P (h) = 0 and we win.  24.48.10. Let f : Y → X be a quasi-compact and quasi-separated morschemes. Assume Y is a normal scheme, any quasi-compact open V ⊂ Y has a finite number of irreducible components. Then the normalization X 0 of X in Y is a normal scheme. Moreover, the morphism Y → X 0 is dominant and induces a bijection of irreducible components. Lemma phism of (1) (2)

Proof. We first prove that X 0 is normal. Let U ⊂ X be an affine open. It suffices to prove that the inverse image of U in X 0 is normal (see Properties, Lemma 23.7.2). By Lemma 24.48.6 we may replace X by U , and hence we may assume X = Spec(A) affine. In this case Y is quasi-compact, and ` hence has a finite number of irreducible components by assumption. Hence Y = i=1,...n Yi is a finite disjoint union of normal integral ` schemes by Properties, Lemma 23.7.5. By Lemma 24.48.7 we see that X 0 = i=1,...,n Xi0 , where Xi0 is the normalization of X in Yi . By Properties, Lemma 23.7.9 we see that Bi = Γ(Yi , OYi ) is a normal domain. Note that Xi0 = Spec(A0i ), where A0i ⊂ Bi is the integral closure of A in Bi , see Lemma 0 24.48.1. ` By0 Algebra, Lemma 7.34.2 we see that Ai ⊂ Bi is a normal domain. Hence 0 X = Xi is a finite union of normal schemes and hence is normal. It is clear from the description of X 0 above that Y → X 0 is dominant and induces a bijection on irreducible components if X is affine. The result in general follows from this by a topological argument (omitted).  Lemma 24.48.11. Let f : X → S be a morphism. Assume that (1) S is a Nagata scheme, (2) f is of finite type13, and (3) X is reduced. Then the normalization ν : S 0 → S of S in X is finite. Proof. There is an immediate reduction to the case S = Spec(R) where R is a Nagata ring. In this case we have to show that the integral closureSA of R in Γ(X, OX ) is finite over R. Since f is of finite type we can write X = i=1,...,n Ui with each Ui affine. Say Ui = Spec(Bi ). Each Bi is a reduced ring of finite type over R (Lemma 24.16.2). Moreover, Γ(X, OX ) ⊂ B = B1 × . . . × Bn . So A is contained in the integral closure A0 of R in B. Note that B is a reduced finite type R-algebra. Since R is Noetherian it suffices to prove that A0 is finite over R. This is Algebra, Lemma 7.145.16.  13The proof shows that the lemma holds if f is quasi-compact and “essentially of finite type”.

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Next, we come to the normalization of a scheme X. We only define/construct it when X has locally finitely many irreducible components. Let X be a scheme such that every quasi-compact open has finitely many irreducible components. Let X (0) ⊂ X be the set of generic points of irreducible components of X. Let a (24.48.11.1) f :Y = Spec(κ(η)) −→ X (0) η∈X

be the inclusion of the generic points into X using the canonical maps of Schemes, Section 21.13. Note that this morphism is quasi-compact by assumption and quasiseparated as Y is separated (see Schemes, Section 21.21). Definition 24.48.12. Let X be a scheme such that every quasi-compact open has finitely many irreducible components. We define the normalization of X as the morphism ν : X ν −→ X which is the normalization of X in the morphism f : Y → X (24.48.11.1) constructed above. Any locally Noetherian scheme has a locally finite set of irreducible components and the definition applies to it. Usually the normalization is defined only for reduced schemes. With the definition above the normalization of X is the same as the normalization of the reduction Xred of X. Lemma 24.48.13. Let X be a scheme such that every quasi-compact open has finitely many irreducible components. The normalization morphism ν factors through the reduction Xred and X ν → Xred is the normalization of Xred . Proof. Let f : Y → X be the morphism (24.48.11.1). We get a factorization Y → Xred → X of f from Schemes, Lemma 21.12.6. By Lemma 24.48.4 we obtain a canonical morphism X ν → Xred and that X ν is the normalization of Xred in Y . The lemma follows as Y → Xred is identical to the morphism (24.48.11.1) constructed for Xred .  If X is reduced, then the normalization of X is the same as the relative spectrum of the integral closure of OX in the sheaf of meromorphic functions KX (see Divisors, Section 26.15). Namely, KX = f∗ OY in this case, see Divisors, Lemma 26.15.7 and its proof. We describe this here explicitly. Lemma 24.48.14. Let X be a reduced scheme such that every quasi-compact open has finitely many irreducible components. Let Spec(A) = U ⊂ X be an affine open. Then (1) A has finitely many minimal primes q1 , . . . , qt , (2) the total ring of fractions Q(A) of A is Q(A/q1 ) × . . . × Q(A/qt ), (3) the integral closure A0 of A in Q(A) is the product of the integral closures of the domains A/qi in the fields Q(A/qi ), and (4) ν −1 (U ) is identified with the spectrum of A0 . Proof. Minimal primes correspond to irreducible components (Algebra, Lemma 7.24.1), hence we have (1) by assumption. Then (0) = q1 ∩ . .Q . ∩ qt because A Q is reduced (Algebra, Lemma 7.16.2). Then we have Q(A) = Aqi = κ(qi ) by Algebra, Lemmas 7.23.2 and 7.24.3. This proves (2). Part (3) follows from

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Algebra, Lemma 7.34.14, or Lemma 24.48.7. Part (4) holds because it is clear that f −1 (U ) → U is the morphism Y  Spec κ(qi ) −→ Spec(A) where f : Y → X is the morphism (24.48.11.1).



Lemma 24.48.15. Let X be a scheme such that every quasi-compact open has finitely many irreducible components. (1) The normalization X ν is a normal scheme. (2) The morphism ν : X ν → X is integral, surjective, and induces a bijection on irreducible components. (3) For any integral, birational14 morphism X 0 → X there exists a factorization X ν → X 0 → X and X ν → X 0 is the normalization of X 0 . (4) For any morphism Z → X with Z a normal scheme such that each irreducible component of Z dominates an irreducible component of X there exists a unique factorization Z → X ν → X. Proof. Let f : Y → X be as in (24.48.11.1). Part (1) follows from Lemma 24.48.10 and the fact that Y is normal. It also follows from the description of the affine opens in Lemma 24.48.14. The morphism ν is integral by Lemma 24.48.4. By Lemma 24.48.10 the morphism Y → X ν induces a bijection on irreducible components, and by construction of Y this implies that X ν → X induces a bijection on irreducible components. By construction f : Y → X is dominant, hence also ν is dominant. Since an integral morphism is closed (Lemma 24.44.7) this implies that ν is surjective. This proves (2). Suppose that α : X 0 → X is integral and birational. Any quasi-compact open U 0 of X 0 maps to a quasi-compact open of X, hence we see that U 0 has only finitely many irreducible components. Let f 0 : Y 0 → X 0 be the morphism (24.48.11.1) constructed starting with X 0 . As α is birational it is clear that Y 0 = Y and f = α ◦ f 0 . Hence the factorization X ν → X 0 → X exists and X ν → X 0 is the normalization of X 0 by Lemma 24.48.4. This proves (3). Let g : Z → X be a morphism whose domain is a normal scheme and such that every irreducible component dominates an irreducible component of X. By Lemma ν 24.48.13 we have X ν = Xred and by Schemes, Lemma 21.12.6 Z → X factors through Xred . Hence we may replace X by Xred and assume X is reduced. Moreover, as the factorization is unique it suffices to construct it locally on Z. Let W ⊂ Z and U ⊂ X be affine opens such that g(W ) ⊂ U . Write U = Spec(A) and W = Spec(B), with g|W given by ϕ : A → B. We will use the results of Lemma 24.48.14 freely. Let p1 , . . . , pt be the minimal primes of A. As Z is normal, we see that B is a normal ring, in particular reduced. Moreover, by assumption any minimal prime q ⊂ B we have thatSϕ−1 (q) is a minimal prime of A. Hence if x ∈ A is a nonzerodivisor, i.e., x 6∈ pi , then ϕ(x) is a nonzerodivisor in B. Thus we obtain a canonical ring map Q(A) → Q(B). As B is normal it is equal to its integral closure in Q(B) (see Algebra, Lemma 7.34.11). Hence we see that the integral closure A0 ⊂ Q(A) of A maps into B via the canonical map Q(A) → Q(B). 14It suffices if X 0 red → Xred is birational.

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Since ν −1 (U ) = Spec(A0 ) this gives the canonical factorization W → ν −1 (U ) → U of ν|W . We omit the verification that it is unique.  Lemma 24.48.16. Let X be an integral, Japanese scheme. The normalization ν : X ν → X is a finite morphism. Proof. Follows from the definitions and Lemma 24.48.14. Namely, in this case the lemma says that ν −1 (Spec(A)) is the spectrum of the integral closure of A in its field of fractions.  Lemma 24.48.17. Let X be a nagata scheme. The normalization ν : X ν → X is a finite morphism. Proof. Note that a Nagata scheme is locally Noetherian, thus Definition 24.48.12 does apply. Write X ν → X as the composition X ν → Xred → X. As Xred → X is a closed immersion it is finite. Hence it suffices to prove the lemma for a reduced Nagata scheme (by Lemma 24.44.5). Let Q Spec(A) = U ⊂ X be an affine open. By Lemma 24.48.14 we have ν −1 (U ) = Spec( A0i ) where A0i is the integral closure of A/qi in its fraction field. As A is a Nagata ring (see Properties, Q Lemma 23.13.6) each of the ring extensions A/qi ⊂ A0i are finite. Hence A → A0i is a finite ring map and we win.  24.49. Zariski’s Main Theorem (algebraic version) This is the version you can prove using purely algebraic methods. Before we can prove more powerful versions (for non-affine morphisms) we need to develop more tools. See Cohomology of Schemes, Section 25.20 and More on Morphisms, Section 33.30. Theorem 24.49.1 (Algebraic version of Zariski’s Main Theorem). Let f : Y → X be an affine morphism of schemes. Assume f is of finite type. Let X 0 be the normalization of X in Y . Picture: Y

/ X0

f0 f

X

~

ν

Then there exists an open subscheme U 0 ⊂ X 0 such that (1) (f 0 )−1 (U 0 ) → U 0 is an isomorphism, and (2) (f 0 )−1 (U 0 ) ⊂ Y is the set of points at which f is quasi-finite. Proof. There is an immediate reduction to the case where X and hence Y are affine. Say X = Spec(R) and Y = Spec(A). Then X 0 = Spec(A0 ), where A0 is the integral closure of R in A, see Definitions 24.48.2 and 24.48.3. By Algebra, Theorem 7.115.13 for every y ∈ Y at which f is quasi-finite, there exists S an open Uy0 ⊂ X 0 such that (f 0 )−1 (Uy0 ) → Uy0 is an isomorphism. Set U 0 = Uy0 where y ∈ Y ranges over all points where f is quasi-finite. It remains to show that f is quasi-finite at all points of (f 0 )−1 (U 0 ). If y ∈ (f 0 )−1 (U 0 ) with image x ∈ X, then we see that Yx → Xx0 is an isomorphism in a neighbourhood of y. Hence there is no point of Yx which specializes to y, since this is true for f 0 (y) in Xx0 , see Lemma 24.44.8. By Lemma 24.21.6 part (3) this implies f is quasi-finite at y. 

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We can use the algebraic version of Zariski’s Main Theorem to show that the set of points where a morphism is quasi-finite is open. Lemma 24.49.2. Let f : X → S be a morphism of schemes. The set of points of X where f is quasi-finite is an open U ⊂ X. The induced morphism U → S is locally quasi-finite. Proof. Suppose f is quasi-finite at x. Let x ∈ U = Spec(R) ⊂ X, V = Spec(A) ⊂ S be affine opens as in Definition 24.21.1. By either Theorem 24.49.1 above or Algebra, Lemma 7.115.14, the set of primes q at which R → A is quasi-finite is open in Spec(A). Since these all correspond to points of X where f is quasi-finite we get the first statement. The second statement is obvious.  We will improve the following lemma to general quasi-finite separated morphisms later, see More on Morphisms, Lemma 33.30.5. Lemma 24.49.3. Let f : Y → X be a morphism of schemes. Assume (1) X and Y are affine, and (2) f is quasi-finite. Then there exists a diagram Y

/Z

j π

~

f

X

with Z affine, π finite and j an open immersion. Proof. This is Algebra, Lemma 7.115.15 reformulated in the laguage of schemes.  Lemma 24.49.4. Let f : Y → X be a quasi-finite morphism of schemes. Let T ⊂ Y be a closed nowhere dense subset of Y . Then f (T ) ⊂ X is a nowhere dense subset of X. Proof. As in the proof of Lemma 24.46.7 this S reduces immediately to the case where the base X is affine. In this case Y = i=1,...,n Yi is a finite union of affine opens (as f is quasi-compact). Since each T ∩ Yi is nowhere dense, and since a finite union of nowhere dense sets is nowhere dense (see Topology, Lemma 5.17.2), it suffices to prove that the image f (T ∩ Yi ) is nowhere dense in X. This reduces us to the case where both X and Y are affine. At this point we apply Lemma 24.49.3 above to get a diagram /Z Y j

f

X

~

π

with Z affine, π finite and j an open immersion. Set T = j(T ) ⊂ Z. By Topology, Lemma 5.17.3 we see T is nowhere dense in Z. Since f (T ) ⊂ π(T ) the lemma follows from the corresponding result in the finite case, see Lemma 24.46.7. 

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24.50. Universally bounded fibres Let X be a scheme over a field k. If X is finite over k, then X = Spec(A) where A is a finite k-algebra. Another way to say this is that X is finite locally free over Spec(k), see Definition 24.46.1. Hence X → Spec(k) has a degree which is an integer d ≥ 0, namely d = dimk (A). We sometime call this the degree of the (finite) scheme X over k. Definition 24.50.1. Let f : X → Y be a morphism of schemes. (1) We say the integer n bounds the degrees of the fibres of f if for all y ∈ Y the fibre Xy is a finite scheme over κ(y) whose degree over κ(y) is ≤ n. (2) We say the fibres of f are universally bounded15 if there exists an integer n which bounds the degrees of the fibres of f . Note that in particular the number of points in a fibre is bounded by n as well. (The converse does not hold, even if all fibres are finite reduced schemes.) Lemma 24.50.2. Let f : X → Y be a morphism of schemes. Let n ≥ 0. The following are equivalent: (1) the integer n bounds the degrees of the fibres of f , and (2) for every morphism Spec(k) → Y , where k is a field, the fibre product Xk = Spec(k) ×Y X is finite over k of degree ≤ n. In this case f is universally bounded and the schemes Xk have at most n points. Proof. The implication (2) ⇒ (1) is trivial. The other implication holds because if the image of Spec(k) → Y is y, then Xk = Spec(k) ×Spec(κ(y)) Xy .  Lemma 24.50.3. A composition of morphisms with universally bounded fibres is a morphism with universally bounded fibres. More precisely, assume that n bounds the degrees of the fibres of f : X → Y and m bounds the degrees of g : Y → Z. Then nm bounds the degrees of the fibres of g ◦ f : X → Z. Proof. Let f : X → Y and g : Y → Z have universally bounded fibres. Say that deg(Xy /κ(y)) ≤ n for all y ∈ Y , and that deg(Yz /κ(z)) ≤ m for all z ∈ Z. Let z ∈ Z be a point. By assumption the scheme Yz is finite over Spec(κ(z)). In particular, the underlying topological space of Yz is a finite discrete set. The fibres of the morphism fz : Xz → Yz are the fibres of f at the corresponding points of Y , which are finite discrete sets by the reasoning above. Hence we conclude that the underlying topological space of Xz is a finite discrete set as well. Thus Xz is an affine scheme (this is a nice exercise; it also follows for example from Properties, Lemma 23.27.1 applied to the set of all points of Xz ). Write Xz = Spec(A), Yz = Spec(B), and k = κ(z). Then k → B → A and we know that (a) dimk (B) ≤ m, and (b) for every maximal ideal m ⊂ B we have dimκ(m) (A/mA) ≤ n. We claim this implies that dimk (A) ≤ nm. Note that B is the product of its localizations Bm , for example because Yz is a disjoint union of 1-pointPschemes, or by Algebra, P Lemmas 7.50.2 and 7.50.8. So we see that dimk (B) = m (Bm ) and dimk (A) = m (Am ) where in both cases m runs over the maximal ideals of B (not of A). By the above, and Nakayama’s Lemma (Algebra, Lemma 7.18.1) we see that each Am is a quotient 15This is probably nonstandard notation.

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⊕n of Bm as a Bm -module. Hence dimk (Am ) ≤ n dimk (Bm ). Putting everything together we see that X X dimk (A) = (Am ) ≤ n dimk (Bm ) = n dimk (B) ≤ nm m

m

as desired.



Lemma 24.50.4. A base change of a morphism with universally bounded fibres is a morphism with universally bounded fibres. More precisely, if n bounds the degrees of the fibres of f : X → Y and Y 0 → Y is any morphism, then the degrees of the fibres of the base change f 0 : Y 0 ×Y X → Y 0 → Y 0 is also bounded by n. Proof. This is clear from the result of Lemma 24.50.2.



Lemma 24.50.5. Let f : X → Y be a morphism of schemes. Let Y 0 → Y be a morphism of schemes, and let f 0 : X 0 = XY 0 → Y 0 be the base change of f . If Y 0 → Y is surjective and f 0 has universally bounded fibres, then f has universally bounded fibres. More precisely, if n bounds the degree of the fibres of f 0 , then also n bounds the degrees of the fibres of f . Proof. Let n ≥ 0 be an integer bounding the degrees of the fibres of f 0 . We claim that n works for f also. Namely, if y ∈ Y is a point, then choose a point y 0 ∈ Y 0 lying over y and observe that Xy0 0 = Spec(κ(y 0 )) ×Spec(κ(y)) Xy . Since Xy0 0 is assumed finite of degree ≤ n over κ(y 0 ) it follows that also Xy is finite of degree ≤ n over κ(y). (Some details omitted.)  Lemma 24.50.6. An immersion has universally bounded fibres. Proof. The integer n = 1 works in the definition.



Lemma 24.50.7. Let f : X → Y be an ´etale morphism of schemes. Let n ≥ 0. The following are equivalent (1) the integer n bounds the degrees of the fibres, (2) for every field k and morphism Spec(k) → Y the base change Xk = Spec(k) ×Y X has at most n points, and (3) for every y ∈ Y and every separable algebraic closure κ(y) ⊂ κ(y)sep the scheme Xκ(y)sep has at most n points. Proof. This follows from Lemma 24.50.2 and the fact that the fibres Xy are disjoint unions of spectra of finite separable field extensions of κ(y), see Lemma 24.37.7.  Having universally bounded fibres is an absolute notion and not a relative notion. This is why the condition in the following lemma is that X is quasi-compact, and not that f is quasi-compact. Lemma 24.50.8. Let f : X → Y be a morphism of schemes. Assume that (1) f is locally quasi-finite, and (2) X is quasi-compact. Then f has universally bounded fibres.

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24. MORPHISMS OF SCHEMES

Proof. Since X is quasi-compact, there exists a finite affine open covering X = S i=1,...,n Ui and affine opens Vi ⊂ Y , i = 1, . . . , n such that f (Ui ) ⊂ Vi . Because of the local nature of “local quasi-finiteness” (see Lemma 24.21.6 part (4)) we see that the morphisms f |Ui : Ui → Vi are locally quasi-finite morphisms S of affines, hence quasi-finite, see Lemma 24.21.9. For y ∈ Y it is clear that Xy = y∈Vi (Ui )y is an open covering. Hence it suffices to prove the lemma for a quasi-finite P morphism of affines (namely, if ni works for the morphism f |Ui : Ui → Vi , then ni works for f ). Assume f : X → Y is a quasi-finite morphism of affines. By Lemma 24.49.3 we can find a diagram /Z X j



f

π

Y with Z affine, π finite and j an open immersion. Since j has universally bounded fibres (Lemma 24.50.6) this reduces us to showing that π has universally bounded fibres (Lemma 24.50.3). This reduces us to a morphism of the form Spec(B) → Spec(A) where A → B is finite. Say B is generated by x1 , . . . , xn over A and say Pi (T ) ∈ A[T ] is a monic polynomial of degree di such that Pi (xi ) = 0 in B (a finite ring extension is integral, see Algebra, Lemma 7.33.3). With these notations it is clear that M X A −→ B, (a(e1 ,...,en ) ) 7−→ a(e1 ,...,en ) xe11 . . . xenn 0≤ei 0. H Proof. Write U = Spec(A) for some ring A. In other words, f1 , . . . , fn are elements f for some A-module M . of A which generate the unit ideal of A. Write F|U = M • ˇ Clearly the Cech complex C (U, F) is identified with the complex Y Y Y Mfi0 → Mfi0 fi1 → Mfi0 fi1 fi2 → . . . i0

i0 i1

i0 i1 i2

We are asked to show that the extended complex Y Y Y (25.2.1.1) 0→M → Mfi0 → Mfi0 fi1 → Mfi0 fi1 fi2 → . . . i0

i0 i1

i0 i1 i2

(whose truncation we have studied in Algebra, Lemma 7.21.2) is exact. It suffices to show that (25.2.1.1) is exact after localizing at a prime p, see Algebra, Lemma 7.22.1. In fact we will show that the extended complex localized at p is homotopic to zero. There exists an index i such that fi 6∈ p. Choose and fix such an element ifix . Note that Mfifix ,p = Mp . Similarly for a localization at a product fi0 . . . fip and p we can drop any fij for which ij = ifix . Let us define a homotopy Y Y h: Mfi0 ...fip+1 ,p −→ Mfi0 ...fip ,p i0 ...ip+1

i0 ...ip

by the rule h(s)i0 ...ip = sifix i0 ...ip (This is “dual” to the Q homotopy in the proof of Cohomology, Lemma 18.10.4.) In other words, h : i0 Mfi0 ,p → M is projection onto the factor Mfifix ,p = Mp 1483

1484

25. COHOMOLOGY OF SCHEMES

and in general the map h equal projection onto the factors Mfifix fi1 ...fip+1 ,p = Mfi1 ...fip+1 ,p . We compute (dh + hd)(s)i0 ...ip

p X = (−1)j h(s)i0 ...ˆij ...ip + d(s)ifix i0 ...ip j=0 p p X X = (−1)j sifix i0 ...ˆij ...ip + si0 ...ip + (−1)j+1 sifix i0 ...ˆij ...ip j=0

j=0

=si0 ...ip This proves the identity map is homotopic to zero as desired.



The following lemma says in particular that for any affine scheme X and any quasicoherent sheaf F on X we have H p (X, F) = 0 for all p > 0. Lemma 25.2.2. Let X be a scheme. Let F be a quasi-coherent OX -module. For any affine open U ⊂ X we have H p (U, F) = 0 for all p > 0. Proof. We are going to apply Cohomology, Lemma 18.11.8. As our basis B for the topology of X we are going to use the affine opens of X. As our set Cov of open coverings we are going to use the standard open coverings of affine opens of X. Next we check that conditions (1), (2) and (3) of Cohomology, Lemma 18.11.8 hold. Note that the intersection of standard opens in an affine is another standard open. Hence property (1) holds. The coverings form a cofinal system of open coverings of any element of B, see Schemes, Lemma 21.5.1. Hence (2) holds. Finally, condition (3) of the lemma follows from Lemma 25.2.1.  Here is a relative version of the vanishing of cohomology of quasi-coherent sheaves on affines. Lemma 25.2.3. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. If f is affine then Ri f∗ F = 0 for all i > 0. Proof. According to Cohomology, Lemma 18.6.3 the sheaf Ri f∗ F is the sheaf associated to the presheaf V 7→ H i (f −1 (V ), F|f −1 (V ) ). By assumption, whenever V is affine we have that f −1 (V ) is affine, see Morphisms, Definition 24.13.1. By Lemma 25.2.2 we conclude that H i (f −1 (V ), F|f −1 (V ) ) = 0 whenever V is affine. Since S has a basis consisting of affine opens we win.  S Lemma 25.2.4. Let X be a scheme. Let U : X = i∈I Ui be an open covering such that Ui0 ...ip is affine open for all p ≥ 0 and all i0 , . . . , ip ∈ I In this case for any quasi-coherent sheaf F we have ˇ p (U, F) = H p (X, F) H as Γ(X, OX )-modules for all p. Proof. In view of Lemma 25.2.2 this is a special case of Cohomology, Lemma 18.11.5. 

25.3. VANISHING OF COHOMOLOGY

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25.3. Vanishing of cohomology We have seen that on an affine scheme the higher cohomology groups of any quasicoherent sheaf vanish (Lemma 25.2.2). It turns out that this also characterizes affine schemes. We give two versions allthough the first covers all conceivable cases. Lemma 25.3.1. Let X be a scheme. Assume that (1) X is quasi-compact, (2) for every quasi-coherent sheaf of ideals I ⊂ OX we have H 1 (X, I) = 0. Then X is affine. Proof. Let x ∈ X be a closed point. Let U ⊂ X be an affine open neighbourhood of x. Write U = Spec(A) and let m ⊂ A be the maximal ideal corresponding to x. Set Z = X \ U and Z 0 = Z ∪ {x}. By Schemes, Lemma 21.12.4 there are quasicoherent sheaves of ideals I, resp. I 0 cutting out the reduced closed subschemes Z, resp. Z 0 . Consider the short exact sequence 0 → I 0 → I → I/I 0 → 0. Since x is a closed point of X and x 6∈ Z we see that I/I 0 is supported at x. In fact, the restriction of I/I 0 to U corresponds to the A-module A/m. Hence we see that Γ(X, I/I 0 ) = A/m. Since by assumption H 1 (X, I 0 ) = 0 we see there exists a global section f ∈ Γ(X, I) which maps to the element 1 ∈ A/m as a section of I/I 0 . Clearly we have x ∈ Xf ⊂ U . This implies that Xf = D(fA ) where fA is the image of f in A = Γ(U, OX ). In particular Xf is affine. S Consider the union W = Xf over all f ∈ Γ(X, OX ) such that Xf is affine. Obviously W is open in X. By the arguments above every closed point of X is contained in W . The closed subset X \W of X is also quasi-compact (see Topology, Lemma 5.9.3). Hence it has a closed point if it is nonempty (see Topology, Lemma 5.9.6). This is a would contradict the fact that all closed points are in W . Hence we conclude X = W . Choose finitely many f1 , . . . , fn ∈ Γ(X, OX ) such that X = Xf1 ∪ . . . ∪ Xfn and such that each Xfi is affine. This is possible as we’ve seen above. By Properties, Lemma 23.25.2 to finish the proof it suffices to show that f1 , . . . , fn generate the unit ideal in Γ(X, OX ). Conisder the short exact sequence 0

/F

/ O⊕n X

f1 ,...,fn

/ OX

/0

The arrow defined by f1 , . . . , fn is surjective since the opens Xfi cover X. We let F be the kernel of this surjective map. Observe that F has a filtration 0 = F0 ⊂ F1 ⊂ . . . ⊂ Fn = F so that each subquotient Fi /Fi−1 is isomorphic to a quasi-coherent sheaf of ideals. Namely we can take Fi to be the intersection of the first i direct summands of ⊕n OX . The assumption of the lemma implies that H 1 (X, Fi /Fi−1 ) = 0 for all i. This implies that H 1 (X, F2 ) = 0 because it is sandwiched between H 1 (X, F1 ) and H 1 (X, F2 /F1 ). Continuing like this we deduce that H 1 (X, F) = 0. Therefore we conclude that the map L f1 ,...,fn / Γ(X, OX ) i=1,...,n Γ(X, OX ) is surjective as desired.



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25. COHOMOLOGY OF SCHEMES

Note that if X is a Noetherian scheme then every quasi-coherent sheaf of ideals is automatically a coherent sheaf of ideals and a finite type quasi-coherent sheaf of ideals. Hence the preceding lemma and the next lemma both apply in this case. Lemma 25.3.2. Let X be a scheme. Assume that (1) X is quasi-compact, (2) X is quasi-separated, and (3) H 1 (X, I) = 0 for every quasi-coherent sheaf of ideals I of finite type. Then X is affine. Proof. By Properties, Lemma 23.20.3 every quasi-coherent sheaf of ideals is a directed colimit of quasi-coherent sheaves of ideals of finite type. By Cohomology, Lemma 18.15.1 taking cohomology on X commutes with directed colimits. Hence we see that H 1 (X, I) = 0 for every quasi-coherent sheaf of ideals on X. In other words we see that Lemma 25.3.1 applies.  25.4. Derived category of quasi-coherent modules In this section we briefly discuss the relationship between quasi-coherent modules and all modules on a scheme S. (This should be elaborated on and generalized.) A reference is [TT90, Appendix B]. By the discussion in Schemes, Section 21.24 the embedding QCoh(OS ) ⊂ Mod(OS ) exhibits QCoh(OS ) as a weak Serre subcategory of the category of OS -modules. Denote DQCoh (OS ) ⊂ D(OS ) the subcategory of complexes whose cohomology sheaves are quasi-coherent, see Derived Categories, Section 11.12. Thus we obtain a canonical functor (25.4.0.1)

D(QCoh(OS )) −→ DQCoh (OS )

see Derived Categories, Equation (11.12.1.1). Lemma 25.4.1. If S = Spec(A) is an affine scheme, then (25.4.0.1) is an equivalence. Proof. The key to this lemma is to prove that the functor RΓ(S, −) gives a quasiinverse. For complexes bounded below this is straightforward using the vanishing of cohomology of Lemma 25.2.2. To prove it also for unbounded complexes we have to be a little bit careful: namely, even if you accept that the unbounded derived functor RΓ(S, −) exists, then it isn’t obvious how to compute it! Let F • be an object of DQCoh (OS )) and denote Hi = H i (F • ) its ith cohomology sheaf. Let B be the set of affine open subsets of S. Then H p (U, Hi ) = 0 for all p > 0, all i ∈ Z, and all U ∈ B, see Lemma 25.2.2. According to Cohomology, Section 18.23 this implies there exists a quasi-isomorphism F • → I • where I • is a K-injective complex, I • = lim In• , each In• is a bounded below complex of injectives, the maps in the system . . . → I2• → I1• are termwise split surjections, and each In• is quasi-isomorphic to τ≥−n F • . In particular, we conclude that RΓ(S, −) is defined at each object of DQCoh (OS )), see Derived Categories, Lemma 11.28.4, with values RΓ(S, F • ) = Γ(S, I • ). This defines an exact functor of triangulated categories (25.4.1.1)

RΓ(S, −) : DQCoh (OS ) −→ D(A),

see Derived Categories, Proposition 11.14.8. In the proof of Cohomology, Lemma 18.23.1 we have seen that H m (Γ(S, I • )) is the limit of the cohomology groups H m (Γ(S, In• )). For n > −m these groups are equal to Γ(S, Hm ) by the vanishing of

25.5. QUASI-COHERENCE OF HIGHER DIRECT IMAGES

1487

higer cohomology and the spectral sequence of Derived Categories, Lemma 11.20.3. Combined with the (assumed) equality ^ Hm = Γ(S, Hm ) we conclude the canonical map of complexes ^ Γ(S, I • ) −→ I • (see Schemes, Lemma 21.7.1) is a quasi-isomorphism. We claim the composition D(A) ∼ = D(QCoh(OS )) −→ DQCoh (OS ) −→ D(A) is isomorphic to the identity functor. Namely, given a complex of A-modules M • , g• , choose F • → I • as above, and finally take Γ(S, I • ). The arguments let F • = M above show that M • = Γ(S, F • ) → Γ(S, I • ) is a quasi-isomorphism. This is functorial in M • , hence we conclude that the composition of functors is isomorphic to the identity functor on D(A). On the other hand, we have seen above that the composition ∼ D(QCoh(OS )) −→ DQCoh (OS ) DQCoh (OS ) −→ D(A) = ^ is isomorphic to the identity functor, via the quasi-isomorphisms Γ(S, I •) → I • above. This finishes the proof.  Actually it is true that the comparison map D(QCoh(OS )) → DQCoh (OS ) is an equivalence for any quasi-compact and (semi-)separated scheme (insert future reference here). 25.5. Quasi-coherence of higher direct images We have seen that the higher cohomology groups of a quasi-coherent module on an affine is zero. For (quasi-)separated quasi-compact schemes X this implies vanishing of cohomology groups of quasi-coherent sheaves beyond a certain degree. However, it may not be the case that X has finite cohomological dimension, because that is defined in terms of vanishing of cohomology of all OX -modules. Lemma 25.5.1. Let X be a quasi-compact separated scheme. Let t = t(X) be the minimal number of affine opens needed to cover X. Then H n (X, F) = 0 for all n ≥ t and all quasi-coherent sheaves F. Proof. First proof. By induction on t. If t = 1 the result follows from Lemma 25.2.2. If t > 1 write X = U ∪ V with V affine open and U = U1 ∪ . . . ∪ Ut−1 a union of t − 1 open affines. Note that in this case U ∩ V = (U1 ∩ V ) ∪ . . . (Ut−1 ∩ V ) is also a union of t − 1 affine open subschemes, see Schemes, Lemma 21.21.8. We apply the Mayer-Vietoris long exact sequence 0 → H 0 (X, F) → H 0 (U, F) ⊕ H 0 (V, F) → H 0 (U ∩ V, F) → H 1 (X, F) → . . . see Cohomology, Lemma 18.8.2. By induction we see that the groups H i (U, F), H i (U, F), H i (U, F) are zero for i ≥ t − 1. It follows immediately that H i (X, F) is zero for i ≥ t. St Second proof. Let U : X = i=1 Ui be a finite affine open covering. Since X is separated the multiple intersections Ui0 ...ip are all affine, see Schemes, Lemma ˇ p (U, F) agree with the 21.21.8. By Lemma 25.2.4 the Cech cohomology groups H cohomology groups. By Cohomology, Lemma 18.17.6 the Cech cohomology groups

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25. COHOMOLOGY OF SCHEMES

• may be computed using the alternating Cech complex Cˇalt (U, F). As the covering p consists of t elements we see immediately that Cˇalt (U, F) = 0 for all p ≥ t. Hence the result follows. 

Lemma 25.5.2. Let X be a quasi-compact quasi-separated scheme. Let X = U1 ∪ . . . ∪ Ut be an affine open covering. Set \ d = maxI⊂{1,...,t} |I| + t( Ui ) i∈I

where t(U ) is the minimal number of affines needed to cover the scheme U . Then H n (X, F) = 0 for all n ≥ d and all quasi-coherent sheaves F. T Proof. Note that since X is quasi-separated the numbers t( i∈I Ui ) are finite. Let St U : X = i=1 Ui . By Cohomology, Lemma 18.11.4 there is a spectral sequence ˇ p (U, H q (F)) E2p,q = H converging to H p+q (U, F). By Cohomology, Lemma 18.17.6 we have E p,q = H p (Cˇ• (U, H q (F)) 2

alt

ˇ The alternating Cech complex with values in the presheaf H q (F) vanishes in high degrees by Lemma 25.5.1, more precisely E2p,q = 0 for p + q ≥ d. Hence the result follows.  Lemma 25.5.3. Let f : X → S be a morphism of schemes. Assume that f is quasi-separated and quasi-compact. (1) For any quasi-coherent OX -module F the higher direct images Rp f∗ F are quasi-coherent on S. (2) If S is quasi-compact, there exists an integer n = n(X, S, f ) such that Rp f∗ F = 0 for all p ≥ n and any quasi-coherent sheaf F on X. (3) In fact, if S is quasi-compact we can find n = n(X, S, f ) such that for every morphism of schemes S 0 → S we have Rp (f 0 )∗ F 0 = 0 for p ≥ n and any quasi-coherent sheaf F 0 on X 0 . Here f 0 : X 0 = S 0 ×S X → S 0 is the base change of f . Proof. We first prove (1). Note that under the hypotheses of the lemma the sheaf R0 f∗ F = f∗ F is quasi-coherent by Schemes, Lemma 21.24.1. Using Cohomology, Lemma 18.6.4 we see that forming higher direct images commutes with restriction to open subschemes. Since being quasi-coherent is local on S we may assume S is affine. Assume S is affine and f quasi-compact and separated. Let t ≥ 1 be the minimal number of affine opens needed to cover X. We will prove this case of (1) by induction on t. If t = 1 then the morphism f is affine by Morphisms, Lemma 24.13.12 and (1) follows from Lemma 25.2.3. If t > 1 write X = U ∪ V with V affine open and U = U1 ∪ . . . ∪ Ut−1 a union of t − 1 open affines. Note that in this case U ∩ V = (U1 ∩ V ) ∪ . . . (Ut−1 ∩ V ) is also a union of t − 1 affine open subschemes, see Schemes, Lemma 21.21.8. We will apply the relative Mayer-Vietoris sequence 0 → f∗ F → a∗ (F|U ) ⊕ b∗ (F|V ) → c∗ (F|U ∩V ) → R1 f∗ F → . . . see Cohomology, Lemma 18.8.3. By induction we see that Rp a∗ F, Rp b∗ F and Rp c∗ F are all quasi-coherent. This implies that each of the sheaves Rp f∗ F is quasi-coherent since it sits in the middle of a short exact sequence with a cokernel

25.5. QUASI-COHERENCE OF HIGHER DIRECT IMAGES

1489

of a map between quasi-coherent sheaves on the left and a kernel of a map between quasi-coherent sheaves on the right. Using the results on quasi-coherent sheaves in Schemes, Section 21.24 we see conclude Rp f∗ F is quasi-coherent. Assume S is affine and f quasi-compact and quasi-separated. Let t ≥ 1 be the minimal number of affine opens needed to cover X. We will prove (1) by induction on t. In case t = 1 the morphism f is separated and we are back in the previous case (see previous paragraph). If t > 1 write X = U ∪ V with V affine open and U a union of t − 1 open affines. Note that in this case U ∩ V is an open subscheme of an affine scheme and hence separated (see Schemes, Lemma 21.21.6). We will apply the relative Mayer-Vietoris sequence 0 → f∗ F → a∗ (F|U ) ⊕ b∗ (F|V ) → c∗ (F|U ∩V ) → R1 f∗ F → . . . see Cohomology, Lemma 18.8.3. By induction and the result of the previous paragraph we see that Rp a∗ F, Rp b∗ F and Rp c∗ F are quasi-coherent. As in the previous paragraph this implies each of sheaves Rp f∗ F is quasi-coherent. Next, we prove (3) and a fortiori (2). Choose a finite affine open covering S = S S −1 S . For each i choose a finite affine open covering f (S ) = j j j=1,...m i=1,...tj Uji . Let \ dj = maxI⊂{1,...,tj } |I| + t( Uji ) i∈I

be the integer found in Lemma 25.5.2. We claim that n(X, S, f ) = max dj works. Namely, let S 0 → S be a morphism of schemes and let F 0 be a quasi-coherent sheaf on X 0 = S 0 ×S X. We want to show that Rp f∗0 F 0 = 0 for p ≥ n(X, S, f ). Since this question is local on S 0 we may assume that S 0 is affine and maps into Sj for some j. Then X 0 = S 0 ×Sj f −1 (Sj ) is covered by the open affines S 0 ×Sj Uji , i = 1, . . . tj and the intersections \ \ S 0 ×Sj Uji = S 0 ×Sj Uji i∈I

i∈I

are covered by the same number of affines as before the base change. Applying Lemma 25.5.2 we get H p (X 0 , F 0 ) = 0. By the first part of the proof we already know that each Rq f∗0 F 0 is quasi-coherent hence has vanishing higher cohomology groups on our affine scheme S 0 , thus we see that H 0 (S 0 , Rp f∗0 F 0 ) = H p (X 0 , F 0 ) = 0 by Cohomology, Lemma 18.12.6. Since Rp f∗0 F 0 is quasi-coherent we conclude that Rp f∗0 F 0 = 0.  Lemma 25.5.4. Let f : X → S be a morphism of schemes. Assume that f is quasi-separated and quasi-compact. Assume S is affine. For any quasi-coherent OX -module F we have H q (X, F) = H 0 (S, Rq f∗ F) for all q ∈ Z. Proof. Consider the Leray spectral sequence E2p,q = H p (S, Rq f∗ F) converging to H p+q (X, F), see Cohomology, Lemma 18.12.4. By Lemma 25.5.3 we see that the sheaves Rq f∗ F are quasi-coherent. By Lemma 25.2.2 we see that E2p,q = 0 when p > 0. Hence the spectral sequence degenerates at E2 and we win. See also Cohomology, Lemma 18.12.6 (2) for the general principle. 

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25.6. Cohomology and base change, I Let f : X → S be a morphism of schemes. Let F be a quasi-coherent sheaf on X. Suppose further that g : S 0 → S is any morphism of schemes. Denote X 0 = XS 0 = S 0 ×S X the base change of X and denote f 0 : X 0 → S 0 the base change of f . Also write g 0 : X 0 → X the projection, and set F 0 = (g 0 )∗ F. Here is a diagram representing the situation: F 0 = (g 0 )∗ F

X0

g0

f0

(25.6.0.1) Rf∗0 F 0

 S0

/X

F

f g

 /S

Rf∗ F

Here is the simplest case of the base change property we have in mind. Lemma 25.6.1. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. Assume f is affine. In this case f∗ F ∼ = Rf∗ F is a quasicoherent sheaf, and for every base change diagram (25.6.0.1) we have g ∗ f∗ F = f∗0 (g 0 )∗ F. Proof. The vanishing of higher direct images is Lemma 25.2.3. The statement is local on S and S 0 . Hence we may assume X = Spec(A), S = Spec(R), S 0 = f for some A-module M . We use Schemes, Lemma 21.7.3 to Spec(R0 ) and F = M describe pullbacks and pushforwards of F. Namely, X 0 = Spec(R0 ⊗R A) and F 0 is the quasi-coherent sheaf associated to (R0 ⊗R A) ⊗A M . Thus we see that the lemma boils down to the equality (R0 ⊗R A) ⊗A M = R0 ⊗R M as R0 -modules.



In many situations it is sufficient to know about the following special case of cohomology and base change. It follows immediately from the stronger results in the next section, but since it is so important it deserves its own proof. Lemma 25.6.2. Let f : X → S be a morphism of schemes. Let F be a quasicoherent sheaf on X. Let g : S 0 → S be a morphism of schemes. Assume that g is flat and that f is quasi-compact and quasi-separated. Then for any i ≥ 0 we have Ri f∗0 F 0 = g ∗ Ri f∗ F with notation as in (25.6.0.1). Moreover, the induced isomorphism is the map given by the base change map of Cohomology, Lemma 18.14.1. Proof. The statement is local on S 0 and hence we may assume S and S 0 are affine. Say S = Spec(A) and S 0 = Spec(B). In this case we are really trying to show that the map H i (X, F) ⊗A B −→ H i (XB , FB ) (given by the reference in the statement of the lemma) is an isomorphism where XB = Spec(B) ×Spec(A) X and FB is the pullback of F to XB . In case X is separated, choose an affine open covering U : X = U1 ∪ . . . ∪ Ut and recall that ˇ p (U, F) = H p (X, F), H

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see Lemma 25.2.4. If UB : XB = (U1 )B ∪ . . . ∪ (Ut )B we obtain by base change, then it is still the case that each (Ui )B is affine and that XB is separated. Thus we obtain ˇ p (UB , FB ) = H p (XB , FB ). H ˇ We have the following relation between the Cech complexes • • ˇ ˇ C (UB , FB ) = C (U, F) ⊗A B as follows from Lemma 25.6.1. Since A → B is flat, the same thing remains true on taking cohomology. In case X is quasi-separated, choose an affine open covering U : X = U1 ∪ . . . ∪ ˇ Ut . We will use the Cech-to-cohomology spectral sequence Cohomology, Lemma 18.11.4. The reader who wishes to avoid this spectral sequence can use MajerVietoris and induction on t as in the proof of Lemma 25.5.3. The spectral sequence ˇ p (U, H q (F)) and converges to H p+q (X, F). Similarly, we has E2 -page E2p,q = H ˇ p (UB , H q (FB )) and converges have a spectral sequence with E2 -page E2p,q = H p+q to H (XB , FB ). Since the intersections Ui0 ...ip are quasi-compact and sepaˇ p (UB , H q (FB )) = rated, the result of the second paragraph of the proof gives H q p ˇ H (U, H (F)) ⊗A B. Using that A → B is flat we conclude that H i (X, F) ⊗A B → H i (XB , FB ) is an isomorphism for all i and we win.  25.7. Colimits and higher direct images General results of this nature can be found in Cohomology, Section 18.15, Sheaves, Lemma 6.29.1, and Modules, Lemma 15.11.6. Lemma 25.7.1. Let f : X → S be a quasi-compact and quasi-separated morphism of schemes. Let F = colim Fi be a filtered colimit of quasi-coherent sheaves on X. Then for any p ≥ 0 we have Rp f∗ F = colim Rp f∗ Fi . Proof. Recall that Rp f∗ F is the sheaf associated to U 7→ H p (f −1 U, F), see Cohomology, Lemma 18.6.3. Recall that the colimit is the sheaf associated to the presheaf colimit (taking colimits over opens). Hence we can apply Cohomology, Lemma 18.15.1 to H p (f −1 U, −) where U is affine to conclude. (Because the basis of affine opens in f −1 U satisfies the assumptions of that lemma.)  25.8. Cohomology and base change, II We would like to prove a little more in situation (25.6.0.1). Namely, if f is quasicompact and quasi-separated we would like to represent Rf∗ F by a complex of quasi-coherent sheaves on S. This can be done in some cases, for example if S is quasi-compact and (semi-)separated, by relating it to the question of whether + DQCoh (S) is equivalent to D+ (QCoh(OS )), see Section 25.4. In this section we will use a different approach which produces a complex having a good base change property. First of all the result is very easy if f and S are separated. Since this is the case which by far the most often used we treat it separately. Lemma 25.8.1. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. Assume X and S are separated and quasi-compact. In this case we can compute Rf∗ F as follows:

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S (1) Choose a finite affine open covering U : X = i=1,...,n Ui . (2) For i0 , . . . , ip ∈ {1, . . . , n} denote fi0 ...ip : Ui0 ...ip → S the restriction of f to the intersection Ui0 ...ip = Ui0 ∩ . . . ∩ Uip . (3) Set Fi0 ...ip equal to the restriction of F to Ui0 ...ip . (4) Set M fi0 ...ip ∗ Fi0 ...ip Cˇp (U, f, F) = i0 ...ip

and define differentials d : Cˇp (U, f, F) → Cˇp+1 (U, f, F) as in Cohomology, Equation (18.9.0.1). Then the complex Cˇ• (U, f, F) is a complex of quasi-coherent sheaves on S which comes equipped with an isomorphism Cˇ• (U, f, F) −→ Rf∗ F in D+ (S). This isomorphism is functorial in the quasi-coherent sheaf F. Proof. Omitted. Hint: Use the resolution F → C• (U, F) of Cohomology, Lemma 18.18.3. Observe that Cˇ• (U, f, F) = f∗ C• (U, F). Also observe that both the inclusion morphisms ji0 ...ip : Ui0 ...ip → X and the morphisms fi0 ...ip : Ui0 ...ip → S are affine because S and X and f : X → S are separated, see Morphisms, Lemma 24.13.11. Hence Rq (ji0 ...ip )∗ Fi0 ...ip as well as Rq (fi0 ...ip )∗ Fi0 ...ip are zero for q > 0. Finally, put all of this information together (e.g. use a spectral sequence, for example by choosing a Cartan-Eilenberg resolution of the complex C• (U, F)).  Next, we are going to consider what happens if we do a base change. Lemma 25.8.2. With notation as in diagram (25.6.0.1). Assume f : X → S and F satisfySthe hypotheses of Lemma 25.8.1. Choose a finite affine open covering U : X = Ui of X. There is a canonical isomorphism g ∗ Cˇ• (U, f, F) −→ Rf∗0 F 0 in D+ (S 0 ). Moreover, if S 0 → S is affine, then in fact g ∗ Cˇ• (U, f, F) = Cˇ• (U 0 , f 0 , F 0 ) S with U 0 : X 0 = Ui0 where Ui0 = (g 0 )−1 (Ui ) = Ui,S 0 is also affine. Proof. In fact we may define Ui0 S = (g 0 )−1 (Ui ) = Ui,S 0 no matter whether S 0 is affine over S or not. Let U 0 : X 0 = Ui0 be the induced covering of X 0 . In this case we claim that g ∗ Cˇ• (U, f, F) = Cˇ• (U 0 , f 0 , F 0 ) with Cˇ• (U 0 , f 0 , F 0 ) defined in exactly the same manner as in Lemma 25.8.1. This is clear from the case of affine morphisms (Lemma 25.6.1) by working locally on S 0 . Moreover, exactly as in the proof of Lemma 25.8.1 one sees that there is an isomorphism Cˇ• (U 0 , f 0 , F 0 ) −→ Rf∗0 F 0 in D+ (S 0 ) since the morphisms Ui0 → X 0 and Ui0 → S 0 are still affine (being base changes of affine morphisms). Details omitted.  The lemma above says that the complex K• = Cˇ• (U, f, F) is a bounded below complex of quasi-coherent sheaves on S which universally computes the higher direct images of f : X → S. This is something about this particular

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complex and it is not preserved by replacing Cˇ• (U, f, F) by a quasi-isomorphic complex in general! In other words, this is not a statement that makes sense in the derived category. The reason is that the pullback g ∗ K• is not equal to the derived pullback Lg ∗ K• of K• in general! Here is a more general case where we can prove this statement. We remark that the condition of S being separated is harmless in most applications, since this is usually used to prove some local property of the total derived image. The proof is significantly more involved and uses hypercoverings; it is a nice example of how you can use them sometimes. Lemma 25.8.3. Let f : X → S be a morphism of schemes. Let F be a quasicoherent sheaf on X. Assume that f is quasi-compact and quasi-separated and that S is quasi-compact and separated. There exists a bounded below complex K• of quasi-coherent OS -modules with the following property: For every morphism g : S 0 → S the complex g ∗ K• is a representative for Rf∗0 F 0 with notation as in diagram (25.6.0.1). Proof. (If f is separated as well, please see Lemma 25.8.2.) The assumptions imply in particular that X is quasi-compact and quasi-separated as a scheme. Let B be the set of affine opens of X. By Hypercoverings, Lemma 20.9.4 we can find a hypercovering K = (I, {Ui }) such that each In is finite and each Ui is an affine open of X. By Hypercoverings, Lemma 20.7.3 there is a spectral sequence with E2 -page ˇ p (K, H q (F)) E p,q = H 2

ˇ p (K, H q (F)) is the pth cohomology group converging to H p+q (X, F). Note that H of the complex Y Y Y H q (Ui , F) → H q (Ui , F) → H q (Ui , F) → . . . i∈I0

i∈I1

i∈I2

Since each Ui is affine we see that this is zero unless q = 0 in which case we obtain Y Y Y F(Ui ) → F(Ui ) → F(Ui ) → . . . i∈I0

i∈I1

i∈I2

Thus we conclude that RΓ(X, F) is computed by this complex. For any n and i ∈ In denote fi : Ui → S the restriction of f to Ui . As S is separated and Ui is affine this morphism is affine. Consider the complex of quasi-coherent sheaves Y Y Y K• = ( fi,∗ F|Ui → fi,∗ F|Ui → fi,∗ F|Ui → . . .) i∈I0

i∈I1

i∈I2

on S. As in Hypercoverings, Lemma 20.7.3 we obtain a map K• → Rf∗ F in D(OS ) by choosing an injective resolution of F (details omitted). Consider any affine scheme V and a morphism g : V → S. Then the base change XV has a hypercovering KV = (I, {Ui,V }) obtained by base change. Moreover, g ∗ fi,∗ F = fi,V,∗ (g 0 )∗ F|Ui,V . Thus the arguments above prove that Γ(V, g ∗ K• ) computes RΓ(XV , (g 0 )∗ F). This finishes the proof of the lemma as it suffices to prove the equality of complexes Zariski locally on S 0 . 

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25.9. Ample invertible sheaves and cohomology Given a ringed space X, an invertible OX -module L, a section s ∈ Γ(X, L) and an OX -module F we get a map F → F ⊗OX L, t 7→ t ⊗ s which we call multiplication by s. We usually denote it t 7→ st. Lemma 25.9.1. Let X be a scheme. Let L be an invertible OX -module. Let s ∈ Γ(X, L) be a section. Let F 0 ⊂ F be quasi-coherent OX -modules. Assume that (1) X is quasi-compact, (2) F is of finite type, and (3) F 0 |Xs = F|Xs . Then there exists an n ≥ 0 such that multiplication by sn on F factors through F 0 . Proof. In other words we claim that sn F ⊂ F 0 ⊗OX L⊗n for some n ≥ 0. If this is true for n0 then it is true for all n ≥ n0 . Hence it suffices to show there is a finite open covering such that the result holds for each of the members of this open covering. Since X is quasi-compact we may therefore assume that X is affine and that L ∼ = OX . Thus the lemma translates into the following algebra problem (use Properties, Lemma 23.16.1): Let A be a ring. Let f ∈ A. Let M 0 ⊂ M be A-modules. Assume M is a finite A-module, and assume that (M 0 )f = Mf . Then there exists an n ≥ 0 such that f n M ⊂ M 0 . The proof of this is omitted.  Let X be a scheme. Let L be an invertible OX -module. Let s ∈ Γ(X, L) be a section. Assume X quasi-compact and quasi-separated. The following lemma says roughly that the category of finitely presented OXs -modules is the category of finitely presented OX -modules where the map multiplication by s has been inverted. Lemma 25.9.2. Let X be a scheme. Let L be an invertible OX -module. Let s ∈ Γ(X, L) be a section. Let F, F 0 be quasi-coherent OX -modules. Let ψ : F|Xs → F 0 |Xs be a map of OXs -modules. Assume that (1) X is quasi-compact and quasi-separated, and (2) F is of finitely presented. Then there exists an n ≥ 0 and a morphism α : F → F 0 ⊗OX L⊗n whose restriction to Xs equals ψ via the identification L⊗n |Xs = OXs coming from s. Moreover, given a pair of solutions (n, α) and (n0 , α0 ) there exists an m ≥ max(n, n0 ) such 0 that sm−n α = sm−n α0 . Proof. If the lemma holds for n0 with map α0 then it holds for allS n ≥ n0 simply by taking α = sn−n0 α0 . Choose a finite affine open covering X = Ui such that S L|Ui is trivial. Choose finite affine open coverings Ui ∩ Ui0 = Uii0 j . Suppose we can prove the lemma when X is affine and L is trivial. Then we can find ni ≥ 0 αi : F|Ui → F 0 |Ui ⊗OUi L⊗ni |Ui satisfying the relation over Ui . By the uniqueness assertion of the lemma, and the finiteness of the number of affines Uii0 j we can find a single large integer m such that the maps sm−ni αi and sm−ni0 αi0 agree over Uii0 j and hence over Ui ∩ Ui0 . Thus the morphisms sm−ni αi glue to give our global map α. Proof of the uniqueness statement is omitted. Assume X affine and that L ∼ = OX . Then the lemma translates into the following algebra problem (use Properties, Lemma 23.16.2): Let A be a ring. Let f ∈ A. Let ψ : Mf → (M 0 )f be a map of Af -modules. Assume M is a finitely presented A-module. Then there exists an n ≥ 0 and an A-module map α : M → M 0 such

25.9. AMPLE INVERTIBLE SHEAVES AND COHOMOLOGY

1495

that α ⊗ 1Af = f n ψ. Moreover, given any second solution (n0 , α0 ) there exists an 0 m ≥ max(n, n0 ) such that f m−n α = f m−n α0 . The proof of this algebraic fact is omitted.  Cohomology is functorial. In particular, given a ringed space X, an invertible OX -module L, a section s ∈ Γ(X, L) we get maps H p (X, F) −→ H p (X, F ⊗OX L),

ξ 7−→ sξ

induced by the map F → F ⊗OX L which is multiplication by s. Lemma 25.9.3. Let X be a scheme. Let L be an invertible OX -module. Let s ∈ Γ(X, L) be a section. Assume that (1) X is quasi-compact and quasi-separated, and (2) Xs is affine. Then for every quasi-coherent OX -module F and every p > 0 and all ξ ∈ H p (X, F) there exists an n ≥ 0 such that sn ξ = 0 in H p (X, F ⊗OX L⊗n ). Proof. You can prove this lemma using a Mayer-Vietoris type argument and induction on the number of affines needed to cover X similar to the proof of Lemma 25.5.3. This may be preferable to the proof that follows. Let F be a quasi-coherent OX -module. Cohomology on X commutes with directed colimits of sheaves of OX -modules, see Cohomology, Lemma 18.15.1. By Properties, Lemma 23.20.6 we can write F as a directed colimit of OX -submodules of finite presentation. Hence every ξ ∈ H p (X, F) is the image of ξ 0 ∈ H p (X, F 0 ) for some OX -submodule of finite presentation. Thus we may replace F by F 0 and assume F is of finite presentation. Let j : Xs → X be the inclusion morphism. Morphisms, Lemma 24.13.10 says that j is an affine morphism. Hence Rq j∗ (j ∗ F) = 0 for all q > 0, see Lemma 25.2.3. Since also H p (Xs , j ∗ F) = 0 by Lemma 25.2.2, we conclude that H p (X, j∗ j ∗ F) = 0 for all p > 0 for example by the Leray spectral sequence ( Cohomology, Lemma 18.12.4). Write j∗ j ∗ F = colimλ∈Λ Fλ as a directed colimit of OX -modules Fλ of finite presentation (Properties, Lemma 23.20.6 again). By Modules, Lemma 15.11.6 there exists a λ ∈ Λ such that F → j∗ j ∗ F factors through Fλ . After shrinking Λ we may assume that we have a compatible collection of morphisms χλ : F → Fλ for all λ ∈ Λ which when taking the colimit gives the canonical map F → j∗ j ∗ F. With these preparations the proof goes as follows. Take ξ ∈ H p (X, F) for some p > 0. It maps to zero in H p (X, j∗ j ∗ F) because we saw above this group is zero. By Cohomology, Lemma 18.15.1 again it follows that ξ maps to zero in H p (X, Fλ ) via the map χλ for some λ. Note that since F → j∗ j ∗ F is an isomorphism over Xs we see that there is an OXs -module map ψ : Fλ |Xs → F|Xs which is a left inverse to χλ : F → Fλ . By Lemma 25.9.2 there exists an n and a map α : Fλ → F ⊗OX L⊗n such that α restricts to ψ on Xs (via L⊗n |Xs ∼ = OXs ). By the uniqueness part of Lemma 25.9.2 applied to α ◦ χλ which restricts to multiplication by sn on Xs we may assume (after increasing n) that the composition F

χλ

/ Fλ

α

/ F ⊗O L⊗n X

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is equal to multiplication by sn on F. Hence we see that sn ξ = 0.



25.10. Cohomology of projective space In this section we compute the cohomology of the twists of the structure sheaf on PnS over a scheme S. Recall that PnS was defined as the fibre product PnS = S ×Spec(Z) PnZ in Constructions, Definition 22.13.2. It was shown to be equal to PnS = ProjS (OS [T0 , . . . , Tn ]) in Constructions, Lemma 22.21.4. In particular, projective space is a particular case of a projective bundle. If S = Spec(R) is affine then we have PnS = PnR = Proj(R[T0 , . . . , Tn ]). All these identifications are compatible and compatible with the constructions of the twisted structure sheaves OPnS (d). Before we state the result we need some notation. Let R be a ring. Recall that R[T0 , . . . , Tn ] is a graded R-algebra where each Ti is homogenous of degree 1. Denote
(R[T0 , . . . , Tn ])d the degree d summand. It is a finite free R-module of rank n+d when d ≥ 0 and zero else. It has a basis consisting of monomials T0e0 . . . Tnen d P with ei = d. We will also use the following notation: R[ T10 , . . . , T1n ] denotes the Z-graded ring with T1i in degree −1. In particular the Z-graded R[ T10 , . . . , T1n ] module 1 1 1 R[ , . . . , ] T0 . . . Tn T0 Tn which shows up in the statement below is zero in degrees ≥ −n,
is free on the 1 n n+d in degree −n − 1 and is free of rank (−1) for d ≤ −n − 1. generator T0 ...T d n Lemma 25.10.1. Let R be a ring. Let n ≥ 0 be an integer. We have  (R[T0 , . . . , Tn ])d if q=0   q n 0 if q 6= 0, n H (P , OPnR (d)) =    1 1 1  if q = n T0 ...Tn R[ T0 , . . . , Tn ] d

as R-modules. Proof. We will use the standard affine open convering [n U : PnR = D+ (Ti ) i=0

to compute the cohomology using the Cech complex. This is permissible by Lemma 25.2.4 since any intersection of finitely many affine D+ (Ti ) is also a standard affine open (see Constructions, Section 22.8). In fact, we can use the alternating or ordered Cech complex according to Cohomology, Lemmas 18.17.3 and 18.17.6. The ordering we will use on {0, . . . , n} is the usual one. Hence the complex we are looking at has terms M 1 p Cˇord (U, OPR (d)) = (R[T0 , . . . , Tn , ])d i0 0, then H i (PnR , F(d)) = 0 for all d large enough. For any k ∈ Z the graded R[T0 , . . . , Tn ]-module M H 0 (PnR , F(d)) d≥k

is a finite R[T0 , . . . , Tn ]-module. Proof. We will use that OPnR (1) is an ample invertible sheaf on the scheme PnR . This follows directly from the definition since PnR covered by the standard affine opens D+ (Ti ). Hence by Properties, Proposition 23.24.12 every finite type quasicoherent OPnR -module is a quotient of a finite direct sum of tensor powers of OPnR (1). On the other hand a coherent sheaves and finite type quasi-coherent sheaves are the same thing on projective space over R by Lemma 25.11.1. Thus we see (1). Projective n-space PnR is covered by n + 1 affines, namely the standard opens D+ (Ti ), i = 0, . . . , n, see Constructions, Lemma 22.13.3. Hence we see that for any quasi-coherent sheaf F on PnR we have H i (PnR , F) = 0 for i ≥ n + 1, see Lemma 25.5.1. Hence (2) holds.

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Let us prove (3) and (4) simultaneously for all coherent sheaves on PnR by descending induction on i. Clearly the result holds for i ≥ n + 1 by (2). Suppose we know the result for i + 1 and we want to show the result for i. (If i = 0, then part (4) is vacuous.) Let F be a coherent sheaf on PnR . Choose a surjection as in (1) and denote G the kernel so that we have a short exact sequence M 0→G→ OPnR (dj ) → F → 0 j=1,...,r

By Lemma 25.11.2 we see that G is coherent. The long exact cohomology sequence gives an exact sequence M H i (PnR , OPnR (dj )) → H i (PnR , F) → H i+1 (PnR , G). j=1,...,r

By induction assumption the right R-module is finite and by Lemma 25.10.1 the left R-module is finite. Since R is Noetherian it follows immediately that H i (PnR , F) is a finite R-module. This proves the induction step for assertion (3). Since OPnR (d) is invertible we see that twisting on PnR is an exact functor (since you get it by tensoring with an invertible sheaf, see Constructions, Definition 22.10.1). This means that for all d ∈ Z the sequence M 0 → G(d) → OPnR (dj + d) → F(d) → 0 j=1,...,r

is short exact. The resulting cohomology sequence is M H i (PnR , OPnR (dj + d)) → H i (PnR , F(d)) → H i+1 (PnR , G(d)). j=1,...,r

By induction assumption we see the module on the right is zero for d  0 and by the computation in Lemma 25.10.1 the module on the left is zero as soon as d ≥ − min{dj } and i ≥ 1. Hence the induction step for assertion (4). This concludes the proof of (3) and (4). In order to prove (5) note that for all sufficiently large d the map M H 0 (PnR , OPnR (dj + d)) → H 0 (PnR , F(d)) j=1,...,r

is surjective by the vanishing of H 1 (PnR , G(d)) we just proved. In other words, the module M Mk = H 0 (PnR , F(d)) d≥k

is for k large enough a quotient of the corresponding module M M Nk = H 0 (PnR , OPnR (dj + d)) d≥k

j=1,...,r

When k is sufficiently small (e.g. k < −dj for all j) then M Nk = R[T0 , . . . , Tn ](dj ) j=1,...,r

by our computations in Section 25.10. In particular it is finitely generated. Suppose k ∈ Z is arbitrary. Choose k−  k  k+ . Consider the diagram Nk− o

Nk+

Mk o

 Mk +

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where the vertical arrow is the surjective map above and the horizontal arrows are the obvious inclusion maps. By what was said above we see that Nk− is a finitely generated R[T0 , . . . , Tn ]-module. Hence Nk+ is a finitely generated R[T0 , . . . , Tn ]module because it is a submodule of a finitely generated module and the ring R[T0 , . . . , Tn ] is Noetherian. Since the vertical arrow is surjective we conclude that Mk+ is a finitely generated R[T0 , . . . , Tn ]-module. The quotient Mk /Mk+ is finite as an R-module since it is a finite direct sum of the finite R-modules H 0 (PnR , F(d)) for k ≤ d < k+ . Note that we use part (3) for i = 0 here. Hence Mk /Mk+ is a fortiori a finite R[T0 , . . . , Tn ]-module. In other words, we have sandwiched Mk between two finite R[T0 , . . . , Tn ]-modules and we win.  Lemma 25.16.2. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. Let L be an invertible sheaf on X. Assume that (1) (2) (3) (4)

S is Noetherian, f is proper, F is coherent, and L is relatively ample on X/S.

Then there exists an n0 such that for all n ≥ n0 we have
Rp f∗ F ⊗OX L⊗n = 0 for all p > 0. Proof. A proper morphism is of finite type S by definition. By Morphisms, Lemma n 24.40.7 there exists an open covering S = Vj and immersions ij : Xj → PVjj , where Xj = f −1 (Vj ) such that i∗j O(1) is a power of L. Since S is quasi-compact we may assume the covering is finite. Clearly, if we solve the question for each of the finitely many systems (Xj → Vj , L|Xj , F|Vj ) then the result follows. Hence we may assume there exists an immersion i : X → PnS such that L⊗d = i∗ O(1) for some d ≥ 1. Repeating the argument above with a finite affine open covering of S we see that we may also assume that S is affine. In this case the vanishing of Rp f∗ (F ⊗ L⊗n ) is equivalent to the vanishing of H p (X, F ⊗ L⊗n ), see Lemma 25.5.4. Since f is proper we see that i is a closed immersion (Morphisms, Lemma 24.42.7). Hence we see that Rp i∗ (F ⊗OX L⊗n ) = 0 for all p ≥ 1 (see Lemma 25.11.8 for example). This implies that H p (X, F ⊗ L⊗n ) = H p (PnS , i∗ (F ⊗ L⊗n )) by the Leray spectral sequence (Cohomology, Lemma 18.12.4). Moreover, by the projection formula (Cohomology, Lemma 18.7.2) we have i∗ (F ⊗OX L⊗n ) = i∗ (F ⊗OX L⊗hnid ) ⊗OPn O(bn/dc) S

for all n ∈ Z where hnid ∈ {0, 1, . . . , d − 1} is the unique element congruent to n module d. The sheaves Fj = i∗ (F ⊗ L⊗j ), j ∈ {0, 1, . . . , d − 1} are coherent by Lemma 25.11.8. Thus we see that for all n large enough the cohomology groups H p (PnS , Fj (n)) vanish by Lemma 25.16.1. Putting everything together this implies the lemma. 

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25.17. Chow’s Lemma In this section we prove Chow’s lemma in the Noetherian case (Lemma 25.17.1). In Limits, Section 27.8 we prove some variants for the non-Noetherian case. Lemma 25.17.1. Let S be a Noetherian scheme. Let f : X → S be a separated morphism of finite type. Then there exists an n ≥ 0 and a diagram Xo

π

X0

/ Pn S

 } S where X 0 → PnS is an immersion, and π : X 0 → X is proper and surjective. Moreover, we may arrange it such that there exists a dense open subscheme U ⊂ X such that π −1 (U ) → U is an isomorphism. Proof. All of the schemes we will encounter during the rest of the proof are going to be of finite type over the Noetherian scheme S and hence Noetherian (see Morphisms, Lemma 24.16.6). All morphisms between them will automatically be quasi-compact, locally of finite type and quasi-separated, see Morphisms, Lemma 24.16.8 and Properties, Lemmas 23.5.4 and 23.5.6. The underlying topological space of X is Noetherian (see Properties, Lemma 23.5.5) and we conclude that X has only finitely many irreducible components (see Topology, Lemma 5.6.2). Say X = X1 ∪. . .∪Xr is the decomposition of X into irreducible components. Let ηi ∈ Xi be the generic point. For every point x ∈ X there exists an affine open Ux ⊂ X which contains x and each of the generic points ηi . See Properties, Lemma 23.27.4. Since X is quasi-compact, we can find a finite affine open covering X = U1 ∪. . .∪Um such that each Ui contains η1 , . . . , ηr . In particular we conclude that the open U = U1 ∩ . . . ∩ Um ⊂ X is a dense open. This and the fact that the Ui are affine opens covering X is all that we will use below. Let X ∗ ⊂ X be the scheme theoretic closure of U → X, see Morphisms, Definition 24.6.2. Let Ui∗ = X ∗ ∩ Ui . Note that Ui∗ is a closed subscheme of Ui . Hence Ui∗ is affine. Since U is dense in X the morphism X ∗ → X is a surjective closed immersion. It is an isomorphism over U . Hence we may replace X by X ∗ and Ui by Ui∗ and assume that U is scheme theoretically dense in X, see Morphisms, Definition 24.7.1. By Morphisms, Lemma 24.40.3 we can find an immersion ji : Ui → PnSi for each i. By Morphisms, Lemma 24.7.7 we can find closed subschemes Zi ⊂ PnSi such that ji : Ui → Zi is a scheme theoretically dense open immersion. Note that Zi → S is proper, see Morphisms, Lemma 24.43.5. Consider the morphism j = (j1 |U , . . . , jn |U ) : U −→ PnS1 ×S . . . ×S PnSn . By the lemma cited above we can find a closed subscheme Z of PnS1 ×S . . . ×S PnSn such that j : U → Z is an open immersion and such that U is scheme theoretically dense in Z. The morphism Z → S is proper. Consider the ith projection pri |Z : Z −→ PnSi . This morphism factors through Zi (see Morphisms, Lemma 24.6.6). Denote pi : Z → Zi the induced morphism. This is a proper morphism, see Morphisms, Lemma

25.17. CHOW’S LEMMA

1517

24.42.7 for example. At this point we have that U ⊂ Ui ⊂ Zi are scheme theoretically dense open immersions. Moreover, we can think of Z as the scheme theoretic image of the “diagonal” morphism U → Z1 ×S . . . ×S Zn . 0 Set Vi = p−1 i (Ui ). Note that pi |Vi : Vi → Ui is proper. Set X = V1 ∪ . . . ∪ Vn . n1 0 By construction X has an immersion into the scheme PS ×S . . . ×S PnSn . Thus by the Segre embedding (see Morphisms, Lemma 24.43.6) we see that X 0 has an immersion into a projective space over S.

We claim that the morphisms pi |Vi : Vi → Ui glue to a morphism X 0 → X. Namely, it is clear that pi |U is the identity map from U to U . Since U ⊂ X 0 is scheme theoretically dense by construction, it is also scheme theoretically dense in the open subscheme Vi ∩ Vj . Thus we see that pi |Vi ∩Vj = pj |Vi ∩Vj as morphisms into the separated S-scheme X, see Morphisms, Lemma 24.7.10. We denote the resulting morphism π : X 0 → X. We claim that π −1 (Ui ) = Vi . Since π|Vi = pi |Vi it follows that Vi ⊂ π −1 (Ui ). Consider the diagram / π −1 (Ui ) Vi pi |Vi

#  Ui

Since Vi → Ui is proper we see that the image of the horizontal arrow is closed, see Morphisms, Lemma 24.42.7. Since Vi ⊂ π −1 (Ui ) is scheme theoretically dense (as it contains U ) we conclude that Vi = π −1 (Ui ) as claimed. This shows that π −1 (Ui ) → Ui is identified with the proper morphism pi |Vi : Vi → S Ui . Hence we see that X has a finite affine covering X = Ui such that the restriction of π is proper on each member of the covering. Thus by Morphisms, Lemma 24.42.3 we see that π is proper. Finally we have to show that π −1 (U ) = U . To see this we argue in the same way as above using the diagram / π −1 (U ) U #  U and using that idU : U → U is proper and that U is scheme theoretically dense in π −1 (U ).  Remark 25.17.2. In the situation of Chow’s Lemma 25.17.1: (1) The morphism π is actually H-projective (hence projective, see Morphisms, Lemma 24.43.3) since the morphism X 0 → PnS ×S X = PnX is a closed immersion (use the fact that π is proper, see Morphisms, Lemma 24.42.7). (2) We may assume that π −1 (U ) is scheme theoretically dense in X 0 . Namely, we can simply replace X 0 by the scheme theoretic closure of π −1 (U ). In this case we can think of U as a scheme theoretically dense open subscheme of X 0 . See Morphisms, Section 24.6. (3) If X is reduced then we may choose X 0 reduced. This is clear from (2).

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25. COHOMOLOGY OF SCHEMES

25.18. Higher direct images of coherent sheaves In this section we prove the fundamental fact that the higher direct images of a coherent sheaf under a proper morphism are coherent. Lemma 25.18.1. Let S be a locally Noetherian scheme. Let f : X → S be a locally projective morphism. Let F be a coherent OX -module. Then Ri f∗ F is a coherent OS -module for all i ≥ 0. Proof. We first remark that a locally projective morphism is proper (Morphisms, Lemma 24.43.5) and hence of finite type. In particular X is locally Noetherian (Morphisms, Lemma 24.16.6) and hence the statement makes sense. Moreover, by Lemma 25.5.3 the sheaves Rp f∗ F are quasi-coherent. Having said this the statement is local on S (for example by Cohomology, Lemma 18.6.4). Hence we may assume S = Spec(R) is the spectrum of a Noetherian ring, and X is a closed subscheme of PnR for some n, see Morphisms, Lemma 24.43.4. In this case, the sheaves Rp f∗ F are the quasi-coherent sheaves associated to the R-modules H p (X, F), see Lemma 25.5.4. Hence it suffices to show that R-modules H p (X, F) are finite R-modules (Lemma 25.11.1). Denote i : X → PnR the closed immersion. Note that Rp i∗ F = 0 by Lemma 25.11.8. Hence the Leray spectral sequence (Cohomology, Lemma 18.12.4) for i : X → PnR degenerates, and we see that H p (X, F) = H p (PnR , i∗ F). Since the sheaf i∗ F is coherent by Lemma 25.11.8 we see that the lemma follows from Lemma 25.16.1.  Here is the general statement. Lemma 25.18.2. Let S be a locally Noetherian scheme. Let f : X → S be a proper morphism. Let F be a coherent OX -module. Then Ri f∗ F is a coherent OS -module for all i ≥ 0. Proof. Since the problem is local on S we may assume that S is a Noetherian scheme. Since a proper morphism is of finite type we see that in this case X is a Noetherian scheme also. Consider the property P of coherent sheaves on X defined by the rule P(F) ⇔ Rp f∗ F is coherent for all p ≥ 0 We are going to use the result of Lemma 25.14.6 to prove that P holds for every coherent sheaf on X. Let 0 → F1 → F2 → F3 → 0 be a short exact sequence of coherent sheaves on X. Consider the long exact sequence of higher direct images Rp−1 f∗ F3 → Rp f∗ F1 → Rp f∗ F2 → Rp f∗ F3 → Rp+1 f∗ F1 Then it is clear that if 2-out-of-3 of the sheaves Fi have property P, then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemma 25.11.2 and 25.11.3. Hence property P holds for the third as well. Let Z ⊂ X be an integral closed subscheme. We have to find a coherent sheaf F on X whose support is contained in Z, whose stalk at the generic point ξ of Z is a 1-dimensional vector space over κ(ξ) such that P holds for F. Denote

25.19. THE THEOREM ON FORMAL FUNCTIONS

1519

g = f |Z : Z → S the restriction of f . Suppose we can find a coherent sheaf G on Z such that (a) Gξ is a 1-dimensional vector space over κ(ξ), (b) Rp g∗ G = 0 for p > 0, and (c) g∗ G is coherent. Then we can consider F = (Z → X)∗ G. As Z → X is a closed immersion we see that (Z → X)∗ G is coherent on X and Rp (Z → X)∗ G = 0 for p > 0 (Lemma 25.11.8). Hence by the relative Leray spectral sequence (Cohomology, Lemma 18.12.8) we will have Rp f∗ F = Rp g∗ G = 0 for p > 0 and f∗ F = g∗ G is coherent. Finally Fξ = ((Z → X)∗ G)ξ = Gξ which verifies the condition on the stalk at ξ. Hence everything depends on finding a coherent sheaf G on Z which has properties (a), (b), and (c). We can apply Chow’s Lemma 25.17.1 to the morphism Z → S. Thus we get a diagram / Pn Z0 Zo π

g

i

g

S

0

  ~ S as in the statement of Chow’s lemma. Also, let U ⊂ Z be the dense open subscheme such that π −1 (U ) → U is an isomorphism. By the discussion in Remark 25.17.2 we see that i0 = (i, π) : PnS ×S Z 0 = PnZ is a closed immersion. Hence L = i∗ OPnX (1) ∼ = (i0 )∗ OPnZ (1) is g 0 -relatively ample and π-relatively ample (for example by Morphisms, Lemma 24.40.7). Hence by Lemma 25.16.2 there exists an n ≥ 0 such that both Rp π∗ L⊗n = 0 for all p > 0 and Rp (g 0 )∗ L⊗n = 0 for all p > 0. Set G = π∗ L⊗n . Property (a) holds because π∗ L⊗ |U is an invertible sheaf (as π −1 (U ) → U is an isomorphism). Properties (b) and (c) hold because by the relative Leray spectral sequence (Cohomology, Lemma 18.12.8) we have E2p,q = Rp g∗ Rq π∗ L⊗n ⇒ Rp+q (g 0 )∗ L⊗n and by choice of n the only nonzero terms in E2p,q are those with q = 0 and the only nonzero terms of Rp+q (g 0 )∗ L⊗n are those with p = q = 0. This implies that Rp g∗ G = 0 for p > 0 and that g∗ G = (g 0 )∗ L⊗n . Finally, applying the previous Lemma 25.18.1 we see that g∗ G = (g 0 )∗ L⊗n is coherent as desired.  Lemma 25.18.3. Let S = Spec(A) with A a Noetherian ring. Let f : X → S be a proper morphism. Let F be a coherent OX -module. Then H i (X, F) is finite A-module for all i ≥ 0. Proof. This is just the affine case of Lemma 25.18.2. Namely, by Lemmas 25.5.3 and 25.5.4 we know that Ri f∗ F is the quasi-coherent sheaf associated to the Amodule H i (X, F) and by Lemma 25.11.1 this is a coherent sheaf if and only if H i (X, F) is an A-module of finite type.  25.19. The theorem on formal functions In this section we study the behaviour of cohomology of sequences of sheaves either of the form {I n F}n≥0 or of the form {F/I n F}n≥0 as n-varies. Here and below we use the following notation. Given a morphism of schemes f : X → Y , a quasi-coherent sheaf F on X, and a quasi-coherent sheaf of ideals

1520

25. COHOMOLOGY OF SCHEMES

I ⊂ OY we denote I n F the quasi-coherent subsheaf generated by products of local sections of f −1 (I n ) and F. In a formula I n F = Im (f ∗ (I n ) ⊗OX F −→ F) . Note that there are natural maps f −1 (I n ) ⊗f −1 OY I m F −→ f ∗ (I n ) ⊗OX I m F −→ I n+m F Hence a section of I n will give rise to a map Rp f∗ (I m F) → Rp f∗ (I n+m F) by functoriality of higher direct images. Localizing and then sheafifying we see that there are OY -module maps I n ⊗OY Rp f∗ (I m F) −→ Rp f∗ (I n+m F). L L In other words we see that n≥0 Rp f∗ (I n F) is a graded n≥0 I n -module. If Y = Spec(A) L and I = Ie we denote I n F simply I n F. TheLmaps introduced above give M = H p (X, I n F) the structure of a graded S = I n -module. If f is proper, A is Noetherian and F is coherent, then this turns out to be a module of finite type. Lemma 25.19.1. Let A be a Noetherian ring. Let I ⊂ A be an ideal. Set S = L n I . Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf n≥0 L p n on X. Then for every p ≥ 0 the graded S-module n≥0 H (X, I F) is a finite S-module. Proof. To prove this we consider the fibre product diagram X 0 = Spec(S) ×Spec(A) X f0

 Spec(S)

/X f

 / Spec(A)

Note that f 0 is a proper morphism, see Morphisms, Lemma 24.42.5. Also, S is a finitely generated A-algebra, and hence Noetherian (Algebra, Lemma 7.29.1). Thus the result will follow from Lemma 25.18.3 if we can show there exists a coherent sheaf F 0 on X 0 whose cohomology groups H p (X 0 , F 0 ) are identified with L p n n≥0 H (X, I F). To do this note that the morphism π : X 0 → X is affine, see Morphisms, Lemma 24.13.8. Hence H p (X 0 , F 0 ) = H p (X, π∗ F 0 ). In other it suffices to construct L words, n a coherent OX 0 -module F 0 such that π∗ F 0 = I F. Note that π∗ OX 0 = n≥0 L L n n I ⊗ O hence the sheaf I F has a natural structure of π∗ OX 0 A X n≥0 n≥0 module. By Morphisms, Lemma 24.13.6 we see that there is a unique quasi-coherent L OX 0 -module F 0 such that π∗ F 0 ∼ = n≥0 I n F as π∗ OX 0 -modules. Finally, we have to show that F 0 is a coherent OX 0 -module. Let Spec(B) = U ⊂ X be any affine open. Say F|U is the coherent OU -module associated to the L finite B-module M . By definition π −1 (U ) = Spec(S ⊗A B). Since 0 B = S ⊗A B = n≥0 I n ⊗A B it is clear that F 0 corresponds to the B 0 -module L n I M which is clearly finitely generated. 

25.19. THE THEOREM ON FORMAL FUNCTIONS

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Lemma 25.19.2. Given a morphism of schemes f : X → Y , a quasi-coherent sheaf F on X, and a quasi-coherent sheaf of ideals I ⊂ OY . Assume Y locally Noetherian, f proper, and F coherent. Then M M= Rp f∗ (I n F) n≥0 L is a graded A = n≥0 I n -module which is quasi-coherent and of finite type. Proof. The statement is local on Y , hence this reduces to the case where Y is affine. In the affine case the result follows from Lemma 25.19.1. Details omitted.  Lemma 25.19.3. Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Then for every p ≥ 0 there exists an integer c ≥ 0 such that (1) the multiplication map I n−c ⊗ H p (X, I c F) → H p (X, I n F) is surjective for all n ≥ c, and (2) the image of H p (X, I n+m F) → H p (X, I n F) is contained in the submodule I m−c H p (X, I n F) for all n ≥ 0, m ≥ c. Proof. ByLLemma 25.19.1 we can find d1 , . . . , dt ≥ 0, and xi L ∈ H p (X, I di F) p n such that n≥0 H (X, I F) is generated by x1 , . . . , xt over S = n≥0 I n . Take c = max{di }. It is clear that (1) holds. For (2) let b = max(0, n − c). Consider the commutative diagram of A-modules I n+m−c−b ⊗ I b ⊗ H p (X, I c F)

/ I n+m−c ⊗ H p (X, I c F)

 I n+m−c−b ⊗ H p (X, I n F)

/ H p (X, I n+m F)  / H p (X, I n F)

By part (1) of the lemma the composition of the horizontal arrows is surjective if n + m ≥ c. On the other hand, it is clear that n + m − c − b ≥ m − c. Hence part (2).  In the situation of Lemmas 25.19.1 and 25.19.3 consider the inverse system F/IF ← F/I 2 F ← F/I 3 F ← . . . We would like to know what happens to the cohomology groups. Here is a first result. Lemma 25.19.4. Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Fix p ≥ 0. (1) There exists a c1 ≥ 0 such that for all n ≥ c1 we have Ker(H p (X, F) → H p (X, F/I n F)) ⊂ I n−c1 H p (X, F). (2) The inverse system (H p (X, F/I n F))n∈N satisfies the Mittag-Leffler condition (see Homology, Definition 10.23.2). (3) In fact for any p and n there exists a c2 (n) ≥ n such that Im(H p (X, F/I k F) → H p (X, F/I n F)) = Im(H p (X, F) → H p (X, F/I n F)) for all k ≥ c2 (n).

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25. COHOMOLOGY OF SCHEMES

Proof. Let c1 = max{cp , cp+1 }, where cp , cp+1 are the integers found in Lemma 25.19.3 for H p and H p+1 . We will use this constant in the proofs of (1), (2) and (3). Let us prove part (1). Consider the short exact sequence 0 → I n F → F → F/I n F → 0 From the long exact cohomology sequence we see that Ker(H p (X, F) → H p (X, F/I n F)) = Im(H p (X, I n F) → H p (X, F)) Hence by our choice of c1 we see that this is contained in I n−c1 H p (X, F) for n ≥ c1 . Note that part (3) implies part (2) by definition of the Mittag-Leffler condition. Let us prove part (3). Fix an n throughout the rest of the proof. Consider the commutative diagram 0

/ I nF O

/F O

/ F/I n F O

/0

0

/ I n+m F

/F

/ F/I n+m F

/0

This gives rise to the following commutative diagram H p (X, I n F) O

/ H p (X, F) O

/ H p (X, F/I n F) O

δ

a

1

H p (X, I n+m F)

/ H p (X, F)

/ H p+1 (X, I n F) O

/ H p (X, F/I n+m F)

/ H p+1 (X, I n+m F)

If m ≥ c1 we see that the image of a is contained in I m−c1 H p+1 (X, I n F). By the Artin-Rees lemma (see Algebra, Lemma 7.48.5) there exists an integer c3 (n) such that   I N H p+1 (X, I n F) ∩ Im(δ) ⊂ δ I N −c3 (n) H p (X, F/I n F) for all N ≥ c3 (n). As H p (X, F/I n F) is annihilated by I n , we see that if m ≥ c3 (n) + c1 + n, then Im(H p (X, F/I n+m F) → H p (X, F/I n F)) = Im(H p (X, F) → H p (X, F/I n F)) In other words, part (3) holds with c2 (n) = c3 (n) + c1 + n.



Theorem 25.19.5 (Theorem on formal functions). Let A be a Noetherian ring. Let I ⊂ A be an ideal. Let f : X → Spec(A) be a proper morphism. Let F be a coherent sheaf on X. Fix p ≥ 0. The system of maps H p (X, F)/I n H p (X, F) −→ H p (X, F/I n F) define an isomorphism of limits H p (X, F)∧ −→ limn H p (X, F/I n F) where the left hand side is the completion of the A-module H p (X, F) with respect to the ideal I, see Algebra, Section 7.91. Moreover, this is in fact a homeomorphism for the limit topologies.

25.19. THE THEOREM ON FORMAL FUNCTIONS

1523

Proof. In fact, this follows immediately from Lemma 25.19.4. We spell out the details. Set M = H p (X, F) and Mn = H p (X, F/I n F). Denote Nn = Im(M → Mn ). By the description of the limit in Homology, Section 10.23 we have Y limn Mn = {(xn ) ∈ Mn | ϕi (xn ) = xn−1 , n = 2, 3, . . .} Pick an element x = (xn ) ∈ limn Mn . By Lemma 25.19.4 part (3) we have xn ∈ Nn for all n since by definition xn is the image of some xn+m ∈ Mn+m for all m. By Lemma 25.19.4 part (1) we see that there exists a factorization M → Nn → M/I n−c1 M of the reduction map. Denote yn ∈ M/I n−c1 M the image of xn for n ≥ c1 . Since for n0 ≥ n the composition M → Mn0 → Mn is the given map M → Mn we see 0 that yn0 maps to yn under the canonical map M/I n −c1 M → M/I n−c1 M . Hence y = (yn+c1 ) defines an element of limn M/I n M . We omit the verification that y maps to x under the map M ∧ = limn M/I n M −→ limn Mn of the lemma. We also omit the verification on topologies.



Lemma 25.19.6. Given a morphism of schemes f : X → Y , a quasi-coherent sheaf F on X, and a quasi-coherent sheaf of ideals I ⊂ OY . Assume (1) Y locally Noetherian, (2) f proper, and (3) F coherent. Let y ∈ Y be a point. Consider the infinitesimal neighbourhoods Xn = Spec(OY,y /mny ) ×Y X fn

in

/X f



Spec(OY,y /mny )

cn

 /Y

of the fibre X1 = Xy and set Fn = i∗n F. Then we have ∧ (Rp f∗ F)y ∼ = limn H p (Xn , Fn ) ∧ as OY,y -modules.

Proof. This is just a reformulation of a special case of the theorem on formal functions, Theorem 25.19.5. Let us spell it out. Note that OY,y is a Noetherian local ring. Consider the canonical morphism c : Spec(OY,y ) → Y , see Schemes, Equation (21.13.1.1). This is a flat morphism as it identifies local rings. Denote momentarily f 0 : X 0 → Spec(OY,y ) the base change of f to this local ring. We see that c∗ Rp f∗ F = Rp f∗0 F 0 by Lemma 25.6.2. Moreover, the infinitesimal neighbourhoods of the fibre Xy and Xy0 are identified (verification omitted; hint: the morphisms cn factor through c). Hence we may assume that Y = Spec(A) is the spectrum of a Noetherian local ring A with maximal ideal m and that y ∈ Y corresponds to the closed point (i.e., to m). In particular it follows that (Rp f∗ F)y = Γ(Y, Rp f∗ F) = H p (X, F).

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25. COHOMOLOGY OF SCHEMES

In this case also, the morphisms cn are each closed immersions. Hence their base changes in are closed immersions as well. Note that in,∗ Fn = in,∗ i∗n F = F/mn F. By the Leray spectral sequence for in , and Lemma 25.11.8 we see that H p (Xn , Fn ) = H p (X, in,∗ F) = H p (X, F/mn F) Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win.  Here is a lemma which we will generalize later to fibres of dimension > 0, namely the next lemma. Lemma 25.19.7. Let f : X → Y be a morphism of schemes. Let y ∈ Y . Assume (1) Y locally Noetherian, (2) f is proper, and (3) f −1 ({y}) is finite. Then for any coherent sheaf F on X we have (Rp f∗ F)y = 0 for all p > 0. Proof. The fibre Xy is finite, and by Morphisms, Lemma 24.21.7 it is a finite discrete space. Moreover, the underlying topological space of each infinitesimal neighourhood Xn is the same. Hence each of the schemes Xn is affine according to Schemes, Lemma 21.11.7. Hence it follows that H p (Xn , Fn ) = 0 for all p > 0. p Hence we see that (Rp f∗ F)∧ y = 0 by Lemma 25.19.6. Note that R f∗ F is coherent p by Lemma 25.18.2 and hence R f∗ Fy is a finite OY,y -module. By Algebra, Lemma 7.91.2 this implies that (Rp f∗ F)y = 0.  Lemma 25.19.8. Let f : X → Y be a morphism of schemes. Let y ∈ Y . Assume (1) Y locally Noetherian, (2) f is proper, and (3) dim(Xy ) = d. Then for any coherent sheaf F on X we have (Rp f∗ F)y = 0 for all p > d. Proof. The fibre Xy is of finite type over Spec(κ(y)). Hence Xy is a Noetherian scheme by Morphisms, Lemma 24.16.6. Hence the underlying topological space of Xy is Noetherian, see Properties, Lemma 23.5.5. Moreover, the underlying topological space of each infinitesimal neighourhood Xn is the same as that of Xy . Hence H p (Xn , Fn ) = 0 for all p > d by Cohomology, Lemma 18.16.5. Hence we see that p (Rp f∗ F)∧ y = 0 by Lemma 25.19.6 for p > d. Note that R f∗ F is coherent by p Lemma 25.18.2 and hence R f∗ Fy is a finite OY,y -module. By Algebra, Lemma 7.91.2 this implies that (Rp f∗ F)y = 0.  25.20. Applications of the theorem on formal functions We will add more here as needed. For the moment we need the following characterization of finite morphisms (in the Noetherian case – for a more general version see the chapter More on Morphisms, Section 33.30). Lemma 25.20.1. (For a more general version see More on Morphisms, Lemma 33.30.6). Let f : X → S be a morphism of schemes. Assume S is locally Noetherian. The following are equivalent (1) f is finite, and (2) f is proper with finite fibres.

25.21. COHOMOLOGY AND BASE CHANGE, III

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Proof. A finite morphism is proper according to Morphisms, Lemma 24.44.10. A finite morphism is quasi-finite according to Morphisms, Lemma 24.44.9. A quasifinite morphism has finite fibres, see Morphisms, Lemma 24.21.10. Hence a finite morphism is proper and has finite fibres. Assume f is proper with finite fibres. We want to show f is finite. In fact it suffices to prove f is affine. Namely, if f is affine, then it follows that f is integral by Morphisms, Lemma 24.44.7 whereupon it follows from Morphisms, Lemma 24.44.4 that f is finite. To show that f is affine we may assume that S is affine, and our goal is to show that X is affine too. Since f is proper we see that X is separated and quasi-compact. Hence we may use the criterion of Lemma 25.3.2 to prove that X is affine. To see this let I ⊂ OX be a finite type ideal sheaf. In particular I is a coherent sheaf on X. By Lemma 25.19.7 we conclude that R1 f∗ Is = 0 for all s ∈ S. In other words, R1 f∗ I = 0. Hence we see from the Leray Spectral Sequence for f that H 1 (X, I) = H 1 (S, f∗ I). Since S is affine, and f∗ I is quasi-coherent (Schemes, Lemma 21.24.1) we conclude H 1 (S, f∗ I) = 0 from Lemma 25.2.2 as desired. Hence H 1 (X, I) = 0 as desired.  As a consequence we have the following useful result. Lemma 25.20.2. (For a more general version see More on Morphisms, Lemma 33.30.7). Let f : X → S be a morphism of schemes. Let s ∈ S. Assume (1) S is locally Noetherian, (2) f is proper, and (3) f −1 ({s}) is a finite set. Then there exists an open neighbourhood V ⊂ S of s such that f |f −1 (V ) : f −1 (V ) → V is finite. Proof. The morphism f is quasi-finite at all the points of f −1 ({s}) by Morphisms, Lemma 24.21.7. By Morphisms, Lemma 24.49.2 the set of points at which f is quasi-finite is an open U ⊂ X. Let Z = X \ U . Then s 6∈ f (Z). Since f is proper the set f (Z) ⊂ S is closed. Choose any open neighbourhood V ⊂ S of s with Z ∩ V = ∅. Then f −1 (V ) → V is locally quasi-finite and proper. Hence it is quasi-finite (Morphisms, Lemma 24.21.9), hence has finite fibres (Morphisms, Lemma 24.21.10), hence is finite by Lemma 25.20.1.  25.21. Cohomology and base change, III In this section we state the simplest case of a very general phenomenon that will be discussed elsewhere (insert future reference here). Please see Remark 25.21.2 for a tranlation of the statement into algebra. Lemma 25.21.1. Let A be a Noetherian ring and set S = Spec(A). Let f : X → S be a proper morphism of schemes. Let F be a coherent OX -module flat over S. Then (1) RΓ(X, F) is a perfect object of D(A), and (2) for any ring map A → A0 the base change map 0 RΓ(X, F) ⊗L A A −→ RΓ(XA0 , FA0 )

is an isomorphism.

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S Proof. Choose a finite affine open covering X = i=1,...,n Ui . By Lemmas 25.8.1 ˇ and 25.8.2 the Cech complex K • = Cˇ • (U, F) satisfies K • ⊗A A0 = RΓ(XA0 , FA0 ) • • ˇ for all ring maps A → A0 . Let Kalt = Cˇalt (U, F) be the alternating Cech complex. • By Cohomology, Lemma 18.17.6 there is a homotopy equivalence Kalt → K • of A-modules. In particular, we have • Kalt ⊗A A0 = RΓ(XA0 , FA0 ) n as well. Since F is flat over A we see that each Kalt is flat over A (see Morphisms, • Lemma 24.26.2). Since moreover Kalt is bounded above (this is why we switched • • 0 ˇ to the alternating Cech complex) Kalt ⊗A A0 = Kalt ⊗L A A by the definition of derived tensor products (see More on Algebra, Section 12.5). By Lemma 25.18.3 the • • ) are finite A-modules. As Kalt is bounded, we conclude cohomology groups H i (Kalt • that Kalt is pseudo-coherent, see More on Algebra, Lemma 12.43.16. Given any A-module M set A0 = A ⊕ M where M is a square zero ideal, i.e., (a, m) · (a0 , m0 ) = 0 • (aa0 , am0 + a0 m). By the above we see that Kalt ⊗L A A has cohomology in degrees • • L has 0, . . . , n. Hence Kalt ⊗A M has cohomology in degrees 0, . . . , n. Hence Kalt finite Tor dimension, see More on Algebra, Definition 12.44.1. We win by More on Algebra, Lemma 12.45.2. 

Remark 25.21.2. A consequence of Lemma 25.21.1 is that there exists a finite complex of finite projective A-modules M • such that we have H i (XA0 , FA0 ) = H i (M • ⊗A A0 ) functorially in A0 . The condition that F is flat over A is essential, see [Har98]. 25.22. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes

(22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces

25.22. OTHER CHAPTERS

(43) Decent Algebraic Spaces (44) Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces (48) More on Morphisms of Spaces (49) Quot and Hilbert Spaces (50) Spaces over Fields (51) Stacks (52) Formal Deformation Theory (53) Groupoids in Algebraic Spaces (54) More on Groupoids in Spaces (55) Bootstrap (56) Examples of Stacks (57) Quotients of Groupoids (58) Algebraic Stacks

(59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

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Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 26

Divisors 26.1. Introduction In this chapter we study some very basic questions related to defining divisors, etc. A basic reference is [DG67]. 26.2. Associated points Let R be a ring and let M be an R-module. Recall that a prime p ⊂ R is associated to M if there exists an element of M whose annihilator is p. See Algebra, Definition 7.61.1. Here is the definition of associated points for quasi-coherent sheaves on schemes as given in [DG67, IV Definition 3.1.1]. Definition 26.2.1. Let X be a scheme. Let F be a quasi-coherent sheaf on X. (1) We say x ∈ X is associated to F if the maximal ideal mx is associated to the OX,x -module Fx . (2) We denote Ass(F) or AssX (F) the set of associated points of F. (3) The associated points of X are the associated points of OX . These definitions are most useful when X is locally Noetherian and F of finite type. For example it may happen that a generic point of an irreducible component of X is not associated to X, see Example 26.2.7. In the non-Noetherian case it may be more convenient to use weakly associated points, see Section 26.5. Let us link the scheme theoretic notion with the algebraic notion on affine opens; note that this correspondence works perfectly only for locally Noetherian schemes. Lemma 26.2.2. Let X be a scheme. Let F be a quasi-coherent sheaf on X. Let Spec(A) = U ⊂ X be an affine open, and set M = Γ(U, F). Let x ∈ U , and let p ⊂ A be the corresponding prime. (1) If p is associated to M , then x is associated to F. (2) If p is finitely generated, then the coverse holds as well. In particular, if X is locally Noetherian, then the equivalence p ∈ Ass(M ) ⇔ x ∈ Ass(F) holds for all pairs (p, x) as above. Proof. This follows from Algebra, Lemma 7.61.14. But we can also argue directly as follows. Suppose p is associated to M . Then there exists an m ∈ M whose annihilator is p. Since localization is exact we see that pAp is the annihilator of m/1 ∈ Mp . Since Mp = Fx (Schemes, Lemma 21.5.4) we conclude that x is associated to F. Conversely, assume that x is associated to F, and p is finitely generated. As x is associated to F there exists an element m0 ∈ Mp whose annihilator is pAp . Write 1529

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m0 = m/f for some f ∈ A, f 6∈ p. The annihilator I of m is an ideal of A such that IAp = pAp . Hence I ⊂ p, and (p/I)p = 0. Since p is finitely generated, there exists a g ∈ A, g 6∈ p such that g(p/I) = 0. Hence the annihilator of gm is p and we win. If X is locally Noetherian, then A is Noetherian (Properties, Lemma 23.5.2) and p is always finitely generated.  Lemma 26.2.3. Let X be a scheme. Let F be a quasi-coherent OX -module. Then Ass(F) ⊂ Supp(F). Proof. This is immediate from the definitions.



Lemma 26.2.4. Let X be a scheme. Let 0 → F1 → F2 → F3 → 0 be a short exact sequence of quasi-coherent sheaves on X. Then Ass(F2 ) ⊂ Ass(F1 ) ∪ Ass(F3 ) and Ass(F1 ) ⊂ Ass(F2 ). Proof. For every point x ∈ X the sequence of stalks 0 → F1,x → F2,x → F3,x → 0 is a short exact sequence of OX,x -modules. Hence the lemma follows from Algebra, Lemma 7.61.3.  Lemma 26.2.5. Let X be a locally Noetherian scheme. Let F be a coherent OX module. Then Ass(F) ∩ U is finite for every quasi-compact open U ⊂ X. Proof. This is true because the set of associated primes of a finite module over a Noetherian ring is finite, see Algebra, Lemma 7.61.5. To translate from schemes to algebra use that U is a finite union of affine opens, each of these opens is the spectrum of a Noetherian ring (Properties, Lemma 23.5.2), F corresponds to a finite module over this ring (Cohomology of Schemes, Lemma 25.11.1), and finally use Lemma 26.2.2.  Lemma 26.2.6. Let X be a locally Noetherian scheme. Let F be a quasi-coherent OX -module. Then F = 0 ⇔ Ass(F) = ∅. Proof. If F = 0, then Ass(F) = ∅ by definition. Conversely, if Ass(F) = ∅, then F = 0 by Algebra, Lemma 7.61.7. To translate from schemes to algebra, restrict to any affine and use Lemma 26.2.2.  Example 26.2.7. Let k be a field. The ring R = R[x1 , x2 , x3 , . . .]/(x2i ) is local with locally nilpotent maximal ideal m. There exists no element of R which has annihilator m. Hence Ass(R) = ∅, and X = Spec(R) is an example of a scheme which has no associated points. Lemma 26.2.8. Let X be a locally Noetherian scheme. Let F be a quasi-coherent OX -module. Let x ∈ Supp(F) be a point in the support of F which is not a specialization of another point of Supp(F). Then x ∈ Ass(F). In particular, any generic point of an irreducible component of X is an associated point of X. Proof. Since x ∈ Supp(F) the module Fx is not zero. Hence Ass(Fx ) ⊂ Spec(OX,x ) is nonempty by Algebra, Lemma 7.61.7. On the other hand, by assumption Supp(Fx ) = {mx }. Since Ass(Fx ) ⊂ Supp(Fx ) (Algebra, Lemma 7.61.2) we see that mx is associated to Fx and we win. 

26.4. EMBEDDED POINTS

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26.3. Morphisms and associated points Lemma 26.3.1. Let f : X → S be a morphism of schemes. Let F be a quasicoherent sheaf on X which is flat over S. Let G be a quasi-coherent sheaf on S. Then we have [ AssX (F ⊗OX f ∗ G) ⊃ AssXs (Fs ) s∈AssS (G)

and equality holds if S is locally Noetherian. Proof. Let x ∈ X and let s = f (x) ∈ S. Set B = OX,x , A = OS,s , N = Fx , and M = Gs . Note that the stalk of F ⊗OX f ∗ G at x is equal to the B-module M ⊗A N . Hence x ∈ AssX (F ⊗OX f ∗ G) if and only if mB is in AssB (M ⊗A N ). Similarly s ∈ AssS (G) and x ∈ AssXs (Fs ) if and only if mA ∈ AssA (M ) and mB /mA B ∈ AssB⊗κ(mA ) (N ⊗κ(mA )). Thus the lemma follows from Algebra, Lemma 7.63.5.  26.4. Embedded points Let R be a ring and let M be an R-module. Recall that a prime p ⊂ R is an embedded associated to M if it is an associated prime of M which is not minimal among the associated primes of M . See Algebra, Definition 7.65.1. Here is the definition of embedded associated points for quasi-coherent sheaves on schemes as given in [DG67, IV Definition 3.1.1]. Definition 26.4.1. Let X be a scheme. Let F be a quasi-coherent sheaf on X. (1) An embedded associated point of F is an associated point which is not maximal among the associated points of F, i.e., it is the specialization of another associated point of F. (2) A point x of X is called an embedded point if x is an embedded associated point of OX . (3) An embedded component of X is an irreducible closed subset Z = {x} where x is an embedded point of X. In the Noetherian case when F is coherent we have the following. Lemma 26.4.2. Let X be a locally Noetherian scheme. Let F be a coherent OX module. Then (1) the generic points of irreducible components of Supp(F) are associated points of F, and (2) an associated point of F is embedded if and only if it is not a generic point of an irreducible component of Supp(F). In particular an embedded point of X is an associated point of X which is not a generic point of an irreducible component of X. Proof. Recall that in this case Z = Supp(F) is closed, see Morphisms, Lemma 24.5.3 and that the generic points of irreducible components of Z are associated points of F, see Lemma 26.2.8. Finally, we have Ass(F) ⊂ Z, by Lemma 26.2.3. These results, combined with the fact that Z is a sober topological space and hence every point of Z is a specialization of a generic point of Z, imply (1) and (2).  Lemma 26.4.3. Let X be a locally Noetherian scheme. Let F be a coherent sheaf on X. Then the following are equivalent: (1) F has no embedded associated points, and (2) F has property (S1 ).

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Proof. This is Algebra, Lemma 7.141.2, combined with Lemma 26.2.2 above.



Lemma 26.4.4. Let X be a locally Noetherian scheme. Let F be a coherent sheaf on X. The set of coherent subsheaves {K ⊂ F | Supp(K) is nowhere dense in Supp(F)} has a maximal element K. Setting F 0 = F/K we have the following (1) Supp(F 0 ) = Supp(F), (2) F 0 has no embedded associated points, and (3) there exists a dense open U ⊂ X such that U ∩ Supp(F) is dense in Supp(F) and F 0 |U ∼ = F|U . Proof. This follows from Algebra, Lemmas 7.65.2 and 7.65.3. Note that U can be taken as the complement of the closure of the set of embedded associated points of F.  Lemma 26.4.5. Let X be a locally Noetherian scheme. Let F be a coherent OX module without embedded associated points. Set I = Ker(OX −→ Hom OX (F, F)). This is a coherent sheaf of ideals which defines a closed subscheme Z ⊂ X without embedded points. Moreover there exists a coherent sheaf G on Z such that (a) F = (Z → X)∗ G, (b) G has no associated embedded points, and (c) Supp(G) = Z (as sets). Proof. Some of the statements we have seen in the proof of Cohomology of Schemes, Lemma 25.11.7. The others follow from Algebra, Lemma 7.65.4.  26.5. Weakly associated points Let R be a ring and let M be an R-module. Recall that a prime p ⊂ R is weakly associated to M if there exists an element m of M such that p is minimal among the primes containing the annihilator of m. See Algebra, Definition 7.64.1. If R is a local ring with maximal ideal m, then m is associated to M if and only if there exists an element m ∈ M whose annihilator has radical m, see Algebra, Lemma 7.64.2. Definition 26.5.1. Let X be a scheme. Let F be a quasi-coherent sheaf on X. (1) We say x ∈ X is weakly associated to F if the maximal ideal mx is weakly associated to the OX,x -module Fx . (2) We denote WeakAss(F) the set of weakly associated points of F. (3) The weakly associated points of X are the weakly associated points of OX . In this case, on any affine open, this corresponds exactly to the weakly associated primes as defined above. Here is the precise statement. Lemma 26.5.2. Let X be a scheme. Let F be a quasi-coherent sheaf on X. Let Spec(A) = U ⊂ X be an affine open, and set M = Γ(U, F). Let x ∈ U , and let p ⊂ A be the corresponding prime. The following are equivalent (1) p is weakly associated to M , and (2) x is weakly associated to F. Proof. This follows from Algebra, Lemma 7.64.2.



26.6. MORPHISMS AND WEAKLY ASSOCIATED POINTS

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Lemma 26.5.3. Let X be a scheme. Let F be a quasi-coherent OX -module. Then Ass(F) ⊂ WeakAss(F) ⊂ Supp(F). Proof. This is immediate from the definitions.



Lemma 26.5.4. Let X be a scheme. Let 0 → F1 → F2 → F3 → 0 be a short exact sequence of quasi-coherent sheaves on X. Then WeakAss(F2 ) ⊂ WeakAss(F1 ) ∪ WeakAss(F3 ) and WeakAss(F1 ) ⊂ WeakAss(F2 ). Proof. For every point x ∈ X the sequence of stalks 0 → F1,x → F2,x → F3,x → 0 is a short exact sequence of OX,x -modules. Hence the lemma follows from Algebra, Lemma 7.64.3.  Lemma 26.5.5. Let X be a scheme. Let F be a quasi-coherent OX -module. Then F = (0) ⇔ WeakAss(F) = ∅ Proof. Follows from Lemma 26.5.2 and Algebra, Lemma 7.64.4



Lemma 26.5.6. Let X be a scheme. Let F be a quasi-coherent OX -module. Let x ∈ Supp(F) be a point in the support of F which is not a specialization of another point of Supp(F). Then x ∈ WeakAss(F). In particular, any generic point of an irreducible component of X is weakly associated to OX . Proof. Since x ∈ Supp(F) the module Fx is not zero. Hence WeakAss(Fx ) ⊂ Spec(OX,x ) is nonempty by Algebra, Lemma 7.64.4. On the other hand, by assumption Supp(Fx ) = {mx }. Since WeakAss(Fx ) ⊂ Supp(Fx ) (Algebra, Lemma 7.64.5) we see that mx is weakly associated to Fx and we win.  Lemma 26.5.7. Let X be a scheme. Let F be a quasi-coherent OX -module. If mx is a finitely generated ideal of OX,x , then x ∈ Ass(F) ⇔ x ∈ WeakAss(F). In particular, if X is locally Noetherian, then Ass(F) = WeakAss(F). Proof. See Algebra, Lemma 7.64.8.



26.6. Morphisms and weakly associated points Lemma 26.6.1. Let f : X → S be an affine morphism of schemes. Let F be a quasi-coherent OX -module. Then we have WeakAssS (f∗ F) ⊂ f (WeakAssX (F)) Proof. We may assume X and S affine, so X → S comes from a ring map A → B. f for some B-module M . By Lemma 26.5.2 the weakly associated Then F = M points of F correspond exactly to the weakly associated primes of M . Similarly, the weakly associated points of f∗ F correspond exactly to the weakly associated primes of M as an A-module. Hence the lemma follows from Algebra, Lemma 7.64.10.  Lemma 26.6.2. Let f : X → S be an affine morphism of schemes. Let F be a quasi-coherent OX -module. If X is locally Noetherian, then we have f (AssX (F)) = AssS (f∗ F) = WeakAssS (f∗ F) = f (WeakAssX (F))

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Proof. We may assume X and S affine, so X → S comes from a ring map A → B. As X is locally Noetherian the ring B is Noetherian, see Properties, Lemma 23.5.2. f for some B-module M . By Lemma 26.2.2 the associated points of F Write F = M correspond exactly to the associated primes of M , and any associated prime of M as an A-module is an associated points of f∗ F. Hence the inclusion f (AssX (F)) ⊂ AssS (f∗ F) follows from Algebra, Lemma 7.61.12. We have the inclusion AssS (f∗ F) ⊂ WeakAssS (f∗ F) by Lemma 26.5.3. We have the inclusion WeakAssS (f∗ F) ⊂ f (WeakAssX (F)) by Lemma 26.6.1. The outer sets are equal by Lemma 26.5.7 hence we have equality everywhere.  Lemma 26.6.3. Let f : X → S be a finite morphism of schemes. Let F be a quasi-coherent OX -module. Then WeakAss(f∗ F) = f (WeakAss(F)). Proof. We may assume X and S affine, so X → S comes from a finite ring map f for some B-module M . By Lemma 26.5.2 the weakly A → B. Write F = M associated points of F correspond exactly to the weakly associated primes of M . Similarly, the weakly associated points of f∗ F correspond exactly to the weakly associated primes of M as an A-module. Hence the lemma follows from Algebra, Lemma 7.64.12.  Lemma 26.6.4. Let f : X → S be a morphism of schemes. Let G be a quasicoherent OS -module. Let x ∈ X with s = f (x). If f is flat at x, the point x is a generic point of the fibre Xs , and s ∈ WeakAssS (G), then x ∈ WeakAss(f ∗ G). Proof. Let A = OS,s , B = OX,x , and M = Gs . Let m ∈ M be an element whose annihilator I = {a ∈ A | am = 0} has radical mA . Then √ m ⊗ 1 has annihilator IB as A → B is faithfully flat. Thus it suffices to see that IB = mB . This follows from the fact that the maximal ideal of B/m √ A B is locally nilpotent (see Algebra, Lemma 7.24.3) and the assumption that I = mA . Some details omitted.  26.7. Relative assassin Definition 26.7.1. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. The relative assassin of F in X over S is the set [ AssX/S (F) = AssXs (Fs ) s∈S

where Fs = (Xs → X)∗ F is the restriction of F to the fibre of f at s. Again there is a caveat that this is best used when the fibres of f are locally Noetherian and F is of finite type. In the general case we should probably use the relative weak assassin (defined in the next section).

26.8. RELATIVE WEAK ASSASSIN

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Lemma 26.7.2. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. Let g : S 0 → S be a morphism of schemes. Consider the base change diagram /X X0 0 g

  g /S S0 and set F 0 = (g 0 )∗ F. Let x0 ∈ X 0 be a point with images x ∈ X, s0 ∈ S 0 and s ∈ S. Assume f locally of finite type. Then x0 ∈ AssX 0 /S 0 (F 0 ) if and only if x ∈ AssX/S (F) and x0 corresponds to a generic point of an irreducible component of Spec(κ(s0 ) ⊗κ(s) κ(x)). Proof. Consider the morphism Xs0 0 → Xs of fibres. As Xs0 = Xs ×Spec(κ(s)) Spec(κ(s0 )) this is a flat morphism. Moreover Fs0 0 is the pullback of Fs via this morphism. As Xs is locally of finite type over the Noetherian scheme Spec(κ(s)) we have that Xs is locally Noetherian, see Morphisms, Lemma 24.16.6. Thus we may apply Lemma 26.3.1 and we see that [ AssXs0 0 (Fs0 0 ) = Ass((Xs0 0 )x ). x∈Ass(Fs )

Thus to prove the lemma it suffices to show that the associated points of the fibre (Xs0 0 )x of the morphism Xs0 0 → Xs over x are its generic points. Note that (Xs0 0 )x = Spec(κ(s0 ) ⊗κ(s) κ(x)) as schemes. By Algebra, Lemma 7.150.1 the ring κ(s0 ) ⊗κ(s) κ(x) is a Noetherian Cohen-Macaulay ring. Hence its associated primes are its minimal primes, see Algebra, Proposition 7.61.6 (minimal primes are associated) and Algebra, Lemma 7.141.2 (no embedded primes).  Remark 26.7.3. With notation and assumptions as in Lemma 26.7.2 we see that it is always the case that (g 0 )−1 (AssX/S (F)) ⊃ AssX 0 /S 0 (F 0 ). If the morphism S 0 → S is locally quasi-finite, then we actually have (g 0 )−1 (AssX/S (F)) = AssX 0 /S 0 (F 0 ) because in this case the field extensions κ(s) ⊂ κ(s0 ) are always finite. In fact, this holds more generally for any morphism g : S 0 → S such that all the field extensions κ(s) ⊂ κ(s0 ) are algebraic, because in this case all prime ideals of κ(s0 ) ⊗κ(s) κ(x) are maximal (and minimal) primes, see Algebra, Lemma 7.33.17. 26.8. Relative weak assassin Definition 26.8.1. Let f : X → S be a morphism of schemes. Let F be a quasicoherent OX -module. The relative weak assassin of F in X over S is the set [ WeakAssX/S (F) = WeakAss(Fs ) s∈S

∗

where Fs = (Xs → X) F is the restriction of F to the fibre of f at s. Lemma 26.8.2. Let f : X → S be a morphism of schemes which is locally of finite type. Let F be a quasi-coherent OX -module. Then WeakAssX/S (F) = AssX/S (F). Proof. This is true becase the fibres of f are locally Noetherian schemes, and associated and weakly associated points agree on locally Noetherian schemes, see Lemma 26.5.7. 

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26.9. Effective Cartier divisors For some reason it seem convenient to define the notion of an effective Cartier divisor before anything else. Definition 26.9.1. Let S be a scheme. (1) A locally principal closed subscheme of S is a closed subscheme whose sheaf of ideals is locally generated by a single element. (2) An effective Cartier divisor on S is a closed subscheme D ⊂ S such that the ideal sheaf ID ⊂ OX is an invertible OX -module. Thus an effective Cartier divisor is a locally principal closed subscheme, but the converse is not always true. Effective Cartier divisors are closed subschemes of pure codimension 1 in the strongest possible sense. Namely they are locally cut out by a single element which is not a zerodivisor. In particular they are nowhere dense. Lemma 26.9.2. Let S be a scheme. Let D ⊂ S be a closed subscheme. The following are equivalent: (1) The subscheme D is an effective Cartier divisor on S. (2) For every x ∈ D there exists an affine open neighbourhood Spec(A) = U ⊂ X of x such that U ∩ D = Spec(A/(f )) with f ∈ A not a zerodivisor. Proof. Assume (1). For every x ∈ D there exists an affine open neighbourhood Spec(A) = U ⊂ X of x such that ID |U ∼ = OU . In other words, there exists a section f ∈ Γ(U, ID ) which freely generates the restriction ID |U . Hence f ∈ A, and the multiplication map f : A → A is injective. Also, since ID is quasi-coherent we see that D ∩ U = Spec(A/(f )). Assume (2). Let x ∈ D. By assumption there exists an affine open neighbourhood Spec(A) = U ⊂ X of x such that U ∩D = Spec(A/(f )) with f ∈ A not a zerodivisor. f) ∼ e∼ Then ID |U ∼ = OU since it is equal to (f =A = OU . Of course ID restricted to the open subscheme S \ D is isomorphic to OX\D . Hence ID is an invertible OS module.  Lemma 26.9.3. Let S be a scheme. Let Z ⊂ S be a locally principal closed subscheme. Let U = S \ Z. Then U → S is an affine morphism. Proof. The question is local on S, see Morphisms, Lemmas 24.13.3. Thus we may assume S = Spec(A) and Z = V (f ) for some f ∈ A. In this case U = D(f ) = Spec(Af ) is affine hence U → S is affine.  Lemma 26.9.4. Let S be a scheme. Let D ⊂ S be an effective Cartier divisor. Let U = S \ D. Then U → S is an affine morphism and U is scheme theoretically dense in S. Proof. Affineness is Lemma 26.9.3. The density question is local on S, see Morphisms, Lemma 24.7.5. Thus we may assume S = Spec(A) and D corresponding to the nonzerodivisor f ∈ A, see Lemma 26.9.2. Thus A ⊂ Af which implies that U ⊂ S is scheme theoretically dense, see Morphisms, Example 24.7.4.  Lemma 26.9.5. Let S be a scheme. Let D ⊂ S be an effective Cartier divisor. Let s ∈ D. If dims (S) < ∞, then dims (D) < dims (S).

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1537

Proof. Assume dims (S) < ∞. Let U = Spec(A) ⊂ S be an affine open neighbourhood of X such that dim(U ) = dims (S) and such that D = V (f ) for some nonzerodivisor f ∈ A (see Lemma 26.9.2). Recall that dim(U ) is the Krull dimension of the ring A and that dim(U ∩ D) is the Krull dimension of the ring A/(f ). Then f is not contained in any minimal prime of A. Hence any maximal chain of primes in A/(f ), viewed as a chain of primes in A, can be extended by adding a minimal prime.  Definition 26.9.6. Let S be a scheme. Given effective Cartier divisors D1 , D2 on S we set D = D1 + D2 equal to the closed subscheme of S corresponding to the quasi-coherent sheaf of ideals ID1 ID2 ⊂ OS . We call this the sum of the effective Cartier divisors D1 and D2 . P It is clear that we may define the sum ni Di given finitely many effective Cartier divisors Di on X and nonnegative integers ni . Lemma 26.9.7. The sum of two effective Cartier divisors is an effective Cartier divisor. Proof. Omitted. Locally f1 , f2 ∈ A are nonzerodivisors, then also f1 f2 ∈ A is a nonzerodivisor.  Lemma 26.9.8. Let X be a scheme. Let D, D0 be two effective Cartier divisors on X. If D ⊂ D0 (as closed subschemes of X), then there exists an effective Cartier divisor D00 such that D0 = D + D00 . Proof. Omitted.



Lemma 26.9.9. Let X be a scheme. Let Z, Y be two closed subschemes of X with ideal sheaves I and J . If IJ defines an effective Cartier divisor D ⊂ X, then Z and Y are effective Cartier divisors and D = Z + Y . Proof. Applying Lemma 26.9.2 we obtain the following algebra situation: A is a ring, I, J ⊂ A ideals and f ∈ A a nonzerodivisor such that IJ = (f ). We have to show P that I and J are locally free A-modules of rank 1. To do this, write f = i=1,...,n xi yi . We can also write xi yi = ai f . Since f is a nonzerodivisor we P see that ai = 1. Thus it suffices to show that each Iai and Jai is free of rank 1 over Aai . After replacing A by Aai we conclude that f = xy for some x ∈ I and y ∈ J. Note that both x and y are nonzerodivisors. We claim that I = (x) and J = (y) which finishes the proof. Namely, if x0 ∈ I, then x0 y = af = axy for some a ∈ A. Hence x0 = ax and we win.  Recall that we have defined the inverse image of a closed subscheme under any morphism of schemes in Schemes, Definition 21.17.7. Lemma 26.9.10. Let f : S 0 → S be a morphism of schemes. Let Z ⊂ S be a locally principal closed subscheme. Then the inverse image f −1 (Z) is a locally principal closed subscheme of S 0 . Proof. Omitted.

 0

Definition 26.9.11. Let f : S → S be a morphism of schemes. Let D ⊂ S be an effective Cartier divisor. We say the pullback of D by f is defined if the closed subscheme f −1 (D) ⊂ S 0 is an effective Cartier divisor. In this case we denote it either f ∗ D or f −1 (D) and we call it the pullback of the effective Cartier divisor.

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The condition that f −1 (D) is an effective Cartier divisor is often satisfied in practice. Here is an example lemma. Lemma 26.9.12. Let f : X → Y be a morphism of schemes. Let D ⊂ Y be an effective Cartier divisor. The pullback of D by f is defined in each of the following cases: (1) X, Y integral and f dominant, (2) X reduced, and for any generic point ξ of any irreducible component of X we have f (ξ) 6∈ D, (3) X is locally Noetherian and for any associated point x of X we have f (x) 6∈ D, (4) X is locally Noetherian, has no embedded points, and for any generic point ξ of any irreducible component of X we have f (ξ) 6∈ D, (5) f is flat, and (6) add more here as needed. Proof. The question is local on X, and hence we reduce to the case where X = Spec(A), Y = Spec(R), f is given by ϕ : R → A and D = Spec(R/(t)) where t ∈ R is not a zerodivisor. The goal in each case is to show that ϕ(t) ∈ A is not a zerodivisor. In case (2) this follows as the intersection of all minimal primes of a ring is the nilradical of the ring, see Algebra, Lemma 7.16.2. Let us prove (3). By Lemma 26.2.2 the associated pointsSof X correspond to the primes p ∈ Ass(A). By Algebra, Lemma 7.61.9 we have p∈Ass(A) p is the set of zerodivisors of A. The hypothesis of (3) is that ϕ(t) 6∈ p for all p ∈ Ass(A). Hence ϕ(t) is a nonzerodivisor as desired. Part (4) follows from (3) and the definitions.



0

Lemma 26.9.13. Let f : S → S be a morphism of schemes. Let D1 , D2 be effective Cartier divisors on S. If the pullbacks of D1 and D2 are defined then the pullback of D = D1 + D2 is defined and f ∗ D = f ∗ D1 + f ∗ D2 . Proof. Omitted.



Definition 26.9.14. Let S be a scheme and let D be an effective Cartier divisor. The invertible sheaf OS (D) associated to D is given by ⊗−1 OS (D) := Hom OS (ID , OS ) = ID .

The canonical section, usually denoted 1 or 1D , is the global section of OS (D) corresponding to the inclusion mapping ID → OS . Lemma 26.9.15. Let S be a scheme. Let D1 , D2 be effective Cartier divisors on S. Let D = D1 + D2 . Then there is a unique isomorphism OS (D1 ) ⊗OS OS (D2 ) −→ OS (D) which maps 1D1 ⊗ 1D2 to 1D . Proof. Omitted.



Definition 26.9.16. Let (X, OX ) be a locally ringed space. Let L be an invertible sheaf on X. A global section s ∈ Γ(X, L) is called a regular section if the map OX → L, f 7→ f s is injective.

26.10. RELATIVE EFFECTIVE CARTIER DIVISORS

1539

Lemma 26.9.17. Let X be a locally ringed space. Let f ∈ Γ(X, OX ). The following are equivalent: (1) f is a regular section, and (2) for any x ∈ X the image f ∈ OX,x is not a zerodivisor. If X is a scheme these are also equivalent to (3) for any affine open Spec(A) = U ⊂ X the image f ∈ A is not a zerodivisor, and S (4) there exists an affine open covering X = Spec(Ai ) such that the image of f in Ai is not a zerodivisor for all i. Proof. Omitted.



Note that a global section s of an invertible OX -module L may be seen as an OX module map s : OX → L. Its dual is therefore a map s : L⊗−1 → OX . (See Modules, Definition 15.21.3 for the definition of the dual invertible sheaf.) Definition 26.9.18. Let X be a scheme. Let L be an invertible sheaf. Let s ∈ Γ(X, L). The zero scheme of s is the closed subscheme Z(s) ⊂ X defined by the quasi-coherent sheaf of ideals I ⊂ OX which is the image of the map s : L⊗−1 → OX . Lemma 26.9.19. Let X be a scheme. Let L be an invertible sheaf. Let s ∈ Γ(X, L). (1) Consider closed immersions i : Z → X such that i∗ s ∈ Γ(Z, i∗ L)) is zero ordered by inclusion. The zero scheme Z(s) is the maximal element of this ordered set. (2) For any morphism of schemes f : Y → X we have f ∗ s = 0 in Γ(Y, f ∗ L) if and only if f factors through Z(s). (3) The zero scheme Z(s) is a locally principal closed subscheme. (4) The zero scheme Z(s) is an effective Cartier divisor if and only if s is a regular section of L. Proof. Omitted.



Lemma 26.9.20. Let S be a scheme. (1) If D ⊂ S is an effective Cartier divisor, then the canonical section 1D of OS (D) is regular. (2) Conversely, if s is a regular section of the invertible sheaf L, then there exists a unique effective Cartier divisor D = Z(s) ⊂ S and a unique isomorphism OS (D) → L which maps 1D to s. The constructions D 7→ (OX (D), 1D ) and (L, s) 7→ Z(s) give mutually inverse maps    pairs (L, s) consisting of an invertible effective Cartier divisors on X ↔ OX -module and a regular global section Proof. Omitted.

 26.10. Relative effective Cartier divisors

The following lemma shows that an effective Cartier divisor which is flat over the base is reall a “family of effective Cartier divisors” over the base. For example the restriction to any fibre is an effective Cartier divisor.

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Lemma 26.10.1. Let f : X → S be a morphism of schemes. Let D ⊂ X be a closed subscheme. Assume (1) D is an effective Cartier divisor, and (2) D → S is a flat morphism. Then for every morphism of schemes g : S 0 → S the pullback (g 0 )−1 D is an effective Cartier divisor on X 0 = S 0 ×S X. Proof. Using Lemma 26.9.2 we translate this as follows into algebra. Let A → B be a ring map and h ∈ B. Assume h is a nonzerodivisor and that B/hB is flat over A. Then h 0→B− → B → B/hB → 0 is a short exact sequence of A-modules with B/hB flat over A. By Algebra, Lemma 7.36.11 this sequence remains exact on tensoring over A with any module, in particular with any A-algebra A0 .  This lemma is the motivation for the following definition. Definition 26.10.2. Let f : X → S be a morphism of schemes. A relative effective Cartier divisor on X/S is an effective Cartier divisor D ⊂ X such that D → S is a flat morphism of schemes. We warn the reader that this may be nonstandard notation. In particular, in [DG67, IV, Section 21.15] the notion of a relative divisor is discussed only when X → S is flat and locally of finite presentation. Our definition is a bit more general. However, it turns out that if x ∈ D then X → S is flat at x in many cases (but not always). Lemma 26.10.3. Let f : X → S be a morphism of schemes. Let D ⊂ X be a relative effective Cartier divisor on X/S. If x ∈ D and OX,x is Noetherian, then f is flat at x. Proof. Set A = OS,f (x) and B = OX,x . Let h ∈ B be an element which generates the ideal of D. Then h is a nonzerodivisor in B such that B/hB is a flat local A-algebra. Let I ⊂ A be a finitely generated ideal. Consider the commutative diagram 0

/B O

0

/ B ⊗A I

h

h

/B O

/ B/hB O

/0

/ B ⊗A I

/ B/hB ⊗A I

/0

The lower sequence is short exact as B/hB is flat over A, see Algebra, Lemma 7.36.11. The right vertical arrow is injective as B/hB is flat over A, see Algebra, Lemma 7.36.4. Hence multiplication by h is surjective on the kernel K of the middle vertical arrow. By Nakayama’s lemma, see Algebra, Lemma 7.18.1 we conclude that K = 0. Hence B is flat over A, see Algebra, Lemma 7.36.4.  The following lemma relies on the algebraic version of openness of the flat locus. The scheme theoretic version can be found in More on Morphisms, Section 33.12. Lemma 26.10.4. Let f : X → S be a morphism of schemes. Let D ⊂ X be a relative effective Cartier divisor. If f is locally of finite presentation, then there

26.10. RELATIVE EFFECTIVE CARTIER DIVISORS

1541

exists an open subscheme U ⊂ X such that D ⊂ U and such that f |U : U → S is flat. Proof. Pick x ∈ D. It suffices to find an open neighbourhood U ⊂ X of x such that f |U is flat. Hence the lemma reduces to the case that X = Spec(B) and S = Spec(A) are affine and that D is given by a nonzerodivisor h ∈ B. By assumption B is a finitely presented A-algebra and B/hB is a flat A-algebra. We are going to use absolute Noetherian approximation. Write B = A[x1 , . . . , xn ]/(g1 , . . . , gm ). Assume h is the image of h0 ∈ A[x1 , . . . , xn ]. Choose a finite type Z-subalgebra A0 ⊂ A such that all the coefficients of the polynomials h0 , g1 , . . . , gm are in A0 . Then we can set B0 = A0 [x1 , . . . , xn ]/(g1 , . . . , gm ) and h0 the image of h0 in B0 . Then B = B0 ⊗A0 A and B/hB = B0 /h0 B0 ⊗A0 A. By Algebra, Lemma 7.151.1 we may, after enlarging A0 , assume that B0 /h0 B0 is flat over A0 . Let K0 = Ker(h0 : B0 → B0 ). As B0 is of finite type over Z we see that K0 is a finitely generated ideal. Let A1 ⊂ A be a finite type Z-subalgebra containing A0 and denote B1 , h1 , K1 the corresponding objects over A1 . By More on Algebra, Lemma 12.24.14 the map K0 ⊗A0 A1 → K1 is surjective. On the other hand, the kernel of h : B → B is zero by assumption. Hence every element of K0 maps to zero in K1 for sufficiently large subrings A1 ⊂ A. Since K0 is finitely generated, we conclude that K1 = 0 for a suitable choice of A1 . Set f1 : X1 → S1 equal to Spec of the ring map A1 → B1 . Set D1 = Spec(B1 /h1 B1 ). Since B = B1 ⊗A1 A, i.e., X = X1 ×S1 S, it now suffices to prove the lemma for X1 → S1 and the relative effective Cartier divisor D1 , see Morphisms, Lemma 24.26.6. Hence we have reduced to the case where A is a Noetherian ring. In this case we know that the ring map A → B is flat at every prime q of V (h) by Lemma 26.10.3. Combined with the fact that the flat locus is open in this case, see Algebra, Theorem 7.121.4 we win.  There is also the following lemma (whose idea is apparantly due to Michael Artin, see [Nob77]) which needs no finiteness assumptions at all. Lemma 26.10.5. Let f : X → S be a morphism of schemes. Let D ⊂ X be a relative effective Cartier divisor on X/S. If f is flat at all points of X \ D, then f is flat. Proof. This translates into the following algebra fact: Let A → B be a ring map and h ∈ B. Assume h is a nonzerodivisor, that B/hB is flat over A, and that the localization Bh is flat over A. Then B is flat over A. The reason is that we have a short exact sequence 0 → B → Bh → colimn (1/hn )B/B → 0 and that the second and third terms are flat over A, which implies that B is flat over A (see Algebra, Lemma 7.36.12). Note that a filtered colimit of flat modules is flat (see Algebra, Lemma 7.36.2) and that by induction on n each (1/hn )B/B ∼ = B/hn B is flat over A since it fits into the short exact sequence h

0 → B/hn−1 B − → B/hn B → B/hB → 0 Some details omitted.



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Example 26.10.6. Here is an example of a relative effective Cartier divisor D where the ambient scheme is not flat in a neighbourhood of D. Namely, let A = k[t] and B = k[t, x, y, x−1 y, x−2 y, . . .]/(ty, tx−1 y, tx−2 y, . . .) Then B is not flat over A but B/xB ∼ = A is flat over A. Moreover x is a nonzerodivisor and hence defines a relative effective Cartier divisor in Spec(B) over Spec(A). If the ambient scheme is flat and locally of finite presentation over the base, then we can characterize a relative effective Cartier divisor in terms of its fibres. See also More on Morphisms, Lemma 33.17.1 for a slightly different take on this lemma. Lemma 26.10.7. Let ϕ : X → S be a flat morphism which is locally of finite presentation. Let Z ⊂ X be a closed subscheme. Let x ∈ Z with image s ∈ S. (1) If Zs ⊂ Xs is a Cartier divisor in a neighbourhood of x, then there exists an open U ⊂ X and a relative effective Cartier divisor D ⊂ U such that Z ∩ U ⊂ D. (2) If Zs ⊂ Xs is a Cartier divisor in a neighbourhood of x, the morphism Z → X is of finite presentation, and Z → S is flat at x, then we can choose U and D such that Z ∩ U = D. (3) If Zs ⊂ Xs is a Cartier divisor in a neighbourhood of x and Z is a locally principal closed subscheme of X in a neighbourhood of x, then we can choose U and D such that Z ∩ U = D. In particular, if Z → S is locally of finite presentation and flat and all fibres Zs ⊂ Xs are effective Cartier divisors, then Z is a relative effective Cartier divisor. Similarly, if Z is a locally principal closed subscheme of X such that all fibres Zs ⊂ Xs are effective Cartier divisors, then Z is a relative effective Cartier divisor. Proof. Choose affine open neighbourhoods Spec(A) of s and Spec(B) of x such that ϕ(Spec(B)) ⊂ Spec(A). Let p ⊂ A be the prime ideal corresponding to s. Let q ⊂ B be the prime ideal corresponding to x. Let I ⊂ B be the ideal corresponding to Z. By the initial assumption of the lemma we know that A → B is flat and of finite presentation. The assumption in (1) means that, after shrinking Spec(B), we may assume I(B ⊗A κ(p)) is generated by a single element which is a nonzerodivisor in B ⊗A κ(p). Say f ∈ I maps to this generator. We claim that after inverting an element g ∈ B, g 6∈ q the closed subscheme D = V (f ) ⊂ Spec(Bg ) is a relative effective Cartier divisor. By Algebra, Lemma 7.151.1 we can find a flat finite type ring map A0 → B0 of Noetherian rings, an element f0 ∈ B0 , a ring map A0 → A and an isomorphism A ⊗A0 B0 ∼ = B. If p0 = A0 ∩ p then we see that B ⊗A κ(p) = (B0 ⊗A0 κ(p0 )) ⊗κ(p0 )) κ(p) hence f0 is a nonzerodivisor in B0 ⊗A0 κ(p0 ). By Algebra, Lemma 7.92.2 we see that f0 is a nonzerodivisor in (B0 )q0 where q0 = B0 ∩ q and that (B0 /f0 B0 )q0 is flat over A0 . Hence by Algebra, Lemma 7.66.8 and Algebra, Theorem 7.121.4 there exists a g0 ∈ B0 , g0 6∈ q0 such that f0 is a nonzerodivisor in (B0 )g0 and such that (B0 /f0 B0 )g0 is flat over A0 . Hence we see that D0 = V (f0 ) ⊂ Spec((B0 )g0 ) is a relative effective Cartier divisor. Since we know that this property is preserved under base change, see Lemma 26.10.1, we obtain the claim mentioned above with g equal to the image of g0 in B.

26.11. THE NORMAL CONE OF AN IMMERSION

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At this point we have proved (1). To see (2) consider the closed immersion Z → D. The surjective ring map u : OD,x → OZ,x is a map of flat local OS,s -algebras which are essentially of finite presentation, and which becomes an isomorphisms after dividing by ms . Hence it is an isomorphism, see Algebra, Lemma 7.120.4. It follows that Z → D is an isomorphism in a neighbourhood of x, see Algebra, Lemma 7.118.6. To see (3), after possibly shrinking U we may assume that the ideal of D is generated by a single nonzerodivisor f and the ideal of Z is generated by an element g. Then f = gh. But g|Us and f |Us cut out the same effective Cartier divisor in a neighbourhood of x. Hence h|Xs is a unit in OXs ,x , hence h is a unit in OX,x hence h is a unit in an open neighbourhood of x. I.e., Z ∩ U = D after shrinking U . The final statements of the lemma follow immediately from parts (2) and (3), combined with the fact that Z → S is locally of finite presentation if and only if Z → X is of finite presentation, see Morphisms, Lemmas 24.22.3 and 24.22.11.  26.11. The normal cone of an immersion Let i : Z → X be a closed immersion. Let I ⊂ OX be the corresponding quasicoherent sheaf of ideals. Consider the quasi-coherent sheaf of graded OX -algebras L n n+1 I /I . Since the sheaves I n /I n+1 are each annihilated by I this graded n≥0 algebra corresponds to a quasi-coherent sheaf of graded OZ -algebras by Morphisms, Lemma 24.4.1. This quasi-coherent gradedL OZ -algebra is called the conormal algebra of Z in X and is often simply denoted n≥0 I n /I n+1 by the abuse of notation mentioned in Morphisms, Section 24.4. Let f : Z → X be an immersion. We define the conormal algebra of f as the conormal sheafL of the closed immersion i : Z → X \ ∂Z, where ∂Z = Z \ Z. It is n n+1 often denoted where I is the ideal sheaf of the closed immersion n≥0 I /I i : Z → X \ ∂Z. Definition 26.11.1. Let f : Z → X be an immersion. The conormal algebra CZ/X,∗ of Z in X or the conormal algebra of f is the quasi-coherent sheaf of graded L OZ -algebras n≥0 I n /I n+1 described above. Thus CZ/X,1 = CZ/X is the conormal sheaf of the immersion. Also CZ/X,0 = OZ and CZ/X,n is a quasi-coherent OZ -module characterized by the property (26.11.1.1)

i∗ CZ/X,n = I n /I n+1

where i : Z → X \ ∂Z and I is the ideal sheaf of i as above. Finally, note that there is a canonical surjective map (26.11.1.2)

Sym∗ (CZ/X ) −→ CZ/X,∗

of quasi-coherent graded OZ -algebras which is an isomorphism in degrees 0 and 1. Lemma 26.11.2. Let i : Z → X be an immersion. The conormal algebra of i has the following properties: (1) Let U ⊂ X be any open such that i(Z) is a closed subset of U . Let I ⊂ OU be the sheaf of ideals corresponding to the closed subscheme i(Z) ⊂ U . Then M  M  CZ/X,∗ = i∗ I n = i−1 I n /I n+1 n≥0

n≥0

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26. DIVISORS

(2) For any affine open Spec(R) = U ⊂ X such thatL Z ∩ U = Spec(R/I) there is a canonical isomorphism Γ(Z ∩ U, CZ/X,∗ ) = n≥0 I n /I n+1 . Proof. Mostly clear from the definitions. Note that given a ring R and an ideal I of R we have I n /I n+1 = I n ⊗R R/I. Details omitted.  Lemma 26.11.3. Let Z

i

/X g

f

  i0 / Z0 X0 be a commutative diagram in the category of schemes. Assume i, i0 immersions. There is a canonical map of graded OZ -algebras f ∗ CZ 0 /X 0 ,∗ −→ CZ/X,∗ characterized by the following property: For every pair of affine opens (Spec(R) = U ⊂ X, Spec(R0 ) = U 0 ⊂ X 0 ) with f (U ) ⊂ U 0 such that Z ∩ U = Spec(R/I) and Z 0 ∩ U 0 = Spec(R0 /I 0 ) the induced map M M Γ(Z 0 ∩ U 0 , CZ 0 /X 0 ,∗ ) = (I 0 )n /(I 0 )n+1 −→ I n /I n+1 = Γ(Z ∩ U, CZ/X,∗ ) n≥0

is the one induced by the ring map f ] : R0 → R which has the property f ] (I 0 ) ⊂ I. Proof. Let ∂Z 0 = Z 0 \ Z 0 and ∂Z = Z \ Z. These are closed
subsets of X 0 and of X. Replacing X 0 by X 0 \ ∂Z 0 and X by X \ g −1 (∂Z 0 ) ∪ ∂Z we see that we may assume that i and i0 are closed immersions. The fact that g ◦ i factors through i0 implies that g ∗ I 0 maps into I under the canonical map g ∗ I 0 → OX , see Schemes, Lemmas 21.4.6 and 21.4.7. Hence we get an induced map of quasi-coherent sheaves g ∗ ((I 0 )n /(I 0 )n+1 ) → I n /I n+1 . Pulling back by i gives i∗ g ∗ ((I 0 )n /(I 0 )n+1 ) → i∗ (I n /I n+1 ). Note that i∗ (I n /I n+1 ) = CZ/X,n . On the other hand, i∗ g ∗ ((I 0 )n /(I 0 )n+1 ) = f ∗ (i0 )∗ ((I 0 )n /(I 0 )n+1 ) = f ∗ CZ 0 /X 0 ,n . This gives the desired map. Checking that the map is locally described as the given map (I 0 )n /(I 0 )n+1 → I n /I n+1 is a matter of unwinding the definitions and is omitted. Another observation is that given any x ∈ i(Z) there do exist affine open neighbourhoods U , U 0 with f (U ) ⊂ U 0 and Z ∩ U as well as U 0 ∩ Z 0 closed such that x ∈ U . Proof omitted. Hence the requirement of the lemma indeed characterizes the map (and could have been used to define it).  Lemma 26.11.4. Let Z f

i

/X g

  i0 / Z0 X0 be a fibre product diagram in the category of schemes with i, i0 immersions. Then the canonical map f ∗ CZ 0 /X 0 ,∗ → CZ/X,∗ of Lemma 26.11.3 is surjective. If g is flat, then it is an isomorphism. Proof. Let R0 → R be a ring map, and I 0 ⊂ R0 an ideal. Set I = I 0 R. Then (I 0 )n /(I 0 )n+1 ⊗R0 R → I n /I n+1 is surjective. If R0 → R is flat, then I n = (I 0 )n ⊗R0 R and we see the map is an isomorphism. 

26.12. REGULAR IDEAL SHEAVES

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Definition 26.11.5. Let i : Z → X be an immersion of schemes. The normal cone CZ X of Z in X is CZ X = SpecZ (CZ/X,∗ ) see Constructions, Definitions 22.7.1 and 22.7.2. The normal bundle of Z in X is the vector bundle NZ X = SpecZ (Sym(CZ/X )) see Constructions, Definitions 22.6.1 and 22.6.2. Thus CZ X → Z is a cone over Z and NZ X → Z is a vector bundle over Z (recall that in our terminology this does not imply that the conormal sheaf is a finite locally free sheaf). Moreover, the canonical surjection (26.11.1.2) of graded algebras defines a canonical closed immersion CZ X −→ NZ X

(26.11.5.1) of cones over Z.

26.12. Regular ideal sheaves In this section we generalize the notion of an effective Cartier divisor to higher codimension. Recall that a sequence of elements f1 , . . . , fr of a ring R is a regular sequence if for each i = 1, . . . , r the element fi is a nonzerodivisor on R/(f1 , . . . , fi−1 ) and R/(f1 , . . . , fr ) 6= 0, see Algebra, Definition 7.66.1. There are three closely related weaker conditions that we can impose. The first is to assume that f1 , . . . , fr is a Koszul-regular sequence, i.e., that Hi (K• (f1 , . . . , fr )) = 0 for i > 0, see More on Algebra, Definition 12.24.1. The sequence is called an H1 -regular sequence if H1 (K• (f1 , . . . , fr )) = 0. Another condition we can impose is that with J = (f1 , . . . , fr ), the map M R/J[T1 , . . . , Tr ] −→ J n /J n+1 n≥0

2

which maps Ti to fi mod J is an isomorphism. In this case we say that f1 , . . . , fr is a quasi-regular sequence, see Algebra, Definition 7.67.1. Given an R-module M there is also a notion of M -regular and M -quasi-regular sequence. We can generalize this to the case of ringed spaces as follows. Let X be a ringed space and let f1 , . . . , fr ∈ Γ(X, OX ). We say that f1 , . . . , fr is a regular sequence if for each i = 1, . . . , r the map (26.12.0.2)

fi : OX /(f1 , . . . , fi−1 ) −→ OX /(f1 , . . . , fi−1 )

is an injective map of sheaves. We say that f1 , . . . , fr is a Koszul-regular sequence if the Koszul complex (26.12.0.3)

K• (OX , f• ),

see Modules, Definition 15.20.2, is acyclic in degrees > 0. We say that f1 , . . . , fr is a H1 -regular sequence if the Koszul complex K• (OX , f• ) is exact in degree 1. Finally, we say that f1 , . . . , fr is a quasi-regular sequence if the map M (26.12.0.4) OX /J [T1 , . . . , Tr ] −→ J d /J d+1 d≥0

is an isomorphism of sheaves where J ⊂ OX is the sheaf of ideals generated by f1 , . . . , fr . (There is also a notion of F-regular and F-quasi-regular sequence for a given OX -module F which we will introduce here if we ever need it.)

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Lemma 26.12.1. Let X be a ringed space. Let f1 , . . . , fr ∈ Γ(X, OX ). We have the following implications f1 , . . . , fr is a regular sequence ⇒ f1 , . . . , fr is a Koszulregular sequence ⇒ f1 , . . . , fr is an H1 -regular sequence ⇒ f1 , . . . , fr is a quasiregular sequence. Proof. Since we may check exactness at stalks, a sequence f1 , . . . , fr is a regular sequence if and only if the maps fi : OX,x /(f1 , . . . , fi−1 ) −→ OX,x /(f1 , . . . , fi−1 ) are injective for all x ∈ X. In other words, the image of the sequence f1 , . . . , fr in the ring OX,x is a regular sequence for all x ∈ X. The other types of regularity can be checked stalkwise as well (details omitted). Hence the implications follow from More on Algebra, Lemmas 12.24.2 and 12.24.5.  Definition 26.12.2. Let X be a ringed space. Let J ⊂ OX be a sheaf of ideals. (1) We say J is regular if for every x ∈ Supp(OX /J ) there exists an open neighbourhood x ∈ U ⊂ X and a regular sequence f1 , . . . , fr ∈ OX (U ) such that J |U is generated by f1 , . . . , fr . (2) We say J is Koszul-regular if for every x ∈ Supp(OX /J ) there exists an open neighbourhood x ∈ U ⊂ X and a Koszul-regular sequence f1 , . . . , fr ∈ OX (U ) such that J |U is generated by f1 , . . . , fr . (3) We say J is H1 -regular if for every x ∈ Supp(OX /J ) there exists an open neighbourhood x ∈ U ⊂ X and a H1 -regular sequence f1 , . . . , fr ∈ OX (U ) such that J |U is generated by f1 , . . . , fr . (4) We say J is quasi-regular if for every x ∈ Supp(OX /J ) there exists an open neighbourhood x ∈ U ⊂ X and a quasi-regular sequence f1 , . . . , fr ∈ OX (U ) such that J |U is generated by f1 , . . . , fr . Many properties of this notion immediately follow from the corresponding notions for regular and quasi-regular sequences in rings. Lemma 26.12.3. Let X be a ringed space. Let J be a sheaf of ideals. We have the following implications: J is regular ⇒ J is Koszul-regular ⇒ J is H1 -regular ⇒ J is quasi-regular. Proof. The lemma immediately reduces to Lemma 26.12.1.



Lemma 26.12.4. Let X be a locally ringed space. Let J ⊂ OX be a sheaf of ideals. Then J is quasi-regular if and only if the following conditions are satisfied: (1) J is an OX -module of finite type, (2) J /J 2 is a finite locally free OX /J -module, and (3) the canonical maps SymnOX /J (J /J 2 ) −→ J n /J n+1 are isomorphisms for all n ≥ 0. Proof. It is clear that if U ⊂ X is an open such that J |U is generated by a quasi-regular sequence f1 , . . . , fr ∈ OX (U ) then J |U is of finite type, J |U /J 2 |U is free with basis f1 , . . . , fr , and the maps in (3) are isomorphisms because they are coordinate free formulation of the degree n part of (26.12.0.4). Hence it is clear that being quasi-regular implies conditions (1), (2), and (3).

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Conversely, suppose that (1), (2), and (3) hold. Pick a point x ∈ Supp(OX /J ). Then there exists a neighbourhood U ⊂ X of x such that J |U /J 2 |U is free of rank r over OU /J |U . After possibly shrinking U we may assume there exist f1 , . . . , fr ∈ J (U ) which map to a basis of J |U /J 2 |U as an OU /J |U -module. In particular we see that the images of f1 , . . . , fr in Jx /Jx2 generate. Hence by Nakayama’s lemma (Algebra, Lemma 7.18.1) we see that f1 , . . . , fr generate the stalk Jx . Hence, since J is of finite type, by Modules, Lemma 15.9.4 after shrinking U we may assume 2 that L f1 , . . . , fr generate J . Finally, from (3) and the isomorphism J |U /J |U = OU /J |U fi it is clear that f1 , . . . , fr ∈ OX (U ) is a quasi-regular sequence.  Lemma 26.12.5. Let (X, OX ) be a locally ringed space. Let J ⊂ OX be a sheaf of ideals. Let x ∈ X and f1 , . . . , fr ∈ Jx whose images give a basis for the κ(x)-vector space Jx /mx Jx . (1) If J is quasi-regular, then there exists an open neighbourhood such that f1 , . . . , fr ∈ OX (U ) form a quasi-regular sequence generating J |U . (2) If J is H1 -regular, then there exists an open neighbourhood such that f1 , . . . , fr ∈ OX (U ) form an H1 -regular sequence generating J |U . (3) If J is Koszul-regular, then there exists an open neighbourhood such that f1 , . . . , fr ∈ OX (U ) form an Koszul-regular sequence generating J |U . Proof. First assume that J is quasi-regular. We may choose an open neighbourhood U ⊂ X of x and a quasi-regular sequence g1 , . . . , gs ∈ OX (U ) which generates J |U . Note that this implies that J /J 2 is free of rank s over OU /J |U (see Lemma 26.12.4 and its proof) and hence r = s. We may shrink U and assume f1 , . . . , fr ∈ J (U ). Thus we may write X fi = aij gj for some aij ∈ OX (U ). By assumption the matrix A = (aij ) maps to an invertible matrix over κ(x). Hence, after shrinking U once more, we may assume that (aij ) is invertible. Thus we see that f1 , . . . , fr give a basis for (J /J 2 )|U which proves that f1 , . . . , fr is a quasi-regular sequence over U . Note that in order to prove (2) and (3) we may, because the assumptions of (2) and (3) are stronger than the assumption in (1), already assume that f1 , . . . , fr ∈ J (U ) P and fi = aij gj with (aij ) invertible as above, where now g1 , . . . , gr is a H1 -regular or Koszul-regular sequence. Since the Koszul complex on f1 , . . . , fr is isomorphic to the Koszul complex on g1 , . . . , gr via the matrix (aij ) (see More on Algebra, Lemma 12.23.4) we conclude that f1 , . . . , fr is H1 -regular or Koszul-regular as desired.  Lemma 26.12.6. Any regular, Koszul-regular, H1 -regular, or quasi-regular sheaf of ideals on a scheme is a finite type quasi-coherent sheaf of ideals. Proof. This follows as such a sheaf of ideals is locally generated by finitely many sections. And any sheaf of ideals locally generated by sections on a scheme is quasi-coherent, see Schemes, Lemma 21.10.1.  Lemma 26.12.7. Let X be a scheme. Let J be a sheaf of ideals. Then J is regular (resp. Koszul-regular, H1 -regular, quasi-regular) if and only if for every x ∈ Supp(OX /J ) there exists an affine open neighbourhood x ∈ U ⊂ X, U = Spec(A) such that J |U = Ie and such that I is generated by a regular (resp. Koszul-regular, H1 -regular, quasi-regular) sequence f1 , . . . , fr ∈ A.

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Proof. By assumption we can find an open neighbourhood U of x over which J is generated by a regular (resp. Koszul-regular, H1 -regular, quasi-regular) sequence f1 , . . . , fr ∈ OX (U ). After shrinking U we may assume that U is affine, say U = Spec(A). Since J is quasi-coherent by Lemma 26.12.6 we see that J |U = Ie for some ideal I ⊂ A. Now we can use the fact that e : ModA −→ QCoh(U ) is an equivalence of categories which preserves exactness. For example the fact that the functions fi generate J means that the fi , seen as elements of A generate I. The fact that (26.12.0.2) is injective (resp. (26.12.0.3) is exact, (26.12.0.3) is exact in degree 1, (26.12.0.4) is an isomorphism) implies the correponding property of the map A/(f1 , . . . , fi−1 ) → L A/(f1 , . . . , fi−1 ) (resp. the complex K• (A, f1 , . . . , fr ), the map A/I[T1 , . . . , Tr ] → I n /I n+1 ). Thus f1 , . . . , fr ∈ A is a regular (resp. Koszul-regular, H1 -regular, quasi-regular) sequence of the ring A.  Lemma 26.12.8. Let X be a locally Noetherian scheme. Let J ⊂ OX be a quasicoherent sheaf of ideals. Let x be a point of the support of OX /J . The following are equivalent (1) Jx is generated by a regular sequence in OX,x , (2) Jx is generated by a Koszul-regular sequence in OX,x , (3) Jx is generated by an H1 -regular sequence in OX,x , (4) Jx is generated by a quasi-regular sequence in OX,x , (5) there exists an affine neighbourhood U = Spec(A) of x such that J |U = Ie and I is generated by a regular sequence in A, and (6) there exists an affine neighbourhood U = Spec(A) of x such that J |U = Ie and I is generated by a Koszul-regular sequence in A, and (7) there exists an affine neighbourhood U = Spec(A) of x such that J |U = Ie and I is generated by an H1 -regular sequence in A, and (8) there exists an affine neighbourhood U = Spec(A) of x such that J |U = Ie and I is generated by a quasi-regular sequence in A, (9) there exists a neighbourhood U of x such that J |U is regular, and (10) there exists a neighbourhood U of x such that J |U is Koszul-regular, and (11) there exists a neighbourhood U of x such that J |U is H1 -regular, and (12) there exists a neighbourhood U of x such that J |U is quasi-regular. In particular, on a locally Noetherian scheme the notions of regular, Koszul-regular, H1 -regular, or quasi-regular ideal sheaf all agree. Proof. It follows from Lemma 26.12.7 that (5) ⇔ (9), (6) ⇔ (10), (7) ⇔ (11), and (8) ⇔ (12). It is clear that (5) ⇒ (1), (6) ⇒ (2), (7) ⇒ (3), and (8) ⇒ (4). We have (1) ⇒ (5) by Algebra, Lemma 7.66.8. We have (9) ⇒ (10) ⇒ (11) ⇒ (12) by Lemma 26.12.3. Finally, (4) ⇒ (1) by Algebra, Lemma 7.67.6. Now all 12 statements are equivalent.  26.13. Regular immersions Let i : Z → X be an immersion of schemes. By definition this means there exists an open subscheme U ⊂ X such that Z is identified with a closed subscheme of U . Let I ⊂ OU be the corresponding quasi-coherent sheaf of ideals. Suppose U 0 ⊂ X is a second such open subscheme, and denote I 0 ⊂ OU 0 the corresponding quasicoherent sheaf of ideals. Then I|U ∩U 0 = I 0 |U ∩U 0 . Moreover, the support of OU /I

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is Z which is contained in U ∩U 0 and is also the support of OU 0 /I 0 . Hence it follows from Definition 26.12.2 that I is a regular ideal if and only if I 0 is a regular ideal. Similarly for being Koszul-regular, H1 -regular, or quasi-regular. Definition 26.13.1. Let i : Z → X be an immersion of schemes. Choose an open subscheme U ⊂ X such that i identifies Z with a closed subscheme of U and denote I ⊂ OU the corresponding quasi-coherent sheaf of ideals. (1) We say i is a regular immersion if I is regular. (2) We say i is a Koszul-regular immersion if I is Koszul-regular. (3) We say i is a H1 -regular immersion if I is H1 -regular. (4) We say i is a quasi-regular immersion if I is quasi-regular. The discussion above shows that this is independent of the choice of U . The conditions are listed in decreasing order of strength, see Lemma 26.13.2. A Koszul-regular closed immersion is smooth locally a regular immersion, see Lemma 26.13.11. In the locally Noetherian case all four notions agree, see Lemma 26.12.8. Lemma 26.13.2. Let i : Z → X be an immersion of schemes. We have the following implications: i is regular ⇒ i is Koszul-regular ⇒ i is H1 -regular ⇒ i is quasi-regular. Proof. The lemma immediately reduces to Lemma 26.12.3.



Lemma 26.13.3. Let i : Z → X be an immersion of schemes. Assume X is locally Noetherian. Then i is regular ⇔ i is Koszul-regular ⇔ i is H1 -regular ⇔ i is quasi-regular. Proof. Follows immediately from Lemma 26.13.2 and Lemma 26.12.8.



Lemma 26.13.4. Let i : Z → X be a regular (resp. Koszul-regular, H1 -regular, quasi-regular) immersion. Let X 0 → X be a flat morphism. Then the base change i0 : Z ×X X 0 → X 0 is a regular (resp. Koszul-regular, H1 -regular, quasi-regular) immersion. Proof. Via Lemma 26.12.7 this translates into the algebraic statements in Algebra, Lemmas 7.66.7 and 7.67.3 and More on Algebra, Lemma 12.24.4.  Lemma 26.13.5. Let i : Z → X be an immersion of schemes. Then i is a quasiregular immersion if and only if the following conditions are satisfied (1) i is locally of finite presentation, (2) the conormal sheaf CZ/X is finite locally free, and (3) the map (26.11.1.2) is an isomorphism. Proof. An open immersion is locally of finite presentation. Hence we may replace X by an open subscheme U ⊂ X such that i identifies Z with a closed subscheme of U , i.e., we may assume that i is a closed immersion. Let I ⊂ OX be the corresponding quasi-coherent sheaf of ideals. Recall, see Morphisms, Lemma 24.22.7 that I is of finite type if and only if i is locally of finite presentation. Hence the equivalence follows from Lemma 26.12.4 and unwinding the definitions.  Lemma 26.13.6. Let Z → Y → X be immersions of schemes. Assume that Z → Y is H1 -regular. Then the canonical sequence of Morphisms, Lemma 24.33.5 0 → i∗ CY /X → CZ/X → CZ/Y → 0 is exact and locally split.

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Proof. Since CZ/Y is finite locally free (see Lemma 26.13.5 and Lemma 26.12.3) it suffices to prove that the sequence is exact. By what was proven in Morphisms, Lemma 24.33.5 it suffices to show that the first map is injective. Working affine locally this reduces to the following question: Suppose that we have a ring A and ideals I ⊂ J ⊂ A. Assume that J/I ⊂ A/I is generated by an H1 -regular sequence. Does this imply that I/I 2 ⊗A A/J → J/J 2 is injective? Note that I/I 2 ⊗A A/J = I/IJ. Hence we are trying to prove that I ∩ J 2 = IJ. This is the result of More on Algebra, Lemma 12.24.7.  A composition of quasi-regular immersions may not be quasi-regular, see Algebra, Remark 7.67.8. The other types of regular immersions are preserved under composition. Lemma (1) (2) (3) (4)

26.13.7. Let i : Z → Y and j : Y → X be immersions of schemes. If i and j are regular immersions, so is j ◦ i. If i and j are Koszul-regular immersions, so is j ◦ i. If i and j are H1 -regular immersions, so is j ◦ i. If i is an H1 -regular immersion and j is a quasi-regular immersion, then j ◦ i is a quasi-regular immersion.

Proof. The algebraic version of (1) is Algebra, Lemma 7.66.9. The algebraic version of (2) is More on Algebra, Lemma 12.24.11. The algebraic version of (3) is More on Algebra, Lemma 12.24.9. The algebraic version of (4) is More on Algebra, Lemma 12.24.8.  Lemma 26.13.8. Let i : Z → Y and j : Y → X be immersions of schemes. Assume that the sequence 0 → i∗ CY /X → CZ/X → CZ/Y → 0 of Morphisms, Lemma 24.33.5 is exact and locally split. (1) If j ◦ i is a quasi-regular immersion, so is i. (2) If j ◦ i is a H1 -regular immersion, so is i. (3) If both j and j ◦ i are Koszul-regular immersions, so is i. Proof. After shrinking Y and X we may assume that i and j are closed immersions. Denote I ⊂ OX the ideal sheaf of Y and J ⊂ OX the ideal sheaf of Z. The conormal sequence is 0 → I/IJ → J /J 2 → J /(I + J 2 ) → 0. Let z ∈ Z and set y = i(z), x = j(y) = j(i(z)). Choose f1 , . . . , fn ∈ Ix which map to a basis of Ix /mz Ix . Extend this to f1 , . . . , fn , g1 , . . . , gm ∈ Jx which map to a basis of Jx /mz Jx . This is possible as we have assumed that the sequence of conormal sheaves is split in a neighbourhood of z, hence Ix /mx Ix → Jx /mx Jx is injective. Proof of (1). By Lemma 26.12.5 we can find an affine open neighbourhood U of x such that f1 , . . . , fn , g1 , . . . , gm forms a quasi-regular sequence generating J . Hence by Algebra, Lemma 7.67.5 we see that g1 , . . . , gm induces a quasi-regular sequence on Y ∩ U cutting out Z. Proof of (2). Exactly the same as the proof of (1) except using More on Algebra, Lemma 12.24.10. Proof of (3). By Lemma 26.12.5 (applied twice) we can find an affine open neighbourhood U of x such that f1 , . . . , fn forms a Koszul-regular sequence generating I and f1 , . . . , fn , g1 , . . . , gm forms a Koszul-regular sequence generating J . Hence by

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More on Algebra, Lemma 12.24.12 we see that g1 , . . . , gm induces a Koszul-regular sequence on Y ∩ U cutting out Z.  Lemma 26.13.9. Let i : Z → Y and j : Y → X be immersions of schemes. Pick z ∈ Z and denote y ∈ Y , x ∈ X the corresponding points. Assume X is locally Noetherian. The following are equivalent (1) i is a regular immersion in a neighbourhood of z and j is a regular immersion in a neighbourhood of y, (2) i and j ◦ i are regular immersions in a neighbourhood of z, (3) j ◦ i is a regular immersion in a neighbourhood of z and the conormal sequence 0 → i∗ CY /X → CZ/X → CZ/Y → 0 is split exact in a neighbourhood of z. Proof. Since X (and hence Y ) is locally Noetherian all 4 types of regular immersions agree, and moreover we may check whether a morphism is a regular immersion on the level of local rings, see Lemma 26.12.8. The implication (1) ⇒ (2) is Lemma 26.13.7. The implication (2) ⇒ (3) is Lemma 26.13.6. Thus it suffices to prove that (3) implies (1). Assume (3). Set A = OX,x . Denote I ⊂ A the kernel of the surjective map OX,x → OY,y and denote J ⊂ A the kernel of the surjective map OX,x → OZ,z . Note that any mimimal sequence of elements generating J in A is a quasi-regular hence regular sequence, see Lemma 26.12.5. By assumption the conormal sequence 0 → I/IJ → J/J 2 → J/(I + J 2 → 0 is split exact as a sequence of A/J-modules. Hence we can pick a minimal system of generators f1 , . . . , fn , g1 , . . . , gm of J with f1 , . . . , fn ∈ I a minimal system of generators of I. As pointed out above f1 , . . . , fn , g1 , . . . , gm is a regular sequence in A. It follows directly from the definition of a regular sequence that f1 , . . . , fn is a regular sequence in A and g 1 , . . . , g m is a regular sequence in A/I. Thus j is a regular immersion at y and i is a regular immersion at z.  Remark 26.13.10. In the situation of Lemma 26.13.9 parts (1), (2), (3) are not equivalent to “j ◦ i and j are regular immersions at z and y”. An example is X = A1k = Spec(k[x]), Y = Spec(k[x]/(x2 )) and Z = Spec(k[x]/(x)). Lemma 26.13.11. Let i : Z → X be a Koszul regular closed immersion. Then there exists a surjective smooth morphism X 0 → X such that the base change i0 : Z ×X X 0 → X 0 of i is a regular immersion. Proof. We may assume that X is affine and the ideal of Z generated by a Koszulregular sequence by replacing X by the members of a suitable affine open covering (affine opens as in Lemma 26.12.7). The affine case is More on Algebra, Lemma 12.24.16.  26.14. Relative regular immersions In this section we consider the base change property for regular immersions. The following lemma does not hold for regular immersions or for Koszul immersions, see Examples, Lemma 66.6.2.

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Lemma 26.14.1. Let f : X → S be a morphism of schemes. Let i : Z ⊂ X be an immersion. Assume (1) i is an H1 -regular (resp. quasi-regular) immersion, and (2) Z → S is a flat morphism. Then for every morphism of schemes g : S 0 → S the base change Z 0 = S 0 ×S Z → X 0 = S 0 ×S X is an H1 -regular (resp. quasi-regular) immersion. Proof. Unwinding the definitions and using Lemma 26.12.7 we translate this into algebra as follows. Let A → B be a ring map and f1 , . . . , fr ∈ B. Assume B/(f1 , . . . , fr )B is flat over A. Consider a ring map A → A0 . Set B 0 = B ⊗A A0 and J 0 = JB 0 . Case I: f1 , . . . , fr is quasi-regular. Set J = (f1 , . . . , fr ). By assumption J n /J n+1 is isomorphic to a direct sum of copies of B/J hence flat over A. By induction and Algebra, Lemma 7.36.12 we conclude that B/J n is flat over A. The ideal (J 0 )n is equal to J n ⊗A A0 , see Algebra, Lemma 7.36.11. Hence (J 0 )n /(J 0 )n+1 = J n /J n+1 ⊗A A0 which clearly implies that f1 , . . . , fr is a quasi-regular sequence in B0. Case II: f1 , . . . , fr is H1 -regular. By More on Algebra, Lemma 12.24.14 the vanishing of the Koszul homology group H1 (K• (B, f1 , . . . , fr )) implies the vanshing of  H1 (K• (B 0 , f10 , . . . , fr0 )) and we win. This lemma is the motivation for the following definition. Definition 26.14.2. Let f : X → S be a morphism of schemes. Let i : Z → X be an immersion. (1) We say i is a relative quasi-regular immersion if Z → S is flat and i is a quasi-regular immersion. (2) We say i is a relative H1 -regular immersion if Z → S is flat and i is an H1 -regular immersion. We warn the reader that this may be nonstandard notation. Lemma 26.14.1 guarantees that relative quasi-regular (resp. H1 -regular) immersions are preserved under any base change. A relative H1 -regular immersion is a relative quasi-regular immersion, see Lemma 26.13.2. Please take a look at Lemma 26.14.5 (or Lemma 26.14.4) which shows that if Z → X is a relative H1 -regular (or quasi-regular) immersion and the ambient scheme is (flat and) locally of finite presentation over S, then Z → X is actually a regular immersion and the same remains true after any base change. Lemma 26.14.3. Let f : X → S be a morphism of schemes. Let Z → X be a relative quasi-regular immersion. If x ∈ Z and OX,x is Noetherian, then f is flat at x. Proof. Let f1 , . . . , fr ∈ OX,x be a quasi-regular sequence cutting out the ideal of Z at x. By Algebra, Lemma 7.67.6 we know that f1 , . . . , fr is a regular sequence. Hence fr is a nonzerodivisor on OX,x /(f1 , . . . , fr−1 ) such that the quotient is a flat OS,f (x) -module. By Lemma 26.10.3 we conclude that OX,x /(f1 , . . . , fr−1 ) is a flat OS,f (x) -module. Continuing by induction we find that OX,x is a flat OS,s module. 

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Lemma 26.14.4. Let X → S be a morphism of schemes. Let Z → X be an immersion. Assume (1) X → S is flat and locally of finite presentation, (2) Z → X is a relative quasi-regular immersion. Then Z → X is a regular immersion and the same remains true after any base change. Proof. Pick x ∈ Z with image s ∈ S. To prove this it suffices to find an affine neighbourhood of x contained in U such that the result holds on that affine open. Hence we may assume that X is affine and there exist a quasi-regular sequence f1 , . . . , fr ∈ Γ(X, OX ) such that Z = V (f1 , . . . , fr ). By Lemma 26.14.1 and its proof the sequence f1 |Xs , . . . , fr |Xs is a quasi-regular sequence in Γ(Xs , OXs ). Since Xs is Noetherian, this implies, possibly after shrinking X a bit, that f1 |Xs , . . . , fr |Xs is a regular sequence, see Algebra, Lemmas 7.67.6 and 7.66.8. By Lemma 26.10.7 it follows that Z1 = V (f1 ) ⊂ X is a relative effective Cartier divisor, again after possibly shrinking X a bit. Applying the same lemma again, but now to Z2 = V (f1 , f2 ) ⊂ Z1 we see that Z2 ⊂ Z1 is a relative effective Cartier divisor. And so on until on reaches Z = Zn = V (f1 , . . . , fn ). Since being a relative effective Cartier divisor is preserved under arbitrary base change, see Lemma 26.10.1, we also see that the final statement of the lemma holds.  Lemma 26.14.5. Let X → S be a morphism of schemes. Let Z → X be a relative H1 -regular immersion. Assume X → S is locally of finite presentation. Then (1) there exists an open subscheme U ⊂ X such that Z ⊂ U and such that U → S is flat, and (2) Z → X is a regular immersion and the same remains true after any base change. Proof. Pick x ∈ Z. To prove (1) suffices to find an open neighbourhood U ⊂ X of x such that U → S is flat. Hence the lemma reduces to the case that X = Spec(B) and S = Spec(A) are affine and that Z is given by an H1 -regular sequence f1 , . . . , fr ∈ B. By assumption B is a finitely presented A-algebra and B/(f1 , . . . , fr )B is a flat A-algebra. We are going to use absolute Noetherian approximation. Write B = A[x1 , . . . , xn ]/(g1 , . . . , gm ). Assume fi is the image of fi0 ∈ A[x1 , . . . , xn ]. Choose a finite type Z-subalgebra A0 ⊂ A such that all the coefficients of the polynomials f10 , . . . , fr0 , g1 , . . . , gm are in A0 . We set B0 = A0 [x1 , . . . , xn ]/(g1 , . . . , gm ) and we denote fi,0 the image of fi0 in B0 . Then B = B0 ⊗A0 A and B/(f1 , . . . , fr ) = B0 /(f0,1 , . . . , f0,r ) ⊗A0 A. By Algebra, Lemma 7.151.1 we may, after enlarging A0 , assume that B0 /(f0,1 , . . . , f0,r ) is flat over A0 . It may not be the case at this point that the Koszul cohomology group H1 (K• (B0 , f0,1 , . . . , f0,r )) is zero. On the other hand, as B0 is Noetherian, it is a finitely generated B0 -module. Let ξ1 , . . . , ξn ∈ H1 (K• (B0 , f0,1 , . . . , f0,r )) be generators. Let A0 ⊂ A1 ⊂ A be a larger finite type Z-subalgebra of A. Denote f1,i the image of f0,i in B1 = B0 ⊗A0 A1 . By More on Algebra, Lemma 12.24.14 the map H1 (K• (B0 , f0,1 , . . . , f0,r )) ⊗A0 A1 −→ H1 (K• (B1 , f1,1 , . . . , f1,r )) is surjective. Furthermore, it is clear that the colimit (over all choices of A1 as above) of the complexes K• (B1 , f1,1 , . . . , f1,r ) is the complex K• (B, f1 , . . . , fr )

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which is acyclic in degree 1. Hence colimA0 ⊂A1 ⊂A H1 (K• (B1 , f1,1 , . . . , f1,r )) = 0 by Algebra, Lemma 7.8.9. Thus we can find a choice of A1 such that ξ1 , . . . , ξn all map to zero in H1 (K• (B1 , f1,1 , . . . , f1,r )). In other words, the Koszul cohomology group H1 (K• (B1 , f1,1 , . . . , f1,r )) is zero. Consider the morphism of affine schemes X1 → S1 equal to Spec of the ring map A1 → B1 and Z1 = Spec(B1 /(f1,1 , . . . , f1,r )). Since B = B1 ⊗A1 A, i.e., X = X1 ×S1 S, and similarly Z = Z1 ×S S1 , it now suffices to prove (1) for X1 → S1 and the relative H1 -regular immersion Z1 → X1 , see Morphisms, Lemma 24.26.6. Hence we have reduced to the case where X → S is a finite type morphism of Noetherian schemes. In this case we know that X → S is flat at every point of Z by Lemma 26.14.3. Combined with the fact that the flat locus is open in this case, see Algebra, Theorem 7.121.4 we see that (1) holds. Part (2) then follows from an application of Lemma 26.14.4.  If the ambient scheme is flat and locally of finite presentation over the base, then we can characterize a relative quasi-regular immersion in terms of its fibres. Lemma 26.14.6. Let ϕ : X → S be a flat morphism which is locally of finite presentation. Let T ⊂ X be a closed subscheme. Let x ∈ T with image s ∈ S. (1) If Ts ⊂ Xs is a quasi-regular immersion in a neighbourhood of x, then there exists an open U ⊂ X and a relative quasi-regular immersion Z ⊂ U such that Zs = Ts ∩ Us and T ∩ U ⊂ Z. (2) If Ts ⊂ Xs is a quasi-regular immersion in a neighbourhood of x, the morphism T → X is of finite presentation, and T → S is flat at x, then we can choose U and Z as in (1) such that T ∩ U = Z. (3) If Ts ⊂ Xs is a quasi-regular immersion in a neighbourhood of x, and T is cut out by c equations in a neighbourhood of x, where c = dimx (Xs ) − dimx (Ts ), then we can choose U and Z as in (1) such that T ∩ U = Z. In each case Z → U is a regular immersion by Lemma 26.14.4. In particular, if T → S is locally of finite presentation and flat and all fibres Ts ⊂ Xs are quasiregular immersions, then T → X is a relative quasi-regular immersion. Proof. Choose affine open neighbourhoods Spec(A) of s and Spec(B) of x such that ϕ(Spec(B)) ⊂ Spec(A). Let p ⊂ A be the prime ideal corresponding to s. Let q ⊂ B be the prime ideal corresponding to x. Let I ⊂ B be the ideal corresponding to T . By the initial assumption of the lemma we know that A → B is flat and of finite presentation. The assumption in (1) means that, after shrinking Spec(B), we may assume I(B ⊗A κ(p)) is generated by a quasi-regular sequence of elements. After possibly localizing B at some g ∈ B, g 6∈ q we may assume there exist f1 , . . . , fr ∈ I which map to a quasi-regular sequence in B ⊗A κ(p) which generates I(B ⊗A κ(p)). By Algebra, Lemmas 7.67.6 and 7.66.8 we may assume after another localization that f1 , . . . , fr ∈ I form a regular sequence in B ⊗A κ(p). By Lemma 26.10.7 it follows that Z1 = V (f1 ) ⊂ Spec(B) is a relative effective Cartier divisor, again after possibly localizing B. Applying the same lemma again, but now to Z2 = V (f1 , f2 ) ⊂ Z1 we see that Z2 ⊂ Z1 is a relative effective Cartier divisor. And so on until one reaches Z = Zn = V (f1 , . . . , fn ). Then Z → Spec(B) is a regular immersion and Z is flat over S, in particular Z → Spec(B) is a relative quasi-regular immersion over Spec(A). This proves (1).

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To see (2) consider the closed immersion Z → D. The surjective ring map u : OD,x → OZ,x is a map of flat local OS,s -algebras which are essentially of finite presentation, and which becomes an isomorphisms after dividing by ms . Hence it is an isomorphism, see Algebra, Lemma 7.120.4. It follows that Z → D is an isomorphism in a neighbourhood of x, see Algebra, Lemma 7.118.6. To see (3), after possibly shrinking U we may assume that the ideal of Z is generated by a regular sequence f1 , . . . , fr (see our construction of Z above) and the ideal of T is generated by g1 , . . . , gc . We claim that c = r. Namely, dimx (Xs ) = dim(OXs ,x ) + trdegκ(s) (κ(x)), dimx (Ts ) = dim(OTs ,x ) + trdegκ(s) (κ(x)), dim(OXs ,x ) = dim(OTs ,x ) + r the first two equalities by Algebra, Lemma 7.108.3 and the second P by r times applying Algebra, Lemma 7.58.11. As T ⊂ Z we see that fi = bij gj . But the ideals of Z and T cut out the same quasi-regular closed subscheme of Xs in a neighbourhood of x. Hence the matrix (bij ) mod mx is invertible (some details omitted). Hence (bij ) is invertible in an open neighbourhood of x. In other words, T ∩ U = Z after shrinking U . The final statements of the lemma follow immediately from part (2), combined with the fact that Z → S is locally of finite presentation if and only if Z → X is of finite presentation, see Morphisms, Lemmas 24.22.3 and 24.22.11.  The following lemma is an enhancement of Morphisms, Lemma 24.35.20. Lemma 26.14.7. Let f : X → S be a smooth morphism of schemes. Let σ : S → X be a section of f . Then σ is a regular immersion. Proof. By Schemes, Lemma 21.21.11 the morphism σ is an immersion. After replacing X by an open neighbourhood of σ(S) we may assume that σ is a closed immersion. Let T = σ(S) be the corresponding closed subscheme of X. Since T → S is an isomorphism it is flat and of finite presentation. Also a smooth morphism is flat and locally of finite presentation, see Morphisms, Lemmas 24.35.9 and 24.35.8. Thus, according to Lemma 26.14.6, it suffices to show that Ts ⊂ Xs is a quasi-regular closed subscheme. This follows immediately from Morphisms, Lemma 24.35.20 but we can also see it directly as follows. Let k be a field and let A be a smooth k-algebra. Let m ⊂ A be a maximal ideal whose residue field is k. Then m is generated by a quasi-regular sequence, possibly after replacing A by Ag for some g ∈ A, g 6∈ m. In Algebra, Lemma 7.130.3 we proved that Am is a regular local ring, hence mAm is generated by a regular sequence. This does indeed imply that m is generated by a regular sequence (after replacing A by Ag for some g ∈ A, g 6∈ m), see Algebra, Lemma 7.66.8.  The following lemma has a kind of converse, see Lemma 26.14.11. Lemma 26.14.8. Let Y

/X

i j



S



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be a commutative diagram of morphisms of schemes. Assume X → S smooth, and i, j immersions. If j is a regular (resp. Koszul-regular, H1 -regular, quasi-regular) immersion, then so is i. Proof. We can write i as the composition Y → Y ×S X → X By Lemma 26.14.7 the first arrow is a regular immersion. The second arrow is a flat base change of Y → S, hence is a regular (resp. Koszul-regular, H1 -regular, quasi-regular) immersion, see Lemma 26.13.4. We conclude by an application of Lemma 26.13.7.  Lemma 26.14.9. Let Y

i

/X



 S be a commutative diagram of morphisms of schemes. Assume that Y → S is syntomic, X → S smooth, and i an immersion. Then i is a regular immersion. Proof. After replacing X by an open neighbourhood of i(Y ) we may assume that i is a closed immersion. Let T = i(Y ) be the corresponding closed subscheme of X. Since T ∼ = Y the morphism T → S is flat and of finite presentation (Morphisms, Lemmas 24.32.6 and 24.32.7). Also a smooth morphism is flat and locally of finite presentation (Morphisms, Lemmas 24.35.9 and 24.35.8). Thus, according to Lemma 26.14.6, it suffices to show that Ts ⊂ Xs is a quasi-regular closed subscheme. As Xs is locally of finite type over a field, it is Noetherian (Morphisms, Lemma 24.16.6). Thus we can check that Ts ⊂ Xs is a quasi-regular immersion at points, see Lemma 26.12.8. Take t ∈ Ts . By Morphisms, Lemma 24.32.9 the local ring OTs ,t is a local complete intersection over κ(s). The local ring OXs ,t is regular, see Algebra, Lemma 7.130.3. By Algebra, Lemma 7.125.7 we see that the kernel of the surjection OXs ,t → OTs ,t is generated by a regular sequence, which is what we had to show.  Lemma 26.14.10. Let Y

i

/X



 S be a commutative diagram of morphisms of schemes. Assume that Y → S is smooth, X → S smooth, and i an immersion. Then i is a regular immersion. Proof. This is a special case of Lemma 26.14.9 because a smooth morphism is syntomic, see Morphisms, Lemma 24.35.7.  Lemma 26.14.11. Let Y

i



/X

 S be a commutative diagram of morphisms of schemes. Assume X → S smooth, and i, j immersions. If i is a Koszul-regular (resp. H1 -regular, quasi-regular) immersion, then so is j. j

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Proof. Let y ∈ Y be any point. Set x = i(y) and set s = j(y). It suffices to prove the result after replacing X, S by open neighbourhoods U, V of x, s and Y by an open neighbourhood of y in i−1 (U ) ∩ j −1 (V ). Hence we may assume that Y , X and S are affine. In this case we can choose a closed immersion h : X → AnS over S for some n. Note that h is a regular immersion by Lemma 26.14.10. Hence h ◦ i is a Koszul-regular (resp. H1 -regular, quasi-regular) immersion, see Lemmas 26.13.7 and 26.13.2. In this way we reduce to the case X = AnS and S affine. After replacing S by an affine open V and replacing Y by j −1 (V ) we may assume that i is a closed immersion and S affine. Write S = Spec(A). Then j : Y → S defines an isomorphism of Y to the closed subscheme Spec(A/I) for some ideal I ⊂ A. The map i : Y = Spec(A/I) → AnS = Spec(A[x1 , . . . , xn ]) corresponds to an A-algebra homomorphism i] : A[x1 , . . . , xn ] → A/I. Choose ai ∈ A which map to i] (xi ) in A/I. Observe that the ideal of the closed immersion i is J = (x1 − a1 , . . . , xn − an ) + IA[x1 , . . . , xn ]. Set K = (x1 − a1 , . . . , xn − an ). We claim the sequence 0 → K/KJ → J/J 2 → J/(K + J 2 ) → 0 is split exact. To see this note that K/K 2 is free with basis xi − ai over the ring A[x1 , . . . , xn ]/K ∼ = A. Hence K/KJ is free with the same basis over the ring A[x1 , . . . , xn ]/J ∼ = A/I. On the other hand, taking derivatives gives a map dA[x1 ,...,xn ]/A : J/J 2 −→ ΩA[x1 ,...,xn ]/A ⊗A[x1 ,...,xn ] A[x1 , . . . , xn ]/J which maps the generators xi − ai to the basis elements dxi of the free module on the right. The claim follows. Moreover, note that x1 − a1 , . . . , xn − an is a regular sequence in A[x1 , . . . , xn ] with quotient ring A[x1 , . . . , xn ]/(x1 − a1 , . . . , xn − an ) ∼ = A. Thus we have a factorization Y → V (x1 − a1 , . . . , xn − an ) → AnS of our closed immersion i where the composition is Koszul-regular (resp. H1 -regular, quasi-regular), the second arrow is a regular immersion, and the associated conormal sequence is split. Now the result follows from Lemma 26.13.8.  26.15. Meromorphic functions and sections See [Kle79] for some possible pitfalls1. Let (X, OX ) be a locally ringed space. For any open U ⊂ X we have defined the set S(U ) ⊂ OX (U ) of regular sections of OX over U , see Definition 26.9.16. The restriction of a regular section to a smaller open is regular. Hence S : U 7→ S(U ) is a subsheaf (of sets) of OX . We sometimes denote S = SX if we want to indicate the dependence on X. Moreover, S(U ) is a multiplicative subset of the ring OX (U ) for each U . Hence we may consider the presheaf of rings U 7−→ S(U )−1 OX (U ), see Modules, Lemma 15.22.1. Definition 26.15.1. Let (X, OX ) be a locally ringed space. The sheaf of meromorphic functions on X is the sheaf KX associated to the presheaf displayed above. A meromorphic function on X is a global section of KX . 1Danger, Will Robinson!

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Since each element of each S(U ) is a nonzerodivisor on OX (U ) we see that the natural map of sheaves of rings OX → KX is injective. Example 26.15.2. Let A = C[x, {yP α }α∈C ]/((x − α)yα , yα yβ ). Any element of A can be written uniquely as f (x) + λα yα with f (x) ∈ C[x] and λα ∈ C. Let X = Spec(A). In this case OX = KX , since on any affine open D(f ) the ring Af any nonzerodivisor is a unit (proof omitted). Definition 26.15.3. Let f : (X, OX ) → (Y, OY ) be a morphism of locally ringed spaces. We say that pulbacks of meromorphic functions are defined for f if for every pair of open U ⊂ X, V ⊂ Y such that f (U ) ⊂ V , and any section s ∈ Γ(V, SY ) the pullback f ] (s) ∈ Γ(U, OX ) is an element of Γ(U, SX ). In this case there is an induced map f ] : f −1 KY → KX , in other words we obtain a commutative diagram of morphisms of ringed spaces (X, KX ) f

 (Y, KY )

/ (X, OX ) f

 / (Y, OX )

We sometimes denote f ∗ (s) = f ] (s) for a section s ∈ Γ(Y, KY ). Lemma 26.15.4. Let f : X → Y be a morphism of schemes. In each of the following cases pullbacks of meromorphic sections are defined. (1) X, Y are integral and f is dominant, (2) X is integral and the generic point of X maps to a generic point of an irreducible component of Y , (3) X is reduced and every generic point of every irreducible component of X maps to the generic point of an irreducible component of Y , (4) X is locally Noetherian, and any associated point of X maps to a generic point of an irreducible component of Y , and (5) X is locally Noetherian, has no embedded points and any generic point of an irreducible component of X maps to the generic point of an irreducible component of Y . Proof. Omitted. Hint: Similar to the proof of Lemma 26.9.12, using the following fact (on Y ): if an element x ∈ R maps to a nonzerodivisor in Rp for a minimal prime p of R, then x 6∈ p. See Algebra, Lemma 7.24.3.  Let (X, OX ) be a locally ringed space. Let F be a sheaf of OX -modules. Consider the presheaf U 7→ S(U )−1 F(U ). Its sheafification is the sheaf F ⊗OX KX , see Modules, Lemma 15.22.2. Definition 26.15.5. Let X be a locally ringed space. Let F be a sheaf of OX modules. (1) We denote KX (F) the sheaf of KX -modules which is the sheafification of the presheaf U 7→ S(U )−1 F(U ). Equivalently KX (F) = F ⊗OX KX (see above). (2) A meromorphic section of F is a global section of KX (F). In particular we have KX (F)x = Fx ⊗OX,x KX,x = Sx−1 Fx

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for any point x ∈ X. However, one has to be careful since it may not be the case that Sx is the set of nonzerodivisors in the local ring OX,x . Namely, there is always an injective map KX,x −→ Q(OX,x ) to the total quotient ring. It is also surjective if and only if Sx is the set of nonzerodivisors in OX,x . Lemma 26.15.6. Let X be a locally Noetherian scheme. (1) For any x ∈ X we have Sx ⊂ OX,x is the set of nonzerodivisors, and KX,x is the total quotient ring of OX,x . (2) For any affine open Spec(A) = U ⊂ X we have that KX (U ) equals the total quotient ring of A. Proof. Let A be a Noetherian ring. Let p ⊂ A be a prime ideal. Let f, g ∈ A, g 6∈ p. Let I = {x ∈ A | f x = 0}. Suppose f /g is a nonzerodivisor in Ap . Then we see that Ip = 0 by exactness of localization. Since A is Noetherian we see that I is finitely generated and hence that g 0 I = 0 for some g 0 ∈ A, g 0 6∈ p. Hence f is a nonzerodivisor in Ag0 , i.e., in a Zariski open neighbourhood of p. This proves (1). Let f ∈ Γ(X, KX,x ) be a meromorphic function on X = Spec(A). Set I = {x ∈ A | xf ∈ A}. For every prime p ⊂ A we can write the image of f in the stalk at p as a/b, a, b ∈ Ap with b ∈ Ap not a zerodivisor. Hence, clearing denominators, we find there exists an element x ∈ I such that x maps to a nonzerodivisor on Ap . Let Ass(A) = {q1 , . . . , qt } be the associated primes of A. By looking at IAqi and using Algebra, Lemma 7.61.14 the above says that S I 6⊂ qi for each i. By Algebra, Lemma 7.14.3 there exists an element x ∈ I, x 6∈ qi . By Algebra, Lemma 7.61.9 we see that x is not a zerodivisor on A. Hence f = (xf )/x is an element of the total ring of fractions of A. This proves (2).  Lemma 26.15.7. Let X be a scheme. Assume X is reduced and any quasi-compact open U ⊂ X has a finite number of irreducible components. (1) The sheaf KX is a quasi-coherent sheaf of OX -algebras. (2) For any x ∈ X we have Sx ⊂ OX,x is the set of nonzerodivisors. In particular KX,x is the total quotient ring of OX,x . (3) For any affine open Spec(A) = U ⊂ X we have that KX (U ) equals the total quotient ring of A. Proof. Let X be as in the lemma. Let X (0) ⊂ X be the set of generic points of irreducible components of X. Let a f :Y = Spec(κ(η)) −→ X (0) η∈X

be the inclusion of the generic points into X using the canonical maps of Schemes, Section 21.13. (This morphism was used in Morphisms, Definition 24.48.12 to define the normalization of X.) We claim that KX = f∗ OY . First note that KY = OY as Y is a disjoint union of spectra of field. Next, note that pullbacks of meromorphic functions are defined for f , by Lemma 26.15.4. This gives a map KX −→ f∗ OY . Let Spec(A) = U ⊂ X be an affine open. Then A is a Q reduced ring Q with finitely many minimal primes q1 , . . . , qt . Then we have Q(A) = Aqi = κ(qi ) by Algebra, Lemmas 7.23.2 and 7.24.3. In other words, already the value of the presheaf

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U 7→ S(U )−1 OX (U ) agrees with f∗ OY (U ) on our affine open U . Hence the displayed map is an isomorphism. Now we are ready to prove (1), (2) and (3). The morphism f is quasi-compact by our assumption that the set of irreducible components of X is locally finite. Hence f is quasi-compact and quasi-separated (as Y is separated). By Schemes, Lemma 21.24.1 f∗ OY is quasi-coherent. This proves (1). Let x ∈ X. Then Y (f∗ OY )x = κ(η) (0) η∈X

, x∈{η}

On the other hand, OX,x is reduced and has finitely minimal primes qi correspondHence by Algebra, Lemmas ing exactly to those η ∈ X (0) such that x ∈ {η}κ(η). Q 7.23.2 and 7.24.3 again we see that Q(OX,x ) = κ(qi ) is the same as (f∗ OY )x . This proves (2). Part (3) we saw during the course of the proof that KX = f∗ OY .  Lemma 26.15.8. Let X be a scheme. Assume X is reduced and any quasi-compact open U ⊂ X has a finite number of irreducible components. Then the normalization morphism ν : X ν → X is the morphism SpecX (O0 ) −→ X where O0 ⊂ KX is the integral closure of OX in the sheaf of meromorphic functions. Proof. Compare the definition of the normalization morphism ν : X ν → X (see Morphisms, Definition 24.48.12) with the result KX = f∗ OY obtained in the proof of Lemma 26.15.7 above.  Lemma 26.15.9. Let X be an integral scheme with generic point η. We have (1) the sheaf of meromorphic functions is isomorphic to the constant sheaf with value the function field (see Morphisms, Definition 24.10.5) of X. (2) for any quasi-coherent sheaf F on X the sheaf KX (F) is isomorphic to the constant sheaf with value Fη . Proof. Omitted.



Definition 26.15.10. Let X be a locally ringed space. Let L be an invertible OX -module. A meromorphic section s of L is said to be regular if the induced map KX → KX (L) is injective. (In other words, this means that s is a regular section of the invertible KX -module KX (L). See Definition 26.9.16.) First we spell out when (regular) meromorphic sections can be pulled back. After that we discuss the existence of regular meromorphic sections and consequences. Lemma 26.15.11. Let f : X → Y be a morphism of locally ringed spaces. Assume that pullbacks of meromorphic functions are defined for f (see Definition 26.15.3). (1) Let F be a sheaf of OY -modules. There is a canonical pullback map f ∗ : Γ(Y, KY (F)) → Γ(X, KX (f ∗ F)) for meromorphic sections of F. (2) Let L be an invertible OX -module. A regular meromorphic section s of L pulls back to a regular meromorphic section f ∗ s of f ∗ L. Proof. Omitted.



In some cases we can show regular meromorphic sections exist. Lemma 26.15.12. Let X be a scheme. Let L be an invertible OX -module. In each of the following cases L has a regular meromorphic section:

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(1) X is integral, (2) X is reduced and any quasi-compact open has a finite number of irreducible components, and (3) X is locally Noetherian and has no embedded points. Proof. In case (1) we have seen in Lemma 26.15.9 that KX (L) is a constant sheaf with value Lη , and hence the result is clear. Suppose X is a scheme. Let G ⊂ X be the set of generic points of irreducible components of X. For each η ∈ G denote jη : η → X the canonical morphism of η = Spec(κ(η)) into X (see Schemes, Lemma 21.13.3). Consider the sheaf Y GX (L) = jη,∗ (Lη ). η∈G

There is a canonical map ϕ : KX (L) −→ GX (L) coming from the maps KX (L)η → Lη and adjunction (see Sheaves, Lemma 6.27.3). We claim that in cases (2) and (3) the map ϕ is an isomorphism for any invertible sheaf L. Before proving this let us show that cases (2) and (3) follow from this. Namely, we can choose sη ∈ Lη which generate Lη , i.e., such that Lη = OX,η sη . SinceQthe claim applied to OX gives KX = GX (OX ) it is clear that the global section s = η∈G sη is regular as desired. To prove that ϕ is an isomorphism we may work locally on X. For example it suffices to show that sections of KX (L) and GX (L) agree over small affine opens U . Say U = Spec(A) and L|U ∼ = OU . By Lemmas 26.15.6 and 26.15.7 we see that Γ(U, KX ) = Q Q(A) is the total ring of fractions of A. On the other hand, Γ(U, GX (OX )) = q⊂A minimal Aq . In both cases we see that the set of minimal primes of A is finite, say q1 , . . . , qt , and that the set of zerodivisors of A is equal to q1 ∪ . . . ∪ qt (see Algebra, Lemma 7.61.9). Hence the result follows from Algebra, Lemma 7.23.2.  Lemma 26.15.13. Let X be a scheme. Let L be an invertible OX -module. Let s be a regular meromorphic section of L. Let us denote I ⊂ OX the sheaf of ideals defined by the rule I(V ) = {f ∈ OZ (V ) | f s ∈ L(V )}. The formula makes sense since L(V ) ⊂ KX (L)(V ). Then I is a quasi-coherent sheaf of ideals and we have injective maps 1 : I −→ OX ,

s : I −→ L

whose cokernels are supported on closed nowhere dense subsets of X. Proof. The question is local on X. Hence we may assume that X = Spec(A), and L = OX . After shrinking furhter we may assume that s = x/y with a, b ∈ A both nonzerodivisors in A. Set I = {x ∈ A | x(a/b) ∈ A}. To show that I is quasi-coherent we have to show that If = {x ∈ Af | x(a/b) ∈ Af } for every f ∈ A. If c/f n ∈ Af , (c/f n )(a/b) ∈ Af , then we see that f m c(a/b) ∈ A for some m, hence c/f n ∈ If . Conversely it is easy to see that If is contained in {x ∈ Af | x(a/b) ∈ Af }. This proves quasi-coherence. Let us prove the final statement. It is clear that (b) ⊂ I. Hence V (I) ⊂ V (b) is a nowhere dense subset as b is a nonzerodivisor. Thus the cokernel of 1 is supported

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in a nowhere dense closed set. The same argument works for the cokerenel of s since s(b) = (a) ⊂ sI ⊂ A.  Definition 26.15.14. Let X be a scheme. Let L be an invertible OX -module. Let s be a regular meromorphic section of L. The sheaf of ideals I constructed in Lemma 26.15.13 is called the ideal sheaf of denominators of s. Here is a lemma which will be used later. Lemma 26.15.15. Suppose given (1) X a locally Noetherian scheme, (2) L an invertible OX -module, (3) s a regular meromorphic section of L, and (4) F coherent on X without embedded associated points and Supp(F) = X. Let I ⊂ OX be the ideal of denominators of s. Let T ⊂ X be the union of the supports of OX /I and L/s(I) which is a nowhere dense closed subset T ⊂ X according to Lemma 26.15.13. Then there are canonical injective maps 1 : IF → F,

s : IF → F ⊗OX L

whose cokernels are supported on T . Proof. Reduce to the affine case with L ∼ = OX , and s = a/b with a, b ∈ A both f. Let I = {x ∈ nonzerodivisors. Proof of reduction step omitted. Write F = M A | x(a/b) ∈ A} so that I = Ie (see proof of Lemma 26.15.13). Note that T = V (I) ∪ V ((a/b)I). For any A-module M consider the map 1 : IM → M ; this is the map that gives rise to the map 1 of the lemma. Consider on the other hand the map σ : IM → Mb , x 7→ ax/b. Since b is not a zerodivisor in A, and since M has support Spec(A) and no embedded primes we see that b is a nonzerodivisor on M also. Hence M ⊂ Mb . By definition of I we have σ(IM ) ⊂ M as submodules of Mb . Hence we get an A-module map s : IM → M (namely the unique map such that s(z)/1 = σ(z) in Mb for all z ∈ IM ). It is injective because a is a nonzerodivisor also (on both A and M ). It is clear that M/IM is annihilated by I and that M/s(IM ) is annihilated by (a/b)I. Thus the lemma follows.  26.16. Relative Proj Some results on relative Proj. First some very basic results. Recall that a relative Proj is always separated over the base, see Constructions, Lemma 22.16.9. Lemma 26.16.1. Let S be a scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. If one of the following holds (1) A is of finite type as a sheaf of A0 -algebras, (2) A is generated by A1 as an A0 -algebra and A1 is a finite type A0 -module, (3) there exists a finite type quasi-coherent A0 -submodule F ⊂ A+ such that A+ /FA is a locally nilpotent sheaf of ideals of A/FA, then p is quasi-compact. Proof. The question is local on the base, see Schemes, Lemma 21.19.2. Thus we may assume S is affine. Say S = Spec(R) and A corresponds to the graded Ralgebra A. Then X = Proj(A), see Constructions, Section 22.15. In case (1) we may after possibly localizing more assume that A is generated by homogeneous elements f1 , . . . , fn ∈ A+ over A0 . Then A+ = (f1 , . . . , fn ) by Algebra, Lemma

26.16. RELATIVE PROJ

1563

f 7.56.1. In P case (3) we see that P F = M for some finite type A0 -module M ⊂ A+ . Say M = A0 fi . Say fi = fi,j is the p decomposition into homogeneous pieces. The condition in (2) signifies that A+ ⊂ (fi,j ). Thus in both cases we conclude that Proj(A) is quasi-compact by Constructions, Lemma 22.8.9. Finally, (2) follows from (1).  Lemma 26.16.2. Let S be a scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. If A is of finite type as a sheaf of OS -algebras, then p is of finite type. Proof. The assumption implies that p is quasi-compact, see Lemma 26.16.1. Hence it suffices to show that p is locally of finite type. Thus the question is local on the base and target, see Morphisms, Lemma 24.16.2. Say S = Spec(R) and A corresponds to the graded R-algebra A. After further localizing on S we may assume that A is a finite type R-algebra. The scheme X is constructed out of glueing the spectra of the rings A(f ) for f ∈ A+ homogeneous. Each of these is of finite type over R by Algebra, Lemma 7.54.9. Thus Proj(A) is of finite type over R.  Lemma 26.16.3. Let S be a scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. If OS → A0 is an integral algebra map2 and A is of finite type as an A0 -algebra, then p is universally closed. Proof. The question is local on the base. Thus we may assume that X = Spec(R) is affine. Let A be the quasi-coherent OX -algebra associated to the graded Ralgebra A. The assumption is that R → A0 is integral and A is of finite type over A0 . Write X → Spec(R) as the composition X → Spec(A0 ) → Spec(R). Since R → A0 is an integral ring map, we see that Spec(A0 ) → Spec(R) is universally closed, see Morphisms, Lemma 24.44.7. The quasi-compact (see Constructions, Lemma 22.8.9) morphism Proj(A) → Proj(A0 ) satisfies the existence part of the valuative criterion by Constructions, Lemma 22.8.11 and hence it is universally closed by Schemes, Proposition 21.20.6. Thus X → Spec(R) is universally closed as a composition of universally closed morphisms.  Lemma 26.16.4. Let S be a scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. The following conditions are equivalent (1) A0 is a finite type OS -module and A is of finite type as an A0 -algebra, (2) A0 is a finite type OS -module and A is of finite type as an OS -algebra If these conditions hold, then p is locally projective and in particular proper. Proof. Assume that A0 is a finite type OS -module. Choose an affine open U = Spec(R) ⊂ X such that A corresponds to a graded R-algebra A with A0 a finite R-module. Condition (1) means that (after possibly localizing further on S) that A is a finite type A0 -algebra and condition (2) means that (after possibly localizing 2In other words, the integral closure of O in A , see Morphisms, Definition 24.48.2, equals 0 S

A0 .

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further on S) that A is a finite type R-algebra. Thus these conditions imply each other by Algebra, Lemma 7.6.2. A locally projective morphism is proper, see Morphisms, Lemma 24.43.5. Thus we may now assume that S = Spec(R) and X = Proj(A) and that A0 is finite over R and A of finite type over R. We will show that X = Proj(A) → Spec(R) is projective. We urge the reader to prove this for themselves, by directly constructing a closed immersion of X into a projective space over R, instead of reading the argument we give below. By Lemma 26.16.2 we see that X is of finite type over Spec(R). Constructions, Lemma 22.10.6 tells us that OX (d) is ample on X for some d ≥ 1 (see Properties, Section 23.24). Hence X → Spec(R) is quasi-projective (by Morphisms, Definition 24.41.1). By Morphisms, Lemma 24.43.12 we conclude that X is isomorphic to an open subscheme of a scheme projective over Spec(R). Therefore, to finish the proof, it suffices to show that X → Spec(R) is universally closed (use Morphisms, Lemma 24.42.7). This follows from Lemma 26.16.3.  Lemma 26.16.5. Let S be a scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. Let i : Z → X be a closed subscheme. Denote I ⊂ A the kernel of the canonical map M A −→ p∗ ((i∗ OZ )(d)) d≥0

If p is quasi-compact, then there is an isomorphism Z = ProjS (A/I). Proof. The morphism p is separated by Constructions, Lemma 22.16.9. If p is quasi-compact, then p∗ transforms quasi-coherent modules into quasi-coherent modules, see Schemes, Lemma 21.24.1. Hence I is a quasi-coherent OS -module. In particular, B = A/I is a quasi-coherent graded OS -algebra. The functoriality morphism Z 0 = ProjS (B) → ProjS (A) is everywhere defined and a closed immersion, see Constructions, Lemma 22.18.3. Hence it suffices to prove Z = Z 0 as closed subschemes of X. Having said this, the question is local on the base and we may assume that S = Spec(R) and that X = Proj(A) for some graded R-algebra A. Assume I = Ie for I ⊂ A a graded ideal. By Constructions, Lemma 22.8.9 there exist f0 , . . . , fn ∈ A+ p S such that A+ ⊂ (f0 , . . . , fn ) in other words X = D+ (fi ). Therefore, it suffices to check that Z ∩ D+ (fi ) = Z 0 ∩ D+ (fi ) for each i. By renumbering we may assume i = 0. Say Z ∩ D+ (f0 ), resp. Z 0 ∩ D+ (f0 ) is cut out by the ideal J, resp. J 0 of A(f0 ) . The inclusion J 0 ⊂ J. Let d be the least common multiple of deg(f0 ), . . . , deg(fn ). nd/ deg(fi ) Note that each of the twists OX (nd) is invertible, trivialized by fi over D+ (fi ), and that for any quasi-coherent module F on X the multiplication maps OX (nd) ⊗OX F(m) → F(nd + m) are isomorphisms, see Constructions, Lemma 22.10.2. Observe that J 0 is the ideal generated by the elements g/f0e where g ∈ I is homogeneous of degree e deg(f0 ) (see proof of Constructions, Lemma 22.11.3). Of course, by replacing g by f0l g for suitable l we may always assume that d|e. Then, since g vanishes as a section of OX (e deg(f0 )) restricted to Z we see that g/f0d is an element of J. Thus J 0 ⊂ J.

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Conversely, suppose that g/f0e ∈ J. Again we may assume d|e. Pick i ∈ {1, . . . , n}. Then Z ∩ D+ (fi ) is cut out by some ideal Ji ⊂ A(fi ) . Moreover, J · A(f0 fi ) = Ji · A(f0 fi ) deg(fi )

The right hand side is the localization of Ji with respect to f0 follows that (e +e) deg(f0 )/ deg(fi ) f0ei g/fi i ∈ Ji

deg(f0 )

/fi

. It

max(e )

i for some ei  0 sufficiently divisible. This proves that f0 g is an element of I, because its restriction to each affine open D+ (fi ) vanishes on the closed subscheme Z ∩ D+ (fi ). Hence g ∈ J 0 and we conclude J ⊂ J 0 as desired. 

In case the closed subscheme is locally cut out by finitely many equations we can define it by a finite type ideal sheaf of A. Lemma 26.16.6. Let S be a quasi-compact and quasi-separated scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. Let i : Z → X be a closed subscheme. If p is quasi-compact and i of finite presentation, then there exists a d > 0 and a quasi-coherent finite type OS -submodule F ⊂ Ad such that Z = ProjS (A/FA). Proof. By Lemma 26.16.5 we know there exists a quasi-coherent graded sheaf of ideals I ⊂ A such that Z = Proj(A/I). Since S is quasi-compact we can choose a finite affine open covering S = U1 ∪ . . . ∪ Un . Say Ui = Spec(Ri ). Let A|Ui correspond to the graded Ri -algebra Ai and I|Ui to the graded ideal Ii ⊂ Ai . Note that p−1 (Ui ) = Proj(Ai ) as schemes over Ri . Since p is quasi-compact we can choose finitely many homogeneous elements fi,j ∈ Ai,+ such that p−1 (Ui ) = D+ (fi,j ). The condition on Z → X means that the ideal sheaf of Z in OX is of finite type, see Morphisms, Lemma 24.22.7. Hence we can find finitely many homogeneous elements hi,j,k ∈ Ii ∩ Ai,+ such that the ideal of Z ∩ D+ (fi,j ) is generated by the elements e hi,j,k /fi,ji,j,k . Choose d > 0 to be a common multiple of all the integers deg(fi,j ) and deg(hi,j,k ). By Properties, Lemma 23.20.7 there exists a finite type F ⊂ Id such that all the local sections (d−deg(hi,j,k ))/ deg(fi,j )

hi,j,k fi,j

are sections of F. By construction F is a solution.



The following version of Lemma 26.16.6 will be used in the proof of Lemma 26.19.2. Lemma 26.16.7. Let S be a quasi-compact and quasi-separated scheme. Let A be a quasi-coherent graded OS -algebra. Let p : X = ProjS (A) → S be the relative Proj of A. Let i : Z → X be a closed subscheme. Let U ⊂ X be an open. Assume that (1) (2) (3) (4) (5)

p is quasi-compact, i of finite presentation, U ∩ p(i(Z)) = ∅, U is quasi-compact, An is a finite type OS -module for all n.

Then there exists a d > 0 and a quasi-coherent finite type OS -submodule F ⊂ Ad with (a) Z = ProjS (A/FA) and (b) the support of Ad /F is disjoint from U .

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Proof. Let I ⊂ A be the sheaf of quasi-coherent graded ideals constructed in Lemma 26.16.5. Then since U ∩ p(i(Z)) = ∅ we see that I|U = A|U . Let Ui , Ri , Ai , Ii , fi,j , hi,j,k , and d be as constructed in the proof of Lemma 26.16.6. Since U ∩p(i(Z)) = ∅ we see that I|U = Ad |U (by our construction of I as a kernel). Since U is quasi-compact we can choose a finite affine open covering U = W1 ∪ . . . ∪ Wm . Since Ad is of finite type we can find finitely many sections gt,s ∈ Ad (Wt ) which generate Ad |Wt = Id |Wt as an OWt -module. To finish the proof, note that by Properties, Lemma 23.20.7 there exists a finite type F ⊂ Id such that all the local sections (d−deg(hi,j,k ))/ deg(fi,j ) hi,j,k fi,j and gt,s are sections of F. By construction F is a solution.  26.17. Blowing up Blowing up is an important tool in algebraic geometry. Definition 26.17.1. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals, and let Z ⊂ X be the closed subscheme corresponding to I, see Schemes, Definition 21.10.2. The blowing up of X along Z, or the blowing up of X in the ideal sheaf I is the morphism M  I n −→ X b : ProjX n≥0

The exceptional divisor of the blow up is the inverse image b−1 (Z). Sometimes Z is called the center of the blowup. We will see later that the exceptional divisor is an effective Cartier divisor. Moreover, the blowing up is characterized as the smallest scheme over X such that the inverse image of Z is an effective Cartier divisor. If b : X 0 → X is the blow up of X in Z, then we often denote OX 0 (n) the twists of the structure sheaf. Note that these are invertible OX 0 -modules and that OX 0 (n) = OX 0 (1)⊗n because X 0 is the relative Proj of a quasi-coherent graded OX -algebra which is generated in degree 1, see Constructions, Lemma 22.16.11. Note that OX 0 (1) is b-relatively very ample, even though b need not be of finite type or even quasi-compact, because X 0 comes equipped with a closed immersion into P(I), see Morphisms, Example 24.39.3. Lemma 26.17.2. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals. Let U = Spec(A) be an affine open subscheme of X and let I ⊂ A be the ideal corresponding to I|U . If b : X 0 → X is the blow up of X in I, then there is a canonical isomorphism M b−1 (U ) = Proj( I d) d≥0

−1

of b (U ) with the homogeneous spectrum of the Rees algebra of I in A. Moreover, b−1 (U ) has an affine open covering by spectra of the affine blowup algebras A[ aI ]. Proof. The first statement is clear from the construction of the relative Proj via glueing, see Constructions, Section 22.15. For a ∈ a(1) the element a LI denote n seen as an element of degree 1 in the Rees algebra n≥0 I . Since these elements L generate the Rees algebra over A we see that Proj( d≥0 I d ) is covered by the affine opens D+ (a(1) ). The affine scheme D+ (a(1) ) is the spectrum of the affine blowup algebra A0 = A[ aI ], see Algebra, Definition 7.55.1. This finishes the proof. 

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Lemma 26.17.3. Let X1 → X2 be a flat morphism of schemes. Let Z2 ⊂ X2 be a closed subscheme. Let Z1 be the inverse image of Z2 in X1 . Let Xi0 be the blow up of Zi in Xi . Then there exists a cartesian diagram X10

/ X20

 X1

 / X2

of schemes. Proof. Let I2 be the ideal sheaf of Z2 in X2 . Denote g : X1 → X2 the given morphism. Then the ideal sheaf of Z1 is the image of g ∗ I2 → OX1 (by definition of the inverse image, see Schemes, Definition 21.17.7). LBy Constructions, Lemma 22.16.10 we see that X1 ×X2 X20 is the relative Proj of n≥0 g ∗ I2n . Because g is flat the map g ∗ I2n → OX1 is injective with image I1n . Thus we see that X1 ×X2 X20 = X10 .  Lemma 26.17.4. Let X be a scheme. Let Z ⊂ X be a closed subscheme. The blowing up b : X 0 → X of Z in X has the following properties: (1) b|b−1 (X\Z) : b−1 (X \ Z) → X \ Z is an isomorphism, (2) the exceptional divisor E = b−1 (Z) is an effective Cartier divisor on X 0 , (3) there is a canonical isomorphism OX 0 (−1) = OX 0 (E) Proof. As blowing up commutes with restrictions to open subschemes (Lemma 26.17.3) the first statement just means that X 0 = X if Z = ∅. In this case we are blowing up in the ideal sheaf I = OX and the result follows from Constructions, Example 22.8.14. The second statement is local on X, hence we may assume X affine. Say X = Spec(A) and Z = Spec(A/I). By Lemma 26.17.2 we see that X 0 is covered by the spectra of the affine blowup algebras A0 = A[ aI ]. Then IA0 = aA0 and a maps to a nonzerodivisor in A0 according to Algebra, Lemma 7.55.2. This proves the lemma as the inverse image of Z in Spec(A0 ) corresponds to Spec(A0 /IA0 ) ⊂ Spec(A0 ). Consider the canonical map ψuniv,1 : b∗ I → OX 0 (1), see discussion following Constructions, Definition 22.16.7. We claim that this factors through an isomorphism IE → OX 0 (1) (which proves the final assertion). Namely, on the affine open corresponding to the blowup algebra A0 = A[ aI ] mentioned above ψuniv,1 corresponds to the A0 -module map  M   I ⊗A A0 −→ I d (1) d≥0

a

1

(1)

where a is as in Algebra, Definition 7.55.1. We omit the verification that this is the map I ⊗A A0 → IA0 = aA0 .  Lemma 26.17.5 (Universal property blowing up). Let X be a scheme. Let Z ⊂ X be a closed subscheme. Let C be the full subcategory of (Sch/X) consisting of Y → X such that the inverse image of Z is an effective Cartier divisor on Y . Then the blowing up b : X 0 → X of Z in X is a final object of C. Proof. We see that b : X 0 → X is an object of C according to Lemma 26.17.4. Let f : Y → X be an object of C. We have to show there exists a unique morphism Y → X 0 over X. Let D = f −1 (Z). Let I ⊂ OX be the ideal sheaf of Z and

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∗ let ID be the ideal sheaf of D. Then f L I → ID isLa surjection to an invertible d OY -module. This extends to a map ψ : f ∗I d → ID of graded OY -algebras. ⊗d d (We observe that ID = ID as D is an effective Cartier divisor.) By the material in Constructions, Section 22.16 the triple (1, f : Y → X, ψ) defines a morphism Y → X 0 over X. The restriction

Y \ D −→ X 0 \ b−1 (Z) = X \ Z is unique. The open Y \ D is scheme theoretically dense in Y according to Lemma 26.9.4. Thus the morphism Y → X 0 is unique by Morphisms, Lemma 24.7.10 (also b is separated by Constructions, Lemma 22.16.9).  Lemma 26.17.6. Let X be a scheme. Let Z ⊂ X be an effective Cartier divisor. The blowup of X in Z is the identity morphism of X. Proof. Immediate from the universal property of blowups (Lemma 26.17.5).



Lemma 26.17.7. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals. If X is integral, then the blow up X 0 of X in I is integral. Proof. Combine Lemma 26.17.2 with Algebra, Lemma 7.55.4.



Lemma 26.17.8. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals. If X is reduced, then the blow up X 0 of X in I is reduced. Proof. Combine Lemma 26.17.2 with Algebra, Lemma 7.55.5.



Lemma 26.17.9. Let X be a scheme. Let b : X 0 → X be a blow up of X in a closed subscheme. For any effective Cartier divisor D on X the pullback b−1 D is defined (see Definition 26.9.11). Proof. By Lemmas 26.17.2 and 26.9.2 this reduces to the following algebra fact: Let A be a ring, I ⊂ A an ideal, a ∈ I, and x ∈ A a nonzerodivisor. Then the image of x in A[ aI ] is a nonzerodivisor. Namely, suppose that x(y/an ) = 0 in A[ aI ]. Then am xy = 0 in A for some m. Hence am y = 0 as x is a nonzerodivisor. Whence y/an is zero in A[ aI ] as desired.  Lemma 26.17.10. Let X be a scheme. Let I ⊂ OX and J be quasi-coherent sheaves of ideals. Let b : X 0 → X be the blowing up of X in I. Let b0 : X 00 → X 0 be the blowing up of X 0 in b−1 J OX 0 . Then X 00 → X is canonically isomorphic to the blowing up of X in IJ . Proof. Let E ⊂ X 0 be the exceptional divisor of b which is an effective Cartier divisor by Lemma 26.17.4. Then (b0 )−1 E is an effective Cartier divisor on X 00 by Lemma 26.17.9. Let E 0 ⊂ X 00 be the exceptional divisor of b0 (also an effective Cartier divisor). Consider the effective Cartier divisor E 00 = E 0 + (b0 )−1 E. By construction the ideal of E 00 is (b ◦ b0 )−1 I(b ◦ b0 )−1 J OX 00 . Hence according to Lemma 26.17.5 there is a canonical morphism from X 00 to the blowup c : Y → X of X in IJ . Conversely, as IJ pulls back to an invertible ideal we see that c−1 IOY defines an effective Cartier divisor, see Lemma 26.9.9. Thus a morphism c0 : Y → X 0 over X by Lemma 26.17.5. Then (c0 )−1 b−1 J OY = c−1 J OY which also defines an effective Cartier divisor. Thus a morphism c00 : Y → X 00 over X 0 . We omit the verification that this morphism is inverse to the morphism X 00 → Y constructed earlier. 

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Lemma 26.17.11. Let X be a scheme. Let I ⊂ OX be a quasi-coherent sheaf of ideals. Let b : X 0 → X be the blowing up of X in the ideal sheaf I If I is of finite type, then (1) b : X 0 → X is a projective morphism, and (2) OX 0 (1) is a b-relatively ample invertible sheaf. Proof. The surjection of graded OX -algebras M Sym∗OX (I) −→

d≥0

Id

defines via Constructions, Lemma 22.18.5 a closed immersion M I d ) −→ P(I). X 0 = ProjX ( d≥0

Hence b is projective, see Morphisms, Definition 24.43.1. The second statement follows for example from the characterization of relatively ample invertible sheaves in Morphisms, Lemma 24.38.4. Some details omitted.  Lemma 26.17.12. Let X be a quasi-compact and quasi-separated scheme. Let Z ⊂ X be a closed subscheme of finite presentation. Let b : X 0 → X be the blowing up with center Z. Let Z 0 ⊂ X 0 be a closed subscheme of finite presentation. Let X 00 → X 0 be the blowing up with center Z 0 . There exists a closed subscheme Y ⊂ X of finite presentation, such that (1) Y = Z ∪ b(Z 0 ) set theoretically, and (2) the composition X 00 → X is isomorphic to the blowing up of X in Y . Proof. The condition that Z → X is of finite presentation means that Z is cut out by a finite type quasi-coherent sheaf of ideals I ⊂ OX , see Morphisms, Lemma L 24.22.7. Write A = n≥0 I n so that X 0 = Proj(A). Note that X \ Z is a quasicompact of X by Properties, Lemma 23.22.1. Since b−1 (X \ Z) → X \ Z is an isomorphism (Lemma 26.17.4) the same result shows that b−1 (X \ Z) \ Z 0 is quasicompact open in X 0 . Hence U = X \ (Z ∪ b(Z 0 )) is quasi-compact open in X. By Lemma 26.16.7 there exist a d > 0 and a finite type OX -submodule F ⊂ I d such that Z 0 = Proj(A/FA) and such that the support of I d /F is contained in X \ U . Since F ⊂ I d is an OX -submodule we may think of F ⊂ I d ⊂ OX as a finite type quasi-coherent sheaf of ideals on X. Let’s denote this J ⊂ OX to prevent confusion. Since I d /J and O/I d are supported on X \ U we see that V (J ) is contained in X \ U . Conversely, as J ⊂ I d we see that Z ⊂ V (J ). Over X \ Z ∼ = X 0 \ b−1 (Z) 0 the sheaf of ideals J cuts out Z (see displayed formula below). Hence V (J ) equals Z ∪ b(Z 0 ). It follows that also V (IJ ) = Z ∪ b(Z 0 ) set theoretically. Moreover, IJ is an ideal of finite type as a product of two such. We claim that X 00 → X is isomorphic to the blowing up of X in IJ which finishes the proof of the lemma by setting Y = V (IJ ). First, recall that the blow up of X in IJ is the same as the blow up of X 0 in b−1 J OX 0 , see Lemma 26.17.10. Hence it suffices to show that the blow up of X 0 in b−1 J OX 0 agrees with the blow up of X 0 in Z 0 . We will show that d b−1 J OX 0 = IE IZ 0 d as ideal sheaves on X 00 . This will prove what we want as IE cuts out the effective Cartier divisor dE and we can use Lemmas 26.17.6 and 26.17.10.

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To see the displayed equality of the ideals we may work locally. With notation A, I, a ∈ I as in Lemma 26.17.2 we see that F corresponds to an R-submodule M ⊂ I d mapping isomorphically to an ideal J ⊂ R. The condition Z 0 = Proj(A/FA) means that Z 0 ∩ Spec(A[ aI ]) is cut out by the ideal generated by the elements m/ad , m ∈ M . Say the element m ∈ M corresponds to the function f ∈ J. Then in the affine blowup algebra A0 = A[ aI ] we see that f = (ad m)/ad = ad (m/ad ). Thus the equality holds. 

26.18. Strict transform In this section we briefly discuss strict transform under blowing up. Let S be a scheme and let Z ⊂ S be a closed subscheme. Let b : S 0 → S be the blowing up of S in Z and denote E ⊂ S 0 the exceptional divisor E = b−1 Z. In the following we will often consider a scheme X over S and form the cartesian diagram pr−1 S0 E  E

/ X ×S S 0 prS 0

 / S0

prX

/X  /S

f

Since E is an effective Cartier divisor (Lemma 26.17.4) we see that pr−1 S0 E ⊂ X ×S ×S S 00 is locally principal (Lemma 26.9.10). Thus the complement of pr−1 S0 E in X ×S S 0 is retrocompact (Lemma 26.9.3). Consequently, for a quasi-coherent OX×S S 0 -module G the subsheaf of sections supported on pr−1 S 0 E is a quasi-coherent submodule, see Properties, Lemma 23.22.5. If G is a quasi-coherent sheaf of algebras, e.g., G = OX×S S 0 , then this subsheaf is an ideal of G. Definition 26.18.1. With Z ⊂ S and f : X → S as above. (1) Given a quasi-coherent OX -module F the strict transform of F with respect to the blowup of S in Z is the quotient F 0 of pr∗X F by the submodule of sections supported on pr−1 S 0 E. (2) The strict transform of X is the closed subscheme X 0 ⊂ X ×S S 0 cut out by the quasi-coherent ideal of sections of OX×S S 0 supported on pr−1 S 0 E. Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism S 0 → S). This notion is often used for closed subschemes of S. It turns out that the strict transform of X is a blowup of X. Lemma 26.18.2. In the situation of Definition 26.18.1. (1) The strict transform X 0 of X is the blowup of X in the closed subscheme f −1 Z of X. (2) For a quasi-coherent OX -module F the strict transform F 0 is canonically isomorphic to the pushfoward along X 0 → X ×S S 0 of the strict transform of F relative to the blowing up X 0 → X.

26.18. STRICT TRANSFORM

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Proof. Let X 00 → X be the blowup of X in f −1 Z. By the universal property of blowing up (Lemma 26.17.5) there exists a commutative diagram X 00

/X

 S0

 /S

whence a morphism X 00 → X ×S S 0 . Thus the first assertion is that this morphism is a closed immersion with image X 0 . The question is local on X. Thus we may assume X and S are affine. Say that S = Spec(A), L X = Spec(B), Land Z is cut out by the ideal I ⊂ A. Set J = IB. The map B ⊗A n≥0 I n → n≥0 J n defines a closed immersion X 00 → X ×S S 0 , see Constructions, Lemmas 22.11.6 and 22.11.5. We omit the verification that this morphism is the same as the one constructed above from the universal property. Pick a ∈ I corresponding to the affine open Spec(A[ aI ]) ⊂ S 0 , see Lemma 26.17.2. The inverse image of Spec(A[ aI ]) in the strict transform X 0 of X is the spectrum of B 0 = (B ⊗A A[ aI ])/a-power-torsion On the other hand, letting b ∈ J be the image of a we see that Spec(B[ Jb ]) is the inverse image of Spec(A[ aI ]) in X 00 . The ring map B ⊗A A[ aI ] −→ B[ Jb ] see Properties, Lemma 23.22.5. is surjective and annihilates a-power torsion as b is a nonzerodivsor in B[ Jb ]. Hence we obtain a surjective map B 0 → B[ Jb ]. To see that the kernel is trivial, we construct anP inverse map. Namely, let z = y/bn be an J n element of B[ b ],P i.e., y ∈ J . Write y = xi bi with xi ∈ I n and bi ∈ B. We map n 0 z to the class of bi ⊗ xi /a in B . This is well defined because an element of the kernel of the map B ⊗A I n → J n is annihilated by an , hence maps to zero in B 0 . This shows that the open Spec(B[ Jb ]) maps isomorphically to the open subscheme I 0 00 0 pr−1 S 0 (Spec(A[ a ])) of X . Thus X → X is an isomorphism. In the notation above, let F correspond to the B-module N . The strict transform of F corresponds to the B ⊗A A[ aI ]-module N 0 = (N ⊗A A[ aI ])/a-power-torsion see Properties, Lemma 23.22.5. The strict transform of F relative to the blowup of X in f −1 Z corresponds to the B[ Jb ]-module N ⊗B B[ Jb ]/b-power-torsion. In exactly the same way as above one proves that these two modules are isomorphic. Details omitted.  Lemma 26.18.3. In the situation of Definition 26.18.1. (1) If X is flat over S at all points lying over Z, then the strict transform of X is equal to the base change X ×S S 0 . (2) Let F be a quasi-coherent OX -module. If F is flat over S at all points lying over Z, then the strict transform F 0 of F is equal to the pullback pr∗X F. Proof. We will prove part (2) as it implies part (1) by the definition of the strict transform of a scheme over S. The question is local on X. Thus we may assume

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that S = Spec(A), X = Spec(B), and that F corresponds to the B-module N . Then F 0 over the open Spec(B ⊗A A[ aI ]) of X ×S S 0 corresponds to the module N 0 = (N ⊗A A[ aI ])/a-power-torsion see Properties, Lemma 23.22.5. Thus we have to show that the a-power-torsion of N ⊗A A[ aI ] is zero. Let y ∈ N ⊗A A[ aI ] with an y = 0. If q ⊂ B is a prime and a 6∈ q, then y maps to zero in (N ⊗A A[ aI ])q . on the other hand, if a ∈ q, then Nq is a flat A-module and we see that Nq ⊗A A[ aI ] = (N ⊗A A[ aI ])q has no a-power torsion (as A[ aI ] doesn’t). Hence y maps to zero in this localization as well. We conclude that y is zero by Algebra, Lemma 7.22.1.  Lemma 26.18.4. Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let b : S 0 → S be the blowing up of Z in S. Let g : X → Y be an affine morphism of schemes over S. Let F be a quasi-coherent sheaf on X. Let g 0 : X ×S S 0 → Y ×S S 0 be the base change of g. Let F 0 be the strict transform of F relative to b. Then g∗0 F 0 is the strict transform of g∗ F. Proof. Observe that g∗0 pr∗X F = pr∗Y g∗ F by Cohomology of Schemes, Lemma 25.6.1. Let K ⊂ pr∗X F be the subsheaf of sections supported in the inverse image of Z in X ×S S 0 . By Properties, Lemma 23.22.6 the pushforward g∗0 K is the subsheaf of sections of pr∗Y g∗ F supported in the inverse image of Z in Y ×S S 0 . As g 0 is affine (Morphisms, Lemma 24.13.8) we see that g∗0 is exact, hence we conclude.  Lemma 26.18.5. Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let D ⊂ S be an effective Cartier divisor. Let Z 0 ⊂ S be the closed subscheme cut out by the product of the ideal sheaves of Z and D. Let S 0 → S be the blowup of S in Z. (1) The blowup of S in Z 0 is isomorphic to S 0 → S. (2) Let f : X → S be a morphism of schemes and let F be a quasi-coherent OX -module. If F has no nonzero local sections supported in f −1 D, then the strict transform of F relative to the blowing up in Z agrees with the strict transform of F relative to the blowing up of S in Z 0 . Proof. The first statement follows on combining Lemmas 26.17.10 and 26.17.6. Using Lemma 26.17.2 this translates into the following algebra problem. Let A be a ring, I ⊂ A an ideal, x ∈ A a nonzerodivisor, and a ∈ I. Let M be an A-module whose x-torsion is zero. To show: the a-power torsion in M ⊗A A[ aI ] is equal to the xa-power torsion. The reason for this is that the kernel and cokernel of the map A → A[ aI ] is a-power torsion, so this map becomes an isomorphism after inverting a. Hence the kernel and cokernel of M → M ⊗A A[ aI ] are a-power torsion too. This implies the result.  Lemma 26.18.6. Let S be a scheme. Let Z ⊂ S be a closed subscheme. Let b : S 0 → S be the blowing up with center Z. Let Z 0 ⊂ S 0 be a closed subscheme. Let S 00 → S 0 be the blowing up with center Z 0 . Let Y ⊂ S be a closed subscheme such that Y = Z ∪ b(Z 0 ) set theoretically and the composition S 00 → S is isomorphic to the blowing up of S in Y . In this situation, given any scheme X over S and F ∈ QCoh(OX ) we have (1) the strict transform of F with respect to the blowing up of S in Y is equal to the strict transform with respect to the blowup S 00 → S 0 in Z 0 of the strict transform of F with respect to the blowup S 0 → S of S in Z, and

26.18. STRICT TRANSFORM

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(2) the strict transform of X with respect to the blowing up of S in Y is equal to the strict transform with respect to the blowup S 00 → S 0 in Z 0 of the strict transform of X with respect to the blowup S 0 → S of S in Z. Proof. Let F 0 be the strict transform of F with respect to the blowup S 0 → S of S in Z. Let F 00 be the strict transform of F 0 with respect to the blowup S 00 → S 0 of S 0 in Z 0 . Let G be the strict transform of F with respect to the blowup S 00 → S of S in Y . We also label the morphisms X ×S S 00 f 00

 S 00

q

/ X ×S S 0

p

f0

 / S0

/X  /S

f

By definition there is a surjection p∗ F → F 0 and a surjection q ∗ F 0 → F 00 which combine by right exactness of q ∗ to a surjection (p ◦ q)∗ F → F 00 . Also we have the surjection (p ◦ q)∗ F → G. Thus it suffices to prove that these two surjections have the same kernel. The kernel of the surjection p∗ F → F 0 is supported on (f ◦ p)−1 Z, so this map is an isomorphism at points in the complement. Hence the kernel of q ∗ p∗ F → q ∗ F 0 is supported on (f ◦ p ◦ q)−1 Z. The kernel of q ∗ F 0 → F 00 is supported on (f 0 ◦ q)−1 Z 0 . Combined we see that the kernel of (p ◦ q)∗ F → F 00 is supported on (f ◦ p ◦ q)−1 Z ∪ (f 0 ◦ q)−1 Z 0 = (f ◦ p ◦ q)−1 Y . By construction of G we see that we obtain a factorization (p ◦ q)∗ F → F 00 → G. To finish the proof it suffices to show that F 00 has no nonzero (local) sections supported on (f ◦ p ◦ q)−1 (Y ) = (f ◦ p ◦ q)−1 Z ∪ (f 0 ◦ q)−1 Z 0 . This follows from Lemma 26.18.5 applied to F 0 on X ×S S 0 over S 0 , the closed subscheme Z 0 and the effective Cartier divisor b−1 Z.  Lemma 26.18.7. In the situation of Definition 26.18.1. Suppose that 0 → F1 → F2 → F3 → 0 is an exact sequence of quasi-coherent sheaves on X which remains exact after any base change T → S. Then the strict transforms of Fi0 relative to any blowup S 0 → S form a short exact sequence 0 → F10 → F20 → F30 → 0 too. Proof. We may localize on S and X and assume both are affine. Then we may push Fi to S, see Lemma 26.18.4. We may assume that our blowup is the morphism 1 : S → S associated to an effective Cartier divisor D ⊂ S. Then the translation into algebra is the following: Suppose that A is a ring and 0 → M1 → M2 → M3 → 0 is a universally exact sequence of A-modules. Let a ∈ A. Then the sequence 0 → M1 /a-power torsion → M2 /a-power torsion → M3 /a-power torsion → 0 is exact too. Namely, surjectivity of the last map and injectivity of the first map are immediate. The problem is exactness in the middle. Suppose that x ∈ M2 maps to zero in M3 /a-power torsion. Then y = an x ∈ M1 for some n. Then y maps to zero in M2 /an M2 . Since M1 → M2 is universally injective we see that y maps to zero in M1 /an M1 . Thus y = an z for some z ∈ M1 . Thus an (x − y) = 0. Hence y maps to the class of x in M2 /a-power torsion as desired. 

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26.19. Admissible blowups To have a bit more control over our blowups we introduce the following standard terminology. Definition 26.19.1. Let X be a scheme. Let U ⊂ X be an open subscheme. A morphism X 0 → X is called a U -admissible blowup if there exists a closed immersion Z → X of finite presentation with Z disjoint from U such that X 0 is isomorphic to the blow up of X in Z. We recall that Z → X is of finite presentation if and only if the ideal sheaf IZ ⊂ OX is of finite type, see Morphisms, Lemma 24.22.7. In particular, a U -admissible blowup is a projective morphism, see Lemma 26.17.11. Note that there can be multiple centers which give rise to the same morphism. Hence the requirement is just the existence of some center disjoint from U which produces X 0 . Finally, as the morphism b : X 0 → X is an isomorphism over U (see Lemma 26.17.4) we will often abuse notation and think of U as an open subscheme of X 0 as well. Lemma 26.19.2. Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open subscheme. Let b : X 0 → X be a U -admissible blowup. Let X 00 → X 0 be a U -admissible blowup. Then the composition X 00 → X is a U -admissible blowup. Proof. Immediate from the more precise Lemma 26.17.12.



Lemma 26.19.3. Let X be a quasi-compact and quasi-separated scheme. Let U, V ⊂ X be quasi-compact open subschemes. Let b : V 0 → V be a U ∩ V -admissible blowup. Then there exists a U -admissible blowup X 0 → X whose restriction to V is V 0 . Proof. Let I ⊂ OV be the finite type quasi-coherent sheaf of ideals such that V (I) is disjoint from U ∩ V and such that V 0 is isomorphic to the blow up of V in I. Let I 0 ⊂ OU ∪V be the quasi-coherent sheaf of ideals whose restriction to U is OU and whose restriction to V is I (see Sheaves, Section 6.33). By Properties, Lemma 23.20.2 there exists a finite type quasi-coherent sheaf of ideals J ⊂ OX whose restriction to U ∪ V is I 0 . The lemma follows.  Lemma 26.19.4. Let X be a quasi-compact and quasi-separated scheme. Let U ⊂ X be a quasi-compact open subscheme. Let bi : Xi → X, i = 1, . . . , n be U admissible blowups. There exists a U -admissible blowup b : X 0 → X such that (a) b factors as X 0 → Xi → X for i = 1, . . . , n and (b) each of the morphismsm X 0 → Xi is a U -admissible blowup. Proof. Let Ii ⊂ OX be the finite type quasi-coherent sheaf of ideals such that V (Ii ) is disjoint from U and such that Xi is isomorphic to the blow up of X in Ii . Set I = I1 · . . . · In and let X 0 be the blowup of X in I. Then X 0 → X factors through bi by Lemma 26.17.10.  Lemma 26.19.5. Let X be a quasi-compact and quasi-separated scheme. Let U, V be quasi-compact disjoint open subschemes of X. Then there exist a U ∪ V admissible blowup b : X 0 → X such that X 0 is a disjoint union of open subschemes X 0 = X10 q X20 with b−1 (U ) ⊂ X10 and b−1 (V ) ⊂ X20 .

26.20. OTHER CHAPTERS

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Proof. Choose a finite type quasi-coherent sheaf of ideals I, resp. J such that X \U = V (I), resp. X \V = V (J ), see Properties, Lemma 23.22.1. Then V (IJ ) = X set theoretically, hence IJ is a locally nilpotent sheaf of ideals. Since I and J are of finite type and X is quasi-compact there exists an n > 0 such that I n J n = 0. We may and do replace I by I n and J by J n . Whence IJ = 0. Let b : X 0 → X be the blowing up in I + J . This is U ∪ V -admissible as V (I + J ) = X \ U ∪ V . We will show that X 0 is a disjoint union of open subschemes X 0 = X10 q X20 such that b−1 I|X20 = 0 and b−1 J |X10 = 0 which will prove the lemma. We will use the description of the blowing up in Lemma 26.17.2. Suppose that U = Spec(A) ⊂ X is an affine open such that I|U , resp. J |U corresponds to the finitely generated ideal I ⊂ A, resp. J ⊂ A. Then b−1 (U ) = Proj(A ⊕ (I + J) ⊕ (I + J)2 ⊕ . . .) I+J This is covered by the affine open subsets A[ I+J x ] and A[ y ] with x ∈ I and y ∈ J. I+J Since x ∈ I is a nonzerodivisor in A[ I+J x ] and IJ = 0 we see that JA[ x ] = 0. I+J I+J Since y ∈ J is a nonzerodivisor in A[ y ] and IJ = 0 we see that IA[ y ] = 0. Moreover, I+J I+J Spec(A[ I+J x ]) ∩ Spec(A[ y ]) = Spec(A[ xy ]) = ∅

because xy is both a nonzero divisor and zero. Thus b−1 (U ) is the disjoint union of the open subscheme U1 defined as the union of the standard opens Spec(A[ I+J x ]) for x ∈ I and the open subscheme U2 which is the union of the affine opens −1 Spec(A[ I+J IOX 0 restricts to zero on U2 y ]) for y ∈ J. We have seen that b −1 0 and b IOX restricts to zero on U1 . We omit the verification that these open subschemes glue to global open subschemes X10 and X20 .  26.20. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes

(22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces

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(43) Decent Algebraic Spaces (44) Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces (48) More on Morphisms of Spaces (49) Quot and Hilbert Spaces (50) Spaces over Fields (51) Stacks (52) Formal Deformation Theory (53) Groupoids in Algebraic Spaces (54) More on Groupoids in Spaces (55) Bootstrap (56) Examples of Stacks (57) Quotients of Groupoids (58) Algebraic Stacks

(59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 27

Limits of Schemes 27.1. Introduction In this chapter we put material related to limits of schemes. We mostly study limits of inverse systems over directed partially ordered sets with affine transition maps. We discuss absolute Noetherian approximation. We characterize schemes locally of finite presentation over a base as those whose associated functor of points is limit preserving. As an application of absolute Noetherian approximation we prove that the image of an affine under an integral morphism is affine. Moreover, we prove some very general variants of Chow’s lemma. A basic reference is [DG67]. 27.2. Directed limits of schemes with affine transition maps In this section we construct the limit. Lemma 27.2.1. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. If all the schemes Si are affine, then the limit S = limi Si exists in the category of schemes. In fact S is affine and S = Spec(colimi Ri ) with Ri = Γ(Si , O). Proof. Just define S = Spec(colimi Ri ). It follows from Schemes, Lemma 21.6.4 that S is the limit even in the category of locally ringed spaces.  Lemma 27.2.2. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. If all the morphisms fii0 : Si → Si0 are affine, then the limit S = limi Si exists in the category of schemes. Moreover, (1) each of the morphisms fi : S → Si is affine, (2) for any i ∈ I and any open subscheme Ui ⊂ Si we have fi−1 (Ui ) = limi0 ≥i fi−1 0 i (Ui ) in the category of schemes. Proof. Choose an element 0 ∈ I. Note that I is nonempty as the limit is directed. For every i ≥ 0 consider the quasi-coherent sheaf of OS0 -algebras Ai = fi0,∗ OSi . Recall that Si = SpecS (Ai ), see Morphisms, Lemma 24.13.3. Set A = colimi≥0 Ai . 0 This is a quasi-coherent sheaf of OS0 -algebras, see Schemes, Section 21.24. Set S = SpecS (A). By Morphisms, Lemma 24.13.5 we get for i ≥ 0 morphisms 0 fi : S → Si compatible with the transition morphisms. Note that the morphisms fi are affine by Morphisms, Lemma 24.13.11 for example. By Lemma 27.2.1 above we see that for any affine open U0 ⊂ S0 the inverse image U = f0−1 (U0 ) ⊂ S is the −1 limit of the system of opens Ui = fi0 (U0 ), i ≥ 0 in the category of schemes. Let T be a scheme. Let gi : T → Si be a compatible system of morphisms. To show that S = limi Si we have to prove there is a unique morphism g : T → S 1577

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with gi = fi ◦ g for all i ∈ I. For every t ∈ T there exists an affine open U0 ⊂ S0 containing g0 (t). Let V ⊂ g0−1 (U0 ) be an affine open neighbourhood containing t. By the remarks above we obtain a unique morphism gV : V → U = f0−1 (U0 ) such that fi ◦ gV = gi |Ui for all i. The open sets V ⊂ T so constructed form a basis for the topology of T . The morphisms gV glue to a morphism g : T → S because of the uniqueness property. This gives the desired morphism g : T → S. We omit the proof of the final statement.



Lemma 27.2.3. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume all the morphisms fii0 : Si → Si0 are affine, Let S = limi Si . (1) We have Sset = limi Si,set where Sset indicates the underlying set of the scheme S. (2) If s, s0 ∈ S and s0 is not a specialization of s then for some i ∈ I the image s0i ∈ Si of s0 is not a specialization of the image si ∈ Si of s. (3) Add more easy facts on topology of S here. (Requirement: whatever is added should be easy in the affine case.) Proof. Proof of (1). Pick i ∈ I. Take Ui ⊂ Si an affine open. Denote Ui0 = fi−1 0 i (Ui ) and U = fi−1 (Ui ). Suppose we can show that Uset = limi0 ≥i Ui0 ,set . Then assertion (1) follows by a simple argument using an affine covering of Si . Hence we may assume all Si and S affine. This reduces us to the following algebra question: Suppose given a system of rings (Ai , ϕii0 ) over I. Set A = colimi Ai with canonical maps ϕi : Ai → A. Then Spec(A) = limi Spec(Ai ) Namely, suppose that we are given primes pi ⊂ Ai such that pi = ϕ−1 ii0 (pi0 ) for all i0 ≥ i. Then we simply set p = {x ∈ A | ∃i, xi ∈ pi with ϕ(xi ) = x} It is clear that this is an ideal and has the property that ϕ−1 i (p) = pi . Then it follows easily that it is a prime ideal as well. This proves (1). Proof of (2). Pick i ∈ I. Pick an affine open Ui ⊂ Si containing fi (s0 ). If fi (s) 6∈ Si then we are done. Hence reduce to the affine case by considering the inverse images of Ui as above. This reduces us to the following algebra question: Suppose given a system of rings (Ai , ϕii0 ) over I. Set A = colimi Ai with canonical maps ϕi : Ai → A. Suppose given primes p, p0 of A. Suppose that p 6⊂ p0 . Then for some i we have −1 0 ϕ−1  i (p) 6⊂ ϕi (p ). This is clear. Lemma 27.2.4. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume all the morphisms fii0 : Si → Si0 are affine, Let S = limi Si . Let i ∈ I. Suppose that Xi is a scheme over Si . Set Xj = Sj ×Si Xi for j ≥ i and set X = S ×Si Xi . Then X = limj≥i Xj Proof. The transition morphisms of the system {Xj }j≥i are affine as they are base changes of the affine morphisms between the Sj , see Morphisms, Lemma 24.13.8. Hence we know the limit of the system {Xj }j≥i exists. There is a canonical morphism X → lim Xj . To see that it is an isomorphism we may work locally. Hence we may assume that Xi = Spec(Bi ) is an affine such that the morphism Xi → Si

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has image contained in an affine open subscheme U of Si . In this case we may also replace each Sj by the inverse image of U in Sj , in other words we may assume all the Sj = Spec(Aj ) are affine. Then we have Xj = Spec(Aj ⊗Ai Bi ). In this case the statement becomes the equality colimj≥i (Aj ⊗Ai Bi ) = (colimj≥i Aj ) ⊗Ai Bi which follows from Algebra, Lemma 7.11.8.



Lemma 27.2.5. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, (2) all the schemes Si are quasi-compact and quasi-separated. Let S = limi Si . Let i ∈ I. Suppose that Fi is a quasi-coherent sheaf on Si . Set ∗ Fj = fji Fi for j ≥ i and set F = fi∗ Fi . Then Γ(S, F) = colimj≥i Γ(Sj , Fj ) Proof. Write Aj = fji,∗ OSj . This is a quasi-coherent sheaf of OSi -algebras (see Morphisms, Lemma 24.13.5) and Sj is the relative spectrum of Aj over Si . In the proof of Lemma 27.2.2 we constructed S as the relative spectrum of A = colimj≥i Aj over Si . Set Mj = Fi ⊗OSi Aj and M = Fi ⊗OSi A. Then we have fji,∗ Fj = Mj and fi,∗ F = M. Since A is the colimit of the sheaves Aj and since tensor product commutes with directed colimits, we conclude that M = colimj≥i Mj . Since Si is quasi-compact and quasi-separated we see that Γ(S, F)

= Γ(Si , M) = Γ(Si , colimj≥i Mj ) =

colimj≥i Γ(Si , Mj )

=

colimj≥i Γ(Sj , Fj )

see Sheaves, Lemma 6.29.1 and Topology, Lemma 5.18.2 for the middle equality.  27.3. Absolute Noetherian Approximation A nice reference for this section is Appendix C of the article by Thomason and Trobaugh [TT90]. See Categories, Section 4.19 for our conventions regarding directed systems. We will use the existence result and properties of the limit from Section 27.2 without further mention. Lemma 27.3.1. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, (2) all the schemes Si are quasi-compact, and (3) all the schemes Si are nonempty. Then the limit S = limi Si is nonempty.

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Proof. Choose i0 ∈ I. Note that I is nonempty as the limit is directed. For convenience write S0 = Si0 and i0 = 0. Choose an affine open covering S0 = S −1 U . Since I is directed there exists a j ∈ {1, . . . , m} such that fi0 (Uj ) 6= ∅ j j=1,...,m −1 for all i ≥ 0. Hence limi≥0 fi0 (Uj ) is not empty since a directed colimit of nonzero −1 rings is nonzero (because 1 6= 0). As limi≥0 fi0 (Uj ) is an open subscheme of the limit we win.  Lemma 27.3.2. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, and (2) all the schemes Si are quasi-compact. Let S = limi Si . Suppose for each i we are given a nonempty closed subset Zi ⊂ Si with fii0 (Zi ) ⊂ Zi0 . Then there exists a point s ∈ S with fi (s) ∈ Zi for all i. Proof. Let Zi ⊂ Si also denote the reduced closed subscheme associated to Zi , see Schemes, Definition 21.12.5. A closed immersion is affine, and a composition of affine morphisms is affine (see Morphisms, Lemmas 24.13.9 and 24.13.7), and hence Zi → Si0 is affine when i ≥ i0 . We conclude that the morphism fii0 : Zi → Zi0 is affine by Morphisms, Lemma 24.13.11. Each of the schemes Zi is quasi-compact as a closed subscheme of a quasi-compact scheme. Hence we may apply Lemma 27.3.1 to see that Z = limi Zi is nonempty. Since there is a canonical morphism Z → S we win.  Lemma 27.3.3. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume all the morphisms fii0 : Si → Si0 are affine. Let S = limi Si . Suppose we are given an i and a morphism T → Si such that (1) T ×Si S = ∅, and (2) T is quasi-compact. Then T ×Si Si0 = ∅ for all sufficiently large i0 . Proof. By Lemma 27.2.4 we see that T ×Si S = limi0 ≥i T ×Si Si0 . Hence the result follows from Lemma 27.3.1.  Lemma 27.3.4. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume all the morphisms fii0 : Si → Si0 are affine, and all the schemes Si are quasi-compact. Let S = limi Si with projection morphisms fi : S → Si . Suppose we are given an i and a locally constructible subset E ⊂ Si such that fi (S) ⊂ E. Then fii0 (Si0 ) ⊂ E for all sufficiently large i0 . Proof. Writing Si as a finite union of open affine subschemes reduces the question to the case that Si is affine and E is constructible, see Lemma 27.2.2 and Properties, Lemma 23.2.1. In this case the complement Si \E is contstructible too. Hence there exists an affine scheme T and a morphism T → Si whose image is Si \E, see Algebra, Lemma 7.27.3. By Lemma 27.3.3 we see that T ×Si Si0 is empty for all sufficiently large i0 , and hence fii0 (Si0 ) ⊂ E for all sufficiently large i0 .  Lemma 27.3.5. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, (2) all the schemes Si are quasi-compact and quasi-separated. Then we have the following:

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(1) Given any quasi-compact open V ⊂ S = limi Si there exists an i ∈ I and a quasi-compact open Vi ⊂ Si such that fi−1 (Vi ) = V . (2) Given Vi ⊂ Si and Vi0 ⊂ Si0 quasi-compact opens such that fi−1 (Vi ) = −1 00 fi−1 ≥ i, i0 such that fi−1 0 (Vi0 ) there exists an index i 00 i (Vi ) = fi00 i0 (Vi0 ). −1 (3) If V1,i , . . . , Vn,i ⊂ Si are quasi-compact opens and S = fi (V1,i ) ∪ . . . ∪ −1 0 fi−1 (Vn,i ) then Si0 = fi−1 0 i (V1,i ) ∪ . . . ∪ fi0 i (Vn,i ) for some i ≥ i. Proof. Choose i0 ∈ I. Note that I is nonempty as the limit is directed. For convenience we write S0 = Si0 and i0 = 0. Choose an affine open covering S0 = U1,0 ∪ . . . ∪ Um,0 . Denote Uj,i ⊂ Si the inverse image of Uj,0 under the transition morphism for i ≥ 0. Denote Uj the inverse image of Uj,0 in S. Note that Uj = limi Uj,i is a limit of affine schemes. We first prove the uniqueness statement: Let Vi ⊂ Si and Vi0 ⊂ Si0 quasi-compact −1 opens such that fi−1 (Vi ) = fi−1 0 (Vi0 ). It suffices to show that fi00 i (Vi ∩ Uj,i00 ) and −1 fi00 i0 (Vi0 ∩ Uj,i00 ) become equal for i00 large enough. Hence we reduce to the case of a limit of affine schemes. In this case write S = Spec(R) and Si = Spec(Ri ) for all i ∈ I. We may write Vi = Si \ VP (h1 , . . . , hmP ) and Vi0 = Si0 \ V (g1 , . . . , gn ). The assumption means thatPthe ideals gj R and P hj R have the same radical in R. This means that gjN = ajj 0 hj 0 and hN bjj 0 gj 0 for some N  0 and ajj 0 j = and bjj 0 in R. Since R = colimi Ri P we can chose an index i00 ≥ i such that the P N N 00 0 0 equations gj = ajj 0 hj 0 and hj P = bjj 0 gj 0 hold P in Ri for some ajj and bjj in Ri00 . This implies that the ideals gj Ri00 and hj Ri00 have the same radical in Ri00 as desired. We prove existence. We may apply the uniqueness statement to the limit of schemes Uj1 ∩ Uj2 = limi Uj1 ,i ∩ Uj2 ,i since these are still quasi-compact due to the fact that the Si were assumed quasi-separated. Hence it is enough to prove existence in the affine case. In this case write S = Spec(R) and Si = Spec(Ri ) for all i ∈ I. Then V = S \ V (g1 , . . . , gn ) for some g1 , . . . , gn ∈ R. Choose any i large enough so that each of the gj comes from an element gj,i ∈ Ri and take Vi = Si \ V (g1,i , . . . , gn,i ). The statement on coverings follows from the uniqueness statement for the opens V1,i ∪ . . . ∪ Vn,i and Si of Si .  Lemma 27.3.6. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, (2) all the schemes Si are quasi-compact and quasi-separated, and (3) the limit S = limi Si is quasi-affine. Then for some i0 ∈ I the schemes Si for i ≥ i0 are quasi-affine. Proof. Choose i0 ∈ I. Note that I is nonempty as the limit is directed. For convenience we write S0 = Si0 and i0 = 0. Let s ∈ S. We may choose an affine open U0 ⊂ S0 containing f0 (s). Since S is quasi-affine we may choose an element a ∈ Γ(S, OS ) such that s ∈ D(a) ⊂ f0−1 (U0 ), and such that D(a) is affine. By Lemma 27.2.5 there exists an i ≥ 0 such that a comes from an element ai ∈ Γ(Si , OSi ). For any index j ≥ i we denote aj the image of ai in the global sections −1 of the structure sheaf of Sj . Consider the opens D(aj ) ⊂ Sj and Uj = fj0 (U0 ). Note that Uj is affine and D(aj ) is a quasi-compact open of Sj , see Properties, Lemma 23.24.4 for example. Hence we may apply Lemma 27.3.5 to the opens Uj

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and Uj ∪ D(aj ) to conclude that D(aj ) ⊂ Uj for some j ≥ i. For such an index j we see that D(aj ) ⊂ Sj is an affine open (because D(aj ) is a standard affine open of the affine open Uj ) containing the image fj (s). We conclude that for every s ∈ S there exist an index i ∈ I, and a global section a ∈ Γ(Si , OSi ) such that D(a) ⊂ Si is an affine open containing fi (s). Because S is quasi-compact we may choose a single index i ∈ I and global sections a1 , . . . , am ∈ Γ(Si , OSi ) such that each D(aj ) ⊂ Si is affine open and such that fi : S → Si has S image contained in the union Wi = j=1,...,m D(aj ). For i0 ≥ i set Wi0 = fi−1 0 i (Wi ). −1 0 Since fi (Wi ) is all of S we see (by Lemma 27.3.5 again) that for a suitable i ≥ i we S have Si0 = Wi0 . Thus we may replace i by i0 and assume that Si = j=1,...,m D(aj ). This implies that OSi is an ample invertible sheaf on Si (see Properties, Definition 23.24.1) and hence that Si is quasi-affine, see Properties, Lemma 23.25.1. Hence we win.  Lemma 27.3.7. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, (2) all the schemes Si are quasi-compact and quasi-separated, and (3) the limit S = limi Si is affine. Then for some i0 ∈ I the schemes Si for i ≥ i0 are affine. Proof. By Lemma 27.3.6 we may assume that Si is quasi-affine for all i. Set Ri = Γ(Si , OSi ). Then Si is a quasi-compact open of Si := Spec(Ri ). Write S = Spec(R). We have R = colimi Ri by Lemma 27.2.5. Hence also S = limi Si . ` Let Zi ⊂ Si be the closed subset such that Si = Zi Si . We have to show that Zi is empty for some i. Assume Zi is nonempty for all i to get a contradiction. By Lemma 27.3.2 there exists a point s of S which maps to a point of Zi for every i. But S = limi Si , and hence we get a contradiction.  Lemma 27.3.8. Let W be a quasi-affine scheme of finite type over Z. Suppose W → Spec(R) is an open immersion into an affine scheme. There exists a finite type Z-algebra A ⊂ R which induces an open immersion W → Spec(A). Moreover, R is the directed colimit of such subalgebras. S Proof. Choose an affine open covering W = i=1,...,n Wi such that each Wi is a standard affine open in Spec(R). In other words, if we write Wi = Spec(Ri ) then Ri = Rfi for some fi ∈ R. Choose finitely many xij ∈ Ri which generate Ri over Z. Pick an N  0 such that each fiN xij comes from an element of R, say yij ∈ R. Set A equal to the Z-algebra generated by the fi and the yij and (optionally) finitely many additional elements of R. Then A works. Details omitted.  Lemma 27.3.9. Suppose given a cartesian diagram of rings BO

s

/R O t

B

0

/R

0

27.3. ABSOLUTE NOETHERIAN APPROXIMATION

1583

Suppose h ∈ B 0 corresponds to g ∈ B and f ∈ R0 such that s(g) = t(f ). Then the diagram Bg O

s

/ Rs(g) = Rt(f ) O t

/ (R0 )f

(B 0 )h is cartesian too.

Proof. Note that B 0 = {(b, r0 ) ∈ B × R0 | s(b) = t(r0 )}. So h = (g, f ) ∈ B 0 . First we show that (B 0 )h maps injectively into Bg × (R0 )f . Namely, suppose that (x, y)/hn maps to zero. This means that (g N x, f N y) is zero for some N . Which clearly implies that x/g n and y/f n are both zero. Next, suppose that x/g n and y/f m are elements which map to the same element of Rs(g) . This means that s(g)N (t(f )m s(x) − s(g)n t(y)) = 0 in R0 for some N  0. We can rewrite this as s(g m+N x) = t(f n+N y). Hence we see that the pair (x/g n , y/f m ) is the image of the element (g m+N x, t(f n+N y)/(g, f )n+m+N of (B 0 )h .  Lemma 27.3.10. Suppose given a cartesian diagram of rings BO

s

/R O t

B0

/ R0

Let W 0 ⊂ Spec(R0 ) be an open of the form W 0 = D(f1 ) ∪ . . . ∪ D(fn ) such that t(fi ) = s(gi ) for some gi ∈ B and Bgi ∼ = Rs(gi ) . Then B 0 → R0 induces an open 0 0 immersion of W into Spec(B ). Proof. Set hi = (gi , fi ) ∈ B 0 . Lemma 27.3.9 above shows that (B 0 )hi ∼ = (R0 )fi as desired.  The following lemma is a precise statement of Noetherian approximation. Lemma 27.3.11. Let S be a quasi-compact and quasi-separated scheme. Let V ⊂ S be a quasi-compact open. Let I be a directed partially ordered set and let (Vi , fii0 ) be an inverse system of schemes over I with affine transition maps, with each Vi of finite type over Z, and with V = lim Vi . Then there exist (1) (2) (3) (4) (5)

a directed partially ordered set J, an inverse system of schemes (Sj , gjj 0 ) over J, an order preserving map α : J → I, open subschemes Vj0 ⊂ Sj , and isomorphisms Vj0 → Vα(j)

such that (1) (2) (3) (4)

the transition morphisms gjj 0 : Sj → Sj 0 are affine, each Sj is of finite type over Z, −1 gjj 0 (Vj 0 ) = Vj , S = lim Sj and V = lim Vj , and

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(5) the diagrams V  Vj0

! / Vα(j)

and

Vj

/ Vα(j)

 Vj 0

 / Vα(j 0 )

are commutative. Proof. Set Z = S \ V . Choose affine opens U1 , . . . , Um ⊂ S such that Z ⊂ S l=1,...,m Ul . Consider the opens [ V ⊂ V ∪ U1 ⊂ V ∪ U1 ∪ U2 ⊂ . . . ⊂ V ∪ Ul = S l=1,...,m

If we can prove the lemma successively for each of the cases V ∪ U1 ∪ . . . ∪ Ul ⊂ V ∪ U1 ∪ . . . ∪ Ul+1 then the lemma will follow for V ⊂ S. In each case we are adding one affine open. Thus we may assume (1) (2) (3) (4)

S =U ∪V, U affine open in S, V quasi-compact open in S, and V = limi Vi with (Vi , fii0 ) an inverse system over a directed set I, each fii0 affine and each Vi of finite type over Z.

Set W = U ∩ V . As S is quasi-separated, this is a quasi-compact open of V . By Lemma 27.3.5 (and after shrinking I) we may assume that there exist opens −1 Wi ⊂ Vi such that fij (Wj ) = Wi and such that fi−1 (Wi ) = W . Since W is a quasi-compact open of U it is quasi-affine. Hence we may assume (after shrinking I again) that Wi is quasi-affine for all i, see Lemma 27.3.6. Write U = Spec(B). Set R = Γ(W, OW ), and Ri = Γ(Wi , OWi ). By Lemma 27.2.5 we have R = colimi Ri . Now we have the maps of rings B

s

/R O ti

Ri We set Bi = {(b, r) ∈ B × Ri | s(b) = ti (t)} so that we have a cartesian diagram BO

s

/R O ti

Bi

/ Ri

for each i. The transition maps Ri → Ri0 induce maps Bi → Bi0 . It is clear that B = colimi Bi . In the next paragraph we show that for all sufficiently large i the composition Wi → Spec(Ri ) → Spec(Bi ) is an open immersion. As W is a quasi-compact open of U = Spec(B) we can find a finitelySmany elements gl ∈ B, l = 1, . . . , m such that D(gl ) ⊂ W and such that W = l=1,...,m D(gl ).

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Note that this implies D(gl ) = Ws(gl ) as open subsets of U , where Ws(gl ) denotes the largest open subset of W on which s(gl ) is invertible. Hence Bgl = Γ(D(gl ), OU ) = Γ(Ws(gl ) , OW ) = Rs(gl ) , where the last equality is Properties, Lemma 23.15.2. Since Ws(gl ) is affine this also implies that D(s(gl )) = Ws(gl ) as open subsets of Spec(R). Since R = colimi Ri we can (after shrinking I) assume there exist gl,i ∈ Ri for all i ∈ I such that s(gl ) = ti (gl,i ). Of course we choose the gl,i such that gl,i maps to gl,i0 under the transition maps Ri → Ri0 . Then, by Lemma 27.3.5 we can (after shrinking I again) assume the corresponding opens D(gl,i ) ⊂ Spec(Ri ) are contained in Wi , j = 1, . . . , m and cover Wi . We conclude that the morphism Wi → Spec(Ri ) → Spec(Bi ) is an open immersion, see Lemma 27.3.10 By Lemma 27.3.8 we can write Bi as a directed colimit of subalgebras Ai,p ⊂ Bi , p ∈ Pi each of finite type over Z and such that Wi is identified with an open subscheme of Spec(Ai,p ). Let Si,p be the scheme obtained by glueing Vi and Spec(Ai,p ) along the open Wi , see Schemes, Section 21.14. Here is the resulting commutative diagram of schemes: V o W u

V i ot

Wi

 t Si,p o

 v Spec(Ai,p )

 So

 U

The morphism S → Si,p arises because the upper right square is a pushout in the category of schemes. Note that Si,p is of finite type over Z since it has a finite affine open covering whose members are spectra of finite type Z-algebras. We ` define a partial ordering on J = i∈I Pi by the rule (i0 , p0 ) ≥ (i, p) if and only if i0 ≥ i and the map Bi → Bi0 maps Ai,p into Ai0 ,p0 . This is exactly the condition needed to define a morphism Si0 ,p0 → Si,p : namely make a commutative diagram as above using the transition morphisms Vi0 → Vi and Wi0 → Wi and the morphism Spec(Ai0 ,p0 ) → Spec(Ai,p ) induced by the ring map Ai,p → Ai0 ,p0 . The relevant commutativities have been built into the constructions. We claim that S is the directed limit of the schemes Si,p . Since by construction the schemes Vi have limit V this boils down to the fact that B is the limit of the rings Ai,p which is true by construction. The map α : J → I is given by the rule j = (i, p) 7→ i. The open subscheme Vj0 is just the image of Vi → Si,p above. The commutativity of the diagrams in (5) is clear from the construction. This finishes the proof of the lemma.  Proposition 27.3.12. Let S be a quasi-compact and quasi-separated scheme. There exist a directed partially ordered set I and an inverse system of schemes (Si , fii0 ) over I such that (1) the transition morphisms fii0 are affine (2) each Si is of finite type over Z, and (3) S = limi Si . Proof. This is a special case of Lemma 27.3.11 with V = ∅.



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27.4. Limits and morphisms of finite presentation The following is a generalization of Algebra, Lemma 7.119.2. Proposition 27.4.1. Let f : X → S be a morphism of schemes. The following are equivalent: (1) The morphism f is locally of finite presentation. (2) For any directed partially ordered set I, and any inverse system (Ti , fii0 ) of S-schemes over I with each Ti affine, we have MorS (limi Ti , X) = colimi MorS (Ti , X) (3) For any directed partially ordered set I, and any inverse system (Ti , fii0 ) of S-schemes over I with each fii0 affine and every Ti quasi-compact and quasi-separated as a scheme, we have MorS (limi Ti , X) = colimi MorS (Ti , X) Proof. It is clear that (3) implies (2). Let us prove that (2) implies (1). Assume (2). Choose any affine opens U ⊂ X and V ⊂ S such that f (U ) ⊂ V . We have to show that OS (V ) → OX (U ) is of finite presentation. Let (Ai , ϕii0 ) be a directed system of OS (V )-algebras. Set A = colimi Ai . According to Algebra, Lemma 7.119.2 we have to show that HomOS (V ) (OX (U ), A) = colimi HomOS (V ) (OX (U ), Ai ) Consider the schemes Ti = Spec(Ai ). They form an inverse system of V -schemes over I with transition morphisms fii0 : Ti → Ti0 induced by the OS (V )-algebra maps ϕi0 i . Set T := Spec(A) = limi Ti . The formula above becomes in terms of morphism sets of schemes MorV (limi Ti , U ) = colimi MorV (Ti , U ). We first observe that MorV (Ti , U ) = MorS (Ti , U ) and MorV (T, U ) = MorS (T, U ). Hence we have to show that MorS (limi Ti , U ) = colimi MorS (Ti , U ) and we are given that MorS (limi Ti , X) = colimi MorS (Ti , X). Hence it suffices to prove that given a morphism gi : Ti → X over S such that the composition T → Ti → X ends up in U there exists some i0 ≥ i such that the composition gi0 : Ti0 → Ti → X ends up in U . Denote Zi0 = gi−1 0 (X \ U ). Assume each Zi0 is nonempty to get a contradiction. By Lemma 27.3.2 there exists a point t of T which is mapped into Zi0 for all i0 ≥ i. Such a point is not mapped into U . A contradiction. Finally, let us prove that (1) implies (3). Assume (1). Let an inverse directed system (Ti , fii0 ) of S-schemes be given. Assume the morphisms fii0 are affine and each Ti is quasi-compact and quasi-separated as a scheme. Let T = limi Ti . Denote fi : T → Ti the projection morphisms. We have to show: (a) Given morphisms gi , gi0 : Ti → X over S such that gi ◦ fi = gi0 ◦ fi , then there exists an i0 ≥ i such that gi ◦ fi0 i = gi0 ◦ fi0 i . (b) Given any morphism g : T → X over S there exists an i ∈ I and a morphism gi : Ti → X such that g = fi ◦ gi .

27.5. FINITE TYPE CLOSED IN FINITE PRESENTATION

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First let us prove the uniqueness part (a). Let gi , gi0 : Ti → X be morphisms such that gi ◦ fi = gi0 ◦ fi . For any i0 ≥ i we set gi0 = gi ◦ fi0 i and gi00 = gi0 ◦ fi0 i . We also set g = gi ◦ fi = gi0 ◦ fi . Consider the morphism (gi , gi0 ) : Ti → X ×S X. Set [ W = U ×V U. U ⊂X affine open,V ⊂S affine open,f (U )⊂V

This is an open in X ×S X, with the property that the morphism ∆X/S factors through a closed immersion into W , see the proof of Schemes, Lemma 21.21.2. Note that the composition (gi , gi0 ) ◦ fi : T → X ×S X is a morphism into W because it factors through the diagonal by assumption. Set Zi0 = (gi0 , gi00 )−1 (X ×S X \ W ). If each Zi0 is nonempty, then by Lemma 27.3.2 there exists a point t ∈ T which maps to Zi0 for all i0 ≥ i. This is a contradiction with the fact that T maps into W . Hence we may increase i and assume that (gi , gi0 ) : Ti → X ×S X is a morphism into W . By construction of W , and since Ti is quasi-compact we can find a finite affine open covering Ti = T1,i ∪ . . . ∪ Tn,i such that (gi , gi0 )|Tj,i is a morphism into U ×V U for some pair (U, V ) as in the definition of W above. Since it suffices to prove that gi0 and gi00 agree on each of the fi−1 0 i (Tj,i ) this reduces us to the affine case. The affine case follows from Algebra, Lemma 7.119.2 and the fact that the ring map OS (V ) → OX (U ) is of finite presentation (see Morphisms, Lemma 24.22.2). Finally, we prove the existence part (b). Let g : T → X be a morphism of schemes over S. We can find a finite affine open covering T = W1 ∪ . . . ∪ Wn such that for each j ∈ {1, . . . , n} there exist affine opens Uj ⊂ X and Vj ⊂ S with f (Uj ) ⊂ Vj and g(Wj ) ⊂ Uj . By Lemmas 27.3.5 and 27.3.7 (after possibly shrinking I) we may assume that there exist affine open coverings Ti = W1,i ∪ . . . ∪ Wn,i compatible with transition maps such that Wj = limi Wj,i . We apply Algebra, Lemma 7.119.2 to the rings corresponding to the affine schemes Uj , Vj , Wj,i and Wj using that OS (Vj ) → OX (Uj ) is of finite presentation (see Morphisms, Lemma 24.22.2). Thus we can find for each j an index ij ∈ I and a morphism gj,ij : Wj,ij → X such that gj,ij ◦ fi |Wj : Wj → Wj,i → X equals g|Wj . By part (a) proved above, using the quasi-compactness of Wj1 ,i ∩ Wj2 ,i which follows as Ti is quasi-separated, we can find an index i0 ∈ I larger than all ij such that gj1 ,ij1 ◦ fi0 ij1 |Wj1 ,i0 ∩Wj2 ,i0 = gj2 ,ij2 ◦ fi0 ij2 |Wj1 ,i0 ∩Wj2 ,i0 for all j1 , j2 ∈ {1, . . . , n}. Hence the morphisms gj,ij ◦ fi0 ij |Wj,i0 glue to given the desired morphism Ti0 → X.  Remark 27.4.2. Let S be a scheme. Let us say that a functor F : (Sch/S)opp → Sets is limit preserving if for every directed inverse system {Ti }i∈I of affine schemes with limit T we have F (T ) = colimi F (Ti ). Let X be a scheme over S, and let hX : (Sch/S)opp → Sets be its functor of points, see Schemes, Section 21.15. In this terminology Proposition 27.4.1 says that a scheme X is locally of finite presentation over S if and only if hX is limit preserving. 27.5. Finite type closed in finite presentation A reference is [Con07]. Lemma 27.5.1. Let f : X → S be a morphism of schemes. Assume: (1) The morphism f is locally of finite type. (2) The scheme X is quasi-compact and quasi-separated.

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Then there exists a morphism of finite presentation f 0 : X 0 → S and an immersion X → X 0 of schemes over S. Proof. By Proposition 27.3.12 we can write X = limi Xi with each Xi of finite type over Z and with transition morphisms fii0 : Xi → Xi0 affine. Consider the commutative diagram X

/ Xi,S

/ Xi

!  S

 / Spec(Z)

Note that Xi is of finite presentation over Spec(Z), see Morphisms, Lemma 24.22.9. Hence the base change Xi,S → S is of finite presentation by Morphisms, Lemma 24.22.4. Thus it suffices to show that the arrow X → Xi,S is an immersion for some i sufficiently large. To do this we choose a finite affine open covering X = V1 ∪. . .∪Vn such that f maps each Vj into an affine open Uj ⊂ S. Let hj,a ∈ OX (Vj ) be a finite set of elements which generate OX (Vj ) as an OS (Uj )-algebra, see Morphisms, Lemma 24.16.2. By Lemmas 27.3.5 and 27.3.7 (after possibly shrinking I) we may assume that there exist affine open coverings Xi = V1,i ∪ . . . ∪ Vn,i compatible with transition maps such that Vj = limi Vj,i . By Lemma 27.2.5 we can choose i so large that each hj,a comes from an element hj,a,i ∈ OXi (Vj,i ). At this point it is clear that Vj −→ Uj ×Spec(Z) Vj,i = (Vj,i )Uj ⊂ (Vj,i )S ⊂ Xi,S is a closed immersion. Since the union of the schemes which appear as the targets of these morphisms form an open of Xi,S we win.  Remark 27.5.2. We cannot do better than this if we do not assume more on S and the morphism f : X → S. For example, in general it will not be possible to find a closed immersion X → X 0 as in the lemma. The reason is that this would imply that f is quasi-compact which may not be the case. An example is to take S to be infinite dimensional affine space with 0 doubled and X to be one of the two infinite dimensional affine spaces. Lemma 27.5.3. Let f : X → S be a morphism of schemes. Assume: (1) The morphism f is of locally of finite type. (2) The scheme X is quasi-compact and quasi-separated, and (3) The scheme S is quasi-separated. Then there exists a morphism of finite presentation f 0 : X 0 → S and a closed immersion X → X 0 of schemes over S. Proof. By Lemma 27.5.1 above there exists a morphism Y → S of finite presentation and an immersion i : X → Y of schemes over S. For every point x ∈ X, there exists an affine open Vx ⊂ Y such that i−1 (Vx ) → Vx is a closed immersion. Since X is quasi-compact we can find finitely may affine opens V1 , . . . , Vn ⊂ Y such that i(X) ⊂ V1 ∪ . . . ∪ Vn and i−1 (Vj ) → Vj is a closed immersion. In other words such that i : X → X 0 = V1 ∪ . . . ∪ Vn is a closed immersion of schemes over S. Since S is quasi-separated and Y is quasi-separated over S we deduce that Y is quasi-separated, see Schemes, Lemma 21.21.13. Hence the open immersion

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X 0 = V1 ∪ . . . ∪ Vn → Y is quasi-compact. This implies that X 0 → Y is of finite presentation, see Morphisms, Lemma 24.22.6. We conclude since then X 0 → Y → S is a composition of morphisms of finite presentation, and hence of finite presentation (see Morphisms, Lemma 24.22.3).  Lemma 27.5.4. Let S be a scheme. Let I be a directed partially ordered set. Let (Xi , fii0 ) be an inverse system of schemes over S indexed by I. Assume (1) the scheme S is quasi-separated, (2) each Xi is locally of finite type over S, (3) all the morphisms fii0 : Xi → Xi0 are affine, (4) all the schemes Xi are quasi-compact and quasi-separated, (5) the morphism X = limi Xi → S is separated. Then Xi → S is separated for all i large enough. Proof. Let i0 ∈ I. Note that I is nonempty as the limit is directed. For convenience write X0 = Xi0 and i0 = 0. As X0 is quasi-compact we can find finitely many affine opens U1 , . . . , Un ⊂ S such that X0 → S maps into U1 ∪. . .∪Un . Denote hi : Xi → S the structure morphism. It suffices to check that for some i ≥ 0 the morphisms h−1 i (Uj ) → Uj are separated for all j = 1, . . . , n. Since S is quasi-separated the morphisms Uj → S are quasi-compact. Hence h−1 i (Uj ) is quasi-compact and quasiseparated. In this way we reduce to the case S affine. Assume S affine. Choose a finite affine open covering X0 = V1,0 ∪ . . . ∪ Vm,0 . As usual we denote Vj,i the inverse image of Vj,0 in Xi for i ≥ 0. We also denote Vj the inverse image of Vj,0 in X. By assumption the intersections Vj1 ,i ∩ Vj2 ,i are quasi-compact opens. Since X is separated we see that Vj1 ∩ Vj2 is affine. Hence we see that Vj1 ,i ∩Vj2 ,i are all affine for i big enough by Lemma 27.3.7. After increasing i0 = 0 we may assume this holds for all i ≥ 0. By Schemes, Lemma 21.21.8 we have to show that for some i big enough the ring map OXi (Vj1 ,i ) ⊗OS (S) OXi (Vj2 ,i ) −→ OXi (Vj1 ,i ∩ Vj2 ,i ) is surjective. Since Vj,i is the inverse image of Vj,0 under the affine transition maps fi0 we see that Vj1 ,i ∩ Vj2 ,i = Vj1 ,i ×Vj1 ,0 (Vj1 ,0 ∩ Vj2 ,0 ) Choose generators xj1 ,j2 ,α ∈ OX0 (Vj1 ,0 ∩ Vj2 ,0 ) as an algebra over OX0 (Vj1 ,0 ). We can choose finitely many of these since OX0 (Vj1 ,0 ∩ Vj2 ,0 ) is a finite type OS (S)algebra, see Morphisms, Lemma 24.16.2. By the displayed equality of fibre products, the images of xj1 ,j2 ,α generate OXi (Vj1 ,i ∩ Vj2 ,i ) as an algebra over OXi (Vj1 ,i ) also. Since X is separated the ring maps OX (Vj1 ) ⊗OS (S) OX (Vj2 ,i ) −→ OX (Vj1 ∩ Vj2 ) are surjective. Hence we can find finite sums X yj1 ,j2 ,α,β ⊗ zj1 ,j2 ,α,β in the left hand side which map to the elements xj1 ,j2 ,α of the right hand side. Using Lemma 27.2.5 we may choose i large enough so that each of the (finitely many) elements yj1 ,j2 ,α,β (resp. zj1 ,j2 ,α,β ) comes from a corresponding element yj1 ,j2 ,α,β,i (resp. zj1 ,j2 ,α,β,i ) of OXi (Vj1 ,i ) (resp. OXi (Vj2 ,i ) and moreover such that the image of X yj1 ,j2 ,α,β,i ⊗ zj1 ,j2 ,α,β,i

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is the image of the element xj1 ,j2 ,α in OXi (Vj1 ,i ∩ Vj2 ,i ). This clearly implies the desired surjectivity and we win.  Remark 27.5.5. Is there an easy example to show that the finite type condition for the morphisms Xi → S is necessary? Email if you have one. A less technical version of the results above is the following. Proposition 27.5.6. Let f : X → S be a morphism of schemes. Assume: (1) The morphism f is of finite type and separated. (2) The scheme S is quasi-compact and quasi-separated. Then there exists a separated morphism of finite presentation f 0 : X 0 → S and a closed immersion X → X 0 of schemes over S. Proof. We have seen that there is a closed immersion X → Y with Y /S of finite presentation. Let I ⊂ OY be the quasi-coherent sheaf of ideals defining X as a closed subscheme of Y . By Properties, Lemma 23.20.3 we can write I as a directed colimit I = colima∈A Ia of its quasi-coherent sheaves of ideals of finite type. Let Xa ⊂ Y be the closed subscheme defined by Ia . These form an inverse system of schemes indexed by A. The transition morphisms Xa → Xa0 are affine because they are closed immersions. Each Xa is quasi-compact and quasi-separated since it is a closed subscheme of Y and Y is quasi-compact and quasi-separated by our assumptions. We have X = lima Xa as follows directly from the fact that I = colima∈A Ia . Each of the morphisms Xa → Y is of finite presentation, see Morphisms, Lemma 24.22.7. Hence the morphisms Xa → S are of finite presentation. Thus it suffices to show that Xa → S is separated for some a ∈ A. This follows from Lemma 27.5.4 as we have assumed that X → S is separated.  We end this section with a variant concerning finite morphisms. Lemma 27.5.7. Let f : X → S be a morphism of schemes. Assume: (1) The morphism f is finite. (2) The scheme S is quasi-compact and quasi-separated. Then there exists a morphism which is finite and of finite presentation f 0 : X 0 → S and a closed immersion X → X 0 of schemes over S. Proof. By Proposition 27.5.6 there is a closed immersion X → Y with g : Y → S separated and of finite presentation. Let I ⊂ OY be the quasi-coherent sheaf of ideals defining X as a closed subscheme of Y . By Properties, Lemma 23.20.3 we can write I as a directed colimit I = colima∈A Ia of its quasi-coherent sheaves of ideals of finite type. Let Xa ⊂ Y be the closed subscheme defined by Ia and denote fa : Xa → S the structure morphism. These form an inverse system of schemes indexed by A. The transition morphisms Xa → Xa0 are affine because they are closed immersions. Each Xa is quasi-compact and separated over S since it is a closed subscheme of Y and Y is quasi-compact and separated over S. We have X = lima Xa as follows directly from the fact that I = colima∈A Ia . Each of the morphisms Xa → Y is of finite presentation, see Morphisms, Lemma 24.22.7. Hence the morphisms Xa → S are of finite presentation. Thus it suffices to show that fa : Xa → S is finite for some a ∈ A. S Choose a finite affine open covering S = j=1,...,n Vj . For each j the scheme f −1 (Vj ) = lima fa−1 (Vj ) is affine (as a finite morphism is affine by definition).

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Hence by Lemma 27.3.7 there exists an a ∈ A such that each fa−1 (Vj ) is affine. In other words, fa : Xa → S is affine, see Morphisms, Lemma 24.13.3. By replacing Y with Xa we may assume g : Y → S is affine. For each j = 1, . . . , m the ring OY (g −1 (Vj )) is a finitely presented OS (Vj )-algebra. Say it is generated by xji , i = 1, . . . , nj . Note that the images of xji in OX (fa−1 (Vj )), resp. OX (f −1 (Vj )) generate over OS (Vj ) as well. Since f : X → S is finite, the image of xji in OX (f −1 (Vj )) satisfies a monic polynomial Pij whose coefficients are elements of OS (Vj ). Since OX (f −1 (Vj )) = colima∈A OXa (fa−1 (Vj )) we see there exists an a ∈ A such that Pji (xij ) maps to zero in OXa (fa−1 (Vj )) for all j, i. It follows from Morphisms, Lemma 24.44.3 that the morphism fa : Xa → S is finite for this a.  27.6. Descending relative objects The following lemma is typical of the type of results in this section. We write out the “standard” proof completely. It may be faster to convince yourself that the result is true than to read this proof. Lemma 27.6.1. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) the morphisms fii0 : Si → Si0 are affine, (2) the schemes Si are quasi-compact and quasi-separated. Let S = limi Si . Then we have the following: (1) For any morphism of finite presentation X → S there exists an index i ∈ I and a morphism of finite presentation Xi → Si such that X ∼ = Xi,S as schemes over S. (2) Given an index i ∈ I, schemes Xi , Yi of finite presentation over Si , and a morphism ϕ : Xi,S → Yi,S over S, there exists an index i0 ≥ i and a morphism ϕi0 : Xi,Si0 → Yi,Si0 whose base change to S is ϕ. (3) Given an index i ∈ I, schemes Xi , Yi of finite presentation over Si and a pair of morphisms ϕi , ψi : Xi → Yi whose base changes ϕi,S = ψi,S are equal, there exists an index i0 ≥ i such that ϕi,Si0 = ψi,Si0 . In other words, the category of schemes of finite presentation over S is the colimit over I of the categories of schemes of finite presentation over Si . Proof. In case each of the schemes Si is affine, and we consider only affine schemes of finite presentation over Si , resp. S this lemma is equivalent to Algebra, Lemma 7.119.6. We claim that the affine case implies the lemma in general. Let us prove (3). Suppose given an index i ∈ I, schemes Xi , Yi of finite presentation over Si and a pair of morphisms ϕi , ψi : Xi → Yi . Assume that the base changes are equal: ϕi,S = ψi,S . We will use the notation Xi0 = Xi,Si0 and Yi0 = Yi,Si0 for i0 ≥ i. We also set X = Xi,S and Y = Yi,S . Note that according to Lemma 27.2.4 we have X = limi0 ≥i Xi0 and similarly for Y . Additionally we denote ϕi0 and ψi0 (resp. ϕ and ψ) the base change of ϕi and ψi to Si0 (resp. S). So our assumption means that ϕ = ψ. Since Yi and Xi are of finite presentation over Si , and since Si is quasi-compact and quasi-separated, also Xi and Yi are quasicompact and quasi-separated (see Morphisms, Lemma 24.22.10). Hence we may S choose a finite affine open covering Yi = Vj,i such that each Vj,i maps into an affine open of S. As above, denote Vj,i0 the inverse image of Vj,i in Yi0 and Vj

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the inverse image in Y . The immersions Vj,i0 → Yi0 are quasi-compact, and the −1 0 inverse images Uj,i0 = ϕ−1 i (Vj,i0 ) and Uj,i0 = ψi (Vj,i0 ) are quasi-compact opens of Xi0 . By assumption the inverse images of Vj under ϕ and ψ in X are equal. 0 Hence by Lemma 27.3.5 there exists an index i0 ≥ i such that of Uj,i0 = Uj,i 0 in S 0 Xi0 . Choose an finite affine open covering Uj,i0 = Uj,i0 = Wj,k,i0 which induce S 0 coverings Uj,i00 = Uj,i Wj,k,i00 for all i00 ≥ i0 . By the affine case there exists 00 = 00 00 an index i such that ϕi |Wj,k,i00 = ψi00 |Wj,k,i00 for all j, k. Then i00 is an index such that ϕi00 = ψi00 and (3) is proved. Let us prove (2). Suppose given an index i ∈ I, schemes Xi , Yi of finite presentation over Si and a morphism ϕ : Xi,S → Yi,S . We will use the notation Xi0 = Xi,Si0 and Yi0 = Yi,Si0 for i0 ≥ i. We also set X = Xi,S and Y = Yi,S . Note that according to Lemma 27.2.4 we have X = limi0 ≥i Xi0 and similarly for Y . Since Yi and Xi are of finite presentation over Si , and since Si is quasi-compact and quasiseparated, also Xi and Yi are quasi-compact and quasi-separated (see Morphisms, S Lemma 24.22.10). Hence we may choose a finite affine open covering Yi = Vj,i such that each Vj,i maps into an affine open of S. As above, denote Vj,i0 the inverse image of Vj,i in Yi0 and Vj the inverse image in Y . The immersions Vj → Y are quasi-compact, and the inverse images Uj = ϕ−1 (Vj ) are quasi-compact opens of X. Hence by Lemma 27.3.5 there exists an index i0 ≥ i and quasi-compact opens Uj,i0 of S Xi0 whose inverse image in X is Uj . Choose an finite S affine open covering 00 Uj,i0 = Wj,k,i0 which induce affine open coverings U = Wj,k,i00 for all i00 ≥ i0 j,i S and an affine open covering Uj = Wj,k . By the affine case there exists an index i00 and morphisms ϕj,k,i00 : Wj,k,i00 → Vj,i00 such that ϕ|Wj,k = ϕj,k,i00 ,S for all j, k. By part (3) proved above, there is a further index i000 ≥ i00 such that ϕj1 ,k1 ,i00 ,Si000 |Wj1 ,k1 ,i000 ∩Wj2 ,k2 ,i000 = ϕj2 ,k2 ,i00 ,Si000 |Wj1 ,k1 ,i000 ∩Wj2 ,k2 ,i000 for all j1 , j2 , k1 , k2 . Then i000 is an index such that there exists a morphism ϕi000 : Xi000 → Yi000 whose base change to S gives ϕ. Hence (2) holds. Let us prove (1). Suppose given a scheme X of finite presentation over S. Since X is of finite presentation over S, and since S is quasi-compact and quasi-separated, also X is quasi-compact and quasi-separated S (see Morphisms, Lemma 24.22.10). Choose a finite affine open covering X = Uj such that each Uj maps into an affine open Vj ⊂ S. Denote Uj1 j2 = Uj1 ∩ Uj2 and Uj1 j2 j3 = Uj1 ∩ Uj2 ∩ Uj3 . By Lemmas 27.3.5 and 27.3.7 we can find an index i1 and affine opens Vj,i1 ⊂ Si1 such that each Vj is the inverse of this in S. Let Vj,i be the inverse image of Vj,i1 in Si for i ≥ i1 . By the affine case we may find an index i2 ≥ i1 and affine schemes Uj,i2 → Vj,i2 such that Uj = S ×Si2 Uj,i2 is the base change. Denote Uj,i = Si ×Si2 Uj,i2 for i ≥ i2 . By Lemma 27.3.5 there exists an index i3 ≥ i2 and open subschemes Wj1 ,j2 ,i3 ⊂ Uj1 ,i3 whose base change to S is equal to Uj1 j2 . Denote Wj1 ,j2 ,i = Si ×Si3 Wj1 ,j2 ,i3 for i ≥ i3 . By part (2) shown above there exists an index i4 ≥ i3 and morphisms ϕj1 ,j2 ,i4 : Wj1 ,j2 ,i4 → Wj2 ,j1 ,i4 whose base change to S gives the identity morphism Uj1 j2 = Uj2 j1 for all j1 , j2 . For all i ≥ i4 denote ϕj1 ,j2 ,i = idS × ϕj1 ,j2 ,i4 the base change. We claim that for some i5 ≥ i4 the system ((Uj,i5 )j , (Wj1 ,j2 ,i5 )j1 ,j2 , (ϕj1 ,j2 ,i5 )j1 ,j2 ) forms a glueing datum as in Schemes, Section 21.14. In order to see this we have to verify that for i large enough we have ϕ−1 j1 ,j2 ,i (Wj1 ,j2 ,i ∩ Wj1 ,j3 ,i ) = Wj1 ,j2 ,i ∩ Wj1 ,j3 ,i

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and that for large enough i the cocycle condition holds. The first condition follows from Lemma 27.3.5 and the fact that Uj2 j1 j3 = Uj1 j2 j3 . The second from part (1) of the lemma proved above and the fact that the cocycle condition holds for the maps id : Uj1 j2 → Uj2 j1 . Ok, so now we can use Schemes, Lemma 21.14.2 to glue the system ((Uj,i5 )j , (Wj1 ,j2 ,i5 )j1 ,j2 , (ϕj1 ,j2 ,i5 )j1 ,j2 ) to get a scheme Xi5 → Si5 . By construction the base change of Xi5 to S is formed by glueing the open affines Uj along the opens Uj1 ← Uj1 j2 → Uj2 . Hence S ×Si5 Xi5 ∼  = X as desired. Lemma 27.6.2. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that ϕi : Xi → Yi is a morphism of schemes of finite presentation over Si . If the base change of ϕi to S is affine, then there exists an index i0 ≥ i such that idSi0 × ϕi : Xi,Si0 → Yi,Si0 is affine. Proof. For i0 ≥ i denote Xi0 = Si0 ×Si Xi and similarly for Yi0 . Denote ϕi0 the base change of ϕi to Si0 .SAlso set X = S ×Si Xi , Y = S ×Si Xi , and ϕ the base change of ϕi to S. Let Yi = Vj,i be a finite affine open covering. Set Uj,i = ϕ−1 i (Vj,i ). For 0 ). Similarly i0 ≥ i we denote Vj,i0 the inverse image of Vj,i in Yi0 and Uj,i0 = ϕ−1 (V 0 j,i i we have Uj = ϕ−1 (Vj ). Then Uj = limi0 ≥i Uj,i0 (see Lemma 27.2.2). Since Uj is affine by assumption we see that each Uj,i0 is affine for i0 large enough, see Lemma 27.3.7. Thus ϕi0 is affine for i0 large enough, see Morphisms, Lemma 24.13.3.  Lemma 27.6.3. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that ϕi : Xi → Yi is a morphism of schemes of finite presentation over Si . If the base change of ϕi to S is flat, then there exists an index i0 ≥ i such that idSi0 × ϕi : Xi,Si0 → Yi,Si0 is flat. Proof. For i0 ≥ i denote Xi0 = Si0 ×Si Xi and similarly for Yi0 . Denote ϕi0 the base change of ϕi to Si0 . Also S set X = S ×Si Xi , Y = S ×Si Xi , and ϕ the base change of ϕi to S. Let Yi = j=1,...,m Vj,i be a finite affine open covering such that −1 each S Vj,i maps into some affine open of Si . For each j =0 1, . . . m let ϕi (Vj,i ) = 0 k=1,...,m(j) Uk,j,i be a finite affine open covering. For i ≥ i we denote Vj,i the inverse image of Vj,i in Yi0 and Uk,j,i0 the inverse image of Uk,j,i in Xi0 . Similarly we have Uk,j ⊂ X and Vj ⊂ Y . Then Uk,j = limi0 ≥i Uk,j,i0 and Vj = limi0 ≥i Vj (see Lemma 27.2.2). Hence we see that the lemma reduces to the case that Xi and Yi are affine and map into an affine open of Si , i.e., we may also assume that S is affine. In the affine case we reduce to the following algebra result. Suppose that R = colimi∈I Ri . For some i ∈ I suppose given a map Ai → Bi of finitely presented Ri algebras. If R ⊗Ri Ai → R ⊗Ri Bi is flat, then for some i0 ≥ i the map Ri0 ⊗Ri Ai → Ri0 ⊗Ri Bi is flat. This follows from Algebra, Lemma 7.151.1 part (3).  Lemma 27.6.4. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that ϕi : Xi → Yi is a morphism of schemes of finite presentation over Si . If the base change of ϕi to S is a finite morphism, then there exists an index i0 ≥ i such that idSi0 × ϕi : Xi,Si0 → Yi,Si0 is a finite morphism. Proof. A finite morphism is affine, see Morphisms, Definition 24.44.1. Hence by Lemma 27.6.2 above we may assume that ϕi is affine. By writing Yi as a finite union of affines we reduce to proving the result when Xi and Yi are affine and map into a common affine Wi ⊂ Si . The corresponding algebra statement follows from Algebra, Lemma 7.151.3. 

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Lemma 27.6.5. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that ϕi : Xi → Yi is a morphism of schemes of finite presentation over Si . If the base change of ϕi to S is a closed immersion, then there exists an index i0 ≥ i such that idSi0 × ϕi : Xi,Si0 → Yi,Si0 is a closed immersion. Proof. A closed immersion is affine, see Morphisms, Lemma 24.13.9. Hence by Lemma 27.6.2 above we may assume that ϕi is affine. By writing Yi as a finite union of affines we reduce to proving the result when Xi and Yi are affine and map into a common affine Wi ⊂ Si . The corresponding algebra statement is a consequence of Algebra, Lemma 7.151.4.  Lemma 27.6.6. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that Xi is a scheme of finite presentation over Si . If the base change of Xi to S is separated over S then there exists an index i0 ≥ i such that Xi,Si0 is separated over Si0 . Proof. Apply Lemma 27.6.5 to the diagonal morphism ∆Xi /Si : Xi → Xi ×Si Xi .  Lemma 27.6.7. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that ϕi : Xi → Yi is a morphism of schemes of finite presentation over Si . If the base change of ϕi to S is finite locally free (of degree d) then there exists an index i0 ≥ i such that the base change of ϕi to Si0 is finite locally free (of degree d). Proof. By Lemmas 27.6.3 and 27.6.4 we see that we may reduce to the case that ϕi is flat and finite. On the other hand, ϕi is locally of finite presentation by Morphisms, Lemma 24.22.11. Hence ϕi is finite locally free by Morphisms, Lemma 24.46.2. If moreover ϕi × S is finite locally free of degree d, then the image of Yi ×Si S → Yi is contained in the open and closed locus Wd ⊂ Yi over which ϕi has degree d. By Lemma 27.3.4 we see that for some i0  i the image of Yi0 → Yi is contained in Wd . Then the base change of ϕi to Si0 will be finite locally free of degree d.  Lemma 27.6.8. With notation and assumptions as in Lemma 27.6.1. Let 0 ∈ I. Suppose that ϕ0 : X0 → Y0 is a morphism of schemes of finite presentation over S0 . If the base change of ϕ0 to S is ´etale then there exists an index i ≥ 0 such that the base change of ϕ0 to Si is ´etale. Proof. Being ´etale is local on the source and the target (Morphisms, Lemma 24.37.2) hence we may assume all Si , Xi , Yi affine. The corresponding algebra fact is Algebra, Lemma 7.151.5.  Lemma 27.6.9. With notation and assumptions as in Lemma 27.6.1. Let 0 ∈ I. Suppose that ϕ0 : X0 → Y0 is a morphism of schemes of finite presentation over S0 . If the base change of ϕ0 to S is a monomorphism then there exists an index i ≥ 0 such that the base change of ϕ0 to Si is a monomorphism. Proof. Recall that a morphism of schemes V → W is a monomorphism if and only if the diagonal V → V ×W V is an isomorphism (Schemes, Lemma 21.23.2). Observe that X0 ×Y0 X0 is of finite presentation over S0 because morphisms of finite presentation are preserved under base change and composition, see Morphisms, Section 24.22. Hence the lemma follows from Lemma 27.6.1 by considering the morphism X0 → X0 ×Y0 X0 . 

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Lemma 27.6.10. With notation and assumptions as in Lemma 27.6.1. Let 0 ∈ I. Suppose that ϕ0 : X0 → Y0 is a morphism of schemes of finite presentation over S0 . If the base change of ϕ0 to S is surjective then there exists an index i ≥ 0 such that the base change of ϕ0 to Si is surjective. Proof. The morphism ϕ0 is of finite presentation, see Morphisms, Lemma 24.22.11. Hence E = ϕ0 (X0 ) is a constructible subset of Y0 , see Morphisms, Lemma 24.23.2. Since ϕi is the base change of ϕ0 by Yi → Y0 we see that the image of ϕi is the inverse image of E in Yi . Moreover, we know that Y → Y0 maps into E. Hence we win by Lemma 27.3.4.  Lemma 27.6.11. Let I be a directed partially ordered set. Let (Si , fii0 ) be an inverse system of schemes over I. Assume (1) all the morphisms fii0 : Si → Si0 are affine, (2) all the schemes Si are quasi-compact and quasi-separated. Let S = limi Si . Then we have the following: (1) For any sheaf of OS -modules F of finite presentation there exists an index i ∈ I and a sheaf of OSi -modules of finite presentation Fi such that F ∼ = fi∗ Ii . (2) Suppose given an index i ∈ I, sheaves of OSi -modules Fi , Gi of finite presentation and a morphism ϕ : fi∗ Fi → fi∗ Gi over S. Then there exists an index i0 ≥ i and a morphism ϕi0 : fi∗0 i Fi → fi∗0 i Gi whose base change to S is ϕ. (3) Suppose given an index i ∈ I, sheaves of OSi -modules Fi , Gi of finite presentation and a pair of morphisms ϕi , ψi : Fi → Gi . Assume that the base changes are equal: fi∗ ϕi = fi∗ ψi . Then there exists an index i0 ≥ i such that fi∗0 i ϕi = fi∗0 i ψi . In other words, the category of modules of finite presentation over S is the colimit over I of the categories modules of finite presentation over Si . Proof. Omitted. Since we have written out completely the proof of Lemma 27.6.1 above it seems wise to use this here and not completely write this proof out also. For example we can use: (1) there is an equivalence of categories between quasi-coherent OS -modules and vector bundles over S, see Constructions, Section 22.6. (2) a vector bundle V(F) → S is of finite presentation over S if and only if F is an OS -module of finite presentation. Then you can descend morphisms in terms of morphisms of the associated vectorbundles. Similarly for objects.  Lemma 27.6.12. With notation and assumptions as in Lemma 27.6.1. Let i ∈ I. Suppose that ϕi : Xi → Yi is a morphism of schemes of finite presentation over Si and that Fi is a quasi-coherent OXi -module of finite presentation. If the pullback of Fi to Xi ×Si S is flat over Yi ×Si S, then there exists an index i0 ≥ i such that the pullback of Fi to Xi ×Si Si0 is flat over Yi ×Si Si0 . Proof. (This lemma is the analogue of Lemma 27.6.3 for modules.) For i0 ≥ i denote Xi0 = Si0 ×Si Xi , Fi0 = (Xi0 → Xi )∗ Fi and similarly for Yi0 . Denote ϕi0 the base change of ϕi to Si0 . Also set X = S ×Si XSi , Y = S ×Si Xi , F = (X → Xi )∗ Fi and ϕ the base change of ϕi to S. Let Yi = j=1,...,m Vj,i be a finite affine open

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covering such that each Vj,i maps into some affine open of Si . For each j = 1, . . . m S 0 let ϕ−1 i (Vj,i ) = k=1,...,m(j) Uk,j,i be a finite affine open covering. For i ≥ i we denote Vj,i0 the inverse image of Vj,i in Yi0 and Uk,j,i0 the inverse image of Uk,j,i in Xi0 . Similarly we have Uk,j ⊂ X and Vj ⊂ YS. Then Uk,j = limi0 ≥i Uk,j,i0 and Vj = limi0 ≥i Vj (see Lemma 27.2.2). Since Xi0 = k,j Uk,j,i0 is a finite open covering it suffices to prove the lemma for each of the morphisms Uk,j,i → Vj,i and the sheaf Fi |Uk,j,i . Hence we see that the lemma reduces to the case that Xi and Yi are affine and map into an affine open of Si , i.e., we may also assume that S is affine. In the affine case we reduce to the following algebra result. Suppose that R = colimi∈I Ri . For some i ∈ I suppose given a map Ai → Bi of finitely presented Ri -algebras. Let Ni be a finitely presented Bi -module. Then, if R ⊗Ri Ni is flat over R ⊗Ri Ai , then for some i0 ≥ i the module Ri0 ⊗Ri Ni is flat over Ri0 ⊗Ri A. This is exactly the result proved in Algebra, Lemma 7.151.1 part (3).  27.7. Characterizing affine schemes If f : X → S is a surjective integral morphism of schemes such that X is an affine scheme then S is affine too. See [Con07, A.2]. Our proof relies on the Noetherian case which we stated and proved in Cohomology of Schemes, Lemma 25.15.3. See also [DG67, II 6.7.1]. Lemma 27.7.1. Let f : X → S be a morphism of schemes. Assume that f is surjective and finite, and assume that X is affine. Then S is affine. Proof. Since f is surjective and X is quasi-compact we see that S is quasi-compact. Consider the commutative diagram X  S

∆

∆

/ X ×X  / S×S

(products over Spec(Z)). Since X is separated the image of the top horizontal arrow is closed. The right vertical arrow is the composition of X × X → X × S → S × S and hence is finite (see Morphisms, Lemmas 24.44.5 and 24.44.6). Hence it is proper (see Morphisms, Lemma 24.44.10). Thus the image of ∆(X) in S × S is closed. But as X → S is surjective we conclude that ∆(S) is closed as well. Hence S is separated. By Lemma 27.5.7 there exists a factorization X → Y → S, with X → Y a closed immersion and Y → S finite and of finite presentation. Let I ⊂ OY be the quasicoherent sheaf of ideals cutting out the closed subscheme X in Y . By Properties, Lemma 23.20.3 we can write I as a directed colimit I = colima∈A Ia of its quasicoherent sheaves of ideals of finite type. Let Xa ⊂ Y be the closed subscheme defined by Ia . These form an inverse system of schemes indexed by A. The transition morphisms Xa → Xa0 are affine because they are closed immersions. Each Xa is quasi-compact and quasi-separated since it is a closed subscheme of Y and Y is quasi-compact and quasi-separated. Each of the morphisms Xa → Y is of finite presentation, see Morphisms, Lemma 24.22.7. Hence the morphisms Xa → S are of finite presentation, and also finite as the composition of a closed immersion and a finite morphism. We have X = lima Xa as follows directly from the fact

27.8. VARIANTS OF CHOW’S LEMMA

1597

that I = colima∈A Ia . Hence by Lemma 27.3.7 we see that Xa is affine for some a ∈ A. Replacing X by Xa we may assume that X → S is surjective, finite, of finite presentation and that X is affine. By Proposition 27.3.12 we may write S = limi∈I Si as a directed limits as schemes of finite type over Z. By Lemma 27.6.1 we can after shrinking I assume there exist schemes Xi → Si of finite presentation such that Xi0 = Xi ×S Si0 for i0 ≥ i and such that X = limi Xi . By Lemma 27.6.4 we may assume that Xi → Si is finite for all i ∈ I as well. By Lemma 27.3.7 once again we may assume that Xi is affine for all i ∈ I. Hence the result follows from the Noetherian case, see Cohomology of Schemes, Lemma 25.15.3.  Proposition 27.7.2. Let f : X → S be a morphism of schemes. Assume that f is surjective and integral, and assume that X is affine. Then S is affine. Proof. Since f is surjective and X is quasi-compact we see that S is quasi-compact. Consider the commutative diagram X  S

∆

∆

/ X ×X  / S×S

(products over Spec(Z)). Since X is separated the image of the top horizontal arrow is closed. The right vertical arrow is the composition of X × X → X × S → S × S and hence is integral (see Morphisms, Lemmas 24.44.5 and 24.44.6). Hence it is universally closed (see Morphisms, Lemma 24.44.7). Thus the image of ∆(X) in S × S is closed. But as X → S is surjective we conclude that ∆(S) is closed as well. Hence S is separated. Consider the sheaf A = f∗ OX . This is a quasi-coherent sheaf of OS -algebras, see Schemes, Lemma 21.24.1. By Properties, Lemma 23.20.3 we can write A = colimi Fi as a filtered colimit of finite type OS -modules. Let Ai ⊂ A be the OS subalgebra generated by Fi . Since the map of algebras OS → A is integral, we see that each Ai is a finite quasi-coherent OS -algebra. Hence Xi = SpecS (Ai ) −→ S is a finite morphism of schemes. It is clear that X = limi Xi . Hence by Lemma 27.3.7 we see that for i sufficiently large the scheme Xi is affine. Moreover, since X → S factors through each Xi we see that Xi → S is surjective. Hence we conclude that S is affine by Lemma 27.7.1.  27.8. Variants of Chow’s Lemma In this section we prove a number of variants of Chow’s lemma. The most interesting version is probably just the Noetherian case, which we stated and proved in Cohomology of Schemes, Section 25.17. Lemma 27.8.1. Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a separated morphism of finite type. Then there exists an n ≥ 0 and a

1598

diagram

27. LIMITS OF SCHEMES

Xo

π

X0

/ Pn S

 } S where X 0 → PnS is an immersion, and π : X 0 → X is proper and surjective. Proof. By Proposition 27.5.6 we can find a closed immersion X → Y where Y is separated and of finite presentation over S. Clearly, if we prove the assertion for Y , then the result follows for X. Hence we may assume that X is of finite presentation over S. Write S = limi Si as a directed limit of Noetherian schemes, see Proposition 27.3.12. By Lemma 27.6.1 we can find an index i ∈ I and a scheme Xi → Si of finite presentation so that X = S ×Si Xi . By Lemma 27.6.6 we may assume that Xi → Si is separated. Clearly, if we prove the assertion for Xi over Si , then the assertion holds for X. The case Xi → Si is treated by Cohomology of Schemes, Lemma 25.17.1.  Here is a variant of Chow’s lemma where we assume the scheme on top has finitely many irreducible components. Lemma 27.8.2. Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a separated morphism of finite type. Assume that X has finitely many irreducible components. Then there exists an n ≥ 0 and a diagram Xo

π

X0

/ Pn S

 } S where X 0 → PnS is an immersion, and π : X 0 → X is proper and surjective. Moreover, there exists an open dense subscheme U ⊂ X such that π −1 (U ) → U is an isomorphism of schemes. Proof. Let X = Z1 ∪ . . . ∪ Zn be the decomposition of X into irreducible components. Let ηj ∈ Zj be the generic point. There are (at least) two ways to proceed with the proof. The first is to redo the proof of Cohomology of Schemes, Lemma 25.17.1 using the general Properties, Lemma 23.27.4 to find suitable affine opens in X. (This is the “standard” proof.) The second is to use absolute Noetherian approximation as in the proof of Lemma 27.8.1 above. This is what we will do here. By Proposition 27.5.6 we can find a closed immersion X → Y where Y is separated and of finite presentation over S. Write S = limi Si as a directed limit of Noetherian schemes, see Proposition 27.3.12. By Lemma 27.6.1 we can find an index i ∈ I and a scheme Yi → Si of finite presentation so that Y = S ×Si Yi . By Lemma 27.6.6 we may assume that Yi → Si is separated. We have the following diagram /Y / Yi /X ηj ∈ Zj   S

 / Si

27.9. APPLICATIONS OF CHOW’S LEMMA

1599

Denote h : X → Yi the composition. For i0 ≥ i write Yi0 = Si0 ×Si Yi . Then Y = limi0 ≥i Yi0 , see Lemma 27.2.4. Choose j, j 0 ∈ {1, . . . , n}, j 6= j 0 . Note that ηj is not a specialization of ηj 0 . By Lemma 27.2.3 we can replace i by a bigger index and assume that h(ηj ) is not a specialization of h(ηj 0 ) for all pairs (j, j 0 ) as above. For such an index, let Y 0 ⊂ Yi be the scheme theoretic image of h : X → Yi , see Morphisms, Definition 24.6.2. The morphism h is quasi-compact as the composition of the quasi-compact morphisms X → Y and Y → Yi (which is affine). Hence by Morphisms, Lemma 24.6.3 the morphism X → Y 0 is dominant. Thus the generic points of Y 0 are all contained in the set {h(η1 ), . . . , h(ηn )}, see Morphisms, Lemma 24.8.3. Since none of the h(ηj ) is the specialization of another we see that the points h(η1 ), . . . , h(ηn ) are pairwise distinct and are each a generic point of Y 0 . We apply Cohomology of Schemes, Lemma 25.17.1 above to the morphism Y 0 → Si . This gives a diagram / Pn Y0 o π Y∗ Si  } Si such that π is proper and surjective and an isomorphism over a dense open subscheme V ⊂ Y 0 . By our choice of i above we know that h(η1 ), . . . , h(ηn ) ∈ V . Consider the commutative diagram X0

X ×Y 0 Y ∗

/ Y∗

 X

 / Y0

 S

  / Si

/ Pn Si

Note that X 0 → X is an isomorphism over the open subscheme U = h−1 (V ) which contains each of the ηj and hence is dense in X. We conclude X ← X 0 → PnS is a solution to the problem posed in the lemma.  27.9. Applications of Chow’s lemma We can use Chow’s lemma to investigate the notions of proper and separated morphisms. As a first application we have the following. Lemma 27.9.1. Let S be a scheme. Let f : X → S be a separated morphism of finite type. The following are equivalent: (1) The morphism f is proper. (2) For any morphism S 0 → S which is locally of finite type the base change XS 0 → S 0 is closed. (3) For every n ≥ 0 the morphism An × X → An × S is closed. Proof. Clearly (1) implies (2), and (2) implies (3), so we just need to show (3) implies (1). First we reduce to the case when S is affine. Assume that (3) implies (1) when the base is affine. Now let f : X → S be a separated morphism of finite

1600

27. LIMITS OF SCHEMES

type. SBeing proper is local on the base (see Morphisms, Lemma 24.42.3), so if S = α Sα is an open affine cover, and if we denote Xα := f −1 (Sα ), then it is enough to show that f |Xα : Xα → Sα is proper for all α. Since Sα is affine, if the map f |Xα satisfies (3), then it will satisfy (1) by assumption, and will be proper. To finish the reduction to the case S is affine, we must show that if f : X → S is separated of finite type satisfying (3), then f |Xα : Xα → Sα is separated of finite type satisfying (3). Separatedness and finite type are clear. To see (3), notice that An × Xα is the open preimage of An × Sα under the map 1 × f . Fix a closed set Z ⊂ An × Xα . Let Z¯ denote the closure of Z in An × X. Then for topological reasons, ¯ ∩ An × Sα = 1 × f (Z). 1 × f (Z) Hence 1 × f (Z) is closed, and we have reduced the proof of (3) ⇒ (1) to the affine case. Assume S affine, and f : X → S separated of finite type. We can apply Chow’s Lemma 27.8.1 to get π : X 0 → X proper surjective and X 0 → PnS an immersion. If X is proper over S, then X 0 → S is proper (Morphisms, Lemma 24.42.4). Since PnS → S is separated, we conclude that X 0 → PnS is proper (Morphisms, Lemma 24.42.7) and hence a closed immersion (Schemes, Lemma 21.10.4). Conversely, assume X 0 → PnS is a closed immersion. Consider the diagram: (27.9.1.1)

/ Pn S

X0 π

 X

f

 /S

All maps are a priori proper except for X → S. Hence we conclude that X → S is proper by Morphisms, Lemma 24.42.8. Therefore, we have shown that X → S is proper if and only if X 0 → PnS is a closed immersion. Assume S is affine and (3) holds, and let n, X 0 , π be as above. Since being a closed morphism is local on the base, the map X × Pn → S × Pn is closed since by (3) X × An → S × An is closed and since projective space is covered by copies of affine n-space, see Constructions, Lemma 22.13.3. By Morphisms, Lemma 24.42.5 the morphism X 0 ×S PnS → X ×S PnS = X × Pn is proper. Since Pn is separated, the projection X 0 ×S PnS = PnX 0 → X 0 will be separated as it is just a base change of a separated morphism. Therefore, the map X 0 → X 0 ×S PnS is proper, since it is a section to a separated map (see Schemes, Lemma 21.21.12). Composing all these proper morphisms X 0 → X 0 ×S PnS → X ×S PnS = X × Pn → S × Pn = PnS we see that the map X 0 → PnS is proper, and hence a closed immersion.



If the base is Noetherian we can show that the valuative criterion holds using only discrete valuation rings. First we state the result concerning separation. We will

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1601

often use solid commutative diagrams of morphisms of schemes having the following shape (27.9.1.2)

Spec(K)

/X ;

 Spec(A)

 /S

with A a valuation ring and K its field of fractions. 27.9.2. Let S be a locally Noetherian scheme. Let f : X → S be a morschemes. Assume f is locally of finite type. The following are equivalent: The morphism f is separated. For any diagram (27.9.1.2) there is at most one dotted arrow. For all diagrams (27.9.1.2) with A a discrete valuation ring there is at most one dotted arrow. (4) For any irreducible component X0 of X with generic point η ∈ X0 , for any discrete valuation ring A ⊂ K = κ(η) with fraction field K and any diagram (27.9.1.2) such that the morphism Spec(K) → X is the canonical one (see Schemes, Section 21.13) there is at most one dotted arrow.

Lemma phism of (1) (2) (3)

Proof. Clearly (1) implies (2), (2) implies (3), and (3) implies (4). It remains to show (4) implies (1). Assume (4). We begin by reducing to S affine. Being separated is a local on the base (see Schemes, Lemma 21.21.8). Hence, as in the proof of Lemma 27.9.1, if we can show that whenever X → S has (4) that the restriction Xα → Sα has (4) where Sα ⊂ S is an (affine) open subset and Xα := f −1 (Sα ), then we will be done. The generic points of the irreducible components of Xα will be the generic points of irreducible components of X, since Xα is open in X. Therefore, any two distinct dotted arrows in the diagram (27.9.2.1)

Spec(K)

/ Xα ;

 Spec(A)

 / Sα

would then give two distinct arrows in diagram (27.9.1.2) via the maps Xα → X and Sα → S, which is a contradiction. Thus we have reduced to the case S is affine. We remark that in the course of this reduction, we prove that if X → S has (4) then the restriction U → V has (4) for opens U ⊂ X and V ⊂ S with f (U ) ⊂ V . We next wish to reduce to the case X → S is finite type. Assume that we know (4) implies (1) when X is finite type. Since S is Noetherian and X is locally of finite type over S we see X is locally Noetherian as well (see Morphisms, Lemma 24.16.6). Thus, X → S is quasi-separated (see Properties, Lemma 23.5.4), and therefore we may apply the valuative criterion to check whether X is separated (see Schemes, S Lemma 21.22.2). Let X = α Xα be an affine open cover of X. Given any two dotted arrows, in a diagram (27.9.1.2), the image of the closed points of Spec A will fall in two sets Xα and Xβ . Since Xα ∪ Xβ is open, for topological reasons it must contain the image of Spec(A) under both maps. Therefore, the two dotted arrows factor through Xα ∪ Xβ → X, which is a scheme of finite type over S. Since Xα ∪ Xβ is an open subset of X, by our previous remark, Xα ∪ Xβ satisfies (4),

1602

27. LIMITS OF SCHEMES

so by assumption, is separated. This implies the two given dotted arrows are the same. Therefore, we have reduced to X → S is finite type. Assume X → S of finite type and assume (4). Since X → S is finite type, and S is an affine Noetherian scheme, X is also Noetherian (see Morphisms, Lemma 24.16.6). Therefore, X → X ×S X will be a quasi-compact immersion of Noetherian schemes. We proceed by contradiction. Assume that X → X ×S X is not closed. Then, there is some y ∈ X ×S X in the closure of the image that is not in the image. As X is Noetherian it has finitely many irreducible components. Therefore, y is in the closure of the image of one of the irreducible components X0 ⊂ X. Give X0 the reduced induced structure. The composition X0 → X → X ×S X factors through the closed subscheme X0 ×S X0 ⊂ X ×S X. Denote the closure of ∆(X0 ) ¯ 0 (again as a reduced closed subscheme). Thus y ∈ X ¯ 0 . Since in X0 ×S X0 by X ¯ X0 → X0 ×S X0 is an immersion, the image of X0 will be open in X0 . Hence X0 ¯ 0 are birational. Since X ¯ 0 is a closed subscheme of a Noetherian scheme, and X it is Noetherian. Thus, the local ring OX¯ 0 ,y is a local Noetherian domain with fraction field K equal to the function field of X0 . By the Krull-Akizuki theorem (see Algebra, Lemma 7.111.11), there exists a discrete valuation ring A dominating OX¯ 0 ,y with fraction field K. This allows to to construct a diagram: (27.9.2.2)

Spec K

/ 8 X0

 A

 / X0 ×S X0

∆

which sends Spec K to the generic point of ∆(X0 ) and the closed point of A to y ∈ X0 ×S X0 (use the material in Schemes, Section 21.13 to construct the arrows). There cannot even exist a set theoretic dotted arrow, since y is not in the image of ∆ by our choice of y. By categorical means, the existence of the dotted arrow in the above diagram is equivalent to the uniqueness of the dotted arrow in the following diagram: (27.9.2.3)

Spec K

/ X0 :

 A

 /S

Therefore, we have non-uniqueness in this latter diagram by the nonexistence in the first. Therefore, X0 does not satisfy uniqueness for discrete valuation rings, and since X0 is an irreducible component of X, we have that X → S does not satisfy (4). Therefore, we have shown (4) implies (1).  Lemma 27.9.3. Let S be a locally Noetherian scheme. Let f : X → S be a morphism of finite type. The following are equivalent: (1) The morphism f is proper. (2) For any diagram (27.9.1.2) there exists exactly one dotted arrow. (3) For all diagrams (27.9.1.2) with A a discrete valuation ring there exists exactly one dotted arrow. (4) For any irreducible component X0 of X with generic point η ∈ X0 , for any discrete valuation ring A ⊂ K = κ(η) with fraction field K and any

27.10. UNIVERSALLY CLOSED MORPHISMS

1603

diagram (27.9.1.2) such that the morphism Spec(K) → X is the canonical one (see Schemes, Section 21.13) there exists exactly one dotted arrow. Proof. (1) implies (2) implies (3) implies (4). We will now show (4) implies (1). As in the proof of Lemma 27.9.2, we can reduce to the case S is affine, since properness is local on the base, and if X → S satisfies (4), then Xα → Sα does as well for open Sα ⊂ S and Xα = f −1 (Sα ). Now S is a Noetherian scheme, and so X is as well, since X → S is of finite type. Now we may use Chow’s lemma (Cohomology of Schemes, Lemma 25.17.1) to get a surjective, proper, birational X 0 → X and an immersion X 0 → PnS . We wish to show X → S is universally closed. As in the proof of Lemma 27.9.1, it is enough to check that X 0 → PnS is a closed immersion. For the sake of contradiction, assume that X 0 → PnS is not a closed immersion. Then there is some y ∈ PnS that is in the closure of the image of X 0 , but is not in the image. So y is in the closure of the ¯ 0 ⊂ Pn image of an irreducible component X00 of X 0 , but not in the image. Let X 0 S 0 0 n be the closure of the image of X0 . As X → PS is an immersion of Noetherian ¯ 0 is open and dense. By Algebra, Lemma 7.111.11 schemes, the morphism X00 → X 0 or Properties, Lemma 23.5.9 we can find a discrete valuation ring A dominating OX¯ 00 ,y and with identical field of fractions K. It is clear that K is the residue field at the generic point of X00 . Thus the solid commutative diagram (27.9.3.1)

Spec K

/ X0 ;

n 6/ P S

 Spec A

 /X

 /S

Note that the closed point of A maps to y ∈ PnS . By construction, there does not exist a set theoretic lift to X 0 . As X 0 → X is birational, the image of X00 in X is an irreducible component X0 of X and K is also identified with the function field of X0 . Hence, as X → S is assumed to satisfy (4), the dotted arrow Spec(A) → X exists. Since X 0 → X is proper, the dotted arrow lifts to the dotted arrow Spec(A) → X 0 (use Schemes, Proposition 21.20.6). We can compose this with the immersion X 0 → PnS to obtain another morphism (not depicted in the diagram) from Spec(A) → PnS . Since PnS is proper over S, it satisfies (2), and so these two morphisms agree. This is a contradiction, for we have constructed the forbidden lift of our original map Spec(A) → PnS to X 0 . 

27.10. Universally closed morphisms In this section we discuss when a quasi-compact but not necessarily separated morphism is universally closed. We first prove a lemma which will allow us to check universal closedness after a base change which is locally of finite presentation. Lemma 27.10.1. Let f : X → S be a quasi-compact morphism of schemes. Let g : T → S be a morphism of schemes. Let t ∈ T be a point and Z ⊂ XT be a closed subscheme such that Z ∩ Xt = ∅. Then there exists an open neighbourhood V ⊂ T

1604

27. LIMITS OF SCHEMES

of t, a commutative diagram V

a

/ T0 b

 T

g

 / S,

and a closed subscheme Z 0 ⊂ XT 0 such that (1) the morphism b : T 0 → S is locally of finite presentation, (2) with t0 = a(t) we have Z 0 ∩ Xt0 = ∅, and (3) Z ∩ XV maps into Z 0 via the morphism XV → XT 0 . Proof. Let s = g(t). During the proof we may always replace T by an open neighbourhood of t. Hence we may also replace S by an open neighbourhood of s. Thus we may and do assume that T and S are affine. Say S = Spec(A), T = Spec(B), g is given by the ring map A → B, and t correspond to the prime ideal q ⊂ B. S As X → S is quasi-compact and S is affine we may write X = i=1,...,n Ui as a finite S union of affine S opens. Write Ui = Spec(Ci ). In particular we have XT = i=1,...,n Ui,T = i=1,...n Spec(Ci ⊗A B). Let Ii ⊂ Ci ⊗A B be the ideal corresponding to the closed subscheme Z ∩ Ui,T . The condition that Z ∩ Xt = ∅ signifies that Ii generates the unit ideal in the ring Ci ⊗A κ(q) = (B \ q)−1 (Ci ⊗A B/qCi ⊗A B) Since Ii (B \ q)−1 (Ci ⊗A B) = (B \ q)−1 Ii this means that 1 = xi /gi for some xi ∈ Ii and gi ∈ B, gi 6∈ q. Thus, clearing denominators we can find a relation of the form X xi + fi,j ci,j = gi j

with xi ∈ Ii , fi,j ∈ q, ci,j ∈ Ci ⊗A B, and gi ∈ B, gi 6∈ q. After replacing B by Bg1 ...gn , i.e., after replacing T by a smaller affine neighbourhood of t, we may assume the equations read X xi + fi,j ci,j = 1 j

with xi ∈ Ii , fi,j ∈ q, ci,j ∈ Ci ⊗A B. To finish the argument write B as a colimit of finitely presented A-algebras Bλ over a directed partially ordered set Λ. For each λ set qλ = (Bλ → B)−1 (q). For sufficiently large λ ∈ Λ we can find (1) an element xi,λ ∈ Ci ⊗A Bλ which maps to xi , (2) elements fi,j,λ ∈ qi,λ mapping to fi,j , and (3) elements ci,j,λ ∈ Ci ⊗A Bλ mapping to ci,j . After increasing λ a bit more the equation X xi,λ + fi,j,λ ci,j,λ = 1 j

0

will hold. Fix such a λ and set T = Spec(Bλ ). Then t0 ∈ T 0 is the point corresponding to the prime qλ . Finally, let Z 0 ⊂ XT 0 be the scheme theoretic closure of Z → XT → XT 0 . As XT → XT 0 is affine, we can compute Z 0 on the affine open pieces Ui,T 0 as the closed subscheme associated to Ker(Ci ⊗A Bλ → Ci ⊗A B/Ii ), see Morphisms, Example 24.6.4. Hence xi,λ is in the ideal defining Z 0 . Thus the last displayed equation shows that Z 0 ∩ Xt0 is empty. 

27.10. UNIVERSALLY CLOSED MORPHISMS

1605

Lemma 27.10.2. Let f : X → S be a quasi-compact morphism of schemes. The following are equivalent (1) f is universally closed, (2) for every morphism S 0 → S which is locally of finite presentation the base change XS 0 → S 0 is closed, and (3) for every n the morphism An × X → An × S is closed. Proof. It is clear that (1) implies (2). Let us prove that (2) implies (1). Suppose that the base change XT → T is not closed for some scheme T over S. By Schemes, Lemma 21.19.8 this means that there exists some specialization t1 t in T and a point ξ ∈ XT mapping to t1 such that ξ does not specialize to a point in the fibre over t. Set Z = {ξ} ⊂ XT . Then Z ∩ Xt = ∅. Apply Lemma 27.10.1. We find an open neighbourhood V ⊂ T of t, a commutative diagram V  T

a

/ T0 b

g

 / S,

and a closed subscheme Z 0 ⊂ XT 0 such that (1) the morphism b : T 0 → S is locally of finite presentation, (2) with t0 = a(t) we have Z 0 ∩ Xt0 = ∅, and (3) Z ∩ XV maps into Z 0 via the morphism XV → XT 0 . Clearly this means that XT 0 → T 0 maps the closed subset Z 0 to a subset of T 0 which contains a(t1 ) but not t0 = a(t). Since a(t1 ) a(t) = t0 we conclude that 0 XT 0 → T is not closed. Hence we have shown that X → S not universally closed implies that XT 0 → T 0 is not closed for some T 0 → S which is locally of finite presentation. In order words (2) implies (1). Assume that An ×X → An ×S is closed for every integer n. We want to prove that XT → T is closed for every scheme T which is locally of finite presentation over S. We may of course assume that T is affine and maps into an affine open V of S (since XT → T being a closed is local on T ). In this case there exists a closed immersion T → An × V because OT (T ) is a finitely presented OS (V )-algebra, see Morphisms, Lemma 24.22.2. Then T → An × S is a locally closed immersion. Hence we get a cartesian diagram / An × X XT fT

 T

fn

 / An × S

of schemes where the horizontal arrows are locally closed immersions. Hence any closed subset Z ⊂ XT can be written as XT ∩Z 0 for some closed subset Z 0 ⊂ An ×X. Then fT (Z) = T ∩ fn (Z 0 ) and we see that if fn is closed, then also fT is closed.  Lemma 27.10.3. Let f : X → S be a finite type morphism of schemes. Assume S is locally Noetherian. Then the following are equivalent (1) f is universally closed, (2) for every n the morphism An × X → An × S is closed, (3) for any diagram (27.9.1.2) there exists some dotted arrow,

1606

27. LIMITS OF SCHEMES

(4) for all diagrams (27.9.1.2) with A a discrete valuation ring there exists some dotted arrow. Proof. The equivalence of (1) and (2) is a special case of Lemma 27.10.2. The equivalence of (1) and (3) is a special case of Schemes, Proposition 21.20.6. Trivially (3) implies (4). Thus all we have to do is prove that (4) implies (2). We will prove that An × X → An × S is closed by the criterion of Schemes, Lemma 21.19.8. Pick n and a specialization z z 0 of points in An ×S and a point y ∈ An ×X lying over z. Note that κ(y) is a finitely generated field extension of κ(z) as An ×X → An ×S is of finite type. Hence by Properties, Lemma 23.5.9 or Algebra, Lemma 7.111.11 implies that there exists a discrete valuation ring A ⊂ κ(y) with fraction field κ(z) dominating the image of OAn ×S,z0 in κ(z). This gives a commutative diagram Spec(κ(y))

/ An × X

/X

 Spec(A)

 / An × S

 /S

Now property (4) implies that there exists a morphism Spec(A) → X which fits into this diagram. Since we already have the morphism Spec(A) → An from the left lower horizontal arrow we also get a morphism Spec(A) → An × X fitting into the left square. Thus the image y 0 ∈ An × X of the closed point is a specialization of y lying over z 0 . This proves that specializations lift along An × X → An × S and we win.  27.11. Limits and dimensions of fibres The following lemma is most often used in the situation of Lemma 27.6.1 to assure that if the fibres of the limit have dimension ≤ d, then the fibres at some finite stage have dimension ≤ d. Lemma 27.11.1. Let I be a directed partially ordered set. Let (fi : Xi → Si ) be an inverse system of morphisms of schemes over I. Assume (1) all the morphisms Si0 → Si are affine, (2) all the schemes Si are quasi-compact and quasi-separated, (3) the morphisms fi are of finite type, and (4) the morphisms Xi0 → Xi ×Si Si0 are closed immersions. Let f : X = limi Xi → S = limi Si be the limit. Let d ≥ 0. If every fibre of f has dimension ≤ d, then for some i every fibre of fi has dimension ≤ d. Proof. For each i let Ui = {x ∈ Xi | dimx ((Xi )fi (x) ) ≤ d}. This is an open subset of Xi , see Morphisms, Lemma 24.29.4. Set Zi = Xi \ Ui (with reduced induced scheme structure). We have to show that Zi = ∅ for some i. If not, then Z = lim Zi 6= ∅, see Lemma 27.3.1. Say z ∈ Z is a point. Note that Z ⊂ X is a closed subscheme. Set s = f (z). For each i let si ∈ Si be the image of s. We remark that Zs is the limit of the schemes (Zi )si and Zs is also the limit of the schemes (Zi )si base changed to κ(s). Moreover, all the morphisms Zs −→ (Zi0 )si0 ×Spec(κ(si0 )) Spec(κ(s)) −→ (Zi )si ×Spec(κ(si )) Spec(κ(s)) −→ Xs are closed immersions by assumption (4). Hence Zs is the scheme theoretic intersection of the closed subschemes (Zi )si ×Spec(κ(si )) Spec(κ(s)) in Xs . Since all the

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irreducible components of the schemes (Zi )si ×Spec(κ(si )) Spec(κ(s)) have dimension > d and contain z we conclude that Zs contains an irreducible component of dimension > d passing through z which contradicts the fact that Zs ⊂ Xs and dim(Xs ) ≤ d.  Lemma 27.11.2. Let S be a quasi-compact and quasi-separated scheme. Let f : X → S be a morphism of finite presentation. Let d ≥ 0 be an integer. If Z ⊂ X be a closed subscheme such that dim(Zs ) ≤ d for all s ∈ S, then there exists a closed subscheme Z 0 ⊂ X such that (1) Z ⊂ Z 0 , (2) Z 0 → X is of finite presentation, and (3) dim(Zs0 ) ≤ d for all s ∈ S. Proof. By Proposition 27.3.12 we can write S = lim Si as the limit of a directed inverse system of Noetherian schemes with affine transition maps. By Lemma 27.6.1 we may assume that there exist a system of morphisms fi : Xi → Si of finite presentation such that Xi0 = Xi ×Si Si0 for all i0 ≥ i and such that X = Xi ×Si S. Let Zi ⊂ Xi be the scheme theoretic image of Z → X → Xi . Then for i0 ≥ i the morphism Xi0 → Xi maps Zi0 into Zi and the induced morphism Zi0 → Zi ×Si Si0 is a closed immersion. By Lemma 27.11.1 we see that the dimension of the fibres of Zi → Si all have dimension ≤ d for a suitable i ∈ I. Fix such an i and set Z 0 = Zi ×Si S ⊂ X. Since Si is Noetherian, we see that Xi is Noetherian, and hence the morphism Zi → Xi is of finite presentation. Therefore also the base change Z 0 → X is of finite presentation. Moreover, the fibres of Z 0 → S are base changes of the fibres of Zi → Si and hence have dimension ≤ d.  27.12. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes

(22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42)

Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces

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(43) Decent Algebraic Spaces (44) Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces (46) Topologies on Algebraic Spaces (47) Descent and Algebraic Spaces (48) More on Morphisms of Spaces (49) Quot and Hilbert Spaces (50) Spaces over Fields (51) Stacks (52) Formal Deformation Theory (53) Groupoids in Algebraic Spaces (54) More on Groupoids in Spaces (55) Bootstrap (56) Examples of Stacks (57) Quotients of Groupoids (58) Algebraic Stacks

(59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 28

Varieties 28.1. Introduction In this chapter we start studying varieties and more generally schemes over a field. A fundamental reference is [DG67]. 28.2. Notation Throughout this chapter we use the letter k to denote the ground field. 28.3. Varieties In the stacks project we will use the following as our definition of a variety. Definition 28.3.1. Let k be a field. A variety is a scheme X over k such that X is integral and the structure morphism X → Spec(k) is separated and of finite type. This definition has the following drawback. Suppose that k ⊂ k 0 is an extension of fields. Suppose that X is a variety over k. Then the base change Xk0 = X ×Spec(k) Spec(k 0 ) is not necessarily a variety over k 0 . This phenomenon (in greater generality) will be discussed in detail in the following sections. The product of two varieties need not be a variety (this is really the same phenomenon). Here is an example. Example 28.3.2. Let k = Q. Let X = Spec(Q(i)) and Y = Spec(Q(i)). Then the product X ×Spec(k) Y of the varieties X and Y is not a variety, since it is reducible. (It is isomorphic to the disjoint union of two copies of X.) If the ground field is algebraically closed however, then the product of varieties is a variety. This follows from the results in the algebra chapter, but there we treat much more general situations. There is also a simple direct proof of it which we present here. Lemma 28.3.3. Let k be an algebraically closed field. Let X, Y be varieties over k. Then X ×Spec(k) Y is a variety over k. Proof. The morphism X ×Spec(k) Y → Spec(k) is of finite type and separated because it is the composition of the morphisms X ×Spec(k) Y → Y → Spec(k) which are separated and of finite type, see Morphisms, Lemmas 24.16.4 and 24.16.3 and Schemes, Lemma 21.21.13. To finish theSproof it suffices to show that X ×Spec(k) Y S is integral. Let X = i=1,...,n Ui , Y = j=1,...,m Vj be finite affine open coverings. If we can show that each Ui ×Spec(k) Vj is integral, then we are done by Properties, Lemmas 23.3.2, 23.3.3, and 23.3.4. This reduces us to the affine case. 1609

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The affine case translates into the following algebra statement: Suppose that A, B are integral domains and finitely generated k-algebras. Then A ⊗k B is an integral domain. To get a contradiction suppose that X X ( ai ⊗ bi )( cj ⊗ dj ) = 0 i=1,...,n

j=1,...,m

in A ⊗k B with both factors nonzero in A ⊗k B. We may assume that b1 , . . . , bn are k-linearly independent in B, and that d1 , . . . , dm are k-linearly independent in B. Of course we may also assume that a1 and c1 are nonzero in A. Hence D(a1 c1 ) ⊂ Spec(A) is nonempty. By the Hilbert Nullstellensatz (Algebra, Theorem 7.31.1) we can find a maximal ideal m ⊂ A contained in D(a1 c1 ) and A/m = k as k is algebraically closed. Denote ai , cj the residue classes of ai , cj in A/m = k. Then equation above becomes X X ( ai bi )( cj dj ) = 0 i=1,...,n

j=1,...,m

which is a contradiction with m ∈ D(a1 c1 ), the linear independence of b1 , . . . , bn and d1 , . . . , dm , and the fact that B is a domain.  28.4. Geometrically reduced schemes If X is a reduced scheme over a field, then it can happen that X becomes nonreduced after extending the ground field. This does not happen for geometrically reduced schemes. Definition 28.4.1. Let k be a field. Let X be a scheme over k. Let x ∈ X be a point. (1) Let x ∈ X be a point. We say X is geometrically reduced at x if for any field extension k ⊂ k 0 and any point x0 ∈ Xk0 lying over x the local ring OXk0 ,x0 is reduced. (2) We say X is geometrically reduced over k if X is geometrically reduced at every point of X. This may seem a little mysterious at first, but it is really the same thing as the notion discussed in the algebra chapter. Here are some basic results explaining the connection. Lemma 28.4.2. Let k be a field. Let X be a scheme over k. Let x ∈ X. The following are equivalent (1) X is geometrically reduced at x, and (2) the ring OX,x is geometrically reduced over k (see Algebra, Definition 7.41.1). Proof. Assume (1). This in particular implies that OX,x is reduced. Let k ⊂ k 0 be a finite purely inseparable field extension. Consider the ring OX,x ⊗k k 0 . By Algebra, Lemma 7.44.2 its spectrum is the same as the spectrum of OX,x . Hence it is a local ring also (Algebra, Lemma 7.17.2). Therefore there is a unique point x0 ∈ Xk0 lying over x and OXk0 ,x0 ∼ = OX,x ⊗k k 0 . By assumption this is a reduced ring. Hence we deduce (2) by Algebra, Lemma 7.42.3. Assume (2). Let k ⊂ k 0 be a field extension. Since Spec(k 0 ) → Spec(k) is surjective, also Xk0 → X is surjective (Morphisms, Lemma 24.11.4). Let x0 ∈ Xk0 be any point lying over x. The local ring OXk0 ,x0 is a localization of the ring OX,x ⊗k k 0 . Hence it is reduced by assumption and (1) is proved. 

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The notion isn’t interesting in characteristic zero. Lemma 28.4.3. Let X be a scheme over a perfect field k (e.g. k has characteristic zero). Let x ∈ X. If OX,x is reduced, then X is geometrically reduced at x. If X is reduced, then X is geometrically reduced over k. Proof. The first statement follows from Lemma 28.4.2 and Algebra, Lemma 7.41.6 and the definition of a perfect field (Algebra, Definition 7.43.1). The second statement follows from the first.  Lemma 28.4.4. Let k be a field of characteristic p > 0. Let X be a scheme over k. The following are equivalent (1) X is geometrically reduced, (2) Xk0 is reduced for every field extension k ⊂ k 0 , (3) Xk0 is reduced for every finite purely inseparable field extension k ⊂ k 0 , (4) Xk1/p is reduced, (5) Xkperf is reduced, (6) Xk¯ is reduced, (7) for every affine open U ⊂ X the ring OX (U ) is geometrically reduced (see Algebra, Definition 7.41.1). Proof. Assume (1). Then for every field extension k ⊂ k 0 and every point x0 ∈ Xk0 the local ring of Xk0 at x0 is reduced. In other words Xk0 is reduced. Hence (2). Assume (2). Let U ⊂ X be an affine open. Then for every field extension k ⊂ k 0 the scheme Xk0 is reduced, hence Uk0 = Spec(O(U ) ⊗k k 0 ) is reduced, hence O(U ) ⊗k k 0 is reduced (see Properties, Section 23.3). In other words O(U ) is geometrically reduced, so (7) holds. Assume (7). For any field extension k ⊂ k 0 the base change Xk0 is gotten by gluing the spectra of the rings OX (U ) ⊗k k 0 where U is affine open in X (see Schemes, Section 21.17). Hence Xk0 is reduced. So (1) holds. This proves that (1), (2), and (7) are equivalent. These are equivalent to (3), (4), (5), and (6) because we can apply Algebra, Lemma 7.42.3 to OX (U ) for U ⊂ X affine open.  Lemma 28.4.5. Let k be a field of characteristic p > 0. Let X be a scheme over k. Let x ∈ X. The following are equivalent (1) X is geometrically reduced at x, (2) OXk0 ,x0 is reduced for every finite purely inseparable field extension k 0 of k and x0 ∈ Xk0 the unique point lying over x, (3) OXk1/p ,x0 is reduced for x0 ∈ Xk0 the unique point lying over x, and (4) OXkperf ,x0 is reduced for x0 ∈ Xkperf the unique point lying over x. Proof. Note that if k ⊂ k 0 is purely inseparable, then Xk0 → X induces a homeomorphism on underlying topological spaces, see Algebra, Lemma 7.44.2. Whence the uniqueness of x0 lying over x mentioned in the statement. Moreover, in this case OXk0 ,x0 = OX,x ⊗k k 0 . Hence the lemma follows from Lemma 28.4.2 above and Algebra, Lemma 7.42.3.  Lemma 28.4.6. Let k be a field. Let X be a scheme over k. Let k 0 /k be a field extension. Let x ∈ X be a point, and let x0 ∈ Xk0 be a point lying over x. The following are equivalent

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(1) X is geometrically reduced at x, (2) Xk0 is geometrically reduced at x0 . In particular, X is geometrically reduced over k if and only if Xk0 is geometrically reduced over k 0 . Proof. It is clear that (1) implies (2). Assume (2). Let k ⊂ k 00 be a finite purely inseparable field extension and let x00 ∈ Xk00 be a point lying over x (actually it is unique). We can find a common field extension k ⊂ k 000 (i.e. with both k 0 ⊂ k 000 and k 00 ⊂ k 000 ) and a point x000 ∈ Xk000 lying over both x0 and x00 . Consider the map of local rings OXk00 ,x00 −→ OXk000 ,x0000 . This is a flat local ring homomorphism and hence faithfully flat. By (2) we see that the local ring on the right is reduced. Thus by Algebra, Lemma 7.147.2 we conlude that OXk00 ,x00 is reduced. Thus by Lemma 28.4.5 we conclude that X is geometrically reduced at x.  Lemma 28.4.7. Let k be a field. Let X, Y be schemes over k. (1) If X is geometrically reduced at x, and Y reduced, then X ×k Y is reduced at every point lying over x. (2) If X geometrically reduced over k and Y reduced. Then X ×k Y is reduced. Proof. Combine, Lemmas 28.4.2 and 28.4.4 and Algebra, Lemma 7.41.5.



Lemma 28.4.8. Let k be a field. Let X be a scheme over k. (1) If x0 x is a specialization and X is geometrically reduced at x, then X is geometrically reduced at x0 . (2) If x ∈ X such that (a) OX,x is reduced, and (b) for each specialization x0 x where x0 is a generic point of an irreducible component of X the scheme X is geometrically reduced at x0 , then X is geometrically reduced at x. (3) If X is reduced and geometrically reduced at all generic points of irreducible components of X, then X is geometrically reduced. Proof. Part (1) follows from Lemma 28.4.2 and the fact that if A is a geometrically reduced k-algebra, then S −1 A is a geometrically reduced k-algebra for any multiplicative subset S of A, see Algebra, Lemma 7.41.3. Let A = OX,x . The assumptions (a) and (b) of (2) imply that A is reduced, and that Aq is geometrically reduced over k for every minimal prime q of A. Hence A is geometrically reduced over k, see Algebra, Lemma 7.41.7. Thus X is geometrically reduced at x, see Lemma 28.4.2. Part (3) follows trivially from part (2).



Lemma 28.4.9. Let k be a field. Let X be a scheme over k. Let x ∈ X. Assume X locally Noetherian and geometrically reduced at x. Then there exists an open neighbourhood U ⊂ X of x which is geometrically reduced over k. Proof. Let R be a Noetherian k-algebra. Let p ⊂ R be a prime. Let I = Ker(R → Rp . Since IRp = Rp and I is finitely generated there exists an f ∈ R, f 6∈ p such that f I = 0. Hence Rf ⊂ Rp .

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Assume X locally Noetherian and geometrically reduced at x. If we apply the above to R = OX (U ) for some affine open neighbourhood of x, and p ⊂ R the prime corresponding to x, then we see that after shrinking U we may assume R ⊂ Rp . By Lemma 28.4.2 the assumption means that Rp is geometrically reduced over k. By Algebra, Lemma 7.41.2 this implies that R is geometrically reduced over k, which in turn implies that U is geometrically reduced.  Example 28.4.10. Let k = Fp (s, t), i.e., a purely transcendental extension of the prime field. Consider the variety X = Spec(k[x, y]/(1 + sxp + ty p )). Let k ⊂ k 0 be any extension such that both s and t have a pth root in k 0 . Then the base change Xk0 is not reduced. Namely, the ring k 0 [x, y]/(1 + sxp + ty p ) contains the element 1 + s1/p x + t1/p y whose pth power is zero but which is not zero (since the ideal (1 + sxp + ty p ) certainly does not contain any nonzero element of degree < p). Lemma 28.4.11. Let k be a field. Let X → Spec(k) be locally of finite type. Assume X has finitely many irreducible components. Then there exists a finite purely inseparable extension k ⊂ k 0 such that (Xk0 )red is geometrically reduced over k0 . Proof. To prove this lemma we may replace X by its reduction Xred . Hence we may assume that X is reduced and locally of finite type over k. Let x1 , . . . , xn ∈ X be the generic points of the irreducible components of X. Note that for every purely inseparable algebraic extension k ⊂ k 0 the morphism (Xk0 )red → X is a homeomorphism, see Algebra, Lemma 7.44.2. Hence the points x01 , . . . , x0n lying over x1 , . . . , xn are the generic points of the irreducible components of (Xk0 )red . As X is reduced the local rings Ki = OX,xi are fields, see Algebra, Lemma 7.24.3. As X is locally of finite type over k the field extensions k ⊂ Ki are finitely generated field extensions. Finally, the local rings OXk0 ,x0i are the fields (Ki ⊗k k 0 )red . By Algebra, Lemma 7.43.3 we can find a finite purely inseparable extension k ⊂ k 0 such that (Ki ⊗k k 0 )red are separable field extensions of k 0 . In particular each (Ki ⊗k k 0 )red is geometrically reduced over k 0 by Algebra, Lemma 7.42.1. At this point Lemma 28.4.8 part (3) implies that (Xk0 )red is geometrically reduced.  28.5. Geometrically connected schemes If X is a connected scheme over a field, then it can happen that X becomes disconnected after extending the ground field. This does not happen for geometrically connected schemes. Definition 28.5.1. Let X be a scheme over the field k. We say X is geometrically connected over k if the scheme Xk0 is connected1 for every field extension k 0 of k. Here is an example of a variety which is not geometrically connected. Example 28.5.2. Let k = Q. The scheme X = Spec(Q(i)) is a variety over Spec(Q). But the base change XC is the spectrum of C ⊗Q Q(i) ∼ = C × C which is the disjoint union of two copies of Spec(C). So in fact, this is an example of a non-geometrically connected variety. Lemma 28.5.3. Let X be a scheme over the field k. Let k ⊂ k 0 be a field extension. Then X is geometrically connected over k if and only if Xk0 is geometrically connected over k 0 . 1An empty topological space is connected.

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Proof. If X is geometrically connected over k, then it is clear that Xk0 is geometrically connected over k 0 . For the converse, note that for any field extension k ⊂ k 00 there exists a common field extension k 0 ⊂ k 000 and k 00 ⊂ k 000 . As the morphism Xk000 → Xk00 is surjective (as a base change of a surjective morphism between spectra of fields) we see that the connectedness of Xk000 implies the connectedness of Xk00 . Thus if Xk0 is geometrically connected over k 0 then X is geometrically connected over k.  Lemma 28.5.4. Let k be a field. Let X, Y be schemes over k. Assume X is geometrically connected over k. Then the projection morphism p : X ×k Y −→ Y induces a bijection between connected components. Proof. The scheme theoretic fibres of p are connected and nonempty, since they are base changes of the geometrically connected scheme X by field extensions. Moreover the scheme theoretic fibres are homeomorphic to the set theoretic fibres, see Schemes, Lemma 21.18.5. By Morphisms, Lemma 24.24.4 the map p is open. Thus we may apply Topology, Lemma 5.4.5 to conclude.  Lemma 28.5.5. Let k be a field. Let A be a k-algebra. Then X = Spec(A) is geometrically connected over k if and only if A is geometrically connected over k (see Algebra, Definition 7.45.3). Proof. Immediate from the definitions.



0

Lemma 28.5.6. Let k ⊂ k be an extension of fields. Let X be a scheme over k. Assume k separably algebraically closed. Then the morphism Xk0 → X induces a bijection of connected components. In particular, X is geometrically connected over k if and only if X is connected. Proof. Since k is separably algebraically closed we see that k 0 is geometrically connected over k, see Algebra, Lemma 7.45.4. Hence Z = Spec(k 0 ) is geometrically connected over k by Lemma 28.5.5 above. Since Xk0 = Z ×k X the result is a special case of Lemma 28.5.4.  Lemma 28.5.7. Let k be a field. Let X be a scheme over k. Let k be a separable algebraic closure of k. Then X is geometrically connected if and only if the base change Xk is connected. Proof. Assume Xk is connected. Let k ⊂ k 0 be a field extension. There exists a 0 0 field extension k ⊂ k such that k 0 embeds into k as an extension of k. By Lemma 28.5.6 we see that Xk0 is connected. Since Xk0 → Xk0 is surjective we conclude that Xk0 is connected as desired.  Lemma 28.5.8. Let k be a field. Let X be a scheme over k. Let A be a k-algebra. Let V ⊂ XA be a quasi-compact open. Then there exists a finitely generated ksubalgebra A0 ⊂ A and a quasi-compact open V 0 ⊂ XA0 such that V = VA0 . Proof. We remark that if X is also quasi-separated this follows from Limits, Lemma 27.3.5. Let U1 , . . . , Un be finitely many affine opens of X such that V ⊂ S Ui,A . Say Ui = Spec(Ri ). Since V is quasi-compact we can find finitely many S S fij ∈ Ri ⊗k A, j = 1, . . . , ni such that V = i j=1,...,ni D(fij ) where D(fij ) ⊂ Ui,A is the corresponding standard open. (We do not claim that V ∩ Ui,A is the union

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of the D(fij ), j = 1, . . . , ni .) It is clear that we can find a finitely generated 0 k-subalgebra A0 ⊂ A such that fij is the image of some fij ∈ Ri ⊗k A0 . Set S 0 0 V = D(fij ) which is a quasi-compact open of XA0 . Denote π : XA → XA0 the 0 canonical morphism. We have π(V ) ⊂ V 0 as π(D(fij )) ⊂ D(fij ). If x ∈ XA with 0 0 0 π(x) ∈ V , then π(x) ∈ D(fij ) for some i, j and we see that x ∈ D(fij ) as fij maps −1 0 to fij . Thus we see that V = π (V ) as desired.  Let k be a field. Let k ⊂ k be a (possibly infinite) Galois extension. For example k could be the separable algebraic closure of k. For any σ ∈ Gal(k/k) we get a corresponding automorphism Spec(σ) : Spec(k) −→ Spec(k). Note that Spec(σ) ◦ Spec(τ ) = Spec(τ ◦ σ). Hence we get an action Gal(k/k)opp × Spec(k) −→ Spec(k) of the opposite group on the scheme Spec(k). Let X be a scheme over k. Since Xk = Spec(k) ×Spec(k) X by definition we see that the action above induces a canonical action (28.5.8.1)

Gal(k/k)opp × Xk −→ Xk .

Lemma 28.5.9. Let k be a field. Let X be a scheme over k. Let k be a (possibly infinite) Galois extension of k. Let V ⊂ Xk be a quasi-compact open. Then (1) there exists a finite subextension k ⊂ k 0 ⊂ k and a quasi-compact open V 0 ⊂ Xk0 such that V = (V 0 )k , (2) there exists an open subgroup H ⊂ Gal(k/k) such that σ(V ) = V for all σ ∈ H. Proof. By Lemma 28.5.8 there exists a finite subextension k ⊂ k 0 ⊂ k and an open V 0 ⊂ Xk0 which pulls back to V . This proves (1). Since Gal(k/k 0 ) is open in  Gal(k/k) part (2) is clear as well. Lemma 28.5.10. Let k be a field. Let k ⊂ k be a (possibly infinite) Galois extension. Let X be a scheme over k. Let T ⊂ Xk have the following properties (1) T is a closed subset of Xk , (2) for every σ ∈ Gal(k/k) we have σ(T ) = T . Then there exists a closed subset T ⊂ X whose inverse image in Xk0 is T . Proof. This lemma immediately reduces to the case where X = Spec(A) is affine. In this case, let I ⊂ A ⊗k k be the radical ideal corresponding to T . Assumption (2) implies that σ(I) = I for all σ ∈ Gal(k/k). Pick x ∈ I. There exists a finite Galois extension k ⊂ k 0 contained in k such that x ∈ A ⊗k k 0 . Set G = Gal(k 0 /k). Set Y P (T ) = (T − σ(x)) ∈ (A ⊗k k 0 )[T ] σ∈G

It is clear that P (T ) is monic and is actually an element of (A ⊗k k 0 )G [T ] = A[T ] (by basic Galois theory). Moreover, if we write P (T ) = T d + a1 T d−1 + . . . + a0 the we see that ai ∈ I := A ∩ I. By Algebra, Lemma 7.35.5 we see that x is contained in the radical of I(A⊗k k). Hence I is the radical of I(A⊗k k) and setting T = V (I) is a solution.  Lemma 28.5.11. Let k be a field. Let X be a scheme over k. The following are equivalent

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(1) X is geometrically connected, (2) for every finite separable field extension k ⊂ k 0 the scheme Xk0 is connected. Proof. It follows immediately from the definition that (1) implies (2). Assume that X is not geometrically connected. Let k ⊂ k be a separable algebraic closure of k. By Lemma 28.5.7 it follows that Xk is disconnected. Say Xk = U q V with U and V open, closed, and nonempty. Suppose that W ⊂ X is any quasi-compact open. Then Wk ∩ U and Wk ∩ V are open and closed in Wk . In particular Wk ∩U and Wk ∩V are quasi-compact, and by Lemma 28.5.9 both Wk ∩ U and Wk ∩ V are defined over a finite subextension and invariant under an open subgroup of Gal(k/k). We will use this without further mention in the following. Pick W0 ⊂ X quasi-compact open such that both W0,k ∩ U and W0,k ∩ V are nonempty. Choose a finite subextension k ⊂ k 0 ⊂ k and a decompostion W0,k0 = U00 q V00 into open and closed subsets such that W0,k ∩ U = (U00 )k and W0,k ∩ V = (V00 )k . Let H = Gal(k/k 0 ) ⊂ Gal(k/k). In particular σ(W0,k ∩ U ) = W0,k ∩ U and similarly for V . Having chosen W0 , k 0 as above, for every quasi-compact open W ⊂ X we set \ [ UW = σ(Wk ∩ U ), VW = σ(Wk ∩ V ). σ∈H

σ∈H

Now, since Wk ∩ U and Wk ∩ V are fixed by an open subgroup of Gal(k/k) we see that the union and intersection above are finite. Hence UW and VW are both open and closed. Also, by construction Wk¯ = UW q VW . We claim that if W ⊂ W 0 ⊂ X are quasi-compact open, then Wk ∩ UW 0 = UW omitted. Hence we see that upon defining and W Sk ∩ VW 0 = VW . Verification S U = W ⊂X UW and V = W ⊂X VW we obtain Xk = U q V is a disjoint union of open and closed subsets. It is clear that V is nonempty as it is constructed by taking unions (locally). On the other hand, U is nonempty since it contains W0 ∩ U by construction. Finally, U, V ⊂ Xk¯ are closed and H-invariant by construction. Hence by Lemma 28.5.10 we have U = (U 0 )k¯ , and V = (V 0 )k¯ for some closed U 0 , V 0 ⊂ Xk0 . Clearly Xk0 = U 0 q V 0 and we see that Xk0 is disconnected as desired.  Lemma 28.5.12. Let k be a field. Let k ⊂ k be a (possibly infinite) Galois extension. Let f : T → X be a morphism of schemes over k. Assume Tk nonempty connected and Xk disconnected. Then X is disconnected. ` Proof. Write Xk = U V with U and V open and closed. Denote f : Tk → Xk the base change of f . Since Tk is connected we see that Tk is contained in either f

−1

(U ) or f

−1

(V ). Say Tk ⊂ f

−1

(U ).

Fix a quasi-compact open W ⊂ X. There exists a finite Galois subextension k ⊂ k 0 ⊂ k such that U ∩` Wk and V ∩ Wk come from quasi-compact opens U 0 , V 0 ⊂ Wk0 . 0 Then also Wk0 = U V 0 . Consider \ [ U 00 = σ(U 0 ), V 00 = σ(V 0 ). 0 0 σ∈Gal(k /k)

σ∈Gal(k /k)

28.5. GEOMETRICALLY CONNECTED SCHEMES

1617

` These are Galois invariant, open and closed, and Wk0 = U 00 V 00 . By Lemma 00 0 28.5.10 we get open and closed ` subsets UW , VW ⊂ W such that U = (UW )k , 00 V = (VW )k0 and W = UW VW . We claim that if W ⊂ W 0 ⊂ X are quasi-compact open, then W ∩ UW 0 = UW and W omitted. Hence we S ∩ VW 0 = VW . Verification S ` see that upon defining U = W ⊂X UW and V = W ⊂X VW we obtain X = U V . It is clear that V is nonempty as it is constructed by taking unions (locally). On the other hand, U is nonempty since it contains f (T ) by construction.  Lemma 28.5.13. Let k be a field. Let T → X be a morphism of schemes over k. Assume T is nonempty and geometrically connected and X connected. Then X is geometrically connected. Proof. This is a reformulation of Lemma 28.5.12.



Lemma 28.5.14. Let k be a field. Let X be a scheme over k. Assume X is connected and has a point x such that k is algebraically closed in κ(x). Then X is geometrically connected. In particular, if X has a k-rational point and X is connected, then X is geometrically connected. Proof. Set T = Spec(κ(x)). Let k ⊂ k be a separable algebraic closure of k. The assumption on k ⊂ κ(x) implies that Tk is irreducible, see Algebra, Lemma 7.44.10. Hence by Lemma 28.5.13 we see that Xk is connected. By Lemma 28.5.7 we conclude that X is geometrically connected.  Lemma 28.5.15. Let k ⊂ K be an extension of fields. Let X be a scheme over k. For every connected component T of X the inverse image TK ⊂ XK is a union of connected components of XK . Proof. This is a purely topological statement. Denote p : XK → X the projection morphism. Let T ⊂ X be a connected component of X. Let t ∈ TK = p−1 (T ). Let C ⊂ XK be a connected component containing t. Then p(C) is a connected subset of X which meets T , hence p(C) ⊂ T . Hence C ⊂ TK .  Lemma 28.5.16. Let k ⊂ K be a finite extension of fields and let X be a scheme over k. Denote by p : XK → X the projection morphism. For every connected component T of XK the image p(T ) is a connected component of X. Proof. The image p(T ) is contained in some connected component X 0 of X. Consider X 0 as a closed subscheme of X in any way. Then T is also a connected 0 component of XK = p−1 (X 0 ) and we may therefore assume that X is connected. The morphism p is open (Morphisms, Lemma 24.24.4), closed (Morphisms, Lemma 24.44.7) and the fibers of p are finite sets (Morphisms, Lemma 24.44.9). Thus we may apply Topology, Lemma 5.4.6 to conclude.  Remark 28.5.17. Let k ⊂ K be an extension of fields. Let X be a scheme over k. Denote p : XK → X the projection morphism. Let T ⊂ XK be a connected component. Is it true that p(T ) is a connected component of X? When k ⊂ K is finite Lemma 28.5.16 tells us the answer is “yes”. In general we do not know the answer. If you do, or if you have a reference, please email [email protected]. Let X be a scheme. We denote π0 (X) the set of connected components of X.

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Lemma 28.5.18. Let k be a field, with separable algebraic closure k. Let X be a scheme over k. There is an action Gal(k/k)opp × π0 (Xk ) −→ π0 (Xk ) with the following properties: (1) An element T ∈ π0 (Xk ) is fixed by the action if and only if there exists a connected component T ⊂ X, which is geometrically connected over k, such that Tk = T . 0 (2) For any field extension k ⊂ k 0 with separable algebraic closure k the diagram Gal(k /k 0 ) × π0 (Xk0 )

/ π0 (X 0 ) k

 Gal(k/k) × π0 (Xk )

 / π0 (X ) k

0

is commutative (where the right vertical arrow is a bijection according to Lemma 28.5.6). Proof. The action (28.5.8.1) of Gal(k/k) on Xk induces an action on its connected components. Connected components are always closed (Topology, Lemma 5.4.3). Hence if T is as in (1), then by Lemma 28.5.10 there exists a closed subset T ⊂ X such that T = Tk . Note that T is geometrically connected over k, see Lemma 28.5.7. To see that T is a connected component of X, suppose that T ⊂ T 0 , T 6= T 0 where T 0 is a connected component of X. In this case Tk0 0 strictly contains T and hence is disconnnected. By Lemma 28.5.12 this means that T 0 is disconnected! Contradiction. We omit the proof of the functoriality in (2).



Lemma 28.5.19. Let k be a field, with separable algebraic closure k. Let X be a scheme over k. Assume (1) X is quasi-compact, and (2) the connected components of Xk are open. Then (a) π0 (Xk ) is finite, and (b) the action of Gal(k/k) on π0 (Xk ) is continuous. Moreover, assumptions (1) and (2) are satisfied when X is of finite type over k. Proof. Since the connected components are open, cover Xk (Topology, Lemma 5.4.3) and Xk is quasi-compact, we conclude that there are only finitely many of them. Thus (a) holds. By Lemma 28.5.8 these connected components are each defined over a finite subextension of k ⊂ k and we get (b). If X is of finite type over k, then Xk is of finite type over k (Morphisms, Lemma 24.16.4). Hence Xk is a Noetherian scheme (Morphisms, Lemma 24.16.6) and has an underlying Noetherian topological space (Properties, Lemma 23.5.5). Thus Xk has finitely many irreducible components (Topology, Lemma 5.6.2) and a fortiori finitely many connected components (which are therefore open). 

28.6. GEOMETRICALLY IRREDUCIBLE SCHEMES

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28.6. Geometrically irreducible schemes If X is an irreducible scheme over a field, then it can happen that X becomes reducible after extending the ground field. This does not happen for geometrically irreducible schemes. Definition 28.6.1. Let X be a scheme over the field k. We say X is geometrically irreducible over k if the scheme Xk0 is irreducible2 for any field extension k 0 of k. Lemma 28.6.2. Let X be a scheme over the field k. Let k ⊂ k 0 be a field extension. Then X is geometrically irreducible over k if and only if Xk0 is geometrically irreducible over k 0 . Proof. If X is geometrically irreducible over k, then it is clear that Xk0 is geometrically irreducible over k 0 . For the converse, note that for any field extension k ⊂ k 00 there exists a common field extension k 0 ⊂ k 000 and k 00 ⊂ k 000 . As the morphism Xk000 → Xk00 is surjective (as a base change of a surjective morphism between spectra of fields) we see that the irreducibility of Xk000 implies the irreducibility of Xk00 . Thus if Xk0 is geometrically irreducible over k 0 then X is geometrically irreducible over k.  Lemma 28.6.3. Let X be a scheme over a separably closed field k. If X is irreducible, then XK is irreducible for any field extension k ⊂ K. I.e., X is geometrically irreducible over k. Proof. Use Properties, Lemma 23.3.3 and Algebra, Lemma 7.44.4.



Lemma 28.6.4. Let k be a field. Let X, Y be schemes over k. Assume X is geometrically irreducible over k. Then the projection morphism p : X ×k Y −→ Y induces a bijection between irreducible components. Proof. First, note that the scheme theoretic fibres of p are irreducible, since they are base changes of the geometrically irreducible scheme X by field extensions. Moreover the scheme theoretic fibres are homeomorphic to the set theoretic fibres, see Schemes, Lemma 21.18.5. By Morphisms, Lemma 24.24.4 the map p is open. Thus we may apply Topology, Lemma 5.5.8 to conclude.  Lemma 28.6.5. Let k be a field. Let X be a scheme over k. The following are equivalent (1) X is geometrically irreducible over k, (2) for every affine open U the k-algebra OX (U ) is geometrically irreducible over k (see Algebra, Definition 7.44.6), S (3) X is irreducible and there exists an affine open covering X = Ui such that each k-algebra OX (Ui ) is geometrically irreducible, and S (4) there exists an open covering X = i∈I Xi such that Xi is geometrically irreducible for each i and such that Xi ∩ Xj 6= ∅ for all i, j ∈ I. Moreover, if X is geometrically irreducible so is every open subscheme of X. 2An irreducible space is nonempty.

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Proof. An affine scheme Spec(A) over k is geometrically irreducible if and only if A is geometrically irreducible over k; this is immediate from the definitions. Recall that if a scheme is irreducible so is every nonempty open subscheme of X, any two nonempty open subsets have a nonempty intersection. Also, if every affine open is irreducible then the scheme is irreducible, see Properties, Lemma 23.3.3. Hence the final statement of the lemma is clear, as well as the implications (1) ⇒ (2), (2) ⇒ (3), and (3) ⇒ (4). If (4) holds, then for any field extension k 0 /k the scheme Xk0 has a covering by irreducible opens which pairwise intersect. Hence Xk0 is irreducible. Hence (4) implies (1).  Lemma 28.6.6. Let X be a geometrically irreducible scheme over the field k. Let ξ ∈ X be its generic point. Then κ(ξ) is a geometrically irreducible over k. Proof. Combining Lemma 28.6.5 and Algebra, Lemma 7.44.8 we see that OX,ξ is geometrically irreducible over k. Since OX,ξ → κ(ξ) is a surjection with locally nilpotent kernel (see Algebra, Lemma 7.24.3) it follows that κ(ξ) is geometrically irreducible, see Algebra, Lemma 7.44.2.  Lemma 28.6.7. Let k ⊂ k 0 be an extension of fields. Let X be a scheme over k. Set X 0 = Xk0 . Assume k separably algebraically closed. Then the morphism X 0 → X induces a bijection of irreducible components. Proof. Since k is separably algebraically closed we see that k 0 is geometrically irreducible over k, see Algebra, Lemma 7.44.7. Hence Z = Spec(k 0 ) is geometrically irreducible over k. by Lemma 28.6.5 above. Since X 0 = Z ×k X the result is a special case of Lemma 28.6.4.  Lemma 28.6.8. Let k be a field. Let X be a scheme over k. Assume X is quasicompact. The following are equivalent: (1) X is geometrically irreducible over k, (2) for every finite separable field extension k ⊂ k 0 the scheme Xk0 is irreducible, and (3) Xk is irreducible, where k ⊂ k is a separable algebraic closure of k. Proof. Assume Xk is irreducible, i.e., assume (3). Let k ⊂ k 0 be a field extension. 0 0 There exists a field extension k ⊂ k such that k 0 embeds into k as an extension of k. By Lemma 28.6.7 we see that Xk0 is irreducible. Since Xk0 → Xk0 is surjective we conclude that Xk0 is irreducible. Hence (1) holds. Let k ⊂ k be a separable algebraic closure of k. Assume not (3), i.e., assume Xk is reducible. Our goal is toSshow that also Xk0 is reducible for some finite subextension k ⊂ k 0 ⊂ k. Let X = i∈I Ui be an affine open covering with Ui not empty. If for some i the scheme Ui is reducible, or if for some pair i 6= j the intersection Ui ∩ Uj is empty, then X is reducible (Properties, Lemma 23.3.3) and we are done. In particular we may assume that Ui,k ∩ Uj,k for all i, j ∈ I is nonempty and we conclude that Ui,k has to be reducible for some i. According to Algebra, Lemma 7.44.5 this means that Ui,k0 is reducible for some finite separable field extension k ⊂ k 0 . Hence also Xk0 is reducible. Thus we see that (2) implies (3). The implication (1) ⇒ (2) is immediate. This proves the lemma.



28.6. GEOMETRICALLY IRREDUCIBLE SCHEMES

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Lemma 28.6.9. Let k ⊂ K be an extension of fields. Let X be a scheme over k. For every irreducible component T of X the inverse image TK ⊂ XK is a union of irreducible components of XK . Proof. Let T ⊂ X be an irreducible component of X. The morphism TK → T is flat, so generalizations lift along TK → T . Hence every ξ ∈ TK which is a generic point of an irreducible component of TK maps to the generic point η of T . If ξ 0 ξ is a specialization in XK then ξ 0 maps to η since there are no points specializing to η in X. Hence ξ 0 ∈ TK and we conclude that ξ = ξ 0 . In other words ξ is the generic point of an irreducible component of XK . This means that the irreducible components of TK are all irreducible components of XK .  For a scheme X we denote IrredComp(X) the set of irreducible components of X. Lemma 28.6.10. Let k ⊂ K be an extension of fields. Let X be a scheme over k. For every irreducible component T ⊂ XK the image of T in X is an irreducible component in X. This defines a canonical map IrredComp(XK ) −→ IrredComp(X) which is surjective. Proof. Consider the diagram XK o

XK

 Xo

 Xk

where K is the separable algebraic closure of K, and where k is the separable algebraic closure of k. By Lemma 28.6.7 the morphism XK → Xk induces a bijection between irreducible components. Hence it suffices to show the lemma for the morphisms Xk → X and XK → XK . In other words we may assume that K = k. The morphism p : Xk → X is integral, flat and surjective. Flatness implies that generalizations lift along p, see Morphisms, Lemma 24.26.8. Hence generic points of irreducible components of Xk map to generic points of irreducible components of X. Integrality implies that p is universally closed, see Morphisms, Lemma 24.44.7. Hence we conclude that the image p(T ) of an irreducible component is a closed irreducible subset which contains a generic point of an irreducible component of X, hence p(T ) is an irreducible component of X. This proves the first assertion. If T ⊂ X is an irreducible component, then p−1 (T ) = TK is a nonempty union of irreducible components, see Lemma 28.6.9. Each of these necessarily maps onto T by the first part. Hence the map is surjective.  Lemma 28.6.11. Let k be a field, with separable algebraic closure k. Let X be a scheme over k. There is an action Gal(k/k)opp × IrredComp(Xk ) −→ IrredComp(Xk ) with the following properties: (1) An element T ∈ IrredComp(Xk ) is fixed by the action if and only if there exists an irreducible component T ⊂ X, which is geometrically irreducible over k, such that Tk = T .

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(2) For any field extension k ⊂ k 0 with separable algebraic closure k the diagram Gal(k /k 0 ) × IrredComp(Xk0 )

/ IrredComp(X 0 ) k

 Gal(k/k) × IrredComp(Xk )

 / IrredComp(X ) k

0

is commutative (where the right vertical arrow is a bijection according to Lemma 28.6.7). Proof. The action (28.5.8.1) of Gal(k/k) on Xk induces an action on its irreducible components. Irreducible components are always closed (Topology, Lemma 5.4.3). Hence if T is as in (1), then by Lemma 28.5.10 there exists a closed subset T ⊂ X such that T = Tk . Note that T is geometrically irreducible over k, see Lemma 28.6.8. To see that T is an irreducible component of X, suppose that T ⊂ T 0 , T 6= T 0 where T 0 is an irreducible component of X. Let η be the generic point of T . It maps to the generic point η of T . Then the generic point ξ ∈ T 0 specializes to η. As Xk → X is flat there exists a point ξ ∈ Xk which maps to ξ and specializes to η. It follows that the closure of the singleton {ξ} is an irreducible closed subset of Xξ which strictly contains T . This is the desired contradiction. We omit the proof of the functoriality in (2).



Lemma 28.6.12. Let k be a field, with separable algebraic closure k. Let X be a scheme over k. The fibres of the map IrredComp(Xk ) −→ IrredComp(X) of Lemma 28.6.10 are exactly the orbits of Gal(k/k) under the action of Lemma 28.6.11. Proof. Let T ⊂ X be an irreducible component of X. Let η ∈ T be its generic point. By Lemmas 28.6.9 and 28.6.10 the generic points of irreducible components of T which map into T map to η. By Algebra, Lemma 7.44.12 the Galois group acts transitively on all of the points of Xk mapping to η. Hence the lemma follows.  Lemma 28.6.13. Let k be a field. Assume X → Spec(k) locally of finite type. In this case (1) the action Gal(k/k)opp × IrredComp(Xk ) −→ IrredComp(Xk ) is continuous if we give IrredComp(Xk ) the discrete topology, (2) every irreducible component of Xk can be defined over a finite extension of k, and (3) given any irreducible component T ⊂ X the scheme Tk is a finite union of irreducible components of Xk which are all in the same Gal(k/k)-orbit. Proof. Let T be an irreducible component of Xk . We may choose an affine open U ⊂ X such that T ∩ Uk is not empty. Write U = Spec(A), so A is a finite type k-algebra, see Morphisms, Lemma 24.16.2. Hence Ak is a finite type k-algebra, and in particular Noetherian. Let p = (f1 , . . . , fn ) be the prime ideal corresponding to

28.7. GEOMETRICALLY INTEGRAL SCHEMES

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T ∩ Uk . Since Ak = A ⊗k k we see that there exists a finite subextension k ⊂ k 0 ⊂ k such that each fi ∈ Ak0 . It is clear that Gal(k/k 0 ) fixes T , which proves (1). Part (2) follows by applying Lemma 28.6.11 (1) to the situation over k 0 which implies the irreducible component T is of the form Tk0 for some irreducible T 0 ⊂ Xk0 . To prove (3), let T ⊂ X be an irreducible component. Choose an irreducible component T ⊂ Xk which maps to T , see Lemma 28.6.10. By the above the orbit of T is finite, say it is T 1 , . . . , T n . Then T 1 ∪ . . . ∪ T n is a Gal(k/k)-invariant closed subset of Xk hence of the form Wk for some W ⊂ X closed by Lemma 28.5.10. Clearly W = T and we win.  Lemma 28.6.14. Let k be a field. Let X → Spec(k) be locally of finite type. Assume X has finitely many irreducible components. Then there exists a finite separable extension k ⊂ k 0 such that every irreducible component of Xk0 is geometrically irreducible over k 0 . Proof. Let k be a separable algebraic closure of k. The assumption that X has finitely many irreducible components combined with Lemma 28.6.13 (3) shows that Xk has finitely many irreducible components T 1 , . . . , T n . By Lemma 28.6.13 (2) there exists a finite extension k ⊂ k 0 ⊂ k and irreducible components Ti ⊂ Xk0 such that T i = Ti,k and we win.  Lemma 28.6.15. Let X be a scheme over the field k. Assume X has finitely many irreducible components which are all geometrically irreducible. Then X has finitely many connected components each of which is geometrically connected. Proof. This is clear because a connected component is a union of irreducible components. Details omitted.  28.7. Geometrically integral schemes If X is an irreducible scheme over a field, then it can happen that X becomes reducible after extending the ground field. This does not happen for geometrically irreducible schemes. Definition 28.7.1. Let X be a scheme over the field k. (1) Let x ∈ X. We say X is geometrically pointwise integral at x if for every field extension k ⊂ k 0 and every x0 ∈ Xk0 lying over x the local ring OXk0 ,x0 is integral. (2) We say X is geometrically pointwise integral if X is geometrically pointwise integral at every point. (3) We say X is geometrically integral over k if the scheme Xk0 is integral for every field extension k 0 of k. The distinction between notions (2) and (3) is necessary. For example if k = R and X = Spec(C[x]), then X is geometrically pointwise integral over R but of course not geometrically integral. Lemma 28.7.2. Let k be a field. Let X be a scheme over k. Then X is geometrically integral over k if and only if X is both geometrically reduced and geometrically irreducible over k. Proof. See Properties, Lemma 23.3.4.



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28.8. Geometrically normal schemes In Properties, Definition 23.7.1 we have defined the notion of a normal scheme. This notion is defined even for non-Noetherian schemes. Hence, contrary to our discussion of “geometrically regular” schemes we consider all field extensions of the ground field. Definition 28.8.1. Let X be a scheme over the field k. (1) Let x ∈ X. We say X is geometrically normal at x if for every field extension k ⊂ k 0 and every x0 ∈ Xk0 lying over x the local ring OXk0 ,x0 is normal. (2) We say X is geometrically normal over k if X is geometrically normal at every x ∈ X. Lemma 28.8.2. Let k be a field. Let X be a scheme over k. Let x ∈ X. The following are equivalent (1) X is geometrically normal at x, (2) for every finite purely inseparable field extension k 0 of k and x0 ∈ Xk0 lying over over x the local ring OXk0 ,x0 is normal, and (3) the ring OX,x is geometrically normal over k (see Algebra, Definition 7.148.2). Proof. It is clear that (1) implies (2). Assume (2). Let k ⊂ k 0 be a finite purely inseparable field extension (for example k = k 0 ). Consider the ring OX,x ⊗k k 0 . By Algebra, Lemma 7.44.2 its spectrum is the same as the spectrum of OX,x . Hence it is a local ring also (Algebra, Lemma 7.17.2). Therefore there is a unique point x0 ∈ Xk0 lying over x and OXk0 ,x0 ∼ = OX,x ⊗k k 0 . By assumption this is a normal ring. Hence we deduce (3) by Algebra, Lemma 7.148.1. Assume (3). Let k ⊂ k 0 be a field extension. Since Spec(k 0 ) → Spec(k) is surjective, also Xk0 → X is surjective (Morphisms, Lemma 24.11.4). Let x0 ∈ Xk0 be any point lying over x. The local ring OXk0 ,x0 is a localization of the ring OX,x ⊗k k 0 . Hence it is normal by assumption and (1) is proved.  Lemma 28.8.3. Let k be a field. Let X be a scheme over k. The following are equivalent (1) (2) (3) (4)

X is geometrically normal, Xk0 is a normal scheme for every field extension k ⊂ k 0 , Xk0 is a normal scheme for every finitely generated field extension k ⊂ k 0 , Xk0 is a normal scheme for every finite purely inseparable field extension k ⊂ k 0 , and (5) for every affine open U ⊂ X the ring OX (U ) is geometrically normal (see Algebra, Definition 7.148.2). Proof. Assume (1). Then for every field extension k ⊂ k 0 and every point x0 ∈ Xk0 the local ring of Xk0 at x0 is normal. By definition this means that Xk0 is normal. Hence (2). It is clear that (2) implies (3) implies (4). Assume (4) and let U ⊂ X be an affine open subscheme. Then Uk0 is a normal scheme for any finite purely inseparable extension k ⊂ k 0 (including k = k 0 ). This

28.9. CHANGE OF FIELDS AND LOCALLY NOETHERIAN SCHEMES

1625

means that k 0 ⊗k O(U ) is a normal ring for all finite purely inseparable extensions k ⊂ k 0 . Hence O(U ) is a geometrically normal k-algebra by definition. Assume (5). For any field extension k ⊂ k 0 the base change Xk0 is gotten by gluing the spectra of the rings OX (U ) ⊗k k 0 where U is affine open in X (see Schemes, Section 21.17). Hence Xk0 is normal. So (1) holds.  Lemma 28.8.4. Let k be a field. Let X be a scheme over k. Let k 0 /k be a field extension. Let x ∈ X be a point, and let x0 ∈ Xk0 be a point lying over x. The following are equivalent (1) X is geometrically normal at x, (2) Xk0 is geometrically normal at x0 . In particular, X is geometrically normal over k if and only if Xk0 is geometrically normal over k 0 . Proof. It is clear that (1) implies (2). Assume (2). Let k ⊂ k 00 be a finite purely inseparable field extension and let x00 ∈ Xk00 be a point lying over x (actually it is unique). We can find a common field extension k ⊂ k 000 (i.e. with both k 0 ⊂ k 000 and k 00 ⊂ k 000 ) and a point x000 ∈ Xk000 lying over both x0 and x00 . Consider the map of local rings OXk00 ,x00 −→ OXk000 ,x0000 . This is a flat local ring homomorphism and hence faithfully flat. By (2) we see that the local ring on the right is normal. Thus by Algebra, Lemma 7.147.3 we conlude that OXk00 ,x00 is normal. By Lemma 28.8.2 we see that X is geometrically normal at x.  Lemma 28.8.5. Let k be a field. Let X be a geometrically normal scheme over k and let Y be a normal scheme over k. Then X ×k Y is a normal scheme. Proof. This reduces to Algebra, Lemma 7.148.4 by Lemma 28.8.3.



28.9. Change of fields and locally Noetherian schemes Let X a locally Noetherian scheme over a field k. It is not always that case that Xk0 is locally Noetherian too. For example if X = Spec(Q) and k = Q, then XQ is the spectrum of Q ⊗Q Q which is not Noetherian. (Hint: It has too many idempotents). But if we only base change using finitely generated field extensions then the Noetherian property is preserved. (Or if X is locally of finite type over k, since this proprety is preserved under base change.) Lemma 28.9.1. Let k be a field. Let X be a scheme over k. Let k ⊂ k 0 be a finitely generated field extension. Then X is locally Noetherian if and only if Xk0 is locally Noetherian. Proof. Using Properties, Lemma 23.5.2 we reduce to the case where X is affine, say X = Spec(A). In this case we have to prove that A is Noetherian if and only if Ak0 is Noetherian. Since A → Ak0 = k 0 ⊗k A is faithfully flat, we see that if Ak0 is Noetherian, then so is A, by Algebra, Lemma 7.147.1. Conversely, if A is Noetherian then Ak0 is Noetherian by Algebra, Lemma 7.29.7. 

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28.10. Geometrically regular schemes A geometrically regular scheme over a field k is a locally Noetherian scheme over k which remains regular upon suitable changes of base field. A finite type scheme over k is geometrically regular if and only if it is smooth over k (see Lemma 28.10.6). The notion of geometric regularity is most interesting in situations where smoothness cannot be used such as formal fibres (insert future reference here). In the following definition we restrict ourselves to locally Noetherian schemes, since the property of being a regular local ring is only defined for Noetherian local rings. By Lemma 28.8.3 above, if we restrict ourselves to finitely generated field extensions then this property is preserved under change of base field. This comment will be used without further reference in this section. In particular the following definition makes sense. Definition 28.10.1. Let k be a field. Let X be a locally Noetherian scheme over k. (1) Let x ∈ X. We say X is geometrically regular at x over k if for every finitely generated field extension k ⊂ k 0 and any x0 ∈ Xk0 lying over x the local ring OXk0 ,x0 is regular. (2) We say X is geometrically regular over k if X is geometrically regular at all of its points. A similar definition works to define geometrically Cohen-Macaulay, (Rk ), and (Sk ) schemes over a field. We will add a section for these separately as needed. Lemma 28.10.2. Let k be a field. Let X be a locally Noetherian scheme over k. Let x ∈ X. The following are equivalent (1) X is geometrically regular at x, (2) for every finite purely inseparable field extension k 0 of k and x0 ∈ Xk0 lying over over x the local ring OXk0 ,x0 is regular, and (3) the ring OX,x is geometrically regular over k (see Algebra, Definition 7.149.2). Proof. It is clear that (1) implies (2). Assume (2). This in particular implies that OX,x is a regular local ring. Let k ⊂ k 0 be a finite purely inseparable field extension. Consider the ring OX,x ⊗k k 0 . By Algebra, Lemma 7.44.2 its spectrum is the same as the spectrum of OX,x . Hence it is a local ring also (Algebra, Lemma 7.17.2). Therefore there is a unique point x0 ∈ Xk0 lying over x and OXk0 ,x0 ∼ = OX,x ⊗k k 0 . By assumption this is a regular ring. Hence we deduce (3) from the definition of a geometrically regular ring. Assume (3). Let k ⊂ k 0 be a field extension. Since Spec(k 0 ) → Spec(k) is surjective, also Xk0 → X is surjective (Morphisms, Lemma 24.11.4). Let x0 ∈ Xk0 be any point lying over x. The local ring OXk0 ,x0 is a localization of the ring OX,x ⊗k k 0 . Hence it is regular by assumption and (1) is proved.  Lemma 28.10.3. Let k be a field. Let X be a locally Noetherian scheme over k. The following are equivalent (1) X is geometrically regular, (2) Xk0 is a regular scheme for every finitely generated field extension k ⊂ k 0 , (3) Xk0 is a regular scheme for every finite purely inseparable field extension k ⊂ k0 ,

28.10. GEOMETRICALLY REGULAR SCHEMES

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(4) for every affine open U ⊂ X the ring OX (U ) is geometrically regular (see Algebra, Definition 7.149.2), and S (5) there exists an affine open covering X = Ui such that each OX (Ui ) is geometrically regular over k. Proof. Assume (1). Then for every finitely generated field extension k ⊂ k 0 and every point x0 ∈ Xk0 the local ring of Xk0 at x0 is regular. By Properties, Lemma 23.9.2 this means that Xk0 is regular. Hence (2). It is clear that (2) implies (3). Assume (3) and let U ⊂ X be an affine open subscheme. Then Uk0 is a regular scheme for any finite purely inseparable extension k ⊂ k 0 (including k = k 0 ). This means that k 0 ⊗k O(U ) is a regular ring for all finite purely inseparable extensions k ⊂ k 0 . Hence O(U ) is a geometrically regular k-algebra and we see that (4) holds. S It is clear that (4) implies (5). Let X = Ui be an affine open covering as in (5). For any field extension k ⊂ k 0 the base change Xk0 is gotten by gluing the spectra of the rings OX (Ui ) ⊗k k 0 (see Schemes, Section 21.17). Hence Xk0 is regular. So (1) holds.  Lemma 28.10.4. Let k be a field. Let X be a scheme over k. Let k 0 /k be a finitely generated field extension. Let x ∈ X be a point, and let x0 ∈ Xk0 be a point lying over x. The following are equivalent (1) X is geometrically regular at x, (2) Xk0 is geometrically regular at x0 . In particular, X is geometrically regular over k if and only if Xk0 is geometrically regular over k 0 . Proof. It is clear that (1) implies (2). Assume (2). Let k ⊂ k 00 be a finite purely inseparable field extension and let x00 ∈ Xk00 be a point lying over x (actually it is unique). We can find a common, finitely generated, field extension k ⊂ k 000 (i.e. with both k 0 ⊂ k 000 and k 00 ⊂ k 000 ) and a point x000 ∈ Xk000 lying over both x0 and x00 . Consider the map of local rings OXk00 ,x00 −→ OXk000 ,x0000 . This is a flat local ring homomorphism of Noetherian local rings and hence faithfully flat. By (2) we see that the local ring on the right is regular. Thus by Algebra, Lemma 7.103.8 we conlude that OXk00 ,x00 is regular. By Lemma 28.10.2 we see that X is geometrically regular at x.  The following lemma is a geometric variant of Algebra, Lemma 7.149.3. Lemma 28.10.5. Let k be a field. Let f : X → Y be a morphism of locally Noetherian schemes over k. Let x ∈ X be a point and set y = f (x). If X is geometrically regular at x and f is flat at x then Y is geometrically regular at y. In particular, if X is geometrically regular over k and f is flat and surjective, then Y is geometrically regular over k. Proof. Let k 0 be finite purely inseparable extension of k. Let f 0 : Xk0 → Yk0 be the base change of f . Let x0 ∈ Xk0 be the unique point lying over x. If we show that Yk0 is regular at y 0 = f 0 (x0 ), then Y is geometrically regular over k at y 0 , see

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Lemma 28.10.3. By Morphisms, Lemma 24.26.6 the morphism Xk0 → Yk0 is flat at x0 . Hence the ring map OYk0 ,y0 −→ OXk0 ,x0 is a flat local homommorphism of local Noetherian rings with right hand side regular by assumption. Hence the left hand side is a regular local ring by Algebra, Lemma 7.103.8.  Lemma 28.10.6. Let k be a field. Let X be a scheme of finite type over k. Let x ∈ X. Then X is geometrically regular at x if and only if X → Spec(k) is smooth at x (Morphisms, Definition 24.35.1). Proof. The question is local around x, hence we may assume that X = Spec(A) for some finite type k-algebra. Let x correspond to the prime p. If A is smooth over k at p, then we may localize A and assume that A is smooth over k. In this case k 0 ⊗k A is smooth over k 0 for all extension fields k 0 /k, and each of these Noetherian rings is regular by Algebra, Lemma 7.130.3. Assume X is geometrically regular at x. Consider the residue field K := κ(x) = κ(p) of x. It is a finitely generated extension of k. By Algebra, Lemma 7.43.3 there exists a finite purely inseparable extension k ⊂ k 0 such that the compositum k 0 K is a separable field extension of k 0 . Let p0 ⊂ A0 = k 0 ⊗k A be a prime ideal lying over p. It is the unique prime lying over p, see Algebra, Lemma 7.44.2. Hence the residue field K 0 := κ(p0 ) is the compositum k 0 K. By assumption the local ring (A0 )p0 is regular. Hence by Algebra, Lemma 7.130.5 we see that k 0 → A0 is smooth at p0 . This in turn implies that k → A is smooth at p by Algebra, Lemma 7.127.18. The lemma is proved.  Example 28.10.7. Let k = Fp (t). It is quite easy to give an example of a regular variety V over k which is not geometrically reduced. For example we can take Spec(k[x]/(xp − t)). In fact, there exists an example of a regular variety V which is geometrically reduced, but not even geometrically normal. Namely, take for p > 2 the scheme V = Spec(k[x, y]/(y 2 − xp + t)). This is a variety as the polynomial y 2 − xp + t ∈ k[x, y] is irreducible. The morphism V → Spec(k) is smooth at all points except at the point v0 ∈ V corresponding to the maximal ideal (y, xp − t) (because 2y is invertible). In particular we see that V is (geometrically) regular at all points, except possibly v0 . The local ring
OV,v0 = k[x, y]/(y 2 − xp + t) (y,xp −t) is a domain of dimension 1. Its maximal ideal is generated by 1 element, namely xp − t. Hence it is a discrete valuation ring and regular. Let k 0 = k[t1/p ]. Denote t0 = t1/p ∈ k 0 , V 0 = Vk0 , v00 ∈ V 0 the unique point lying over v0 . Over k 0 we can write xp − t = (x − t0 )p , but the polynomial y 2 − (x − t0 )p is still irreducible and V 0 is still a variety. But the element y ∈ f.f.(OV 0 ,v00 ) x − t0 is integral over OV 0 ,v00 (just compute its square) and not contained in it, so V 0 is not normal at v00 . This concludes the example.

28.13. ALGEBRAIC SCHEMES

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28.11. Change of fields and the Cohen-Macaulay property The following lemma says that it does not make sense to define geometrically CohenMacaulay schemes, since these would be the same as Cohen-Macaulay schemes. Lemma 28.11.1. Let X be a locally Noetherian scheme over the field k. Let k ⊂ k 0 be a finitely generated field extension. Let x ∈ X be a point, and let x0 ∈ Xk0 be a point lying over x. Then we have OX,x is Cohen-Macaulay ⇔ OXk0 ,x0 is Cohen-Macaulay If X is locally of finite type over k, the same holds for any field extension k ⊂ k 0 . Proof. The first case of the lemma follows from Algebra, Lemma 7.150.2. The second case of the lemma is equivalent to Algebra, Lemma 7.122.6.  28.12. Change of fields and the Jacobson property A scheme locally of finite type over a field has plenty of closed points, namely it is Jacobson. Moreover, the residue fields are finite extensions of the ground field. Lemma 28.12.1. Let X be a scheme which is locally of finite type over k. Then (1) for any closed point x ∈ X the extension k ⊂ κ(x) is algebraic, and (2) X is a Jacobson scheme (Properties, Definition 23.6.1). Proof. A scheme is Jacobson if and only if it has an affine open covering by Jacobson schemes, see Properties, Lemma 23.6.3. The property on residue fields at closed points is also local on X. Hence we may assume that X is affine. In this case the result is a consequence of the Hilbert Nullstellenstaz, see Algebra, Theorem 7.31.1. It also follows from a combination of Morphisms, Lemmas 24.17.8, 24.17.9, and 24.17.10.  It turns out that if X is not locally of finite type, then we can achieve the same result after making a suitably large base field extension. Lemma 28.12.2. Let X be a scheme over a field k. For any field extension k ⊂ K whose cardinality is large enough we have (1) for any closed point x ∈ XK the extension K ⊂ κ(x) is algebraic, and (2) XK is a Jacobson scheme (Properties, Definition 23.6.1). S Proof. Choose an affine open covering X = Ui . By Algebra, Lemma 7.32.12 and Properties, Lemma 23.6.2 there exist cardinals κi such that Ui,K has the desired properties over K if #(K) ≥ κi . Set κ = max{κi }. Then if the cardinality of K is larger than κ we see that each Ui,K satisfies the conclusions of the lemma. Hence XK is Jacobson by Properties, Lemma 23.6.3. The statement on residue fields at closed points of XK follows from the corresponding statements for residue fields of closed points of the Ui,K .  28.13. Algebraic schemes The following definition is taken from [DG67, I Definition 6.4.1]. Definition 28.13.1. Let k be a field. An algebraic k-scheme is a scheme X over k such that the structure morphism X → Spec(k) is of finite type. A locally algebraic k-scheme is a scheme X over k such that the structure morphism X → Spec(k) is locally of finite type.

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Note that every (locally) algebraic k-scheme is (locally) Noetherian, see Morphisms, Lemma 24.16.6. The category of algebraic k-schemes has all products and fibre products (unlike the category of varieties over k). Similarly for the category of locally algebraic k-schemes. Lemma 28.13.2. Let k be a field. Let X be a locally algebraic k-scheme of dimension 0. Then X is a disjoint union of spectra of local Artinian k-algebras A with dimk (A) < ∞. If X is an algebraic k-scheme of dimension 0, then in addition X is affine and the morphism X → Spec(k) is finite. Proof. Let X be a locally algebraic k-scheme of dimension 0. Let U = Spec(A) ⊂ X be an affine open subscheme. Since dim(X) = 0 we see that dim(A) = 0. By Noether normalization, see Algebra, Lemma 7.107.4 we see that there exists a finite injection k → A, i.e., dimk (A) < ∞. Hence A is Artinian, see Algebra, Lemma 7.50.2. This implies that A = A1 × . . . × Ar is a product of finitely many Artinian local rings, see Algebra, Lemma 7.50.8. Of course dimk (Ai ) < ∞ for each i as the sum of these dimensions equals dimk (A). The arguments above show that X has an open covering whose members are finite discrete topological spaces. Hence X is a discrete topological space. It follows that X is isomorphic to the disjoint union of its connected components each of which is a singleton. Since a singleton scheme is affine we conclude (by the results of the paragraph above) that each of these singletons is the spectrum of a local Artinian k-algebra A with dimk (A) < ∞. Finally, if X is an algebraic k-scheme of dimension 0, then X is quasi-compact hence is a finite disjoint union X = Spec(A1 ) q . . . q Spec(Ar ) hence affine (see Schemes, Lemma 21.6.8) and we have seen the finiteness of X → Spec(k) in the first paragraph of the proof.  28.14. Closures of products Some results on the relation between closure and products. Lemma 28.14.1. Let k be a field. Let X, Y be schemes over k, and let A ⊂ X, B ⊂ Y be subsets. Set AB = {z ∈ X ×k Y | prX (γ) ∈ A, prY (γ) ∈ B} ⊂ X ×k Y Then set theoretically we have A ×k B = AB Proof. The inclusion AB ⊂ A ×k B is immediate. We may replace X and Y by the reduced closed subschemes A and B. Let W ⊂ X ×k Y be a nonempty open subset. By Morphisms, Lemma 24.24.4 the subset U = prX (W ) is nonempty open in X. Hence A ∩ U is nonempty. Pick a ∈ A ∩ U . Denote Yκ(a) = {a} ×k Y the fibre of prX : X ×k Y → X over a. By Morphisms, Lemma 24.24.4 again the morphism Ya → Y is open as Spec(κ(a)) → Spec(k) is universally open. Hence the nonempty open subset Wa = W ×X×k Y Ya maps to a nonempty open subset of Y . We conclude there exists a b ∈ B in the image. Hence AB ∩ W 6= ∅ as desired.  Lemma 28.14.2. Let k be a field. Let f : A → X, g : B → Y be morphisms of schemes over k. Then set theoretically we have f (A) ×k g(B) = (f × g)(A ×k B)

28.15. SCHEMES SMOOTH OVER FIELDS

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Proof. This follows from Lemma 28.14.1 as the image of f × g is f (A)g(B) in the notation of that lemma.  Lemma 28.14.3. Let k be a field. Let f : A → X, g : B → Y be quasi-compact morphisms of schemes over k. Let Z ⊂ X be the scheme theoretic image of f , see Morphisms, Definition 24.6.2. Similarly, let Z 0 ⊂ Y be the scheme theoretic image of g. Then Z ×k Z 0 is the scheme theoretic image of f × g. Proof. Recall that Z is the smallest closed subscheme of X through which f factors. Similarly for Z 0 . Let W ⊂ X ×k Y be the scheme theoretic image of f × g. As f × g factors through Z ×k Z 0 we see that W ⊂ Z ×k Z 0 . To prove the other inclusion let U ⊂ X and V ⊂ Y be affine opens. By Morphisms, Lemma 24.6.3 the scheme Z ∩ U is the scheme theoretic image of f |f −1 (U ) : f −1 (U ) → U , and similarly for Z 0 ∩ V and W ∩ U ×k V . Hence we may S assume X and Y affine. As f and g S are quasi-compact this implies that A = Ui is a finite union ` of affines and`B = Vj is a finite union of affines. Then we may replace A by Ui and B by Vj , i.e., we may assume that A and B are affine as well. In this case Z is cut out by Ker(Γ(X, OX ) → Γ(A, OA )) and similarly for Z 0 and W . Hence the result follows from the equality Γ(A ×k B, OA×k B ) = Γ(A, OA ) ⊗k Γ(B, OB ) which holds as A and B are affine. Details omitted.



28.15. Schemes smooth over fields Here are two lemmas characterizing smooth schemes over fields. Lemma 28.15.1. Let k be a field. Let X be a scheme over k. Assume (1) X is locally of finite type over k, (2) ΩX/k is locally free, and (3) k has characteristic zero. Then the structure morphism X → Spec(k) is smooth. Proof. This follows from Algebra, Lemma 7.130.7.



In positive characteristic there exist nonreduced schemes of finite type whose sheaf of differentials is free, for example Spec(Fp [t]/(tp )) over Spec(Fp ). If the ground field k is nonperfect of characteristic p, there exist reduced schemes X/k with free ΩX/k which are nonsmooth, for example Spec(k[t]/(tp − a) where a ∈ k is not a pth power. Lemma 28.15.2. Let k be a field. Let X be a scheme over k. Assume (1) X is locally of finite type over k, (2) ΩX/k is locally free, (3) X is reduced, and (4) k is perfect. Then the structure morphism X → Spec(k) is smooth. Proof. Let x ∈ X be a point. As X is locally Noetherian (see Morphisms, Lemma 24.16.6) there are finitely many irreducible components X1 , . . . , Xn passing through x (see Properties, Lemma 23.5.5 and Topology, Lemma 5.6.2). Let ηi ∈ Xi be the generic point. As X is reduced we have OX,ηi = κ(ηi ), see Algebra, Lemma

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7.24.3. Moreover, κ(ηi ) is a finitely generated field extension of the perfect field k hence separably generated over k (see Algebra, Section 7.40). It follows that ΩX/k,ηi = Ωκ(ηi )/k is free of rank the transcendence degree of κ(ηi ) over k. By Morphisms, Lemma 24.29.1 we conclude that dimηi (Xi ) = rankηi (ΩX/k ). Since x ∈ X1 ∩ . . . ∩ Xn we see that rankx (ΩX/k ) = rankηi (ΩX/k ) = dim(Xi ). Therefore dimx (X) = rankx (ΩX/k ), see Algebra, Lemma 7.106.5. It follows that X → Spec(k) is smooth at x for example by Algebra, Lemma 7.130.3.  Lemma 28.15.3. Let X → Spec(k) be a smooth morphism where k is a field. Then X is a regular scheme. Proof. (See also Lemma 28.10.6.) By Algebra, Lemma 7.130.3 every local ring OX,x is regular. And because X is locally of finite type over k it is locally Noetherian. Hence X is regular by Properties, Lemma 23.9.2.  Lemma 28.15.4. Let X → Spec(k) be a smooth morphism where k is a field. Then X is geometrically regular, geometrically normal, and geometrically reduced over k. Proof. (See also Lemma 28.10.6.) Let k 0 be a finite purely inseparable extension of k. It suffices to prove that Xk0 is regular, normal, reduced, see Lemmas 28.10.3, 28.8.3, and 28.4.5. By Morphisms, Lemma 24.35.5 the morphism Xk0 → Spec(k 0 ) is smooth too. Hence it suffices to show that a scheme X smooth over a field is regular, normal, and reduced. We see that X is regular by Lemma 28.15.3. Hence Properties, Lemma 23.9.4 guarantees that X is normal.  Lemma 28.15.5. Let k be a field. Let d ≥ 0. Let W ⊂ Adk be nonempty open. Then there exists a closed point w ∈ W such that k ⊂ κ(w) is finite separable. Proof. After possible shrinking W we may assume that W = Adk \ V (f ) for some f ∈ k[x1 , . . . , xn ]. If the lemma is wrong then f (a1 , . . . , an ) = 0 for all (a1 , . . . , an ) ∈ (k sep )n . This is absurd as k sep is an infinite field.  Lemma 28.15.6. Let k be a field. If X is smooth over Spec(k) then the set {x ∈ X closed such that k ⊂ κ(x) is finite separable} is dense in X. Proof. It suffices to show that given a nonempty smooth X over k there exists at least one closed point whose residue field is finite separable over k. To see this, choose a diagram π / Ad Xo U k

with π ´etale, see Morphisms, Lemma 24.37.20. The morphism π : U → Adk is open, see Morphisms, Lemma 24.37.13. By Lemma 28.15.5 we may choose a closed point w ∈ π(V ) whose residue field is finite separable over k. Pick any x ∈ V with π(x) = w. By Morphisms, Lemma 24.37.7 the field extension κ(w) ⊂ κ(x) is finite separable. Hence k ⊂ κ(x) is finite separable. The point x is a closed point of X by Morphisms, Lemma 24.21.2. 

28.16. TYPES OF VARIETIES

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Lemma 28.15.7. Let X be a scheme over a field k. If X is locally of finite type and geometrically reduced over k then X contains a dense open which is smooth over k. Proof. The problem is local on X, hence we may assume X is quasi-compact. Let S X = X1 ∪ . . . ∪ Xn be the irreducible components of X. Then Z = i6=j Xi ∩ Xj is nowhere dense in X. Hence we may replace X by X \Z. As X \Z is a disjoint union of irreducible schemes, this reduces us to the case where X is irreducible. As X is irreducible and reduced, it is integral, see Properties, Lemma 23.3.4. Let η ∈ X be its generic point. Then the function field K = k(X) = κ(η) is geometrically reduced over k, hence separable over k, see Algebra, Lemma 7.42.1. Let U = Spec(A) ⊂ X be any nonempty affine open so that K = f.f.(A) = A(0) . Apply Algebra, Lemma 7.130.5 to conclude that A is smooth at (0) over k. By definition this means that some principal localization of A is smooth over k and we win.  Lemma 28.15.8. Let k be a field. Let f : X → Y be a morphism of schemes locally of finite type over k. Let x ∈ X be a point and set y = f (x). If X → Spec(k) is smooth at x and f is flat at x then Y → Spec(k) is smooth at y. In particular, if X is smooth over k and f is flat and surjective, then Y is smooth over k. Proof. It suffices to show that Y is geometrically regular at y, see Lemma 28.10.6. This follows from Lemma 28.10.5 (and Lemma 28.10.6 applied to (X, x)).  28.16. Types of varieties Short section discussion some elementary global properties of varieties. Definition 28.16.1. Let k be a field. Let X be a variety over k. (1) We say X is an affine variety if X is an affine scheme. This is equivalent to requiring X it be isomorphic to a closed subscheme of Ank for some n. (2) We say X is a projective variety if the structure morphism X → Spec(k) is projective. By Morphisms, Lemma 24.43.4 this is true if and only if X is isomorphic to a closed subscheme of Pnk for some n. (3) We say X is a quasi-projective variety if the structure morphism X → Spec(k) is quasi-projective. By Morphisms, Lemma 24.41.4 this is true if and only if X is isomorphic to a locally closed subscheme of Pnk for some n. (4) A proper variety is a variety such that the morphism X → Spec(k) is proper. Note that a projective variety is a proper variety, see Morphisms, Lemma 24.43.5. Also, an affine variety is quasi-projective as Ank is isomorphic to an open subscheme of Pnk , see Constructions, Lemma 22.13.3. Lemma 28.16.2. Let X be a proper variety over k. Then Γ(X, OX ) is a field which is a finite extension of the field k. Proof. By Cohomology of Schemes, Lemma 25.18.2 we see that Γ(X, OX ) is a finite dimensional k-vector space. It is also a k-algebra without zero-divisors. Hence it is a field, see Algebra, Lemma 7.33.17. 

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28.17. Groups of invertible functions It is often (but not always) the case that O∗ (X)/k ∗ is a finitely generated abelian group if X is a variety over k. We show this by a series of lemmas. Everything rests on the following special case. Lemma 28.17.1. Let k be an algebraically closed field. Let X be a proper variety over k. Let X ⊂ X be an open subscheme. Assume X is normal. Then O∗ (X)/k ∗ is a finitely generated abelian group. Proof. We will use without further mention that for any affine open U of X the ring O(U ) is a finitely generated k-algebra, which is Noetherian, a domain and normal, see Algebra, Lemma 7.29.1, Properties, Definition 23.3.1, Properties, Lemmas 23.5.2 and 23.7.2, Morphisms, Lemma 24.16.2. Let ξ1 , . . . , ξr be the generic points of the complement of X in X. There are finitely many since X has a Noetherian underlying topological space (see Morphisms, Lemma 24.16.6, Properties, Lemma 23.5.5, and Topology, Lemma 5.6.2). For each i the local ring Oi = OX,ξi is a normal Noetherian local domain (as a localization of a Noetherian normal domain). Let J ⊂ {1, . . . , r} be the set of indices i such that dim(Oi ) = 1. For j ∈ J the local ring Oj is a discrete valuation ring, see Algebra, Lemma 7.111.6. Hence we obtain a valuation vj : k(X)∗ −→ Z with the property that vj (f ) ≥ 0 ⇔ f ∈ Oj . Think of O(X) as a sub k-algebra of k(X) = k(X). We claim that the kernel of the map Y Y O(X)∗ −→ Z, f 7−→ vj (f ) j∈J

is k ∗ . It is clear that this claim proves the lemma. Namely, suppose that f ∈ O(X) is an element of the kernel. Let U = Spec(B) ⊂ X be any affine open. Then B is a Noetherian normal domain. For every height one prime q ⊂ B with corresponding point ξ ∈ X we see that either ξ = ξj for some j ∈ J or that ξ ∈ X. The reason is that codim({ξ}, X) = 1 by Properties, Lemma 23.11.4 and hence if ξ ∈ X \ X it must be a generic point of X \ X, hence equal to some ξj , j ∈ J. We conclude T that f ∈ OX,ξ = Bq in either case as f is in the kernel of the map. Thus f ∈ ht(q)=1 Bq = B, see Algebra, Lemma 7.141.6. In other words, we see that f ∈ Γ(X, OX ). But since k is algebraically closed we conclude that f ∈ k by Lemma 28.16.2.  Next, we generalize the case above by some elementary arguments, still keeping the field algebraically closed. Lemma 28.17.2. Let k be an algebraically closed field. Let X be an integral scheme locally of finite type over k. Then O∗ (X)/k ∗ is a finitely generated abelian group. Proof. As X is integral the restriction mapping O(X) → O(U ) is injective for any nonempty open subscheme U ⊂ X. Hence we may assume that X is affine. Choose a closed immersion X → Ank and denote X the closure of X in Pnk via the usual immersion Ank → Pnk . Thus we may assume that X is an affine open of a projective variety X.

28.17. GROUPS OF INVERTIBLE FUNCTIONS

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ν

Let ν : X → X be the normalization morphism, see Morphisms, Definition ν 24.48.12. We know that ν is finite, dominant, and that X is a normal irreducible scheme, see Morphisms, Lemmas 24.48.15, 24.48.16, and 24.19.2. It follows that ν X is a proper variety, because X → Spec(k) is proper as a composition of a finite and a proper morphism (see results in Morphisms, Sections 24.42 and 24.44). It also follows that ν is a surjective morphism, because the image of ν is closed and contains the generic point of X. Hence setting X ν = ν −1 (X) we see that it suffices to prove the result for X ν . In other words, we may assume that X is a nonempty  open of a normal proper variety X. This case is handled by Lemma 28.17.1. The preceding lemma implies the following slight generalization. Lemma 28.17.3. Let k be an algebraically closed field. Let X be a connected reduced scheme which is locally of finite type over k with finitely many irreducible components. Then O∗ (X)/k ∗ is a finitely generated abelian group. S Proof. Let X = Xi be the irreducible components. By Lemma 28.17.2 we see that O(Xi )∗ /k ∗ is a finitely generated abelian group. Let f ∈ O(X)∗ be in the kernel of the map Y O(X)∗ −→ O(Xi )∗ /k ∗ . Then for each i there exists an element λi ∈ k such that f |Xi = λi . By restricting to Xi ∩ Xj we conclude that λi = λj if Xi ∩ Xj 6= ∅. Since X is connected we conclude that all λi agree and hence that f ∈ k ∗ . This proves that Y O(X)∗ /k ∗ ⊂ O(Xi )∗ /k ∗ and the lemma follows as on the right we have a product of finitely many finitely generated abelian groups.  Lemma 28.17.4. Let k be a field. Let X be a scheme over k which is connected and reduced. Then the integral closure of k in Γ(X, OX ) is a field. Proof. Let k 0 ⊂ Γ(X, OX ) be the integral closure of k. Then X → Spec(k) factors through Spec(k 0 ), see Schemes, Lemma 21.6.4. As X is reduced we see that k 0 has no nonzero nilpotent elements. As k → k 0 is integral we see that every prime ideal of k 0 is both a maximal ideal and a minimal prime, and Spec(k 0 ) is totally disconnected, see Algebra, Lemmas 7.33.18 and 7.24.5. As X is connected the morphism X → Spec(k 0 ) is constant, say with image the point corresponding to p ⊂ k 0 . Then any f ∈ k 0 , f 6∈ p maps to an invertible element of OX . By definition of k 0 this then forces f to be a unit of k 0 . Hence we see that k 0 is local with maximal ideal p, see Algebra, Lemma 7.17.2. Since we’ve already seen that k 0 is reduced this implies that k 0 is a field, see Algebra, Lemma 7.24.3.  Proposition 28.17.5. Let k be a field. Let X be a scheme over k. Assume that X is locally of finite type over k, connected, reduced, and has finitely many irreducible components. Then O(X)∗ /k ∗ is a finitely generated abelian group if in addition to the conditions above at least one of the following conditions is satisfied: (1) the integral closure of k in Γ(X, OX ) is k, (2) X has a k-rational point, or (3) X is geometrically integral.

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Proof. Let k be an algebraic closure of k. Let Y be a connected component of (Xk )red . Note that the canonical morphism p : Y → X is open (by Morphisms, Lemma 24.24.4) and closed (by Morphisms, Lemma 24.44.7). Hence p(Y ) = X as X was assumed connected. In particular, as X is reduced this implies O(X) ⊂ O(Y ). By Lemma 28.6.13 we see that Y has finitely many irreducible components. Thus Lemma 28.17.3 applies to Y . This implies that if O(X)∗ /k ∗ is not a finitely generated abelian group, then there exist elements f ∈ O(X), f 6∈ k which map to an element of k via the map O(X) → O(Y ). In this case f is algebraic over k, hence integral over k. Thus, if condition (1) holds, then this cannot happen. To finish the proof we show that conditions (2) and (3) imply (1). Let k ⊂ k 0 ⊂ Γ(X, OX ) be the integral closure of k in Γ(X, OX ). By Lemma 28.17.4 we see that k 0 is a field. If e : Spec(k) → X is a k-rational point, then e] : Γ(X, OX ) → k is a section to the inclusion map k → Γ(X, OX ). In particular the restriction of e] to k 0 is a field map k 0 → k over k, which clearly shows that (2) implies (1). If the integral closure k 0 of k in Γ(X, OX ) is not trivial, then we see that X is either not geometrically connected (if k ⊂ k 0 is not purely inseparable) or that X is not geometrically reduced (if k ⊂ k 0 is nontrivial purely inseparable). Details omitted. Hence (3) implies (1).  Lemma 28.17.6. Let k be a field. Let X be a variety over k. The group O(X)∗ /k ∗ is a finitely generated abelian group provided at least one of the following conditions holds: (1) k is integrally closed in Γ(X, OX ), (2) k is algebraically closed in k(X), (3) X is geometrically integral over k, or (4) k is the “intersection” of the field extensions k ⊂ κ(x) where x runs over the closed points of x. Proof. We see that (1) is enough by Proposition 28.17.5. We omit the verification that each of (2), (3), (4) implies (1).  28.18. Uniqueness of base field The phrase “let X be a scheme over k” means that X is a scheme which comes equipped with a morphism X → Spec(k). Now we can ask whether the field k is uniquely determined by the scheme X. Of course this is not the case, since for example A1C which we ordinarily consider as a scheme over the field C of complex numbers, could also be considered as a scheme over Q. But what if we ask that the morphism X → Spec(k) does not factor as X → Spec(k 0 ) → Spec(k) for any nontrivial field extension k ⊂ k 0 ? In other words we ask that k is somehow maximal such that X lives over k. An example to show that this still does not garantee uniqueness of k is the scheme    1 , P ∈ Q[y], P 6= 0 X = Spec Q(x)[y] P (y) At first sight this seems to be a scheme over Q(x), but on a second look it is clear that it is also a scheme over Q(y). Moreover, the fields Q(x) and Q(y) are subfields of R = Γ(X, OX ) which are maximal among the subfields of R (details omitted).

28.18. UNIQUENESS OF BASE FIELD

1637

In particular, both Q(x) and Q(y) are maximal in the sense above. Note that both morphisms X → Spec(Q(x)) and X → Spec(Q(y)) are “essentially of finite type” (i.e., the corresponding ring map is essentially of finite type). Hence X is a Noetherian scheme of finite dimension, i.e., it is not completely pathological. Another issue that can prevent uniqueness is that the scheme X may be nonreduced. In that case there can be many different morphisms from X to the spectrum of a given field. As an explicit example consider the dual numbers D = C[y]/(y 2 ) = C ⊕ C. Given any derivation θ : C → C over Q we get a ring map C −→ D,

c 7−→ c + θ(c).

The subfield of C on which all of these maps are the same is the algebraic closure of Q. This means that taking the intersection of all the fields that X can live over may end up being a very small field if X is nonreduced. One observation in this regard is the following: given a field k and two subfields k1 , k2 of k such that k is finite over k1 and over k2 , then in general it is not the case that k is finite over k1 ∩ k2 . An example is the field k = Q(t) and its subfields k1 = Q(t2 ) and Q((t + 1)2 ). Namely we have k1 ∩ k2 = Q in this case. So in the following we have to be careful when taking intersections of fields. Having said all of this we now show that if X is locally of finite type over a field, then some uniqueness holds. Here is the precise result. Proposition 28.18.1. Let X be a scheme. Let a : X → Spec(k1 ) and b : X → Spec(k2 ) be morphisms from X to spectra of fields. Assume a, b are locally of finite type, and X is reduced, and connected. Then we have k10 = k20 , where ki0 ⊂ Γ(X, OX ) is the integral closure of ki in Γ(X, OX ). Proof. First, assume the lemma holds in case X is quasi-compact (we will do the quasi-compact case below). As X is locally of finite type over a field, it is locally Noetherian, see Morphisms, Lemma 24.16.6. In particular this means that it is locally connected, connected components of open subsets are open, and intersections of quasi-compact opens are quasi-compact, see Properties, Lemma 23.5.5, Topology, Lemma 5.4.9, S Topology, Section 5.6, and Topology, Lemma 5.11.1. Pick an open covering X = i∈I Ui such that each Ui is quasi-compact and connected. For each i let Ki ⊂ OX (Ui ) be the integral closure of k1 and of k2 . For each pair i, j ∈ I we decompose a Ui ∩ Uj = Ui,j,l into its finitely many connected components. Write Ki,j,l ⊂ O(Ui,j,l ) for the integral closure of k1 and of k2 . By Lemma 28.17.4 the rings Ki and Ki,j,l are fields. Now we claim that k10 and k20 both equal the kernel of the map Y Y Ki −→ Ki,j,l , (xi )i 7−→ xi |Ui,j,l − xj |Ui,j,l which proves what we want. Namely, it is clear that k10 is contained in this kernel. On the other hand, suppose that (xi )i is in the kernel. By the sheaf condition (xi )i corresponds to f ∈ O(X). Pick some i0 ∈ I and let P (T ) ∈ k1 [T ] be a monic polynomial with P (xi0 ) = 0. Then we claim that P (f ) = 0 which proves that f ∈ k1 . To prove this we have to show that P (xi ) = 0 for all i. Pick i ∈ I. As X is connected there exists a sequence i0 , i1 , . . . , in = i ∈ I such that Uit ∩ Uit+1 6= ∅. Now this means that for each t there exists an lt such that xit and xit+1 map to

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28. VARIETIES

the same element of the field Ki,j,l . Hence if P (xit ) = 0, then P (xit+1 ) = 0. By induction, starting with P (xi0 ) = 0 we deduce that P (xi ) = 0 as desired. To finish the proof of the lemma we prove the lemma under the additional hypothesis that X is quasi-compact. By Lemma 28.17.4 after replacing ki by ki0 we may assume that ki is integrally closed in Γ(X, OX ). This implies that O(X)∗ /ki∗ is a finitely generated abelian group, see Proposition 28.17.5. Let k12 = k1 ∩ k2 as a subring of O(X). Note that k12 is a field. Since ∗ k1∗ /k12 −→ O(X)∗ /k2∗ ∗ we see that k1∗ /k12 is a finitely generated abelian group as well. Hence there exist ∗ α1 , . . . , αn ∈ k1 such that every element λ ∈ k1 has the form

λ = cα1e1 . . . αnen for some ei ∈ Z and c ∈ k12 . In particular, the ring map 1 ] −→ k1 , xi 7−→ αi k12 [x1 , . . . , xn , x1 . . . xn is surjective. By the Hilbert Nullstellensatz, Algebra, Theorem 7.31.1 we conclude that k1 is a finite extension of k12 . In the same way we conclude that k2 is a finite extension of k12 . In particular both k1 and k2 are contained in the integral closure 0 0 of k12 in Γ(X, OX ). But since k12 is a field by Lemma 28.17.4 and since we k12 chose ki to be integrally closed in Γ(X, OX ) we conclude that k1 = k12 = k2 as desired.  28.19. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes

(24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44)

Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces (45) Limits of Algebraic Spaces

28.19. OTHER CHAPTERS

(46) (47) (48) (49) (50) (51) (52) (53) (54) (55) (56) (57) (58) (59) (60)

Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability

(61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

1639

Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 29

Chow Homology and Chern Classes 29.1. Introduction In this chapter we discuss Chow homology groups and the construction of chern classes of vector bundles as elements of operational Chow cohomology groups (everything with Z-coefficients). We follow the first few chapters of [Ful98], except that we have been less precise about the supports of the cycles involved. More classical discussions of Chow groups can be found in [Sam56], [Che58a], and [Che58b]. Of course there are many others. To make the material a little bit more challenging we decided to treat a somewhat more general case than is usually done. Namely we assume our schemes X are locally of finite type over a fixed locally Noetherian base scheme which is universally catenary and has a given dimension function. This seems to be all that is needed to be able to define the Chow homology groups A∗ (X) and the action of capping with chern classes on them. This is an indication that we should be able to define these also for algebraic stacks locally of finite type over such a base. In another chapter we will define the intersection products on A∗ (X) using Serre’s Tor-formula in case X is nonsingular (see [Ser00], or [Ser65]) and we have a good moving lemma. See (insert future reference here).

29.2. Determinants of finite length modules The material in this section is related to the material in the paper [KM76] and to the material in the thesis [Ros09]. If you have a good reference then please email [email protected]. Given any field κ and any finite dimensional κ-vector space V we set detκ (V ) equal to detκ (V ) = ∧n (V ) where n = dimκ (V ). We want to generalize this slightly. Definition 29.2.1. Let R be a local ring with maximal ideal m and residue field κ. Let M be a finite length R-module. Say l = lengthR (M ). (1) Given elements x1 , . . . , xr ∈ M we denote hx1 , . . . , xr i = Rx1 + . . . + Rxr the R-submodule of R generated by x1 , . . . , xr . (2) We will say an l-tuples of elements (e1 , . . . , el ) of M is admissible if mei ∈ he1 , . . . , ei−1 i for i = 1, . . . , l. (3) A symbol [e1 , . . . , el ] will mean (e1 , . . . , el ) is an admissible l-tuple. (4) An admissible relation between symbols is one of the following: (a) if (e1 , . . . , el ) is an admissible sequence and for some 1 ≤ a ≤ l we have ea ∈ he1 , . . . , ea−1 i, then [e1 , . . . , el ] = 0, 1641

1642

29. CHOW HOMOLOGY AND CHERN CLASSES

(b) if (e1 , . . . , el ) is an admissible sequence and for some 1 ≤ a ≤ l we have ea = λe0a + x with λ ∈ R∗ , and x ∈ he1 , . . . , ea−1 i, then [e1 , . . . , el ] = λ[e1 , . . . , ea−1 , e0a , ea+1 , . . . , el ] where λ ∈ κ∗ is the image of λ in the residue field, and (c) if (e1 , . . . , el ) is an admissible sequence and mea ⊂ he1 , . . . , ea−2 i then [e1 , . . . , el ] = −[e1 , . . . , ea−2 , ea , ea−1 , ea+1 , . . . , el ]. (5) We define the determinant of the finite length R-module to be   κ-vector space generated by symbols detκ (M ) = κ-linear combinations of admissible relations We stress that always l = lengthR (M ). We also stress that it does not follow that the symbol [e1 , . . . , el ] is additive in the entries (this will typically not be the case). Before we can show that the determinant detκ (M ) actually has dimension 1 we have to show that it has dimension at most 1. Lemma 29.2.2. With notations as above we have dimκ (detκ (M )) ≤ 1. Proof. Fix an admissible sequence (f1 , . . . , fl ) of M such that lengthR (hf1 , . . . , fi i) = i for i = 1, . . . , l. Such an admissible sequence exists exactly because M has length l. We will show that any element of detκ (M ) is a κ-multiple of the symbol [f1 , . . . , fl ]. This will prove the lemma. Let (e1 , . . . , el ) be an admissible sequence of M . It suffices to show that [e1 , . . . , el is a multiple of [f1 , . . . , fl ]. First assume that he1 , . . . , el i 6= M . Then there exists an i ∈ [1, . . . , l] such that ei ∈ he1 , . . . , ei−1 i. It immediately follows from the first admissible relation that [e1 , . . . , en ] = 0 in detκ (M ). Hence we may assume that he1 , . . . , el i = M . In particular there exists a smallest index i ∈ {1, . . . , l} such that f1 ∈ he1 , . . . , ei i. This means that ei = λf1 + x with x ∈ he1 , . . . , ei−1 i and λ ∈ R∗ . By the second admissible relation this means that [e1 , . . . , el ] = λ[e1 , . . . , ei−1 , f1 , ei+1 , . . . , el ]. Note that mf1 = 0. Hence by applying the third admissible relation i − 1 times we see that [e1 , . . . , el ] = (−1)i−1 λ[f1 , e1 , . . . , ei−1 , ei+1 , . . . , el ]. Note that it is also the case that hf1 , e1 , . . . , ei−1 , ei+1 , . . . , el i = M . By induction suppose we have proven that our original symbol is equal to a scalar times [f1 , . . . , fj , ej+1 , . . . , el ] for some admissible sequence (f1 , . . . , fj , ej+1 , . . . , el ) whose elements generate M , i.e., with hf1 , . . . , fj , ej+1 , . . . , el i = M . Then we find the smallest i such that fj+1 ∈ hf1 , . . . , fj , ej+1 , . . . , ei i and we go through the same process as above to see that [f1 , . . . , fj , ej+1 , . . . , el ] = (scalar)[f1 , . . . , fj , fj+1 , ej+1 , . . . , eˆi , . . . , el ] Continuing in this vein we obtain the desired result.



Before we show that detκ (M ) always has dimension 1, let us show that it agree with the usual top exterior power in the case the module is a vector space over κ.

29.2. DETERMINANTS OF FINITE LENGTH MODULES

1643

Lemma 29.2.3. Let R be a local ring with maximal ideal m and residue field κ. Let M be a finite length R-module which is annihilated by m. Let l = n = dimκ (M ). Then the map detκ (M ) −→ ∧lκ (M ),

[e1 , . . . , el ] 7−→ e1 ∧ . . . ∧ el

is an isomorphism. Proof. It is clear that the rule described in the lemma gives a κ-linear map since all of the admissible relations are satisfied by the usual symbols e1 ∧ . . . ∧ el . It is also clearly a surjective map. Since by Lemma 29.2.2 the left hand side has dimension at most one we see that the map is an isomorphism.  Lemma 29.2.4. Let R be a local ring with maximal ideal m and residue field κ. Let M be a finite length R-module. The determinant detκ (M ) defined above is a κ-vector space of dimension 1. It is generated by the symbol [f1 , . . . , fl ] for any admissible sequence such that hf1 , . . . fl i = M . Proof. We know detκ (M ) has dimension at most 1, and in fact that it is generated by [f1 , . . . , fl ], by Lemma 29.2.2 and its proof. We will show by induction on l = length(M ) that it is nonzero. For l = 1 it follows from Lemma 29.2.3. Choose a nonzero element f ∈ M with mf = 0. Set M = M/hf i, and denote the quotient map x 7→ x. We will define a surjective map ψ : detk (M ) → detκ (M ) which will prove the lemma since by induction the determinant of M is nonzero. We define ψ on symbols as follows. Let (e1 , . . . , el ) be an admissible sequence. If f 6∈ he1 , . . . , el i then we simply set ψ([e1 , . . . , el ]) = 0. If f ∈ he1 , . . . , el i then we choose an i minimal such that f ∈ he1 , . . . , ei i and write ei = λf + x for some λ ∈ R and x ∈ he1 , . . . , ei−1 i. In this case we set ψ([e1 , . . . , el ]) = λ[e1 , . . . , ei−1 , ei+1 , . . . , el ]. Note that it is indeed the case that (e1 , . . . , ei−1 , ei+1 , . . . , el ) is an admissible sequence in M , so this makes sense. Let us show that extending this rule κ-linearly to linear combinations of symbolds does indeed lead to a map on determinants. To do this we have to show that the admissible relations are mapped to zero. Type (a) relations. Suppose we have (e1 , . . . , el ) an admissible sequence and for some 1 ≤ a ≤ l we have ea ∈ he1 , . . . , ea−1 i. Suppose that f ∈ he1 , . . . , ei i with i minimal. Then it is immediate that i 6= a. Since it is also the case that ea ∈ he1 , . . . , eˆi , . . . , ea−1 i we see immediately that the same admissible relation for detκ (M ) forces the symbol [e1 , . . . , ei−1 , ei+1 , . . . , el ] to be zero as desired. Type (b) relations. Suppose we have (e1 , . . . , el ) an admissible sequence and for some 1 ≤ a ≤ l we have ea = λe0a + x with λ ∈ R∗ , and x ∈ he1 , . . . , ea−1 i. Suppose that f ∈ he1 , . . . , ei i with i minimal. Say ei = µf + y with y ∈ he1 , . . . , ei−1 i. If i < a then the desired equality is λ[e1 , . . . , ei−1 , ei+1 , . . . , el ] = λ[e1 , . . . , ei−1 , ei+1 , . . . , ea−1 , e0a , ea+1 , . . . , el ] which follows from ea = λe0a + x and the corresponding admissible relation for detκ (M ). If i > a then the desired equality is λ[e1 , . . . , ei−1 , ei+1 , . . . , el ] = λ[e1 , . . . , ea−1 , e0a , ea+1 , . . . , ei−1 , ei+1 , . . . , el ]

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29. CHOW HOMOLOGY AND CHERN CLASSES

which follows from ea = λe0a + x and the corresponding admissible relation for detκ (M ). The interesting case is when i = a. In this case we have ea = λe0a + x = µf + y. Hence also e0a = λ−1 (µf + y − x). Thus we see that ψ([e1 , . . . , el ]) = µ[e1 , . . . , ei−1 , ei+1 , . . . , el ] = ψ(λ[e1 , . . . , ea−1 , e0a , ea+1 , . . . , el ]) as desired. Type (c) relations. Suppose that (e1 , . . . , el ) is an admissible sequence and mea ⊂ he1 , . . . , ea−2 i. Suppose that f ∈ he1 , . . . , ei i with i minimal. Say ei = λf + x with x ∈ he1 , . . . , ei−1 i. If i < a − 1, then the desired equality is λ[e1 , . . . , ei−1 , ei+1 , . . . , el ] = λ[e1 , . . . , ei−1 , ei+1 , . . . , ea−2 , ea , ea−1 , ea+1 , . . . , el ] which follows from the type (c) admissible relation for detκ (M ). Similarly, if i > a, then the desired equality is λ[e1 , . . . , ei−1 , ei+1 , . . . , el ] = λ[e1 , . . . , ea−2 , ea , ea−1 , ea+1 , . . . , ei−1 , ei+1 , . . . , el ] which follows from the type (c) admissible relation for detκ (M ). If i = a, then the desired equality is λ[e1 , . . . , ea−1 , ea+1 , . . . , el ] = λ[e1 , . . . , ea−2 , ea−1 , ea+1 , . . . , el ] which is tautological. Finally, the interesting case is i = a − 1. This case itself splits into two cases as to whether f ∈ he1 , . . . , ea−2 , ea i or not. If not, then we see that the desired equality is λ[e1 , . . . , ea−2 , ea , . . . , el ] = λ[e1 , . . . , ea−2 , ea , ea+1 , . . . , el ] which is tautological since after switching ea−1 and ea the smallest index such that f is in the becomes equal to i0 = a and it is again ea which is removed. On the other hand, suppose that f ∈ he1 , . . . , ea−2 , ea i. In this case we see that we can, besides the equality ea−1 = λf + x of above, also write ea = µf + y with y ∈ he1 , . . . , ea−2 i. Clearly this means that both ea ∈ he1 , . . . , ea−1 i and ea−1 ∈ he1 , . . . , ea−2 , ea i. Thus we can use relations of type (a) and the compatibility of ψ with these shown above to see that both ψ([e1 , . . . , el ])

and ψ([e1 , . . . , ea−2 , ea , ea−1 , ea+1 , . . . , el ])

are zero, as desired. At this point we have shown that ψ is well defined, and all that remains is to show that it is surjective. To see this let (f 2 , . . . , f l ) be an admissible sequence in M . We can choose lifts f2 , . . . , fl ∈ M , and then (f, f2 , . . . , fl ) is an admissible sequence in M . Since ψ([f, f2 , . . . , fl ]) = [f2 , . . . , fl ] we win.  Let R be a local ring with maximal ideal m and residue field κ. Note that if ϕ : M → N is an isomorphism of finite length R-modules, then we get an isomorphism detκ (ϕ) : detκ (M ) → detκ (N ) simply by the rule detκ (ϕ)([e1 , . . . , el ]) = [ϕ(e1 ), . . . , ϕ(el )] for any symbol [e1 , . . . , el ] for M . Hence we see that detκ is a functor     finite length R-modules 1-dimensional κ-vector spaces (29.2.4.1) −→ with isomorphisms with isomorphisms

29.2. DETERMINANTS OF FINITE LENGTH MODULES

1645

This is typical for a “determinant functor” (see [Knu02]), as is the following additivity property. Lemma 29.2.5. Let (R, m, κ) be a local ring. For every short exact sequence 0→K→L→M →0 of finite length R-modules there exists a canonical isomorphism γK→L→M : detκ (K) ⊗κ detκ (M ) −→ detκ (L) defined by the rule on nonzero symbols [e1 , . . . , ek ] ⊗ [f 1 , . . . , f m ] −→ [e1 , . . . , ek , f1 , . . . , fm ] with the following properties: (1) For every isomorphism of short exact sequences, i.e., for every commutative diagram /L /M /0 /K 0 u

0

 / K0

v

w

 / L0

 / M0

/0

with short exact rows and isomorphisms u, v, w we have γK 0 →L0 →M 0 ◦ (detκ (u) ⊗ detκ (w)) = detκ (v) ◦ γK→L→M , (2) for every commutative square of finite length R-modules with exact rows and columns 0

0

0

0

 /A

 /B

 /C

/0

0

 /D

 /E

 /F

/0

0

 /G

 /H

 /I

/0

 0

 0

 0

the following diagram is commutative detκ (A) ⊗ detκ (C) ⊗ detκ (G) ⊗ detκ (I)

γA→B→C ⊗γG→H→I

/ detκ (B) ⊗ detκ (H) 

detκ (E) O



 detκ (A) ⊗ detκ (G) ⊗ detκ (C) ⊗ detκ (I)

γB→E→H

γD→E→F γA→D→G ⊗γC→F →I

/ detκ (D) ⊗ detκ (F )

where  is the switch of the factors in the tensor product times (−1)cg with c = lengthR (C) and g = lengthR (G), and

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29. CHOW HOMOLOGY AND CHERN CLASSES

(3) the map γK→L→M agrees with the usual isomorphism if 0 → K → L → M → 0 is actually a short exact sequence of κ-vector spaces. Proof. The significance of taking nonzero symbols in the explicit description of the map γK→L→M is simply that if (e1 , . . . , el ) is an admissible sequence in K, and (f 1 , . . . , f m ) is an admissible sequence in M , then it is not garanteed that (e1 , . . . , el , f1 , . . . , fm ) is an admissible sequence in L (where of course fi ∈ L signifies a lift of f i ). However, if the symbol [e1 , . . . , el ] is nonzero in detκ (K), then necessarily K = he1 , . . . , ek i (see proof of Lemma 29.2.2), and in this case it is true that (e1 , . . . , ek , f1 , . . . , fm ) is an admissible sequence. Moreover, by the admissible relations of type (b) for detκ (L) we see that the value of [e1 , . . . , ek , f1 , . . . , fm ] in detκ (L) is independent of the choice of the lifts fi in this case also. Given this remark, it is clear that an admissible relation for e1 , . . . , ek in K translates into an admissible relation among e1 , . . . , ek , f1 , . . . , fm in L, and similarly for an admissible relation among the f 1 , . . . , f m . Thus γ defines a linear map of vector spaces as claimed in the lemma. By Lemma 29.2.4 we know detκ (L) is generated by any single symbol [x1 , . . . , xk+m ] such that (x1 , . . . , xk+m ) is an admissible sequence with L = hx1 , . . . , xk+m i. Hence it is clear that the map γK→L→M is surjective and hence an isomorphism. Property (1) holds because detκ (v)([e1 , . . . , ek , f1 , . . . , fm ]) =

[v(e1 ), . . . , v(ek ), v(f1 ), . . . , v(fm )]

= γK 0 →L0 →M 0 ([u(e1 ), . . . , u(ek )] ⊗ [w(f1 ), . . . , w(fm )]). Property (2) means that given a symbol [α1 , . . . , αa ] generating detκ (A), a symbol [γ1 , . . . , γc ] generating detκ (C), a symbol [ζ1 , . . . , ζg ] generating detκ (G), and a symbol [ι1 , . . . , ιi ] generating detκ (I) we have [α1 , . . . , αa , γ˜1 , . . . , γ˜c , ζ˜1 , . . . , ζ˜g , ˜ι1 , . . . , ˜ιi ] =

(−1)cg [α1 , . . . , αa , ζ˜1 , . . . , ζ˜g , γ˜1 , . . . , γ˜c , ˜ι1 , . . . , ˜ιi ]

(for suitable lifts x ˜ in E) in detκ (E). This holds because we may use the admissible relations of type (c) cg times in the following order: move the ζ˜1 past the elements γ˜c , . . . , γ˜1 (allowed since mζ˜1 ⊂ A), then move ζ˜2 past the elements γ˜c , . . . , γ˜1 (allowed since mζ˜2 ⊂ A + Rζ˜1 ), and so on. Part (3) of the lemma is obvious. This finishes the proof.



We can use the maps γ of the lemma to define more general maps γ as follows. Suppose that (R, m, κ) is a local ring. Let M be a finite length R-module and suppose we are given a finite filtration (see Homology, Definition 10.13.1) M = F n ⊃ F n+1 ⊃ . . . ⊃ F m−1 ⊃ F m = 0. Then there is a canonical isomorphism O γ(M,F ) : detκ (F i /F i+1 ) −→ detκ (M ) i

well defined up to sign(!). One can make the sign explicit either by giving a well defined order of the terms in the tensor product (starting with higher indices unfortunately), and by thinking of the target category for the functor detκ as the category of 1-dimensional super vector spaces. See [KM76, Section 1].

29.2. DETERMINANTS OF FINITE LENGTH MODULES

1647

Here is another typical result for determinant functors. It is not hard to show. The tricky part is usually to show the existence of a determinant functor. Lemma 29.2.6. Let (R, m, κ) be any local ring. The functor     finite length R-modules 1-dimensional κ-vector spaces detκ : −→ with isomorphisms with isomorphisms endowed with the maps γK→L→M is characterized by the following properties (1) its restriction to the subcategory of modules annihilated by m is isomorphic to the usual determinant functor (see Lemma 29.2.3), and (2) (1), (2) and (3) of Lemma 29.2.5 hold. Proof. Omitted.



Lemma 29.2.7. Let (R, m, κ) be a local ring. Let I ⊂ m be an ideal, and set R0 = R/I. Let detR,κ denote the determinant functor on the category ModfR finite length R-modules and denote detR0 ,κ the determinant on the category ModfR0 of finite length R0 -modules. Then ModfR0 ⊂ ModfR is a full subcategory and there exists an isomorphism of functors detR,κ |Modf = detR0 ,κ R0

compatible with the isomorphisms γK→L→M for either of these functors. Proof. This can be shown by using the characterization of the pair (detR0 ,κ , γ) in Lemma 29.2.6. But really the isomorphism is obtained by mapping a symbol [x1 , . . . , xl ] ∈ detR,κ (M ) to the corresponding symbol [x1 , . . . , xl ] ∈ detR0 ,κ (M ) which “obviously” works.  Here is a case where we can compute the determinant of a linear map. In fact there is nothing mysterious about this in any case, see Example 29.2.9 for a random example. Lemma 29.2.8. Let R be a local ring with residue field κ. Let u ∈ R∗ be a unit. Let M be a module of finite length over R. Denote uM : M → M the map multiplication by u. Then detκ (uM ) : detκ (M ) −→ detκ (M ) l is multiplication by u where l = lengthR (M ) and u ∈ κ∗ is the image of u. Proof. Denote fM ∈ κ∗ the element such that detκ (uM ) = fM iddetκ (M ) . Suppose that 0 → K → L → M → 0 is a short exact sequence of finite R-modules. Then we see that uk , uL , uM give an isomorphism of short exact sequences. Hence by Lemma 29.2.5 (1) we conclude that fK fM = fL . This means that by induction on length it suffices to prove the lemma in the case of length 1 where it is trivial.  Example 29.2.9. Consider the local ring R = Zp . Set M = Zp /(p2 ) ⊕ Zp /(p3 ). Let u : M → M be the map given by the matrix   a b u= pc d where a, b, c, d ∈ Zp , and a, d ∈ Z∗p . In this case detκ (u) equals multiplication by a2 d3 mod p ∈ F∗p . This can easily be seen by consider the effect of u on the symbol [p2 e, pe, pf, e, f ] where e = (0, 1) ∈ M and f = (1, 0) ∈ M .

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29.3. Periodic complexes Of course there is a very general notion of periodic complexes. We can require periodicity of the maps, or periodicity of the objects. We will add these here as needed. For the moment we only need the following cases. Definition 29.3.1. Let R be a ring. (1) A 2-periodic complex over R is given by a quadruple (M, N, ϕ, ψ) consisting of R-modules M , N and R-module maps ϕ : M → N , ψ : N → M such that ...

/M

ϕ

/N

ψ

/M

ϕ

/N

/ ...

is a complex. In this setting we define the cohomology modules of the complex to be the R-modules H 0 (M, N, ϕ, ψ) = Ker(ϕ)/Im(ψ),

and H 1 (M, N, ϕ, ψ) = Ker(ψ)/Im(ϕ).

We say the 2-periodic complex is exact if the cohomology groups are zero. (2) A (2, 1)-periodic complex over R is given by a triple (M, ϕ, ψ) consisting of an R-module M and R-module maps ϕ : M → M , ψ : M → M such that ...

/M

ϕ

/M

ψ

/M

ϕ

/M

/ ...

is a complex. Since this is a special case of a 2-periodic complex we have its cohomology modules H 0 (M, ϕ, ψ), H 1 (M, ϕ, ψ) and a notion of exactness. In the following we will use any result proved for 2-periodic complexes without further mention for (2, 1)-periodic complexes. It is clear that the collection of 2-periodic complexes (resp. (2, 1)-periodic complexes) forms a category with morphisms (f, g) : (M, N, ϕ, ψ) → (M 0 , N 0 , ϕ0 , ψ 0 ) pairs of morphisms f : M → M 0 and g : N → N 0 such that ϕ0 ◦ f = f ◦ ϕ and ψ 0 ◦ g = g ◦ ψ. In fact it is an abelian category, with kernels and cokernels as in Homology, Lemma 10.10.3. Also, note that a special case are the (2, 1)-periodic complexes of the form (M, 0, ψ). In this special case we have H 0 (M, 0, ψ) = Coker(ψ),

and H 1 (M, 0, ψ) = Ker(ψ).

Definition 29.3.2. Let R be a local ring. Let (M, N, ϕ, ψ) be a 2-periodic complex over R whose cohomology groups have finite length over R. In this case we define the multiplicity of (M, N, ϕ, ψ) to be the integer eR (M, N, ϕ, ψ) = lengthR (H 0 (M, N, ϕ, ψ)) − lengthR (H 1 (M, N, ϕ, ψ)) We will sometimes (especially in the case of a (2, 1)-periodic complex with ϕ = 0) call this the Herbrand quotient1. Lemma 29.3.3. Let R be a local ring. (1) If (M, N, ϕ, ψ) is a 2-periodic complex such that M , N have finite length. Then eR (M, N, ϕ, ψ) = lengthR (M ) − lengthR (N ). (2) If (M, ϕ, ψ) is a (2, 1)-periodic complex such that M has finite length. Then eR (M, ϕ, ψ) = 0. 1If the residue field of R is finite with q elements it is customary to call the Herbrand quotient h(M, N, ϕ, ψ) = q eR (M,N,ϕ,ψ) which is equal to the number of elements of H 0 divided by the number of elements of H 1 .

29.3. PERIODIC COMPLEXES

1649

(3) Suppose that we have a short exact sequence of (2, 1)-periodic complexes 0 → (M1 , N1 , ϕ1 , ψ1 ) → (M2 , N2 , ϕ2 , ψ2 ) → (M3 , N3 , ϕ3 , ψ3 ) → 0 If two out of three have cohomology modules of finite length so does the third and we have eR (M2 , N2 , ϕ2 , ψ2 ) = eR (M1 , N1 , ϕ1 , ψ1 ) + eR (M3 , N3 , ϕ3 , ψ3 ). Proof. Proof of (3). Abbreviate A = (M1 , N1 , ϕ1 , ψ1 ), B = (M2 , N2 , ϕ2 , ψ2 ) and C = (M3 , N3 , ϕ3 , ψ3 ). We have a long exact cohomology sequence . . . → H 1 (C) → H 0 (A) → H 0 (B) → H 0 (C) → H 1 (A) → H 1 (B) → H 1 (C) → . . . This gives a finite exact sequence 0 → I → H 0 (A) → H 0 (B) → H 0 (C) → H 1 (A) → H 1 (B) → K → 0 with 0 → K → H 1 (C) → I → 0 a filtration. By additivity of the length function (Algebra, Lemma 7.49.3) we see the result. The proofs of (1) and (2) are omitted.  Let R be a local ring with residue field κ. Let (M, ϕ, ψ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, ϕ, ψ) is exact. We are going to use the determinant construction to define an invariant of this situation. See Section 29.2. Let us abbreviate Kϕ = Ker(ϕ), Iϕ = Im(ϕ), Kψ = Ker(ψ), and Iψ = Im(ψ). The short exact sequences 0 → Kϕ → M → Iϕ → 0,

0 → Kψ → M → Iψ → 0

give isomorphisms γϕ : detκ (Kϕ ) ⊗ detκ (Iϕ ) −→ detκ (M ),

γψ : detκ (Kψ ) ⊗ detκ (Iψ ) −→ detκ (M ),

see Lemma 29.2.5. On the other hand the exactness of the complex gives equalities Kϕ = Iψ , and Kψ = Iϕ and hence an isomorphism σ : detκ (Kϕ ) ⊗ detκ (Iϕ ) −→ detκ (Kψ ) ⊗ detκ (Iψ ) by switching the factors. Using this notation we can define our invariant. Definition 29.3.4. Let R be a local ring with residue field κ. Let (M, ϕ, ψ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, ϕ, ψ) is exact. The determinant of (M, ϕ, ψ) is the element detκ (M, ϕ, ψ) ∈ κ∗ such that the composition −1 γψ ◦σ◦γϕ

detκ (M ) −−−−−−→ detκ (M ) is multiplication by (−1)lengthR (Iϕ )lengthR (Iψ ) detκ (M, ϕ, ψ). Remark 29.3.5. Here is a more down to earth description of the determinant introduced above. Let R be a local ring with residue field κ. Let (M, ϕ, ψ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, ϕ, ψ) is exact. Let us abbreviate Iϕ = Im(ϕ), Iψ = Im(ψ) as above. Assume that lengthR (Iϕ ) = a and lengthR (Iψ ) = b, so that a + b = lengthR (M ) by exactness. Choose admissible sequences x1 , . . . , xa ∈ Iϕ and y1 , . . . , yb ∈ Iψ such that the symbol [x1 , . . . , xa ] generates detκ (Iϕ ) and the symbol [x1 , . . . , xb ] generates detκ (Iψ ).

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29. CHOW HOMOLOGY AND CHERN CLASSES

Choose x ˜i ∈ M such that ϕ(˜ xi ) = xi . Choose y˜j ∈ M such that ψ(˜ yj ) = yj . Then detκ (M, ϕ, ψ) is characterized by the equality [x1 , . . . , xa , y˜1 , . . . , y˜b ] = (−1)ab detκ (M, ϕ, ψ)[y1 , . . . , yb , x ˜1 , . . . , x ˜a ] in detκ (M ). This also explains the sign. Lemma 29.3.6. Let R be a local ring with residue field κ. Let (M, ϕ, ψ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, ϕ, ψ) is exact. Then detκ (M, ϕ, ψ) detκ (M, ψ, ϕ) = 1. Proof. Omitted.



Lemma 29.3.7. Let R be a local ring with residue field κ. Let (M, ϕ, ϕ) be a (2, 1)-periodic complex over R. Assume that M has finite length and that (M, ϕ, ψ) is exact. Then lengthR (M ) = 2lengthR (Im(ϕ)) and 1

detκ (M, ϕ, ψ) = (−1)lengthR (Im(ϕ)) = (−1) 2 lengthR (M ) Proof. Follows directly from the sign rule in the definitions.



Lemma 29.3.8. Let R be a local ring with residue field κ. Let M be a finite length R-module. (1) if ϕ : M → M is an isomorphism then detκ (M, ϕ, 0) = detκ (ϕ). (2) if ψ : M → M is an isomorphism then detκ (M, 0, ψ) = detκ (ψ)−1 . Proof. Let us prove (1). Set ψ = 0. Then we may, with notation as above Definition 29.3.4, identify Kϕ = Iψ = 0, Iϕ = Kψ = M . With these identifications, the map γϕ : κ ⊗ detκ (M ) = detκ (Kϕ ) ⊗ detκ (Iϕ ) −→ detκ (M ) is identified with detκ (ϕ−1 ). On the other hand the map γψ is identified with the identity map. Hence γψ ◦ σ ◦ γϕ−1 is equal to detκ (ϕ) in this case. Whence the result. We omit the proof of (2).  Lemma 29.3.9. Let R be a local ring with residue field κ. Suppose that we have a short exact sequence of (2, 1)-periodic complexes 0 → (M1 , ϕ1 , ψ1 ) → (M2 , ϕ2 , ψ2 ) → (M3 , ϕ3 , ψ3 ) → 0 with all Mi of finite length, and each (M1 , ϕ1 , ψ1 ) exact. Then detκ (M2 , ϕ2 , ψ2 ) = detκ (M1 , ϕ1 , ψ1 ) detκ (M3 , ϕ3 , ψ3 ). in κ∗ .

29.3. PERIODIC COMPLEXES

1651

Proof. Let us abbreviate Iϕ,i = Im(ϕi ), Kϕ,i = Ker(ϕi ), Iψ,i = Im(ψi ), and Kψ,i = Ker(ψi ). Observe that we have a commutative square 0

0

0

0

 / Kϕ,1

 / Kϕ,2

 / Kϕ,3

/0

0

 / M1

 / M2

 / M3

/0

0

 / Iϕ,1

 / Iϕ,2

 / Iϕ,3

/0

 0

 0

 0

of finite length R-modules with exact rows and columns. The top row is exact since it can be identified with the sequence Iψ,1 → Iψ,2 → Iψ,3 → 0 of images, and similarly for the bottom row. There is a similar diagram involving the modules Iψ,i and Kψ,i . By definition detκ (M2 , ϕ2 , ψ2 ) corresponds, up to a sign, to the composition of the left vertical maps in the following diagram detκ (M1 ) ⊗ detκ (M3 )

/ detκ (M2 )

γ

γ −1

γ −1 ⊗γ −1

 detκ (Kϕ,1 ) ⊗ detκ (Iϕ,1 ) ⊗ detκ (Kϕ,3 ) ⊗ detκ (Iϕ,3 )

γ⊗γ

 / detκ (Kϕ,2 ) ⊗ detκ (Iϕ,2 )

γ⊗γ

 / detκ (Kψ,2 ) ⊗ detκ (Iψ,2 )

σ

σ⊗σ

 detκ (Kψ,1 ) ⊗ detκ (Iψ,1 ) ⊗ detκ (Kψ,3 ) ⊗ detκ (Iψ,3 )

γ

γ⊗γ

 detκ (M1 ) ⊗ detκ (M3 )

γ

 / detκ (M2 )

The top and bottom squares are commutative up to sign by applying Lemma 29.2.5 (2). The middle square is trivially commutative (we are just switching factors). Hence we see that detκ (M2 , ϕ2 , ψ2 ) =  detκ (M1 , ϕ1 , ψ1 ) detκ (M3 , ϕ3 , ψ3 ) for some sign . And the sign can be worked out, namely the outer rectangle in the diagram above commutes up to 

=

(−1)length(Iϕ,1 )length(Kϕ,3 )+length(Iψ,1 )length(Kψ,3 )

=

(−1)length(Iϕ,1 )length(Iψ,3 )+length(Iψ,1 )length(Iϕ,3 )

(proof omitted). It follows easily from this that the signs work out as well. 2



Example 29.3.10. Let k be a field. Consider the ring R = k[T ]/(T ) of dual numbers over k. Denote t the class of T in R. Let M = R and ϕ = ut, ψ = vt with u, v ∈ k ∗ . In this case detk (M ) has generator e = [t, 1]. We identify Iϕ = Kϕ = Iψ = Kψ = (t). Then γϕ (t ⊗ t) = u−1 [t, 1] (since u−1 ∈ M is a lift of t ∈ Iϕ ) and γψ (t ⊗ t) = v −1 [t, 1] (same reason). Hence we see that detk (M, ϕ, ψ) = −u/v ∈ k ∗ .

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29. CHOW HOMOLOGY AND CHERN CLASSES

Example 29.3.11. Let R = Zp and let M = Zp /(pl ). Let ϕ = pb u and ϕ = pa v with a, b ≥ 0, a + b = l and u, v ∈ Z∗p . Then a computation as in Example 29.3.10 shows that detFp (Zp /(pl ), pb u, pa v)

=

(−1)ab ua /v b mod p

=

(−1)ordp (α)ordp (β)

αordp (β) mod p β ordp (α)

with α = pb u, β = pa v ∈ Zp . See Lemma 29.4.10 for a more general case (and a proof). Example 29.3.12. Let R = k be a field. Let M = k ⊕a ⊕ k ⊕b be l = a + b dimensional. Let ϕ and ψ be the following diagonal matrices ϕ = diag(u1 , . . . , ua , 0, . . . , 0),

ψ = diag(0, . . . , 0, v1 , . . . , vb )

∗

with ui , vj ∈ k . In this case we have u1 . . . ua . v1 . . . vb This can be seen by a direct computation or by computing in case l = 1 and using the additivity of Lemma 29.3.9. detk (M, ϕ, ψ) =

Example 29.3.13. Let R = k be a field. Let M = k ⊕a ⊕k ⊕a be l = 2a dimensional. Let ϕ and ψ be the following block matrices     0 U 0 V ϕ= , ψ= , 0 0 0 0 with U, V ∈ Mat(a × a, k) invertible. In this case we have det(U ) . det(V ) This can be seen by a direct computation. The case a = 1 is similar to the computation in Example 29.3.10. detk (M, ϕ, ψ) = (−1)a

Example 29.3.14. Let R = k be  0 0 0 u1 0 0  ϕ= 0 0 0 0 0 u2

a field. Let M = k ⊕4 . Let    0 0 0 0 0   0  ϕ =  0 0 v2 0  0 0 0 0 0 0 v1 0 0 0

with u1 , u2 , v1 , v2 ∈ k ∗ . Then we have detk (M, ϕ, ψ) = −

u1 u2 . v1 v2

Next we come to the analogue of the fact that the determinant of a composition of linear endomorphisms is the product of the determinants. To avoid very long formulae we write Iϕ = Im(ϕ), and Kϕ = Ker(ϕ) for any R-module map ϕ : M → M . We also denote ϕψ = ϕ ◦ ψ for a pair of morphisms ϕ, ψ : M → M . Lemma 29.3.15. Let R be a local ring with residue field κ. Let M be a finite length R-module. Let α, β, γ be endomorphisms of M . Assume that (1) Iα = Kβγ , and similarly for any permutation of α, β, γ, (2) Kα = Iβγ , and similarly for any permutation of α, β, γ. Then

29.3. PERIODIC COMPLEXES

(1) (2) (3) (4)

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The triple (M, α, βγ) is an exact (2, 1)-periodic complex. The triple (Iγ , α, β) is an exact (2, 1)-periodic complex. The triple (M/Kβ , α, γ) is an exact (2, 1)-periodic complex. We have detκ (M, α, βγ) = detκ (Iγ , α, β) detκ (M/Kβ , α, γ).

Proof. It is clear that the assumptions imply part (1) of the lemma. To see part (1) note that the assumptions imply that Iγα = Iαγ , and similarly for kernels and any other pair of morphisms. Moreover, we see that Iγβ = Iβγ = Kα ⊂ Iγ and similarly for any other pair. In particular we get a short exact sequence α

0 → Iβγ → Iγ − → Iαγ → 0 and similarly we get a short exact sequence β

0 → Iαγ → Iγ − → Iβγ → 0. This proves (Iγ , α, β) is an exact (2, 1)-periodic complex. Hence part (2) of the lemma holds. To see that α, γ give well defined endomorphisms of M/Kβ we have to check that α(Kβ ) ⊂ Kβ and γ(Kβ ) ⊂ Kβ . This is true because α(Kβ ) = α(Iγα ) = Iαγα ⊂ Iαγ = Kβ , and similarly in the other case. The kernel of the map α : M/Kβ → M/Kβ is Kβα /Kβ = Iγ /Kβ . Similarly, the kernel of γ : M/Kβ → M/Kβ is equal to Iα /Kβ . Hence we conclude that (3) holds. We introduce r = lengthR (Kα ), s = lengthR (Kβ ) and t = lengthR (Kγ ). By the exact sequences above and our hypotheses we have lengthR (Iα ) = s + t, lengthR (Iβ ) = r + t, lengthR (Iγ ) = r + s, and length(M ) = r + s + t. Choose (1) an admissible sequence x1 , . . . , xr ∈ Kα generating Kα (2) an admissible sequence y1 , . . . , ys ∈ Kβ generating Kβ , (3) an admissible sequence z1 , . . . , zt ∈ Kγ generating Kγ , (4) elements x ˜i ∈ M such that βγ x ˜ i = xi , (5) elements y˜i ∈ M such that αγ y˜i = yi , (6) elements z˜i ∈ M such that βα˜ zi = zi . With these choices the sequence y1 , . . . , ys , α˜ z1 , . . . , α˜ zt is an admissible sequence in Iα generating it. Hence, by Remark 29.3.5 the determinant D = detκ (M, α, βγ) is the unique element of κ∗ such that [y1 , . . . , ys , α˜ z1 , . . . , α˜ zs , x ˜1 , . . . , x ˜r ] r(s+t)

= (−1)

D[x1 , . . . , xr , γ y˜1 , . . . , γ y˜s , z˜1 , . . . , z˜t ]

By the same remark, we see that D1 = detκ (M/Kβ , α, γ) is characterized by [y1 , . . . , ys , α˜ z1 , . . . , α˜ zt , x ˜1 , . . . , x ˜r ] = (−1)rt D1 [y1 , . . . , ys , γ x ˜1 , . . . , γ x ˜r , z˜1 , . . . , z˜t ] By the same remark, we see that D2 = detκ (Iγ , α, β) is characterized by [y1 , . . . , ys , γ x ˜1 , . . . , γ x ˜r , z˜1 , . . . , z˜t ] = (−1)rs D2 [x1 , . . . , xr , γ y˜1 , . . . , γ y˜s , z˜1 , . . . , z˜t ] Combining the formulas above we see that D = D1 D2 as desired.



Lemma 29.3.16. Let R be a local ring with residue field κ. Let α : (M, ϕ, ψ) → (M 0 , ϕ0 , ψ 0 ) be a morphism of (2, 1)-periodic complexes over R. Assume (1) M , M 0 have finite length,

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(2) (M, ϕ, ψ), (M 0 , ϕ0 , ψ 0 ) are exact, (3) the maps ϕ, ψ induce the zero map on K = Ker(α), and (4) the maps ϕ, ψ induce the zero map on Q = Coker(α). Denote N = α(M ) ⊂ M 0 . We obtain two short exact sequences of (2, 1)-periodic complexes 0 → (N, ϕ0 , ψ 0 ) → (M 0 , ϕ0 , ψ 0 ) → (Q, 0, 0) → 0 0 → (K, 0, 0) → (M, ϕ, ψ) → (N, ϕ0 , ψ 0 ) → 0 which induce two isomorphisms αi : Q → K, i = 0, 1. Then detκ (M, ϕ, ψ) = detκ (α0−1 ◦ α1 ) detκ (M 0 , ϕ0 , ψ 0 ) In particular, if α0 = α1 , then detκ (M, ϕ, ψ) = detκ (M 0 , ϕ0 , ψ 0 ). Proof. There are (at least) two ways to prove this lemma. One is to produce an enormous commutative diagram using the properties of the determinants. The other is to use the characterization of the determinants in terms of admissible sequences of elements. It is the second approach that we will use. First let us explain precisely what the maps αi are. Namely, α0 is the composition α0 : Q = H 0 (Q, 0, 0) → H 1 (N, ϕ0 , ψ 0 ) → H 2 (K, 0, 0) = K and α1 is the composition α1 : Q = H 1 (Q, 0, 0) → H 2 (N, ϕ0 , ψ 0 ) → H 3 (K, 0, 0) = K coming from the boundary maps of the short exact sequences of complexes displayed in the lemma. The fact that the complexes (M, ϕ, ψ), (M 0 , ϕ0 , ψ 0 ) are exact implies these maps are isomorphisms. We will use the notation Iϕ = Im(ϕ), Kϕ = Ker(ϕ) and similarly for the other maps. Exactness for M and M 0 means that Kϕ = Iψ and three similar equalities. We introduce k = lengthR (K), a = lengthR (Iϕ ), b = lengthR (Iψ ). Then we see that lengthR (M ) = a + b, and lengthR (N ) = a + b − k, lengthR (Q) = k and lengthR (M 0 ) = a+b. The exact sequences below will show that also lengthR (Iϕ0 ) = a and lengthR (Iψ0 ) = b. The assumption that K ⊂ Kϕ = Iψ means that ϕ factors through N to give an exact sequence ϕα−1

0 → α(Iψ ) → N −−−→ Iψ → 0. −1 0 Here ϕα (x ) = y means x0 = α(x) and y = ϕ(x). Similarly, we have ψα−1

0 → α(Iϕ ) → N −−−→ Iϕ → 0. The assumption that ψ 0 induces the zero map on Q means that Iψ0 = Kϕ0 ⊂ N . This means the quotient ϕ0 (N ) ⊂ Iϕ0 is identified with Q. Note that ϕ0 (N ) = α(Iϕ ). Hence we conclude there is an isomorphism ϕ0 : Q → Iϕ0 /α(Iϕ ) simply described by ϕ0 (x0 mod N ) = ϕ0 (x0 ) mod α(Iϕ ). In exactly the same way we get ψ 0 : Q → Iψ0 /α(Iψ ) Finally, note that α0 is the composition Q

ϕ0

/ Iϕ0 /α(Iϕ )

ψα−1 |I

/α(Iϕ ) ϕ0

/K

29.3. PERIODIC COMPLEXES

1655

and similarly α1 = ϕα−1 |Iψ0 /α(Iψ ) ◦ ψ 0 . To shorten the formulas below we are going to write αx instead of α(x) in the following. No confusion should result since all maps are indicated by greek letters and elements by roman letters. We are going to choose (1) an admissible sequence z1 , . . . , zk ∈ K generating K, (2) elements zi0 ∈ M such that ϕzi0 = zi , (3) elements zi00 ∈ M such that ψzi00 = zi , (4) elements xk+1 , . . . , xa ∈ Iϕ such that z1 , . . . , zk , xk+1 , . . . , xa is an admissible sequence generating Iϕ , (5) elements x ˜i ∈ M such that ϕ˜ xi = xi , (6) elements yk+1 , . . . , yb ∈ Iψ such that z1 , . . . , zk , yk+1 , . . . , yb is an admissible sequence generating Iψ , (7) elements y˜i ∈ M such that ψ y˜i = yi , and (8) elements w1 , . . . , wk ∈ M 0 such that w1 mod N, . . . , wk mod N are an admissible sequence in Q generating Q. By Remark 29.3.5 the element D = detκ (M, ϕ, ψ) ∈ κ∗ is characterized by [z1 , . . . , zk , xk+1 , . . . , xa , z100 , . . . , zk00 , y˜k+1 , . . . , y˜b ] =

(−1)ab D[z1 , . . . , zk , yk+1 , . . . , yb , z10 , . . . , zk0 , x ˜k+1 , . . . , x ˜a ]

Note that by the discussion above αxk+1 , . . . , αxa , ϕw1 , . . . , ϕwk is an admissible sequence generating Iϕ0 and αyk+1 , . . . , αyb , ψw1 , . . . , ψwk is an admissible sequence generating Iψ0 . Hence by Remark 29.3.5 the element D0 = detκ (M 0 , ϕ0 , ψ 0 ) ∈ κ∗ is characterized by [αxk+1 , . . . , αxa , ϕ0 w1 , . . . , ϕ0 wk , α˜ yk+1 , . . . , α˜ yb , w1 , . . . , wk ] =

(−1)ab D0 [αyk+1 , . . . , αyb , ψ 0 w1 , . . . , ψ 0 wk , α˜ xk+1 , . . . , α˜ xa , w1 , . . . , wk ]

Note how in the first, resp. second displayed formula the the first, resp. last k entries of the symbols on both sides are the same. Hence these formulas are really equivalent to the equalities [αxk+1 , . . . , αxa , αz100 , . . . , αzk00 , α˜ yk+1 , . . . , α˜ yb ] =

(−1)ab D[αyk+1 , . . . , αyb , αz10 , . . . , αzk0 , α˜ xk+1 , . . . , α˜ xa ]

and [αxk+1 , . . . , αxa , ϕ0 w1 , . . . , ϕ0 wk , α˜ yk+1 , . . . , α˜ yb ] =

(−1)ab D0 [αyk+1 , . . . , αyb , ψ 0 w1 , . . . , ψ 0 wk , α˜ xk+1 , . . . , α˜ xa ]

in detκ (N ). Note that ϕ0 w1 , . . . , ϕ0 wk and αz100 , . . . , zk00 are admissible sequences generating the module Iϕ0 /α(Iϕ ). Write [ϕ0 w1 , . . . , ϕ0 wk ] = λ0 [αz100 , . . . , αzk00 ] in detκ (Iϕ0 /α(Iϕ )) for some λ0 ∈ κ∗ . Similarly, write [ψ 0 w1 , . . . , ψ 0 wk ] = λ1 [αz10 , . . . , αzk0 ] in detκ (Iψ0 /α(Iψ )) for some λ1 ∈ κ∗ . On the one hand it is clear that αi ([w1 , . . . , wk ]) = λi [z1 , . . . , zk ] for i = 0, 1 by our description of αi above, which means that detκ (α0−1 ◦ α1 ) = λ1 /λ0

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and on the other hand it is clear that λ0 [αxk+1 , . . . , αxa , αz100 , . . . , αzk00 , α˜ yk+1 , . . . , α˜ yb ] =

[αxk+1 , . . . , αxa , ϕ0 w1 , . . . , ϕ0 wk , α˜ yk+1 , . . . , α˜ yb ]

and λ1 [αyk+1 , . . . , αyb , αz10 , . . . , αzk0 , α˜ xk+1 , . . . , α˜ xa ] =

[αyk+1 , . . . , αyb , ψ 0 w1 , . . . , ψ 0 wk , α˜ xk+1 , . . . , α˜ xa ]

which imply λ0 D = λ1 D0 . The lemma follows.



29.4. Symbols The correct generality for this construction is perhaps the situation of the following lemma. Lemma 29.4.1. Let A be a Noetherian local ring. Let M be a finite A-module of dimension 1. Assume ϕ, ψ : M → M are two injective A-module maps, and assume ϕ(ψ(M )) = ψ(ϕ(M )), for example if ϕ and ψ commute. Then lengthR (M/ϕψM ) < ∞ and (M/ϕψM, ϕ, ψ) is an exact (2, 1)-periodic complex. Proof. Let q be a minimal prime of the support of M . Then Mq is a finite length Aq -module, see Algebra, Lemma 7.60.11. Hence both ϕ and ψ induce isomorphisms Mq → Mq . Thus the support of M/ϕψM is {mA } and hence it has finite length (see lemma cited above). Finally, the kernel of ϕ on M/ϕψM is clearly ψM/ϕψM , and hence the kernel of ϕ is the image of ψ on M/ϕψM . Similarly the other way since M/ϕψM = M/ψϕM by assumption.  Lemma 29.4.2. Let A be a Noetherian local ring. Let a, b ∈ A. (1) if M is a finite A-module of dimension 1 such that a, b are nonzerodivisors on M , then lengthA (M/abM ) < ∞ and (M/abM, a, b) is a (2, 1)-periodic exact complex. (2) if a, b are nonzerodivisors and dim(A) = 1 then lengthA (A/(ab)) < ∞ and (A/(ab), a, b) is a (2, 1)-periodic exact complex. In particular, in these case detκ (M/abM, a, b) ∈ κ∗ , resp. detκ (A/(ab), a, b) ∈ κ∗ are defined. Proof. Follows from Lemma 29.4.1.



Definition 29.4.3. Let A be a Noetherian local ring with residue field κ. Let a, b ∈ A. Let M be a finite A-module of dimension 1 such that a, b are nonzerodivisors on M . We define the symbol associated to M, a, b to be the element dM (a, b) = detκ (M/abM, a, b) ∈ κ∗ Lemma 29.4.4. Let A be a Noetherian local ring. Let a, b, c ∈ A. Let M be a finite A-module with dim(M ) = 1. Assume a, b, c are nonzerodivisors on M . Then dM (a, bc) = dM (a, b)dM (a, c) and dM (a, b)dM (b, a) = 1. Proof. The first statement is immediate from Lemma 29.3.15 above. The second comes from Lemma 29.3.6. 

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Definition 29.4.5. Let A be a Noetherian local domain of dimension 1 with residue field κ. Let K be the fraction field of A. We define the tame symbol of A to be the map K ∗ × K ∗ −→ κ∗ , (x, y) 7−→ dA (x, y) where dA (x, y) is extended to K ∗ × K ∗ by the multiplicativity of Lemma 29.4.4. It is clear that we may extend more generally dM (−, −) to certain rings of fractions of A (even if A is not a domain). Lemma 29.4.6. Let A be a Noetherian local ring. Let M be a finite A-module of dimension 1. Let b ∈ A be a nonzerodivisor on M , and let u ∈ A∗ . Then dM (u, b) = ulengthM (M/bM ) mod mA . In particular, if M = A, then dA (u, b) = uordA (b) mod mA . Proof. Note that in this case M/ubM = M/bM on which multiplication by b is zero. Hence dM (u, b) = detκ (u|M/bM ) by Lemma 29.3.8. The lemma then follows from Lemma 29.2.8.  Lemma 29.4.7. Let A be a Noetherian local ring. Let a, b ∈ A. Let 0 → M → M 0 → M 00 → 0 be a short exact sequence of A-modules of dimension 1 such that a, b are nonzerodivisors on all three A-modules. Then dM 0 (a, b) = dM (a, b)dM 00 (a, b) in κ∗ . Proof. It is easy to see that this leads to a short exact sequence of exact (2, 1)periodic complexes 0 → (M/abM, a, b) → (M 0 /abM 0 , a, b) → (M 00 /abM 00 , a, b) → 0 Hence the lemma follows from Lemma 29.3.9.



Lemma 29.4.8. Let A be a Noetherian local ring. Let α : M → M 0 be a homomorphism of finite A-modules of dimension 1. Let a, b ∈ A. Assume (1) a, b are nonzerodivisors on both M and M 0 , and (2) dim(Ker(α)), dim(Coker(α)) ≤ 0. Then dM (a, b) = dM 0 (a, b). Proof. If a ∈ A∗ , then the equality follows from the equality length(M/bM ) = length(M 0 /bM 0 ) and Lemma 29.4.6. Similarly if b is a unit the lemma holds as well (by the symmetry of Lemma 29.4.4). Hence we may assume that a, b ∈ mA . This in particular implies that m is not an associated prime of M , and hence α : M → M 0 is injective. This permits us to think of M as a submodule of M 0 . By assumption M 0 /M is a finite A-module with support {mA } and hence has finite length. Note that for any third module M 00 with M ⊂ M 00 ⊂ M 0 the maps M → M 00 and M 00 → M 0 satisfy the assumptions of the lemma as well. This reduces us, by induction on the length of M 0 /M , to the case where lengthA (M 0 /M ) = 1. Finally, in this case consider the map α : M/abM −→ M 0 /abM 0 .

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By construction the cokernel Q of α has length 1. Since a, b ∈ mA , they act trvially on Q. It also follows that the kernel K of α has length 1 and hence also a, b act trivially on K. Hence we may apply Lemma 29.3.16. Thus it suffices to see that the two maps αi : Q → K are the same. In fact, both maps are equal to the map  q = x0 mod Im(α) 7→ abx0 ∈ K. We omit the verification. Lemma 29.4.9. Let A be a Noetherian local ring. Let M be a finite A-module with dim(M ) = 1. Let a, b ∈ A nonzerodivisors on M . Let q1 , . . . , qt be the minimal primes in the support of M . Then Y lengthAq (Mqi ) i dM (a, b) = dA/qi (a, b) i=1,...,t

∗

as elements of κ . Proof. Choose a filtration by A-submodules 0 = M0 ⊂ M1 ⊂ . . . ⊂ Mn = M such that each quotient Mj /Mj−1 is isomorphic to A/pj for some prime ideal pj of A. See Algebra, Lemma 7.60.1. For each j we have either pj = qi for some i, or pj = mA . Moreover, for a fixed i, the number of j such that pj = qi is equal to lengthAq (Mqi ) by Algebra, Lemma 7.60.13. Hence dMj (a, b) is defined for each j i and  dMj−1 (a, b)dA/qi (a, b) if pj = qi dMj (a, b) = dMj−1 (a, b) if pj = mA by Lemma 29.4.7 in the first instance and Lemma 29.4.8 in the second. Hence the lemma.  Lemma 29.4.10. Let A be a discrete valuation ring with fraction field K. For nonzero x, y ∈ K we have dA (x, y) = (−1)ordA (x)ordA (y)

xordA (y) mod mA , y ordA (x)

in other words the symbol is equal to the usual tame symbol. Proof. By multiplicativity it suffices to prove this when x, y ∈ A. Let t ∈ A be a uniformizer. Write x = tb u and y = tb v for some a, b ≥ 0 and u, v ∈ A∗ . Set l = a + b. Then tl−1 , . . . , tb is an admissible sequence in (x)/(xy) and tl−1 , . . . , ta is an admissible sequence in (y)/(xy). Hence by Remark 29.3.5 we see that dA (x, y) is characterized by the equation [tl−1 , . . . , tb , v −1 tb−1 , . . . , v −1 ] = (−1)ab dA (x, y)[tl−1 , . . . , ta , u−1 ta−1 , . . . , u−1 ]. Hence by the admissible relations for the symbols [x1 , . . . , xl ] we see that dA (x, y) = (−1)ab ua /v b mod mA as desired.



We add the following lemma here. It is very similar to Algebra, Lemma 7.111.2. Lemma 29.4.11. Let R be a local Noetherian domain of dimension 1 with maximal ideal m. Let a, b ∈ m be nonzero. There exists a finite ring extension R ⊂ R0 with same field of fractions, and t, a0 , b0 ∈ R0 such that a = ta0 and b = tb0 and R0 = a0 R0 + b0 R0 .

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1659

Proof. Set I = (a, b). The idea is to blow up R in I as in the proof of Algebra, Lemma 7.111.2. Instead of doing the algebraic argument we work geometrically. Let L X = Proj( I d /I d+1 ). By Divisors, Lemma 26.17.7 this is an integral scheme. The morphism X → Spec(R) is projective by Divisors, Lemma 26.17.11. By Algebra, Lemma 7.105.2 and the fact that X is quasi-compact we see that the fibre of X → Spec(R) over m is finite. By Properties, Lemma 23.27.5 there exists an affine open U ⊂ X containing this fibre. Hence X = U because X → Spec(R) is closed. In other words X is affine, say X = Spec(R0 ). By Morphisms, Lemma 24.16.2 we see that R → R0 is of finite type. Since X → Spec(R) is proper and affine it is integral (see Morphisms, Lemma 24.44.7). Hence R → R0 is of finite type and integral, hence finite (Algebra, Lemma 7.33.5). By Divisors, Lemma 26.17.4 we see that IR0 is a locally principal ideal. Since R0 is semi-local we see that IR0 is principal, see Algebra, Lemma 7.73.6, say IR0 = (t). Then we have a = a0 t and  b = b0 t and everything is clear. Lemma 29.4.12. Let A be a Noetherian local ring. Let a, b ∈ A. Let M be a finite A-module of dimension 1 on which each of a, b, b − a are nonzerodivisors. Then dM (a, b − a)dM (b, b) = dM (b, b − a)dM (a, b) ∗

in κ . Proof. By Lemma 29.4.9 it suffices to show the relation when M = A/q for some prime q ⊂ A with dim(A/q) = 1. In case M = A/q we may replace A by A/q and a, b by their images in A/q. Hence we may assume A = M and A a local Noetherian domain of dimension 1. The reason is that the residue field κ of A and A/q are the same and that for any A/qmodule M the determinant taken over A or over A/q are canonically identified. See Lemma 29.2.7. It suffices to show the relation when both a, b are in the maximal ideal. Namely, the case where one or both are units follows from Lemma 29.4.6. Choose an extension A ⊂ A0 and factorizations a = ta0 , b = tb0 as in Lemma 29.4.11. Note that also b − a = t(b0 − a0 ) and that A0 = (a0 , b0 ) = (a0 , b0 − a0 ) = (b0 − a0 , b0 ). Here and in the following we think of A0 as an A-module and a, b, a0 , b0 , t as A0 0 module endomorphisms of A0 . We will use the notation dA A0 (a , b ) and so on to indicate 0 0 0 0 0 0 0 0 dA A0 (a , b ) = detκ (A /a b A , a , b ) which is defined by Lemma 29.4.1. The upper index A is used to distinguish this from the already defined symbol dA0 (a0 , b0 ) which is different (for example because it has values in the residue field of A0 which may be different from κ). By Lemma 29.4.8 we see that dA (a, b) = dA A0 (a, b), and similarly for the other combinations. Using this and multiplicativity we see that it suffices to prove 0 0 0 A 0 0 A 0 0 0 A 0 0 dA A0 (a , b − a )dA0 (b , b ) = dA0 (b , b − a )dA0 (a , b )

Now, since (a0 , b0 ) = A0 and so on we have A0 /(a0 (b0 − a0 )) A0 /(b0 (b0 − a0 )) A0 /(a0 b0 )

∼ = A0 /(a0 ) ⊕ A0 /(b0 − a0 ) ∼ = A0 /(b0 ) ⊕ A0 /(b0 − a0 ) ∼ A0 /(a0 ) ⊕ A0 /(b0 ) =

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Moreover, note that multiplication by b0 − a0 on A/(a0 ) is equal to multiplication by b0 , and that multiplication by b0 − a0 on A/(b0 ) is equal to multiplication by −a0 . Using Lemmas 29.3.8 and 29.3.9 we conclude 0 0 0 dA A0 (a , b − a ) A 0 0 0 dA0 (b , b − a ) 0 0 dA A0 (a , b )

= detκ (b0 |A0 /(a0 ) )−1 detκ (a0 |A0 /(b0 −a0 ) ) = detκ (−a0 |A0 /(b0 ) )−1 detκ (b0 |A0 /(b0 −a0 ) ) = detκ (b0 |A0 /(a0 ) )−1 detκ (a0 |A0 /(b0 ) )

Hence we conclude that 0

0

0 0 0 A 0 0 0 A 0 0 (−1)lengthA (A /(b )) dA A0 (a , b − a ) = dA0 (b , b − a )dA0 (a , b )

the sign coming from the −a0 in the second equality above. On the other hand, by 0 0 lengthA (A0 /(b0 )) Lemma 29.3.7 we have dA , and the lemma is proved. A0 (b , b ) = (−1)  The tame symbol is a Steinberg symbol. Lemma 29.4.13. Let A be a Noetherian local domain of dimension 1. Let K = f.f.(A). For x ∈ K \ {0, 1} we have dA (x, 1 − x) = 1 Proof. Write x = a/b with a, b ∈ A. The hypothesis implies, since 1−x = (b−a)/b, that also b − a 6= 0. Hence we compute dA (x, 1 − x) = dA (a, b − a)dA (a, b)−1 dA (b, b − a)−1 dA (b, b) Thus we have to show that dA (a, b − a)dA (b, b) = dA (b, b − a)dA (a, b). This is Lemma 29.4.12.  29.5. Lengths and determinants In this section we use the determinant to compare lattices. The key lemma is the following. Lemma 29.5.1. Let R be a noetherian local ring. Let q ⊂ R be a prime with dim(R/q) = 1. Let ϕ : M → N be a homomorphism of finite R-modules. Assume there exist x1 , . . . , xl ∈ M and y1 , . . . , yl ∈ M with the following properties (1) M = hx1 , . . . , xl i, (2) hx1 , . . . , xi i/hx1 , . . . , xi−1 i ∼ = R/q for i = 1, . . . , l, (3) N = hy1 , . . . , yl i, and (4) hy1 , . . . , yi i/hy1 , . . . , yi−1 i ∼ = R/q for i = 1, . . . , l. Then ϕ is injective if and only if ϕq is an isomorphism, and in this case we have lengthR (Coker(ϕ)) = ordR/q (f ) where f ∈ κ(q) is the element such that [ϕ(x1 ), . . . , ϕ(xl )] = f [y1 , . . . , yl ] in detκ(q) (Nq ). Proof. First, note that the lemma holds in case l = 1. Namely, in this case x1 is a basis of M over R/q and y1 is a basis of N over R/q and we have ϕ(x1 ) = f y1 for some f ∈ R. Thus ϕ is injective if and only if f 6∈ q. Moreover, Coker(ϕ) = R/(f, q) and hence the lemma holds by definition of ordR/q (f ) (see Algebra, Definition 7.113.2).

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1661

In fact, suppose more generally that ϕ(xi ) = fi yi for some fi ∈ R, fi 6∈ q. Then the induced maps hx1 , . . . , xi i/hx1 , . . . , xi−1 i −→ hy1 , . . . , yi i/hy1 , . . . , yi−1 i are all injective and have cokernels isomorphic to R/(fi , q). Hence we see that X lengthR (Coker(ϕ)) = ordR/q (fi ). On the other hand it is clear that [ϕ(x1 ), . . . , ϕ(xl )] = f1 . . . fl [y1 , . . . , yl ] in this case from the admissible relation (b) for symbols. Hence we see the result holds in this case also. We prove the general case by induction on l. Assume l > 1. Let i ∈ {1, . . . , l} be minimal such that ϕ(x1 ) ∈ hy1 , . . . , yi i. We will argue by induction on i. If i = 1, then we get a commutative diagram 0

/ hx1 i

/ hx1 , . . . , xl i

/ hx1 , . . . , xl i/hx1 i

/0

0

 / hy1 i

 / hy1 , . . . , yl i

 / hy1 , . . . , yl i/hy1 i

/0

and the lemma follows from the snake lemma and induction on l. Assume now that i > 1. Write ϕ(x1 ) = a1 y1 + . . . + ai−1 yi−1 + ayi with aj , a ∈ R and a 6∈ q (since otherwise i was not minimal). Set   xj if j = 1 yj if j < i 0 0 xj = and yj = axj if j ≥ 2 ayj if j ≥ i 0 Let M 0 = hx01 , . . . , x0l i and N 0 = hy10 , . . . , yl0 i. Since ϕ(x01 ) = a1 y10 +. . .+ai−1 yi−1 +yi0 0 0 0 by construction and since for j > 1 we have ϕ(xj ) = aϕ(xi ) ∈ hy1 , . . . , yl i we get a commutative diagram of R-modules and maps

M0  M

/ N0

ϕ0

 /N

ϕ

By the result of the second paragraph of the proof we know that lengthR (M/M 0 ) = (l − 1)ordR/q (a) and similarly lengthR (M/M 0 ) = (l − i + 1)ordR/q (a). By a diagram chase this implies that lengthR (Coker(ϕ0 )) = lengthR (Coker(ϕ)) + i ordR/q (a). On the other hand, it is clear that writing [ϕ(x1 ), . . . , ϕ(xl )] = f [y1 , . . . , yl ],

[ϕ0 (x01 ), . . . , ϕ(x0l )] = f 0 [y10 , . . . , yl0 ]

we have f 0 = ai f . Hence it suffices to prove the lemma for the case that ϕ(x1 ) = a1 y1 + . . . ai−1 yi−1 + yi , i.e., in the case that a = 1. Next, recall that [y1 , . . . , yl ] = [y1 , . . . , yi−1 , a1 y1 + . . . ai−1 yi−1 + yi , yi+1 , . . . , yl ] by the admissible relations for symbols. The sequence y1 , . . . , yi−1 , a1 y1 + . . . + ai−1 yi−1 + yi , yi+1 , . . . , yl satisfies the conditions (3), (4) of the lemma also. Hence,

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we may actually assume that ϕ(x1 ) = yi . In this case, note that we have qx1 = 0 which implies also qyi = 0. We have [y1 , . . . , yl ] = −[y1 , . . . , yi−2 , yi , yi−1 , yi+1 , . . . , yl ] by the third of the admissible relations defining detκ(q) (Nq ). Hence we may replace y1 , . . . , yl by the sequence y10 , . . . , yl0 = y1 , . . . , yi−2 , yi , yi−1 , yi+1 , . . . , yl (which also satisfies conditions (3) and (4) of the lemma). Clearly this decreases the invariant i by 1 and we win by induction on i.  To use the previous lemma we show that often sequences of elements with the required properties exist. Lemma 29.5.2. Let R be a local Noetherian ring. Let q ⊂ R be a prime ideal. Let M be a finite R-module such that q is one of the minimal primes of the support of M . Then there exist x1 , . . . , xl ∈ M such that (1) the support of M/hx1 , . . . , xl i does not contain q, and (2) hx1 , . . . , xi i/hx1 , . . . , xi−1 i ∼ = R/q for i = 1, . . . , l. Moreover, in this case l = lengthRq (Mq ). Proof. The condition that q is a minimal prime in the support of M implies that l = lengthRq (Mq ) is finite (see Algebra, Lemma 7.60.11). Hence we can find y1 , . . . , yl ∈ Mq such that hy1 , . . . , yi i/hy1 , . . . , yi−1 i ∼ = κ(q) for i = 1, . . . , l. We can find fi ∈ R, fi 6∈ q such that fi yi is the image of some element zi ∈ M . Moreover, as R is Noetherian we can write q = (g1 , . . . , gt ) for some gj ∈ R. By assumption gj yi ∈ hy1 , . . . , yi−1 i inside the module Mq . By our choice of zi we can find some further elements fji ∈ R, fij 6∈ q such that fij gj zi ∈ hz1 , . . . , zi−1 i (equality in the module M ). The lemma follows by taking x1 = f11 f12 . . . f1t z1 ,

x2 = f11 f12 . . . f1t f21 f22 . . . f2t z2 ,

and so on. Namely, since all the elements fi , fij are invertible in Rq we still have that Rq x1 +. . .+Rq xi /Rq x1 +. . .+Rq xi−1 ∼ = κ(q) for i = 1, . . . , l. By construction, qxi ∈ hx1 , . . . , xi−1 i. Thus hx1 , . . . , xi i/hx1 , . . . , xi−1 i is an R-module generated by one element, annihilated q such that localizing at q gives a q-dimensional vector space over κ(q). Hence it is isomorphic to R/q.  Here is the main result of this section. We will see below the various different consequences of this proposition. The reader is encouraged to first prove the easier Lemma 29.5.4 his/herself. Proposition 29.5.3. Let R be a local Noetherian ring with residue field κ. Suppose that (M, ϕ, ψ) is a (2, 1)-periodic complex over R. Assume (1) M is a finite R-module, (2) the cohomology modules of (M, ϕ, ψ) are of finite length, and (3) dim(Supp(M )) = 1. Let qi , i = 1, . . . , t be the minimal primes of the support of M . Then we have2 X
−eR (M, ϕ, ψ) = ordR/qi detκ(qi ) (Mqi , ϕqi , ψqi ) i=1,...,t

2 Obviously we could get rid of the minus sign by redefining det (M, ϕ, ψ) as the inverse of κ its current value, see Definition 29.3.4.

29.5. LENGTHS AND DETERMINANTS

1663

Proof. We first reduce to the case t = 1 in the following way. Note that Supp(M ) = {m, q1 , . . . , qt }, where m ⊂ R is the maximal ideal. Let Mi denote the image of M → Mqi , so Supp(Mi ) = {m, qi }. The map ϕ (resp. ψ) induces an R-module map ϕi : Mi → Mi (resp. ψi : Mi → Mi ). Thus we get a morphism of (2, 1)-periodic complexes M (M, ϕ, ψ) −→ (Mi , ϕi , ψi ). i=1,...,t

The kernel and cokernel of this map have support equal to {m} (or are zero). Hence by Lemma 29.3.3 these (2, 1)-periodic complexes have multiplicity 0. In other words we have X eR (M, ϕ, ψ) = eR (Mi , ϕi , ψi ) i=1,...,t

On the other hand we clearly have Mqi = Mi,qi , and hence the terms of the right hand side of the formula of the lemma are equal to the expressions
ordR/qi detκ(qi ) (Mi,qi , ϕi,qi , ψi,qi ) In other words, if we can prove the lemma for each of the modules Mi , then the lemma holds. This reduces us to the case t = 1. Assume we have a (2, 1)-periodic complex (M, ϕ, ψ) over a Noetherian local ring with M a finite R-module, Supp(M ) = {m, q}, and finite length cohomology modules. The proof in this case follows from Lemma 29.5.1 and careful bookkeeping. Denote Kϕ = Ker(ϕ), Iϕ = Im(ϕ), Kψ = Ker(ψ), and Iψ = Im(ψ). Since R is Noetherian these are all finite R-modules. Set a = lengthRq (Iϕ,q ) = lengthRq (Kψ,q ),

b = lengthRq (Iψ,q ) = lengthRq (Kϕ,q ).

Equalities because the complex becomes exact after localizing at q. Note that l = lengthRq (Mq ) is equal to l = a + b. We are going to use Lemma 29.5.2 to choose sequences of elements in finite Rmodules N with support contained in {m, q}. In this case Nq has finite length, say n ∈ N. Let us call a sequence w1 , . . . , wn ∈ N with properties (1) and (2) of Lemma 29.5.2 a “good sequence”. Note that the quotient N/hw1 , . . . , wn i of N by the submodule generated by a good sequence has support (contained in) {m} and hence has finite length (Algebra, Lemma 7.60.11). Moreover, the symbol [w1 , . . . , wn ] ∈ detκ(q) (Nq ) is a generator, see Lemma 29.2.4. Having said this we choose good sequences x1 , . . . , x b y1 , . . . , ya

in Kϕ , in Iϕ ∩ ht1 , . . . ta i,

t 1 , . . . , ta s1 , . . . , sb

in Kψ , in Iψ ∩ hx1 , . . . , xb i.

We will adjust our choices a little bit as follows. Choose lifts y˜i ∈ M of yi ∈ Iϕ and s˜i ∈ M of si ∈ Iψ . It may not be the case that q˜ y1 ⊂ hx1 , . . . , xb i and it may not be the case that q˜ s1 ⊂ ht1 , . . . , ta i. However, using that q is finitely generated (as in the proof of Lemma 29.5.2) we can find a d ∈ R, d 6∈ q such that qd˜ y1 ⊂ hx1 , . . . , xb i and qd˜ s1 ⊂ ht1 , . . . , ta i. Thus after replacing yi by dyi , y˜i by d˜ yi , si by dsi and s˜i by d˜ si we see that we may assume also that x1 , . . . , xb , y˜1 , . . . , y˜b and t1 , . . . , ta , s˜1 , . . . , s˜b are good sequences in M . Finally, we choose a good sequence z1 , . . . , zl in the finite R-module hx1 , . . . , xb , y˜1 , . . . , y˜a i ∩ ht1 , . . . , ta , s˜1 , . . . , s˜b i. Note that this is also a good sequence in M .

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29. CHOW HOMOLOGY AND CHERN CLASSES

Since Iϕ,q = Kψ,q there is a unique element h ∈ κ(q) such that [y1 , . . . , ya ] = h[t1 , . . . , ta ] inside detκ(q) (Kψ,q ). Similarly, as Iψ,q = Kϕ,q there is a unique element h ∈ κ(q) such that [s1 , . . . , sb ] = g[x1 , . . . , xb ] inside detκ(q) (Kϕ,q ). We can also do this with the three good sequences we have in M . All in all we get the following identities [y1 , . . . , ya ]

= h[t1 , . . . , ta ]

[s1 , . . . , sb ]

= g[x1 , . . . , xb ]

[z1 , . . . , zl ]

= fϕ [x1 , . . . , xb , y˜1 , . . . , y˜a ]

[z1 , . . . , zl ]

= fψ [t1 , . . . , ta , s˜1 , . . . , s˜b ]

for some g, h, fϕ , fψ ∈ κ(q). Having set up all this notation let us compute detκ(q) (M, ϕ, ψ). Namely, consider the element [z1 , . . . , zl ]. Under the map γψ ◦ σ ◦ γϕ−1 of Definition 29.3.4 we have [z1 , . . . , zl ]

=

fϕ [x1 , . . . , xb , y˜1 , . . . , y˜a ]

7→

fϕ [x1 , . . . , xb ] ⊗ [y1 , . . . , ya ]

7→

fϕ h/g[t1 , . . . , ta ] ⊗ [s1 , . . . , sb ]

7→

fϕ h/g[t1 , . . . , ta , s˜1 , . . . , s˜b ]

=

fϕ h/fψ g[z1 , . . . , zl ]

This means that detκ(q) (Mq , ϕq , ψq ) is equal to fϕ h/fψ g up to a sign. We abbreviate the following quantities kϕ

=

lengthR (Kϕ /hx1 , . . . , xb i)

kψ

=

lengthR (Kψ /ht1 , . . . , ta i)

iϕ

=

lengthR (Iϕ /hy1 , . . . , ya i)

iψ

=

lengthR (Iψ /hs1 , . . . , sa i)

mϕ

=

lengthR (M/hx1 , . . . , xb , y˜1 , . . . , y˜a i)

mψ

=

lengthR (M/ht1 , . . . , ta , s˜1 , . . . , s˜b i)

δϕ

=

lengthR (hx1 , . . . , xb , y˜1 , . . . , y˜a ihz1 , . . . , zl i)

δψ

=

lengthR (ht1 , . . . , ta , s˜1 , . . . , s˜b ihz1 , . . . , zl i)

Using the exact sequences 0 → Kϕ → M → Iϕ → 0 we get mϕ = kϕ + iϕ . Similarly we have mψ = kψ + iψ . We have δϕ + mϕ = δψ + mψ since this is equal to the colength of hz1 , . . . , zl i in M . Finally, we have δϕ = ordR/q (fϕ ),

δψ = ordR/q (fψ )

by our first application of the key Lemma 29.5.1.

29.5. LENGTHS AND DETERMINANTS

1665

Next, let us compute the multiplicity of the periodic complex eR (M, ϕ, ψ)

=

lengthR (Kϕ /Iψ ) − lengthR (Kψ /Iϕ )

=

lengthR (hx1 , . . . , xb i/hs1 , . . . , sb i) + kϕ − iψ −lengthR (ht1 , . . . , ta i/hy1 , . . . , ya i) − kψ + iϕ

=

ordR/q (g/h) + kϕ − iψ − kψ + iϕ

=

ordR/q (g/h) + mϕ − mψ

=

ordR/q (g/h) + δψ − δϕ

=

ordR/q (fψ g/fϕ h)

where we used the key Lemma 29.5.1 twice in the third equality. By our computation of detκ(q) (Mq , ϕq , ψq ) this proves the proposition.  In most applications the following lemma suffices. Lemma 29.5.4. Let R be a Noetherian local ring with maximal ideal m. Let M be a finite R-module, and let ψ : M → M be an R-module map. Assume that (1) Ker(ψ) and Coker(ψ) have finite length, and (2) dim(Supp(M )) ≤ 1. Write Supp(M ) = {m, q1 , . . . , qt } and denote fi ∈ κ(qi )∗ the element such that detκ(qi ) (ψqi ) : detκ(qi ) (Mqi ) → detκ(qi ) (Mqi ) is multiplication by fi . Then we have X lengthR (Coker(ψ)) − lengthR (Ker(ψ)) = ordR/qi (fi ). i=1,...,t

0

1

Proof. Recall that H (M, 0, ψ) = Coker(ψ) and H (M, 0, ψ) = Ker(ψ), see remarks above Definition 29.3.2. The lemma follows by combining Proposition 29.5.3 with Lemma 29.3.8. Alternative proof. Reduce to the case Supp(M ) = {m, q} as in the proof of Proposition 29.5.3. Then directly combine Lemmas 29.5.1 and 29.5.2 to prove this specific case of Proposition 29.5.3. There is much less bookkeeping in this case, and the reader is encouraged to work this out. Details omitted.  Lemma 29.5.5. Let R be a Noetherian local ring with maximal ideal m. Let M be a finite R-module. Let x ∈ R. Assume that (1) dim(Supp(M )) ≤ 1, and (2) dim(M/xM ) ≤ 0. Write Supp(M ) = {m, q1 , . . . , qt }. Then X lengthR (Mx ) − lengthR (x M ) = ordR/qi (x)lengthRq (Mqi ). i=1,...,t

i

where Mx = M/xM and x M = Ker(x : M → M ). Proof. This is a special case of Lemma 29.5.4. To see that fi = x Lemma 29.2.8.

lengthRq (Mqi ) i

see 

Lemma 29.5.6. Let R be a Noetherian local ring with maximal ideal m. Let I ⊂ R be an ideal and let x ∈ R. Assume x is a nonzerodivisor on R/I and that dim(R/I) = 1. Then X lengthR (R/(x, I)) = lengthR (R/(x, qi ))lengthRq ((R/I)qi ) i

i

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29. CHOW HOMOLOGY AND CHERN CLASSES

where q1 , . . . , qn are the minimal primes over I. More generally if M is any finite Cohen-Macaulay module of dimension 1 over R and dim(M/xM ) = 0, then X lengthR (M/xM ) = lengthR (R/(x, qi ))lengthRq (Mqi ). i

i

where q1 , . . . , qt are the minimal primes of the support of M . Proof. These are special cases of Lemma 29.5.5.



Lemma 29.5.7. Let A be a Noetherian local ring. Let M be a finite A-module. Let a, b ∈ A. Assume (1) dim(A) = 1, (2) both a and b are nonzerodivisors in A, (3) A has no embedded primes, (4) M has no embedded associated primes, (5) Supp(M ) = Spec(A). Let I = {x ∈ A | x(a/b) ∈ A}. Let q1 , . . . , qt be the minimal primes of A. Then (a/b)IM ⊂ M and X lengthA (M/(a/b)IM ) − lengthA (M/IM ) = lengthAq (Mqi )ordA/qi (a/b) i

i

Proof. Since M has no embedded associated primes, and since the support of M is Spec(A) we see that Ass(M ) = {q1 , . . . , qt }. Hence a, b are nonzerodivisors on M . Note that lengthA (M/(a/b)IM ) = lengthA (bM/aIM ) = lengthA (M/aIM ) − lengthA (M/bM ) = lengthA (M/aM ) + lengthA (aM/aIM ) − lengthA (M/bM ) = lengthA (M/aM ) + lengthA (M/IM ) − lengthA (M/bM ) as the injective map b : M → bM maps (a/b)IM to aIM and the injective map a : M → aM maps IM to aIM . Hence the left hand side of the equation of the lemma is equal to lengthA (M/aM ) − lengthA (M/bM ). Applying the second formula of Lemma 29.5.6 with x = a, b respectively and using Algebra, Definition 7.113.2 of the ord-functions we get the result.  29.6. Application to tame symbol In this section we apply the results above to show the following lemma. Lemma 29.6.1. Let A be a 2-dimensional Noetherian local domain. Let K = f.f.(A). Let f, g ∈ K ∗ . Let q1 , . . . , qt be the height 1 primes q of A such that either f or g is not an element of A∗q . Then we have X ordA/qi (dAqi (f, g)) = 0 i=1,...,t

We can also write this as X height(q)=1

ordA/q (dAq (f, g)) = 0

since at any height one prime q of A where f, g ∈ A∗q we have dAq (f, g) = 1 by Lemma 29.4.6.

29.7. SETUP

1667

Proof. Since the tame symbols dAq (f, g) are additive (Lemma 29.4.4) and the order functions ordA/q are additive (Algebra, Lemma 7.113.1) it suffices to prove the formula when f = a ∈ A and g = b ∈ A. In this case we see that we have to show X ordA/q (detκ (Aq /(ab), a, b)) = 0 height(q)=1

By Proposition 29.5.3 this is equivalent to showing that eA (A/(ab), a, b) = 0. a

b

a

Since the complex A/(ab) − → A/(ab) → − A/(ab) − → A/(ab) is exact we win.



29.7. Setup We will throughout work over a locally Noetherian universally catenary base S endowed with a dimension function δ. Allthough it is likely possible to generalize (parts of) the discussion in the chapter, it seems that this is a good first approximation. We usually do not assume our schemes are separated or quasi-compact. Many interesting algebraic stacks are non-separated and/or non-quasi-compact and this is a good case study to see how to develop a reasonable theory for those as well. In order to reference these hypotheses we give it a number. Situation 29.7.1. Here S is a locally Noetherian, and universally catenary scheme. Moreover, we assume S is endowed with a dimension function δ : S −→ Z. See Morphisms, Definition 24.18.1 for the notion of a universally catenary scheme, and see Topology, Definition 5.16.1 for the notion of a dimension function. Recall that any locally Noetherian catenary scheme locally has a dimension function, see Properties, Lemma 23.11.3. Moreover, there are lots of schemes which are universally catenary, see Morphisms, Lemma 24.18.4. Let (S, δ) be as in Situation 29.7.1. Any scheme X locally of finite type over S is locally Noetherian and catenary. In fact, X has a canonical dimension function δ = δX/S : X −→ Z associated to (f : X → S, δ) given by the rule δX/S (x) = δ(f (x)) + trdegκ(f (x)) κ(x). See Morphisms, Lemma 24.31.2. Moreover, if h : X → Y is a morphism of schemes locally of finite type over S, and x ∈ X, y = h(x), then obviously δX/S (x) = δY /S (y) + trdegκ(y) κ(x). We will freely use this function and its properties in the following. Here are the basic examples of setups as above. In fact, the main interest lies in the case where the base is the spectrum of a field, or the case where the base is the spectrum of a Dedekind ring (e.g. Z, or a discrete valuation ring). Example 29.7.2. Here S = Spec(k) and k is a field. We set δ(pt) = 0 where pt indicates the unique point of S. The pair (S, δ) is an example of a situation as in Situation 29.7.1 by Morphisms, Lemma 24.18.4. Example 29.7.3. Here S = Spec(A), where A is a Noetherian domain of dimension 1. For example we could consider A = Z. We set δ(p) = 0 if p is a maximal ideal and δ(p) = 1 if p = (0) corresponds to the generic point. This is an example of Situation 29.7.1 by Morphisms, Lemma 24.18.4. In good cases δ corresponds to the dimension function.

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29. CHOW HOMOLOGY AND CHERN CLASSES

Lemma 29.7.4. Let (S, δ) be as in Situation 29.7.1. Assume in addition S is a Jacobson scheme, and δ(s) = 0 for every closed point s of S. Let X be locally of finite type over S. Let Z ⊂ X be an integral closed subscheme and let ξ ∈ Z be its generic point. The following integers are the same: (1) δX/S (ξ), (2) dim(Z), and (3) dim(OZ,z ) where z is a closed point of Z. Proof. Let X → S, ξ ∈ Z ⊂ X be as in the lemma. Since X is locally of finite type over S we see that X is Jacobson, see Morphisms, Lemma 24.17.9. Hence closed points of X are dense in every closed subset of Z and map to closed points of S. Hence given any chain of irreducible closed subsets of Z we can end it with a closed point of Z. It follows that dim(Z) = supz (dim(OZ,z ) (see Properties, Lemma 23.11.4) where z ∈ Z runs over the closed points of Z. Note that dim(OZ,z ) = δ(ξ) − δ(z)) by the properties of a dimension function. For each closed z ∈ Z the field extension κ(z) ⊃ κ(f (z)) is finite, see Morphisms, Lemma 24.17.8. Hence δX/S (z) = δ(f (z)) = 0 for z ∈ Z closed. It follows that all three integers are equal.  In the situation of the lemma above the value of δ at the generic point of a closed irreducible subset is the dimension of the irreducible closed subset. However, in general we cannot expect the equality to hold. For example if S = Spec(C[[t]]) and X = Spec(C((t))) then we would get δ(x) = 1 for the unique point of X, but dim(X) = 0. Still we want to think of δX/S as giving the dimension of the irreducible closed subschemes. Thus we introduce the following terminology. Definition 29.7.5. Let (S, δ) as in Situation 29.7.1. For any scheme X locally of finite type over S and any irreducible closed subset Z ⊂ X we define dimδ (Z) = δ(ξ) where ξ ∈ Z is the generic point of Z. We will call this the δ-dimension of Z. If Z is a closed subscheme of X, then we define dimδ (Z) as the supremum of the δ-dimensions of its irreducible components. 29.8. Cycles Since we are not assuming our schemes are quasi-compact we have to be a little careful when defining cycles. We have to allow infinite sums because a rational function may have infinitely many poles for example. In any case, if X is quasicompact then a cycle is a finite sum as usual. Definition 29.8.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let k ∈ Z. (1) A collection of closed subschemes {Zi }i∈I of X is said to be locally finite if for every quasi-compact open U ⊂ X the set #{i ∈ I | Zi ∩ U 6= ∅} is finite. (2) A cycle on X is a formal sum X α= nZ [Z]

29.9. CYCLE ASSOCIATED TO A CLOSED SUBSCHEME

1669

where the sum is over integral closed subschemes Z ⊂ X, each nZ ∈ Z, and the collection {Z; nZ 6= 0} is locally finite. (3) A k-cycle, on X is a cycle X α= nZ [Z] where nZ 6= 0 ⇒ dimδ (Z) = k. (4) The abelian group of all k-cycles on X is denoted Zk (X). In other words, a k-cycle on X is a locally finite formal Z-linear combination of P integral closed subschemes of δ-dimension k. Addition of k-cycles α = n [Z] Z P and β = mZ [Z] is given by X α+β = (nZ + mZ )[Z], i.e., by adding the coefficients. 29.9. Cycle associated to a closed subscheme Lemma 29.9.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. (1) The collection of irreducible components of Z is locally finite. (2) Let Z 0 ⊂ Z be an irreducible component and let ξ ∈ Z 0 be its generic point. Then lengthOX,ξ OZ,ξ < ∞ (3) If dimδ (Z) ≤ k and ξ ∈ Z with δ(ξ) = k, then ξ is a generic point of an irreducible component of Z. Proof. Let U ⊂ X be a quasi-compact open subscheme. Then U is a Noetherian scheme, and hence has a Noetherian underlying topological space (Properties, Lemma 23.5.5). Hence every subspace is Noetherian and has finitely many irreducible components (see Topology, Lemma 5.6.2). This proves (1). Let Z 0 ⊂ Z, ξ ∈ Z 0 be as in (2). Then dim(OZ,ξ ) = 0 (for example by Properties, Lemma 23.11.4). Hence OZ,ξ is Noetherian local ring of dimension zero, and hence has finite length over itself (see Algebra, Proposition 7.58.6). Hence, it also has finite length over OX,ξ , see Algebra, Lemma 7.49.12. Assume ξ ∈ Z and δ(ξ) = k. Consider the closure Z 0 = {ξ}. It is an irreducible closed subscheme with dimδ (Z 0 ) = k by definition. Since dimδ (Z) = k it must be an irreducible component of Z. Hence we see (3) holds.  Definition 29.9.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. (1) For any irreducible component Z 0 ⊂ Z with generic point ξ the integer mZ 0 ,Z = lengthOX,ξ OZ,ξ (Lemma 29.9.1) is called the multiplicity of Z 0 in Z. (2) Assume dimδ (Z) ≤ k. The k-cycle associated to Z is X [Z]k = mZ 0 ,Z [Z 0 ] where the sum is over the irreducible components of Z of δ-dimension k. (This is a k-cycle by Lemma 29.9.1.)

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It is important to note that we only define [Z]k if the δ-dimension of Z does not exceed k. In other words, by convention, if we write [Z]k then this implies that dimδ (Z) ≤ k. 29.10. Cycle associated to a coherent sheaf Lemma 29.10.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let F be a coherent OX -module. (1) The collection of irreducible components of the support of F is locally finite. (2) Let Z 0 ⊂ Supp(F) be an irreducible component and let ξ ∈ Z 0 be its generic point. Then lengthOX,ξ Fξ < ∞ (3) If dimδ (Supp(F)) ≤ k and ξ ∈ Z with δ(ξ) = k, then ξ is a generic point of an irreducible component of Supp(F). Proof. By Cohomology of Schemes, Lemma 25.11.7 the support Z of F is a closed subset of X. We may think of Z as a reduced closed subscheme of X (Schemes, Lemma 21.12.4). Hence (1) and (3) follow immediately by applying Lemma 29.9.1 to Z ⊂ X. Let ξ ∈ Z 0 be as in (2). In this case for any specialization ξ 0 ξ in X we have Fξ0 = 0. Recall that the non-maximal primes of OX,ξ correspond to the points of X specializing to ξ (Schemes, Lemma 21.13.2). Hence Fξ is a finite OX,ξ -module whose support is {mξ }. Hence it has finite length by Algebra, Lemma 7.60.11.  Definition 29.10.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let F be a coherent OX -module. (1) For any irreducible component Z 0 ⊂ Supp(F) with generic point ξ the integer mZ 0 ,F = lengthOX,ξ Fξ (Lemma 29.10.1) is called the multiplicity of Z 0 in F. (2) Assume dimδ (Supp(F)) ≤ k. The k-cycle associated to F is X [F]k = mZ 0 ,F [Z 0 ] where the sum is over the irreducible components of Supp(F) of δ-dimension k. (This is a k-cycle by Lemma 29.10.1.) It is important to note that we only define [F]k if F is coherent and the δ-dimension of Supp(F) does not exceed k. In other words, by convention, if we write [F]k then this implies that F is coherent on X and dimδ (Supp(F)) ≤ k. Lemma 29.10.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let Z ⊂ X be a closed subscheme. If dimδ (Z) ≤ k, then [Z]k = [OZ ]k . Proof. This is because in this case the multiplicities mZ 0 ,Z and mZ 0 ,OZ agree by definition.  Lemma 29.10.4. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let 0 → F → G → H → 0 be a short exact sequence of coherent sheaves on X. Assume that the δ-dimension of the supports of F, G, and H is ≤ k. Then [G]k = [F]k + [H]k . Proof. Follows immediately from additivity of lengths, see Algebra, Lemma 7.49.3. 

29.12. PROPER PUSHFORWARD

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29.11. Preparation for proper pushforward Lemma 29.11.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume X, Y integral and dimδ (X) = dimδ (Y ). Then either f (X) is contained in a proper closed subscheme of Y , or f is dominant and the extension of function fields R(Y ) ⊂ R(X) is finite. Proof. The closure f (X) ⊂ Y is irreducible as X is irreducible. If f (X) 6= Y , then we are done. If f (X) = Y , then f is dominant and by Morphisms, Lemma 24.8.5 we see that the generic point ηY of Y is in the image of f . Of course this implies that f (ηX ) = ηY , where ηX ∈ X is the generic point of X. Since δ(ηX ) = δ(ηY ) we see that R(Y ) = κ(ηY ) ⊂ κ(ηX ) = R(X) is an extension of transcendence degree 0. Hence Morphisms, Lemma 24.47.4 applies.  Lemma 29.11.2. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is quasi-compact, and {Zi }i∈I is a locally finite collection of closed subsets of X. Then {f (Zi )}i∈I is a locally finite collection of closed subsets of Y . Proof. Let V ⊂ Y be a quasi-compact open subset. Since f is quasi-compact the open f −1 (V ) is quasi-compact. Hence the set {i ∈ I | Zi ∩ f −1 (V ) 6= ∅} is finite by assumption. Since this is the same as the set {i ∈ I | f (Zi ) ∩ V 6= ∅} we win.  29.12. Proper pushforward Definition 29.12.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is proper. (1) Let Z ⊂ X be an integral closed subscheme with dimδ (Z) = k. We define  0 if dimδ (f (Z)) < k, f∗ [Z] = deg(Z/f (Z))[f (Z)] if dimδ (f (Z)) = k. Here we think of f (Z) ⊂ Y as an integral closed subscheme. The degree of Z overPf (Z) is finite if dimδ (f (Z)) = dimδ (Z) by Lemma 29.11.1. (2) Let α = nZ [Z] be a k-cycle on X. The pushforward of α as the sum X f∗ α = nZ f∗ [Z] where each f∗ [Z] is defined as above. The sum is locally finite by Lemma 29.11.2 above. By definition the proper pushforward of cycles f∗ : Zk (X) −→ Zk (Y ) is a homomorphism of abelian groups. It turns X 7→ Zk (X) into a covariant functor on the category of schemes locally of finite type over S with morphisms equal to proper morphisms. Lemma 29.12.2. Let (S, δ) be as in Situation 29.7.1. Let X, Y , and Z be locally of finite type over S. Let f : X → Y and g : Y → Z be proper morphisms. Then g∗ ◦ f∗ = (g ◦ f )∗ as maps Zk (X) → Zk (Z).

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Proof. Let W W 0 = f (Z) ⊂ W 0 (resp. W 00 ) that g∗ (f∗ [W ]) dimδ (W 00 ) = k,

⊂ X be an integral closed subscheme of dimension k. Consider Y and W 00 = g(f (Z)) ⊂ Z. Since f , g are proper we see that is an integral closed subscheme of Y (resp. Z). We have to show = (f ◦ g)∗ [W ]. If dimδ (W 00 ) < k, then both sides are zero. If then we see the induced morphisms W −→ W 0 −→ W 00

both satisfy the hypotheses of Lemma 29.11.1. Hence g∗ (f∗ [W ]) = deg(W/W 0 ) deg(W 0 /W 00 )[W 00 ],

(f ◦ g)∗ [W ] = deg(W/W 00 )[W 00 ].

Then we can apply Morphisms, Lemma 24.47.6 to conclude.



Lemma 29.12.3. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is proper. (1) Let Z ⊂ X be a closed subscheme with dimδ (Z) ≤ k. Then f∗ [Z]k = [f∗ OZ ]k . (2) Let F be a coherent sheaf on X such that dimδ (Supp(F)) ≤ k. Then f∗ [F]k = [f∗ F]k . Note that the statement makes sense since f∗ F and f∗ OZ are coherent OY -modules by Cohomology of Schemes, Lemma 25.18.2. Proof. Part (1) follows from (2) and Lemma 29.10.3. Let F be a coherent sheaf on X. Assume that dimδ (Supp(F)) ≤ k. By Cohomology of Schemes, Lemma 25.11.7 there exists a closed subscheme i : Z → X and a coherent OZ -module G such that i∗ G ∼ = F and such that the support of F is Z. Let Z 0 ⊂ Y be the scheme theoretic image of f |Z : Z → Y .Consider the commutative diagram of schemes Z

i

f |Z

 Z0

0

i

/X  /Y

f

We have f∗ F = f∗ i∗ G = i0∗ (f |Z )∗ G by going around the diagram in two ways. Suppose we know the result holds for closed immersions and for f |Z . Then we see that f∗ [F]k = f∗ i∗ [G]k = (i0 )∗ (f |Z )∗ [G]k = (i0 )∗ [(f |Z )∗ G]k = [(i0 )∗ (f |Z )∗ G]k = [f∗ F]k as desired. The case of a closed immersion is straightforward (omitted). Note that f |Z : Z → Z 0 is a dominant morphism (see Morphisms, Lemma 24.6.3). Thus we have reduced to the case where dimδ (X) ≤ k and f : X → Y is proper and dominant. Assume dimδ (X) ≤ k and f : X → Y is proper and dominant. Since f is dominant, for every irreducible component Z ⊂ Y with generic point η there exists a point ξ ∈ X such that f (ξ) = η. Hence δ(η) ≤ δ(ξ) ≤ k. Thus we see that in the expressions X X f∗ [F]k = nZ [Z], and [f∗ F]k = mZ [Z]. whenever nZ 6= 0, or mZ 6= 0 the integral closed subscheme Z is actually an irreducible component of Y of δ-dimension k. Pick such an integral closed subscheme Z ⊂ Y and denote η its generic point. Note that for any ξ ∈ X with f (ξ) = η we

29.13. PREPARATION FOR FLAT PULLBACK

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have δ(ξ) ≥ k and hence ξ is a generic point of an irreducible component of X of δ-dimension k as well (see Lemma 29.9.1). Since f is quasi-compact and X is locally Noetherian, there can be only finitely many of these and hence f −1 ({η}) is finite. By Morphisms, Lemma 24.47.1 there exists an open neighbourhood η ∈ V ⊂ Y such that f −1 (V ) → V is finite. Replacing Y by V and X by f −1 (V ) we reduce to the case where Y is affine, and f is finite. Write Y = Spec(R) and X = Spec(A) (possible as a finite morphism is affine). f for Then R and A are Noetherian rings and A is finite over R. Moreover F = M some finite A-module M . Note that f∗ F corresponds to M viewed as an R-module. Let p ⊂ R be the minimal prime corresponding to η ∈ Y . The coefficient of Z in [f∗ F]k is clearly lengthRp (Mp ). Let qi , i = 1, . . . , t be the primes of A lying over p. Q Then Ap = Aqi since Ap is an Artinian ring being finite over the dimension zero local Noetherian ring Rp . Clearly the coefficient of Z in f∗ [F]k is X [κ(qi ) : κ(p)]lengthAq (Mqi ) i=1,...,t

i

Hence the desired equality follows from Algebra, Lemma 7.49.12.



29.13. Preparation for flat pullback Recall that a morphism f : X → Y which is locally of finite type is said to have relative dimension r if every nonempty fibre is equidimensional of dimension r. See Morphisms, Definition 24.30.1. Lemma 29.13.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is flat of relative dimension r. For any closed subset Z ⊂ Y we have dimδ (f −1 (Z)) = dimδ (Z) + r. If Z is irreducible and Z 0 ⊂ f −1 (Z) is an irreducible component, then Z 0 dominates Z and dimδ (Z 0 ) = dimδ (Z) + r. Proof. It suffices to prove the final statement. We may replace Y by the integral closed subscheme Z and X by the scheme theoretic inverse image f −1 (Z) = Z ×Y X. Hence we may assume Z = Y is integral and f is a flat morphism of relative dimension r. Since Y is locally Noetherian the morphism f which is locally of finite type, is actually locally of finite presentation. Hence Morphisms, Lemma 24.26.9 applies and we see that f is open. Let ξ ∈ X be a generic point of an irreducible component of X. By the openness of f we see that f (ξ) is the generic point η of Z = Y . Note that dimξ (Xη ) = r by assumption that f has relative dimension r. On the other hand, since ξ is a generic point of X we see that OX,ξ = OXη ,ξ has only one prime ideal and hence has dimension 0. Thus by Morphisms, Lemma 24.29.1 we conclude that the transcendence degree of κ(ξ) over κ(η) is r. In other words, δ(ξ) = δ(η) + r as desired.  Here is the lemma that we will use to prove that the flat pullback of a locally finite collection of closed subschemes is locally finite. Lemma 29.13.2. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume {Zi }i∈I is a locally finite collection of closed subsets of Y . Then {f −1 (Zi )}i∈I is a locally finite collection of closed subsets of Y .

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Proof. Let U ⊂ X be a quasi-compact open subset. Since the image f (U ) ⊂ Y is a quasi-compact subset there exists a quasi-compact open V ⊂ Y such that f (U ) ⊂ V . Note that {i ∈ I | f −1 (Zi ) ∩ U 6= ∅} ⊂ {i ∈ I | Zi ∩ V 6= ∅}. Since the right hand side is finite by assumption we win.



29.14. Flat pullback In the following we use f −1 (Z) to denote the scheme theoretic inverse image of a closed subscheme Z ⊂ Y for a morphism of schemes f : X → Y . We recall that the scheme theoretic inverse image is the fibre product f −1 (Z)

/X

 Z

 /Y

and it is also the closed subscheme of X cut out by the quasi-coherent sheaf of ideals f −1 (I)OX , if I ⊂ OY is the quasi-coherent sheaf of ideals corresponding to Z in Y . (This is discussed in Schemes, Section 21.4 and Lemma 21.17.6 and Definition 21.17.7.) Definition 29.14.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a morphism. Assume f is flat of relative dimension r. (1) Let Z ⊂ Y be an integral closed subscheme of δ-dimension k. We define f ∗ [Z] to be the (k + r)-cycle on X to the scheme theoretic inverse image f ∗ [Z] = [f −1 (Z)]k+r . −1 This makes P sense since dimδ (f (Z)) = k + r by Lemma 29.13.1. (2) Let α = ni [Zi ] be a k-cycle on Y . The flat pullback of α by f is the sum X f ∗α = ni f ∗ [Zi ]

where each f ∗ [Zi ] is defined as above. The sum is locally finite by Lemma 29.13.2. (3) We denote f ∗ : Zk (Y ) → Zk+r (Y ) the map of abelian groups so obtained. An open immersion is flat. This is an important though trivial special case of a flat morphism. If U ⊂ X is open then sometimes the pullback by j : U → X of a cycle is called the restriction of the cycle to U . Note that in this case the maps j ∗ : Zk (X) −→ Zk (U ) are all surjective. The reason is that given any integral closed subscheme Z 0 ⊂ U , we can take the closure of Z of Z 0 in X and think of it as a reduced closed subscheme of X (see Schemes, Lemma 21.12.4). And clearly Z ∩ U = Z 0 , in other words j ∗ [Z] = [Z 0 ] whence the surjectivity. In fact a little bit more is true. Lemma 29.14.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let U ⊂ X be an open subscheme, and denote i : Y = X \ U → X as a reduced closed subscheme of X. For every k ∈ Z the sequence Zk (Y )

i∗

/ Zk (X)

j∗

/ Zk (U )

/0

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is an exact complex of abelian groups. Proof. By the description above the basis elements [Z] of the free abelian group Zk (X) map either to (distinct) basis elements [Z ∩ U ] or to zero if Z ⊂ Y . Hence the lemma is clear.  Lemma 29.14.3. Let (S, δ) be as in Situation 29.7.1. Let X, Y, Z be locally of finite type over S. Let f : X → Y and g : Y → Z be flat morphisms of relative dimensions r and s. Then g ◦ f is flat of relative dimension r + s and f ∗ ◦ g ∗ = (g ◦ f )∗ as maps Zk (Z) → Zk+r+s (X). Proof. The composition is flat of relative dimension r + s by Morphisms, Lemma 24.30.3. Suppose that (1) W ⊂ Z is a closed integral subscheme of δ-dimension k, (2) W 0 ⊂ Y is a closed integral subscheme of δ-dimension k + s with W 0 ⊂ g −1 (W ), and (3) W 00 ⊂ Y is a closed integral subscheme of δ-dimension k + s + r with W 00 ⊂ f −1 (W 0 ). We have to show that the coefficient n of [W 00 ] in (g ◦ f )∗ [W ] agrees with the coefficient m of [W 00 ] in f ∗ (g ∗ [W ]). That it suffices to check the lemma in these cases follows from Lemma 29.13.1. Let ξ 00 ∈ W 00 , ξ 0 ∈ W 0 and ξ ∈ W be the generic points. Consider the local rings A = OZ,ξ , B = OY,ξ0 and C = OX,ξ00 . Then we have local flat ring maps A → B, B → C and moreover n = lengthC (C/mA C),

and m = lengthC (C/mB C)lengthB (B/mA B)

Hence the equality follows from Algebra, Lemma 7.49.14.



Lemma 29.14.4. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. (1) Let Z ⊂ Y be a closed subscheme with dimδ (Z) ≤ k. Then we have dimδ (f −1 (Z)) ≤ k + r and [f −1 (Z)]k+r = f ∗ [Z]k in Zk+r (X). (2) Let F be a coherent sheaf on Y with dimδ (Supp(F)) ≤ k. Then we have dimδ (Supp(f ∗ F)) ≤ k + r and f ∗ [F]k = [f ∗ F]k+r in Zk+r (X). Proof. Part (1) follows from part (2) by Lemma 29.10.3 and the fact that f ∗ OZ = Of −1 (Z) . Proof of (2). As X, Y are locally Noetherian we may apply Cohomology of Schemes, Lemma 25.11.1 to see that F is of finite type, hence f ∗ F is of finite type (Modules, Lemma 15.9.2), hence f ∗ F is coherent (Cohomology of Schemes, Lemma 25.11.1 again). Thus the lemma makes sense. Let W ⊂ Y be an integral closed subscheme of δ-dimension k, and let W 0 ⊂ X be an integral closed subscheme of dimension k + r mapping into W under f . We have to show that the coefficient n of [W ] in f ∗ [F]k agrees with the coefficient m of [W ] in [f ∗ F]k+r . Let ξ ∈ W and ξ 0 ∈ W 0 be the generic points. Let A = OY,ξ , B = OX,ξ0 and set M = Fξ as an A-module. (Note that M has finite length by our dimension assumptions, but we actually do

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not need to verify this. See Lemma 29.10.1.) We have f ∗ Fξ0 = B ⊗A M . Thus we see that n = lengthB (B ⊗A M )

and m = lengthA (M )lengthB (B/mA B)

Thus the equality follows from Algebra, Lemma 7.49.13.



29.15. Push and pull In this section we verify that proper pushforward and flat pullback are compatible when this makes sense. By the work we did above this is a consequence of cohomology and base change. Lemma 29.15.1. Let (S, δ) be as in Situation 29.7.1. Let X0

g0

f0

 Y0

g

/X  /Y

f

be a fibre product diagram of schemes locally of finite type over S. Assume f : X → Y proper and g : Y 0 → Y flat of relative dimension r. Then also f 0 is proper and g 0 is flat of relative dimension r. For any k-cycle α on X we have g ∗ f∗ α = f∗0 (g 0 )∗ α in Zk+r (Y 0 ). Proof. The assertion that f 0 is proper follows from Morphisms, Lemma 24.42.5. The assertion that g 0 is flat of relative dimension r follows from Morphisms, Lemmas 24.30.2 and 24.26.7. It suffices to prove the equality of cycles when α = [W ] for some integral closed subscheme W ⊂ X of δ-dimension k. Note that in this case we have α = [OW ]k , see Lemma 29.10.3. By Lemmas 29.12.3 and 29.14.4 it therefore suffices to show that f∗0 (g 0 )∗ OW is isomorphic to g ∗ f∗ OW . This follows from cohomology and base change, see Cohomology of Schemes, Lemma 25.6.2.  Lemma 29.15.2. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a finite locally free morphism of degree d (see Morphisms, Definition 24.46.1). Then f is both proper and flat of relative dimension 0, and f∗ f ∗ α = dα for every α ∈ Zk (Y ). Proof. A finite locally free morphism is flat and finite by Morphisms, Lemma 24.46.2, and a finite morphism is proper by Morphisms, Lemma 24.44.10. We omit showing that a finite morphism has relative dimension 0. Thus the formula makes sense. To prove it, let Z ⊂ Y be an integral closed subscheme of δ-dimension k. It suffices to prove the formula for α = [Z]. Since the base change of a finite locally free morphism is finite locally free (Morphisms, Lemma 24.46.4) we see that f∗ f ∗ OZ is a finite locally free sheaf of rank d on Z. Hence f∗ f ∗ [Z] = f∗ f ∗ [OZ ]k = [f∗ f ∗ OZ ]k = d[Z] where we have used Lemmas 29.14.4 and 29.12.3.



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29.16. Preparation for principal divisors Recall that if Z is an irreducible closed subset of a scheme X, then the codimension of Z in X is equal to the dimension of the local ring OX,ξ , where ξ ∈ Z is the generic point. See Properties, Lemma 23.11.4. Definition 29.16.1. Let X be a locally Noetherian scheme. Assume X is integral. Let f ∈ R(X)∗ . For every integral closed subscheme Z ⊂ X of codimension 1 we define the order of vanishing of f along Z as the integer ordZ (f ) = ordOX,ξ (f ) where the right hand side is the notion of Algebra, Definition 7.113.2 and ξ is the generic point of Z. Of course it can happen that ordZ (f ) < 0. In this case we say that f has a pole along Z and that −ordZ (f ) > 0 is the order of pole of f along Z. Note that for f, g ∈ R(X)∗ we have ordZ (f g) = ordZ (f ) + ordZ (g). Lemma 29.16.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral. If Z ⊂ X is an integral closed subscheme of codimension 1, then dimδ (Z) = dimδ (X) − 1. Proof. This is more or less the defining property of a dimension function.



Lemma 29.16.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral. Let f ∈ R(X)∗ . Then the set {Z ⊂ X | Z is integral, closed of codimension 1 and ordZ (f ) 6= 0} is locally finite in X. Proof. This is true simply because there exists a nonempty open subscheme U ⊂ X ∗ ), and hence the codimension 1 such that f corresponds to a section of Γ(U, OX irreducibles which can occur in the set of the lemma are all irreducible components of X \ U . Hence Lemma 29.9.1 gives the desired result.  Lemma 29.16.4. Let f : X → Y be a morphism of schemes. Let ξ ∈ Y be a point. Assume that (1) X, Y are integral, (2) X is locally Noetherian (3) f is proper, dominant and R(X) ⊂ R(Y ) is finite, and (4) dim(OY,ξ ) = 1. Then there exists an open neighbourhood V ⊂ Y of ξ such that f |f −1 (V ) : f −1 (V ) → V is finite. Proof. By Cohomology of Schemes, Lemma 25.20.2 it suffices to prove that f −1 ({ξ}) is finite. We replace Y by an affine open, say Y = Spec(R). Note that R is Noetherian, as X is assumed locally Noetherian. Since f is proper it is quasi-compact. Hence we can find a finite affine open covering X = U1 ∪ . . . ∪ Un with each Ui = Spec(Ai ). Note that R → Ai is a finite type injective homomorphism of domains with f.f.(R) ⊂ f.f.(Ai ) finite. Thus the lemma follows from Algebra, Lemma 7.105.2. 

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29.17. Principal divisors The following definition is the key to everything that follows. Definition 29.17.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral with dimδ (X) = n. Let f ∈ R(X)∗ . The principal divisor associated to f is the (n − 1)-cycle X div(f ) = divX (f ) = ordZ (f )[Z] where the sum is over integral closed subschemes of codimension 1 and ordZ (f ) is as in Definition 29.16.1. This makes sense by Lemmas 29.16.2 and 29.16.3. Lemma 29.17.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral with dimδ (X) = n. Let f, g ∈ R(X)∗ . Then div(f g) = div(f ) + div(g) in Zn−1 (X). Proof. This is clear from the additivity of the ord functions.



An important role in the discussion of principal divisors is played by the “universal” principal divisor [0] − [∞] on P1S . To make this more precise, let us denote D0 , D∞ ⊂ P1S = ProjS (OS [X0 , X1 ]) the closed subscheme cut out by the section X1 , resp. X0 of O(1). These are effective Cartier divisors, see Divisors, Definition 26.9.1 and Lemma 26.9.20. The following lemma says that loosely speaking we have “div(X1 /X0 ) = [D0 ] − [D1 ]” and that this is the universal principal divisor. Lemma 29.17.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let f ∈ R(X)∗ . Let ∗ ). U ⊂ X be a nonempty open such that f corresponds to a section f ∈ Γ(U, OX 1 1 Let Y ⊂ X ×S PS be the closure of the graph of f : U → PS . Then (1) the projection morphism p : Y → X is proper, (2) p|p−1 (U ) : p−1 (U ) → U is an isomorphism, (3) the pullbacks q −1 D0 and q −1 D∞ via the morphism q : Y → P1S are effective Cartier divisors on Y , (4) we have divY (f ) = [q −1 D0 ]n−1 − [q −1 D∞ ]n−1 (5) we have divX (f ) = p∗ divY (f ) −1

(6) if we view Y0 = q D0 , and Y∞ = q −1 D∞ as closed subschemes of X via the morphism p then we have divX (f ) = [Y0 ]n−1 − [Y∞ ]n−1 Proof. Since X is integral, we see that U is integral. Hence Y is integral, and (1, f )(U ) ⊂ Y is an open dense subscheme. Also, note that the closed subscheme Y ⊂ X ×S P1S does not depend on the choice of the open U , since after all it is the closure of the one point set {η 0 } = {(1, f )(η)} where η ∈ X is the generic point. Having said this let us prove the assertions of the lemma.

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For (1) note that p is the composition of the closed immersion Y → X ×S P1S = P1X with the proper morphism P1X → X. As a composition of proper morphisms is proper (Morphisms, Lemma 24.42.4) we conclude. It is clear that Y ∩ U ×S P1S = (1, f )(U ). Thus (2) follows. It also follows that dimδ (Y ) = n. Note that q(η 0 ) = f (η) is not contained in D0 or D∞ since f ∈ R(X)∗ . Hence q −1 D0 and q −1 D∞ are effective Cartier divisors on Y by Divisors, Lemma 26.9.12. Thus we see (3). It also follows that dimδ (q −1 D0 ) = n − 1 and dimδ (q −1 D∞ ) = n − 1. Consider the effective Cartier divisor q −1 D0 . At every point ξ ∈ q −1 D0 we have f ∈ OY,ξ and the local equation for q −1 D0 is given by f . In particular, if δ(ξ) = n−1 so ξ is the generic point of a integral closed subscheme Z of δ-dimension n − 1, then we see that the coefficient of [Z] in divY (f ) is ordZ (f ) = lengthOY,ξ (OY,ξ /f OY,ξ ) = lengthOY,ξ (Oq−1 D0 ,ξ ) which is the coefficient of [Z] in [q −1 D0 ]n−1 . A similar argument using the rational function 1/f shows that −[q −1 D∞ ] agrees with the terms with negative coefficients in the expression for divY (f ). Hence (4) follows. Note that D0 → S is an isomorphism. Hence we see that X ×S D0 → X is an isomorphism as well. Clearly we have q −1 D0 = Y ∩ X ×S D0 (scheme theoretic intersection) inside X ×S P1S . Hence it is really the case that Y0 → X is a closed immersion. By the same token we see that p∗ Oq−1 D0 = OY0 and hence by Lemma 29.12.3 we have p∗ [q −1 D0 ]n−1 = [Y0 ]n−1 . Of course the same is true for D∞ and Y∞ . Hence to finish the proof of the lemma it suffices to prove the last assertion. Let Z ⊂ X be an integral closed subscheme of δ-dimension n − 1. We want to show that the coefficient of [Z] in div(f ) is the same as the coefficient of [Z] in [Y0 ]n−1 − [Y∞ ]n−1 . We may apply Lemma 29.16.4 to the morphism p : Y → X and the generic point ξ ∈ Z. Hence we may replace X by an affine open neighbourhood of ξ and assume that p : Y → X is finite. Write X = Spec(R) and Y = Spec(A) with p induced by a finite homomorphism R → A of Noetherian domains which induces an isomorphism f.f.(R) ∼ = f.f.(A) of fraction fields. Now we have f ∈ f.f.(R) and a prime p ⊂ R with dim(Rp ) = 1. The coefficient of [Z] in divX (f ) is ordRp (f ). The coefficient of [Z] in p∗ divY (f ) is X [κ(q) : κ(p)]ordAq (f ) q lying over p

The desired equality therefore follows from Algebra, Lemma 7.113.8.



This lemma will be superseded by the more general Lemma 29.20.1. Lemma 29.17.4. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Assume X, Y are integral and n = dimδ (Y ). Let f : X → Y be a flat morphism of relative dimension r. Let g ∈ R(Y )∗ . Then f ∗ (divY (g)) = divX (g) in Zn+r−1 (X).

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29. CHOW HOMOLOGY AND CHERN CLASSES

Proof. Note that since f is flat it is dominant so that f induces an embedding R(Y ) ⊂ R(X), and hence we may think of g as an element of R(X)∗ . Let Z ⊂ X be an integral closed subscheme of δ-dimension n + r − 1. Let ξ ∈ Z be its generic point. If dimδ (f (Z)) > n − 1, then we see that the coefficient of [Z] in the left and right hand side of the equation is zero. Hence we may assume that Z 0 = f (Z) is an intral closed subscheme of Y of δ-dimension n − 1. Let ξ 0 = f (ξ). It is the generic point of Z 0 . Set A = OY,ξ0 , B = OX,ξ . The ring map A → B is a flat local homomorphism of Noetherian local domains of dimension 1. We have g ∈ f.f.(A). What we have to show is that ordA (g)lengthB (B/mA B) = ordB (g). This follows from Algebra, Lemma 7.49.13 (details omitted).



29.18. Two fun results on principal divisors The first lemma implies that the pushforward of a principal divisor along a generically finite morphism is a principal divisor. Lemma 29.18.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Assume X, Y are integral and n = dimδ (X) = dimδ (Y ). Let p : X → Y be a dominant proper morphism. Let f ∈ R(X)∗ . Set g = NmR(X)/R(Y ) (f ). Then we have p∗ div(f ) = div(g). Proof. Let Z ⊂ Y be an integral closed subscheme of δ-dimension n − 1. We want to show that the coefficient of [Z] in p∗ div(f ) and div(g) are equal. We may apply Lemma 29.16.4 to the morphism p : X → X and the generic point ξ ∈ Z. Hence we may replace X by an affine open neighbourhood of ξ and assume that p : Y → X is finite. Write X = Spec(R) and Y = Spec(A) with p induced by a finite homomorphism R → A of Noetherian domains which induces an finite field extension f.f.(R) ⊂ f.f.(A) of fraction fields. Now we have f ∈ f.f.(A), g = Nm(f ) ∈ f.f.(R), and a prime p ⊂ R with dim(Rp ) = 1. The coefficient of [Z] in divY (g) is ordRp (g). The coefficient of [Z] in p∗ divX (f ) is X [κ(q) : κ(p)]ordAq (f ) q lying over p

The desired equality therefore follows from Algebra, Lemma 7.113.8.



The following lemma says that the degree of a principal divisor on a proper curve is zero. Lemma 29.18.2. Let K be any field. Let X be a 1-dimensional integral scheme endowed with a proper morphism c : X → Spec(K). Let f ∈ K(X)∗ be an invertible rational function. Then X [κ(x) : K]ordOX,x (f ) = 0 x∈X closed

where ord is as in Algebra, Definition 7.113.2. In other words, c∗ div(f ) = 0.

29.19. RATIONAL EQUIVALENCE

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Proof. Consider the diagram Y

p

q

 P1K

/X c

c

0

 / Spec(K)

that we constructed in Lemma 29.17.3 starting with X and the rational function f over S = Spec(K). We will use all the results of this lemma without further mention. We have to show that c∗ divX (f ) = p∗ c∗ divY (f ) = 0. This is the same as proving that c0∗ q∗ divY (f ) = 0. If q(Y ) is a closed point of P1K then we see that divX (f ) = 0 and the lemma holds. Thus we may assume that q is dominant. Since divY (f ) = [q −1 D0 ]0 − [q −1 D∞ ]0 we see (by definition of flat pullback) that divY (f ) = q ∗ ([D0 ]0 − [D∞ ]0 ). Suppose we can show that q : Y → P1K is finite locally free of degree d (see Morphisms, Definition 24.46.1). Then byy Lemma 29.15.2 we get q∗ divY (f ) = d([D0 ]0 − [D∞ ]0 ). Since clearly c0∗ [D0 ]0 = c0∗ [D∞ ]0 we win. It remains to show that q is finite locally free. (It will automatically have some given degree as P1K is connected.) Since dim(P1K ) = 1 we see that q is finite for example by Lemma 29.16.4. All local rings of P1K at closed points are regular local rings of dimension 1 (in other words discrete valuation rings), since they are localizations of K[T ] (see Algebra, Lemma 7.106.1). Hence for y ∈ Y closed the local ring OY,y will be flat over OP1K ,q(y) as soon as it is torsion free. This is obviously the case as OY,y is a domain and q is dominant. Thus q is flat. Hence q is finite locally free by Morphisms, Lemma 24.46.2.  29.19. Rational equivalence In this section we define rational equivalence on k-cycles. We will allow locally finite sums of images of principal divisors (under closed immersions). This leads to some pretty strange phenomena, see Example 29.19.3. However, if we do not allow these then we do not know how to prove that capping with chern classes of line bundles factors through rational equivalence. Definition 29.19.1. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Let k ∈ Z. (1) Given any locally finite collection {Wj ⊂ X} of integral closed subschemes with dimδ (Wj ) = k + 1, and any fj ∈ R(Wj )∗ we may consider X (ij )∗ div(fj ) ∈ Zk (X) where ij : ` Wj →` X is the inclusion morphism. This makes sense as the morphism ij : Wj → X is proper. (2) We say that α ∈ Zk (X) is rationally equivalent to zero if α is a cycle of the form displayed above. (3) We say α, β ∈ Zk (X) are rationally equivalent and we write α ∼rat β if α − β is rationally equivalent to zero. (4) We define Ak (X) = Zk (X)/ ∼rat to be the Chow group of k-cycles on X. This is sometimes called the Chow group of k-cycles module rational equivalence on X.

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There are many other interesting (adequate) equivalence relations. Rational equivalence is the coarsest one of them all. A very simple but important lemma is the following. Lemma 29.19.2. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Let U ⊂ X be an open subscheme, and denote i : Y = X \U → X as a reduced closed subscheme of X. Let k ∈ Z. Suppose α, β ∈ Zk (X). If α|U ∼rat β|U then there exist a cycle γ ∈ Zk (Y ) such that α ∼rat β + i∗ γ. In other words, the sequence Ak (Y )

i∗

/ Ak (X)

j∗

/ Ak (U )

/0

is an exact complex of abelian groups. Proof. Let {Wj }j∈J be a locally finite collection of integral closed subschemes of δ-dimension k + 1, and let fj ∈ R(Wj )∗ be elements such that (α − β)|U = P (ij )∗ div(fj ) as in the definition. Set Wj0 ⊂ X equal to the closure of Wj . Suppose that V ⊂ X is a quasi-compact open. Then also V ∩ U is quasi-compact open in U as V is Noetherian. Hence the set {j ∈ J | Wj ∩ V 6= ∅} = {j ∈ J | Wj0 ∩ V 6= ∅} is finite since {Wj } is locally finite. In other words we see that {Wj0 } is also locally finite. Since R(Wj ) = R(Wj0 ) we see that X α−β− (i0j )∗ div(fj ) is a cycle supported on Y and the lemma follows (see Lemma 29.14.2).



Example 29.19.3. Here is a “strange” example. Suppose that S is the spectrum of a field k with δ as in Example 29.7.2. Suppose that X = C1 ∪C2 ∪. . . is an infinite union of curves Cj ∼ = P1k glued together in the following way: The point ∞ ∈ Cj is glued transversally to the point 0 ∈ Cj+1 for j = 1, 2, 3, . . .. Take the point 0 ∈ C1 . This gives a zero cycle [0] ∈ Z0 (X). The “strangeness” in this situation is that actually [0] ∼rat 0! Namely we can choose the rational function fj ∈ R(Cj ) to be the function which has a simple zero at 0P and a simple pole at ∞ and no other zeros or poles. Then we see that the sum (ij )∗ div(fj ) is exactly the 0-cycle [0]. In fact it turns out that A0 (X) = 0 in this example. If you find this too bizarre, then you can just make sure your spaces are always quasi-compact (so X does not even exist for you). Remark 29.19.4. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Suppose we have infinite collections αi , βi ∈ Zk (X), i ∈ I of kcycles on X. Suppose that the P supports Pof αi and βi form locally finite collections of closed subsets of X so that αi and βi are defined asPcycles. Moreover, assume P that αi ∼rat βi for each i. Then it is not clear that αi ∼rat βi . Namely, the problem is that the rational equivalences may be given by locally finite families {Wi,j , fi,j ∈ R(Wi,j )∗ }j∈Ji but the union {Wi,j }i∈I,j∈Ji may not be locally finite. In many cases in practice, one has a locally finite family of closed subsets {Ti }i∈I such that αi , βi are supported on Ti and such that αi = βi in Ak (Ti ), in other words, ∗ the families {Wi,j , P fi,j ∈ R(WP i,j ) }j∈Ji consist of subschemes Wi,j ⊂ Ti . In this case it is true that αi ∼rat βi on X, simply because the family {Wi,j }i∈I,j∈Ji is automatically locally finite in this case.

29.20. PROPERTIES OF RATIONAL EQUIVALENCE

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29.20. Properties of rational equivalence Lemma 29.20.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be schemes locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Let α ∼rat β be rationally equivalent k-cycles on Y . Then f ∗ α ∼rat f ∗ β as (k + r)-cycles on X. Proof. What do we have to show? Well, suppose we are given a collection ij : Wj −→ Y of closed immersions, with each Wj integral of δ-dimension k + 1 and rational functions fj ∈ R(Wj )∗ . Moreover, assume that the collection {ij (Wj )}j∈J is locally finite on Y . Then we have to show that X f ∗( ij,∗ div(fj )) is rationally equivalent to zero on X. Consider the fibre products i0j : Wj0 = Wj ×Y X −→ X. 0 0 For each j, consider the collection {Wj,l }l∈Lj of irreducible components Wj,l ⊂ Wj0 having δ-dimension k + 1. We may write X 0 [Wj0 ]k+1 = nj,l [Wj,l ]k+1 l∈Lj

0 for some nj,l > 0. By Lemma 29.13.1 we see that Wj,l → Wj is dominant and 0 ∗ hence we can let fj,l ∈ R(Wj,l ) denote the image of fj under the map of fields 0 R(Wj ) → R(Wj,l ). We claim that 0 (1) the collection {Wj,l }j∈J,l∈Lj is locally finite on X, and P P n (2) with obvious notation f ∗ ( ij,∗ div(fj )) = i0j,l,∗ div(fj,lj,l ).

Clearly this claim implies the lemma. To show (1), note that {Wj0 } is a locally finite collection of closed subschemes of X by Lemma 29.13.2. Hence if U ⊂ X is quasi-compact, then U meets only finitely many Wj0 . By Lemma 29.9.1 the collection of irreducible components of each Wj is 0 locally finite as well. Hence we see only finitely many Wj,l meet U as desired. Let Z ⊂ X be an integral closed subscheme of δ-dimension k + r. We have to show P ∗ that the coefficient n of [Z] in f ( i div(f j,∗ j )) is equal to the coefficient m of P 0 n [Z] in ij,l,∗ div(fj,lj,l ). Let Z 0 be the closure of f (Z) which is an integral closed subscheme of Y . By Lemma 29.13.1 we have dimδ (Z 0 ) ≥ k. If dimδ (Z 0 ) > k, then the coefficients n and m are both zero, since the generic point of Z will not be 0 contained in any Wj0 or Wj,l . Hence we may assume that dimδ (Z 0 ) = k. We are going to translate the equality of n and m into algebra. Namely, let ξ 0 ∈ Z 0 and ξ ∈ Z be the generic points. Set A = OY,ξ0 and B = OX,ξ . Note that A, B are Noetherian, A → B is flat, local, and that mA B is an ideal of definition of the local ring B. There are finitely many j such that Wj passes through ξ 0 , and these correspond to prime ideals p1 , . . . , pT ⊂ A

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with the property that dim(A/pt ) = 1 for each t = 1, . . . , T . The rational functions fj correspond to elements ft ∈ κ(pt )∗ . Say pt corresponds to Wj . By construction, 0 the closed subschemes Wj,l which meet ξ correspond 1 − 1 with minimal primes pt B ⊂ qt,1 , . . . , qt,St ⊂ B over pt B. The integers nj,l correspond to the integers nt,s = lengthBqt,s ((B/pt B)Bqt,s ) The rational functions fj,l correspond to the images ft,s ∈ κ(qt,s )∗ of the elements ft ∈ κ(pt )∗ . Putting everything together we see that X n= ordA/pt (ft )lengthB (B/mA B) and that m=

X

ordB/qt,s (ft,s )lengthBqt,s ((B/pt B)Bqt,s )

Note that it suffices to prove the equality for each t ∈ {1, . . . , T } separately. Writing ft = x/y for some nonzero x, y ∈ A/pt coming from x, y ∈ A we see that it suffices to prove lengthA/pt (A/(pt , x))lengthB (B/mA B) = lengthB (B/(x, pt )B) (equality uses Algebra, Lemma 7.49.13) equals X ordB/qt,s (B/(x, qt,s ))lengthBqt,s ((B/pt B)Bqt,s ) s=1,...,St

and similarly for y. Note that as x 6∈ pt we see that x is a nonzerodivisor on A/pt . As A → B is flat it follows that x is a nonzerodivisor on the module M = B/pt B. Hence the equality above follows from Lemma 29.5.6.  Lemma 29.20.2. Let (S, δ) be as in Situation 29.7.1. Let X, Y be schemes locally of finite type over S. Let p : X → Y be a proper morphism. Suppose α, β ∈ Zk (X) are rationally equivalent. Then p∗ α is rationally equivalent to p∗ β. Proof. What do we have to show? Well, suppose we are given a collection ij : Wj −→ X of closed immersions, with each Wj integral of δ-dimension k + 1 and rational functions fj ∈ R(Wj )∗ . Moreover, assume that the collection {ij (Wj )}j∈J is locally finite on X. Then we have to show that X  p∗ ij,∗ div(fj ) is rationally equivalent to zero on X. Note that the sum is equal to X

p∗ ij,∗ div(fj ).

Let Wj0 ⊂ Y be the integral closed subscheme which is the image of p ◦ ij . The collection {Wj0 } is locally finite in Y by Lemma 29.11.2. Hence it suffices to show, for a given j, that either p∗ ij,∗ div(fj ) = 0 or that it is equal to i0j,∗ div(gj ) for some gj ∈ R(Wj0 )∗ .

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The arguments above therefore reduce us to the case of a since integral closed subscheme W ⊂ X of δ-dimension k + 1. Let f ∈ R(W )∗ . Let W 0 = p(W ) as above. We get a commutative diagram of morphisms W

i

p0

 W0

i0

/X  /Y

p

Note that p∗ i∗ div(f ) = i0∗ (p0 )∗ div(f ) by Lemma 29.12.2. As explained above we have to show that (p0 )∗ div(f ) is the divisor of a rational function on W 0 or zero. There are three cases to distinguish. The case dimδ (W 0 ) < k. In this case automatically (p0 )∗ div(f ) = 0 and there is nothing to prove. The case dimδ (W 0 ) = k. Let us show that (p0 )∗ div(f ) = 0 in this case. Let η ∈ W 0 be the generic point. Note that c : Wη → Spec(K) is a proper integral curve over K = κ(η) whose function field K(Wη ) is identified with R(W ). Here is a diagram Wη c

 Spec(K)

/W p

 / W0

Let us denote fη ∈ K(Wη )∗ the rational function corresponding to f ∈ R(W )∗ . Moreover, the closed points ξ of Wη correspond 1 − 1 to the closed integral subschemes Z = Zξ ⊂ W of δ-dimension k with f (Z) = W 0 . Note that the multiplicity of Zξ in div(f ) is equal to ordOWη ,ξ (fη ) simply because the local rings OWη ,ξ and OW,ξ are identified (as subrings of their fraction fields). Hence we see that the multiplicity of [W 0 ] in (p0 )∗ div(f ) is equal to the multiplicity of [Spec(K)] in c∗ div(fη ). By Lemma 29.18.2 this is zero. The case dimδ (W 0 ) = k + 1. In this case Lemma 29.18.1 applies, and we see that  indeed p0∗ div(f ) = div(g) for some g ∈ R(W 0 )∗ as desired. 29.21. Different characterizations of rational equivalence Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Given any closed subscheme Z ⊂ X ×S P1S = X × P1 we let Z0 , resp. Z∞ be the −1 scheme theoretic closed subscheme Z0 = pr−1 2 (D0 ), resp. Z∞ = pr2 (D∞ ). Here D0 , D∞ are as defined just above Lemma 29.17.3. Lemma 29.21.1. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Let W ⊂ X ×S P1S be an integral closed subscheme of δ-dimension k + 1. Assume W 6= W0 , and W 6= W∞ . Then (1) W0 , W∞ are effective Cartier divisors of W , (2) W0 , W∞ can be viewed as closed subschemes of X and [W0 ]k ∼rat [W∞ ]k , (3) for any locally finite family of integral closed subschemes Wi ⊂ X ×S 1 P PS of δ-dimension k + 1 with Wi 6= (Wi )0 and Wi 6= (Wi )∞ we have ([(Wi )0 ]k − [(Wi )∞ ]k ) ∼rat 0 on X, and

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(4) for any α ∈ Zk (X) with α ∼rat 0 there exists a locally finite family 1 of P integral closed subschemes Wi ⊂ X ×S PS as above such that α = ([(Wi )0 ]k − [(Wi )∞ ]k ). Proof. Part (1) follows from Divisors, Lemma 26.9.12 since the generic point of W is not mapped into D0 or D∞ under the projection X ×S P1S → P1S by assumtion. Since X ×S D0 → X is an isomorphism we see that W0 is isomorphic to a closed subscheme of X. Similarly for W∞ . Consider the morphism p : W → X. It is proper and on W we have [W0 ]k ∼rat [W∞ ]k . Hence part (2) follows from Lemma 29.20.2 as clearly p∗ [W0 ]k = [W0 ]k and similarly for W∞ . The only content of statement (3) is, given parts (1) and (2), that the collection {(Wi )0 , (Wi )∞ } is a locally finite collection of closed subschemes of X. This is clear. Suppose that α ∼rat 0. By definition this means there exist integral closed sub∗ schemes Vi ⊂ X of δ-dimension k + 1 and rational functions P fi ∈ R(Vi ) such that the family {Vi }i∈I is locally finite in X and such that α = (Vi → X)∗ div(fi ). Let Wi ⊂ Vi ×S P1S ⊂ X ×S P1S be the closure of the graph of the rational map fi as in Lemma 29.17.3. Then we have that (Vi → X)∗ div(fi ) is equal to [(Wi )0 ]k − [(Wi )∞ ]k by that same lemma. Hence the result is clear.  Lemma 29.21.2. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Let Z be a closed subscheme of X × P1 . Assume dimδ (Z) ≤ k + 1, dimδ (Z0 ) ≤ k, dimδ (Z∞ ) ≤ k and assume any embedded point ξ (Divisors, Definition 26.4.1) of Z has δ(ξ) < k. Then [Z0 ]k ∼rat [Z∞ ]k as k-cycles on X. Proof. Let {Wi }i∈I be the collection of irreducible components of Z which have δ-dimension k + 1. Write X [Z]k+1 = ni [Wi ] with ni > 0 as per definition. Note that {Wi } is a locally finite collection of closed subsets of X ×S P1S by Lemma 29.9.1. We claim that X [Z0 ]k = ni [(Wi )0 ]k and similarly for [Z∞ ]k . If we prove this then the lemma follows from Lemma 29.21.1. Let Z 0 ⊂ X be an integral closed subscheme of δ-dimension k. To prove the equality above it suffices to showPthat the coefficient n of [Z 0 ] in [Z0 ]k is the same as the coefficient m of [Z 0 ] in ni [(Wi )0 ]k . Let ξ 0 ∈ Z 0 be the generic point. 0 1 Set ξ = (ξ , 0) ∈ X ×S PS . Consider the local ring A = OX×S P1S ,ξ . Let I ⊂ A be the ideal cutting out Z, in other words so that A/I = OZ,ξ . Let t ∈ A be the element cutting out X ×S D0 (i.e., the coordinate of P1 at zero pulled back). By our choice of ξ 0 ∈ Z 0 we have δ(ξ) = k and hence dim(A/I) = 1. Since ξ is not an embedded point by definition we see that A/I is Cohen-Macaulay. Since dimδ (Z0 ) = k we see that dim(A/(t, I)) = 0 which implies that t is a nonzerodivisor on A/I. Finally, the irreducible closed subschemes Wi passing through ξ correspond

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to the minimal primes I ⊂ qi over I. The multiplicities ni correspond to the lengths lengthAq (A/I)qi . Hence we see that i

n = lengthA (A/(t, I)) and m=

X

lengthA (A/(t, qi ))lengthAq (A/I)qi i

Thus the result follows from Lemma 29.5.6.



Lemma 29.21.3. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Let F be a coherent sheaf on X × P1 . Let i0 , i∞ : X → X × P1 be the closed immersion such that it (x) = (x, t). Denote F0 = i∗0 F and F∞ = i∗∞ F. Assume (1) dimδ (Supp(F)) ≤ k + 1, (2) dimδ (Supp(F0 )) ≤ k, dimδ (Supp(F∞ )) ≤ k, and (3) any nonmaximal associated point (insert future reference here) ξ ∈ Supp(F) of F has δ(ξ) < k. Then [F0 ]k ∼rat [F∞ ]k as k-cycles on X. Proof. Let {Wi }i∈I be the collection of irreducible components of Supp(F) which have δ-dimension k + 1. Write X [F]k+1 = ni [Wi ] with ni > 0 as per definition. Note that {Wi } is a locally finite collection of closed subsets of X ×S P1S by Lemma 29.10.1. We claim that X [F0 ]k = ni [(Wi )0 ]k and similarly for [F∞ ]k . If we prove this then the lemma follows from Lemma 29.21.1. Let Z 0 ⊂ X be an integral closed subscheme of δ-dimension k. To prove the equality above it suffices to show n of [Z 0 ] in [F0 ]k is the same P that the coefficient 0 0 as the coefficient m of [Z ] in ni [(Wi )0 ]k . Let ξ ∈ Z 0 be the generic point. Set 0 1 ξ = (ξ , 0) ∈ X ×S PS . Consider the local ring A = OX×S P1S ,ξ . Let M = Fξ as an A-module. Let t ∈ A be the element cutting out X ×S D0 (i.e., the coordinate of P1 at zero pulled back). By our choice of ξ 0 ∈ Z 0 we have δ(ξ) = k and hence dim(M ) = 1. Since ξ is not an associated point of F by definition we see that M is Cohen-Macaulay module. Since dimδ (Supp(F0 )) = k we see that dim(M/tM ) = 0 which implies that t is a nonzerodivisor on M . Finally, the irreducible closed subschemes Wi passing through ξ correspond to the minimal primes qi of Ass(M ). The multiplicities ni correspond to the lengths lengthAq Mqi . Hence we see that i

n = lengthA (M/tM ) and m=

X

lengthA (A/(t, qi )A)lengthAq Mqi

Thus the result follows from Lemma 29.5.6.

i



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29.22. Rational equivalence and K-groups In this section we compare the cycle groups Zk (X) and the Chow groups Ak (X) with certain K0 -groups of abelian categories of coherent sheaves on X. We avoid having to talk about K1 (A) for an abelian category A by dint of Homology, Lemma 10.8.3. In particular, the motivation for the precise form of Lemma 29.22.4 is that lemma. Let us introduce the following notation. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. We denote Coh(X) = Coh(OX ) the category of coherent sheaves on X. It is an abelian category, see Cohomology of Schemes, Lemma 25.11.2. For any k ∈ Z we let Coh≤k (X) be the full subcategory of Coh(X) consisting of those coherent sheaves F having dimδ (Supp(F)) ≤ k. Lemma 29.22.1. Let us introduce the following notation. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. The categories Coh≤k (X) are Serre subcategories of the abelian category Coh(X). Proof. Omitted. The definition of a Serre subcateory is Homology, Definition 10.7.1.  Lemma 29.22.2. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. There are maps Zk (X) −→ K0 (Coh≤k (X)/Coh≤k−1 (X)) −→ Zk (X) whose composition is the identity. The first is the map hM i hM X ⊕nZ nZ [Z] 7→ OZ − nZ >0

nZ 0 OZ since the family {Z | nZ > 0} is locally finite on X. The map F → [F]k is additive on Coh≤k (X), see Lemma 29.10.4. And [F]k = 0 if F ∈ Coh≤k−1 (X). This implies we have the left map as shown in the lemma. It is clear that their composition is the identity. In case X is quasi-compact we will show that the right arrow is injective. Suppose that q ∈ K0 (Coh≤k (X)/Coh≤k+1 (X)) maps to zero in Zk (X). By Homology, Lemma 10.8.3 we can find a q˜ ∈ K0 (Coh≤k (X)) mapping to q. Write q˜ = [F] − [G] for some F, G ∈ K0 (Coh≤k (X)). Since X is quasi-compact we may apply Cohomology of Schemes, Lemma 25.14.3. This shows that there exist integral closed subschemes Zj , Ti ⊂ X and (nonzero) ideal sheaves Ij ⊂ OZj , Ii ⊂ OTi such that F, resp. G have filtrations whose succesive quotients are the sheaves Ij , resp. Ii . In particular we see that dimδ (Zj ), dimδ (Ti ) ≤ k. In other words we have X X [F] = [Ij ], [G] = [Ii ], j i P P in K0 (Coh≤k (X)). Our assumption is that j [Ij ]k − i [Ii ]k = 0. It is clear that we may throw out the indices j, resp. i such that dimδ (Zj ) < k, resp. dimδ (Ti ) < k, since the corresponding sheaves are in Cohk−1 (X) and also do not contribute to the cycle. Moreover, the exact sequences 0 → Ij → OZj → OZj /Ij → 0 and

29.22. RATIONAL EQUIVALENCE AND K-GROUPS

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0 → Ii → OTi → OZi /Ii → 0 show similarly that we may replace Ij , resp. Ii by OZj , resp. OTi . OK, and finally, at this point it is clear that our assumption X X [OZj ]k − [OTi ]k = 0 j i P P implies that in K0 (Cohk (X)) we have also j [OZj ] − i [OTi ] = 0 as desired.  Remark 29.22.3. It seems likely that the arrows of Lemma 29.22.2 are not isomorphisms if`X is not quasi-compact. For example, suppose X is an infinite disjoint union X = n∈N P1k over a field k. Let F, resp. G be the coherent sheaf on X whose restriction to the nth summand is equal to the skyscraper sheaf at 0 associated to OP1k ,0 /mn0 , resp. κ(0)⊕n . The cycle associated to F is equal to the cycle P associated to G, namely both are equal to n[0n ] where 0n ∈ X denotes 0 on the nth component of X. But there seems to be no way to show that [F] = [G] in K0 (Coh(X)) since any proof we can envision uses infinitely many relations. Lemma 29.22.4. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Let F be a coherent sheaf on X. Let ...

/F

ϕ

/F

ψ

/F

ϕ

/F

/ ...

be a complex as in Homology, Equation (10.8.2.1). Assume that (1) dimδ (Supp(F)) ≤ k + 1. (2) dimδ (Supp(H i (F, ϕ, ψ))) ≤ k for i = 0, 1. Then we have [H 0 (F, ϕ, ψ)]k ∼rat [H 1 (F, ϕ, ψ)]k as k-cycles on X. Proof. Let {Wj }j∈J be the collection of irreducible components of Supp(F) which have δ-dimension k+1. Note that {Wj } is a locally finite collection of closed subsets of X by Lemma 29.10.1. For every j, let ξj ∈ Wj be the generic point. Set fj = detκ(ξj ) (Fξj , ϕξj , ψξj ) ∈ R(Wj )∗ . See Definition 29.3.4 for notation. We claim that X −[H 0 (F, ϕ, ψ)]k + [H 1 (F, ϕ, ψ)]k = (Wj → X)∗ div(fj ) If we prove this then the lemma follows. Let Z ⊂ X be an integral closed subscheme of δ-dimension k. To prove the equality above it suffices to show that the coefficient n P of [Z] in [H 0 (F, ϕ, ψ)]k − [H 1 (F, ϕ, ψ)]k is the same as the coefficient m of [Z] in (Wj → X)∗ div(fj ). Let ξ ∈ Z be the generic point. Consider the local ring A = OX,ξ . Let M = Fξ as an A-module. Denote ϕ, ψ : M → M the action of ϕ, ψ on the stalk. By our choice of ξ ∈ Z we have δ(ξ) = k and hence dim(M ) = 1. Finally, the integral closed subschemes Wj passing through ξ correspond to the minimal primes qi of Supp(M ). In each case the element fj ∈ R(Wj )∗ corresponds to the element detκ(qi ) (Mqi , ϕ, ψ) in κ(qi )∗ . Hence we see that n = −eA (M, ϕ, ψ) and m=

X

ordA/qi (detκ(qi ) (Mqi , ϕ, ψ))

Thus the result follows from Proposition 29.5.3.



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Lemma 29.22.5. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Denote Bk (X) the image of the map K0 (Coh≤k (X)/Coh≤k−1 (X)) −→ K0 (Coh≤k+1 (X)/Coh≤k−1 (X)). There is a commutative diagram   Coh≤k (X) K0 Coh≤k−1 (X)  Zk (X)

/ Bk (X)  

/ K0



Coh≤k+1 (X) Coh≤k−1 (X)



 / Ak (X)

where the left vertical arrow is the one from Lemma 29.22.2. If X is quasi-compact then both vertical arrows are isomorphisms. Proof. Suppose we have an element [A] − [B] of K0 (Coh≤k (X)/Coh≤k−1 (X)) which maps to zero in Bk (X), i.e., in K0 (Coh≤k+1 (X)/Coh≤k−1 (X)). Suppose [A] = [A] and [B] = [B] for some coherent sheaves A, B on X supported in δ-dimension ≤ k. The assumption that [A] − [B] maps to zero in the group K0 (Coh≤k+1 (X)/Coh≤k−1 (X)) means that there exists coherent sheaves A0 , B 0 on X supported in δ-dimension ≤ k − 1 such that [A ⊕ A0 ] − [B ⊕ B 0 ] is zero in K0 (Cohk+1 (X)) (use part (1) of Homology, Lemma 10.8.3). By part (2) of Homology, Lemma 10.8.3 this means there exists a (2, 1)-periodic complex (F, ϕ, ψ) in the category Coh≤k+1 (X) such that A ⊕ A0 = H 0 (F, ϕ, ψ) and B ⊕ B 0 = H 1 (F, ϕ, ψ). By Lemma 29.22.4 this implies that [A ⊕ A0 ]k ∼rat [B ⊕ B 0 ]k This proves that [A] − [B] maps to zero via the composition K0 (Coh≤k (X)/Coh≤k−1 (X)) −→ Zk (X) −→ Ak (X). In other words this proves the commutative diagram exists. Next, assume that X is quasi-compact. By Lemma 29.22.2 the left vertical arrow is bijective. Hence it suffices to show any α ∈ Zk (X) which is rationally equivalent to zero maps to zero in Bk (X) via the inverse of the left vertical P arrow composed with the horizontal arrow. By Lemma 29.21.1 we see that α = ([(Wi )0 ]k − [(Wi )∞ ]k ) for some closed integral subschemes Wi ⊂ X ×S P1S of δ-dimension k + 1. Moreover the family {Wi } is finite because X is quasi-compact. Note that the ideal sheaves Ii , Ji ⊂ OWi of the effective Cartier divisors (Wi )0 , (Wi )∞ are isomorphic (as OWi modules). This is true because the ideal sheaves of D0 and D∞ on P1 are isomorphic and Ii , Ji are the pullbacks of these. (Some details omitted.) Hence we have short exact sequences 0 → Ii → OWi → O(Wi )0 → 0,

0 → Ji → OWi → O(Wi )∞ → 0

of coherent OWi -modules. Also, since [(Wi )0 ]k = [p∗ O(Wi )0 ]k in Zk (X) we see that the inverse of the left vertical arrow maps [(Wi )0 ]k to the element [p∗ O(Wi )0 ] in K0 (Coh≤k (X)/Coh≤k−1 (X)). Thus we have X α = ([(Wi )0 ]k − [(Wi )∞ ]k ) X
7→ [p∗ O(Wi )0 ] − [p∗ O(Wi )∞ ] X = ([p∗ OWi ] − [p∗ Ii ] − [p∗ OWi ] + [p∗ Ji ])

29.23. PREPARATION FOR THE DIVISOR ASSOCIATED TO AN INVERTIBLE SHEAF1691

in K0 (Coh≤k+1 (X)/Coh≤k−1 (X)). By what was said above this is zero, and we win.  Remark 29.22.6. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Assume X is quasi-compact. The result of Lemma 29.22.5 in particular gives a map Ak (X) −→ K0 (Coh(X)/Coh≤k−1 (X)). We have not been able to find a statement or conjecture in the literature as to whether this map is should be injective or not. If X is connected nonsingular, then, using the isomorphism K0 (X) = K 0 (X) (see insert future reference here) and chern classes (see below), one can show that the map is an isomorphism up to (p − 1)!-torsion where p = dimδ (X) − k. 29.23. Preparation for the divisor associated to an invertible sheaf For the following remarks, see Divisors, Section 26.15. Let X be a scheme. Let L be an invertible OX -module. Let ξ ∈ X be a point. If sξ , s0ξ ∈ Lξ generate Lξ as ∗ OX,ξ -module, then there exists a unit u ∈ OX,ξ such that sξ = us0ξ . The stalk of the sheaf of meromorphic sections KX (L) of L at x is equal to KX,x ⊗OX,x Lx . Thus the image of any meromorphic section s of L in the stalk at x can be written as s = f sξ with f ∈ KX,x . Below we will abbreviate this by saying f = s/sξ . Also, if X is integral we have KX,x = R(X) is equal to the function field of X, so s/sξ ∈ R(X). If s is a regular meromorphic section (see Divisors, Definition 26.15.10), then actually f ∈ R(X)∗ . (On an integral scheme a regular meromorphic section is the same thing as a nonzero meromorphic section.) Hence the following definition makes sense. Definition 29.23.1. Let X be a locally Noetherian scheme. Assume X is integral. Let L be an invertible OX -module. Let s ∈ Γ(X, KX (L)) be a regular meromorphic section of L. For every integral closed subscheme Z ⊂ X of codimension 1 we define the order of vanishing of s along Z as the integer ordZ,L (s) = ordOX,ξ (s/sξ ) where the right hand side is the notion of Algebra, Definition 7.113.2, ξ ∈ Z is the generic point, and sξ ∈ Lξ is a generator. Lemma 29.23.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral. Let L be an invertible OX -module. Let s ∈ KX (L) be a regular (i.e., nonzero) meromorphic section of L. Then the set {Z ⊂ X | Z is irreducible, closed of codimension 1 and ordZ,L (s) 6= 0} is locally finite in X. Proof. This is true simply because there exists a nonempty open subscheme U ⊂ X such that s corresponds to a section of Γ(U, L) which generates L over U . Hence the codimension 1 irreducibles which can occur in the set of the lemma are all irreducible components of X \ U . Hence Lemma 29.9.1 gives the desired result. 

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29. CHOW HOMOLOGY AND CHERN CLASSES

Lemma 29.23.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let L be an invertible OX module. Let s, s0 ∈ KX (L) be nonzero meromorphic sections of L. Then f = s/s0 is an element of R(X)∗ and we have X X ordZ,L (s)[Z] = ordZ,L (s0 )[Z] + div(f ) (where the sums are over integral closed subschemes Z ⊂ X of δ-dimension n − 1) as elements of Zn−1 (X). Proof. This is clear from the definitions. Note that Lemma 29.23.2 garantees that the sums are indeed elements of Zn−1 (X).  29.24. The divisor associated to an invertible sheaf The material above allows us to define the divisor associated to an invertible sheaf. Definition 29.24.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let L be an invertible OX -module. (1) For any nonzero meromorphic section s of L we define the Weil divisor associated to s as X divL (s) := ordZ,L (s)[Z] ∈ Zn−1 (X) where the sum is over integral closed subschemes Z ⊂ X of δ-dimension n − 1. (2) We define Weil divisor associated to L c1 (L) ∩ [X] = class of divL (s) ∈ An−1 (X) where s is any nonzero meromorphic section of L over X. This is well defined by Lemma 29.23.3. There are some cases where it is easy to compute the Weil divisor associated to an invertible sheaf. Lemma 29.24.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let L be an invertible OX -module. Let s ∈ Γ(X, L) be a nonzero global section. Then divL (s) = [Z(s)]n−1 in Zn−1 (X) and c1 (L) ∩ [X] = [Z(s)]n−1 in An−1 (X). Proof. Let Z ⊂ X be an integral closed subscheme of δ-dimension n − 1. Let ξ ∈ Z be its generic point. Choose a generator sξ ∈ Lξ . Write s = f sξ for some f ∈ OX,ξ . By definition of Z(s), see Divisors, Definition 26.9.18 we see that Z(s) is cut out by a quasi-coherent sheaf of ideals I ⊂ OX such that Iξ = (f ). Hence lengthOX,x (OZ(s),ξ ) = lengthOX,x (OX,ξ /(f )) = ordOX,x (f ) as desired.  Lemma 29.24.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let L, N be invertible OX -modules. Then

29.25. INTERSECTING WITH CARTIER DIVISORS

1693

(1) Let s, resp. t be a nonzero meromorphic section of L, resp. N . Then st is a nonzero meromorphic section of L ⊗ N , and divL⊗N (st) = divL (s) + divN (t) in Zn−1 (X). (2) We have c1 (L) ∩ [X] + c1 (N ) ∩ [X] = c1 (L ⊗OX N ) ∩ [X] in An−1 (X). Proof. Let s, resp. t be a nonzero meromorphic section of L, resp. N . Then st is a nonzero meromorphic section of L⊗N . Let Z ⊂ X be an integral closed subscheme of δ-dimension n − 1. Let ξ ∈ Z be its generic point. Choose generators sξ ∈ Lξ , and tξ ∈ Nξ . Then sξ tξ is a generator for (L ⊗ N )ξ . So st/(sξ tξ ) = (s/sξ )(t/tξ ). Hence we see that divL⊗N ,Z (st) = divL,Z (s) + divN ,Z (t) by the additivity of the ordZ function.



The following lemma will be superseded by the more general Lemma 29.25.4. Lemma 29.24.4. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Assume X, Y are integral and n = dimδ (Y ). Let L be an invertible OY -module. Let f : X → Y be a flat morphism of relative dimension r. Let L be an invertible sheaf on Y . Then f ∗ (c1 (L) ∩ [Y ]) = c1 (f ∗ L) ∩ [X] in An+r−1 (X). Proof. Let s be a nonzero meromorphic section of L. We will show that actually f ∗ divL (s) = divf ∗ L (f ∗ s) and hence the lemma holds. To see this let ξ ∈ Y be a point and let sξ ∈ Lξ be a generator. Write s = gsξ with g ∈ R(X)∗ . Then there is an open neighbourhood V ⊂ Y of ξ such that sξ ∈ L(V ) and such that sξ generates L|V . Hence we see that divL (s)|V = div(g)|V . In exactly the same way, since f ∗ sξ generates L over f −1 (V ) and since f ∗ s = gf ∗ sξ we also have divL (f ∗ s)|f −1 (V ) = div(g)|f −1 (V ) . Thus the desired equality of cycles over f −1 (V ) follows from the corresponding result for pullbacks of principal divisors, see Lemma 29.17.4.  29.25. Intersecting with Cartier divisors Definition 29.25.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L be an invertible OX -module. We define, for every integer k, an operation c1 (L) ∩ − : Zk+1 (X) → Ak (X) called intersection with the first chern class of L. (1) Given an integral closed subscheme i : W → X with dimδ (W ) = k + 1 we define c1 (L) ∩ [W ] = i∗ (c1 (i∗ L) ∩ [W ]) where the right hand side is defined in Definition 29.24.1.

1694

29. CHOW HOMOLOGY AND CHERN CLASSES

P (2) For a general (k + 1)-cycle α = ni [Wi ] we set X c1 (L) ∩ α = ni c1 (L) ∩ [Wi ] P Write each c1 (L) ∩ Wi = j ni,j [Zi,j ] with {Zi,j }j a locally finite sum of integral closed subschemes of Wi . Since {Wi } is a locally finite collection of integral closed subschemes on X, it follows easily that {Zi,j }P i,j is a locally finite collection of closed subschemes of X. Hence c1 (L) ∩ α = ni ni,j [Zi,j ] is a cycle. Another, ` more convenient, way to think about this is to observe that the morphism Wi → X is`proper. QHence c1 (L) ∩ α can be viewed as the pushforward of a class in Ak ( W i ) = Ak (Wi ). This also explains why the result is well defined up to rational equivalence on X. The main goal for the next few sections is to show that intersecting with c1 (L) factors through rational equivalence, and is commutative. This is not a trviality. Lemma 29.25.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L, N be an invertible sheaves on X. Then c1 (L) ∩ α + c1 (N ) ∩ α = c1 (L ⊗OX N ) ∩ α in Ak (X) for every α ∈ Zk−1 (X). Moreover, c1 (OX ) ∩ α = 0 for all α. Proof. The additivity follows directly from Lemma 29.24.3 and the definitions. To see that c1 (OX ) ∩ α = 0 consider the section 1 ∈ Γ(X, OX ). This restricts to an everywhere nonzero section on any integral closed subscheme W ⊂ X. Hence c1 (OX ) ∩ [W ] = 0 as desired.  The following lemma is a useful result in order to compute the intersection product of the c1 of an invertible sheaf and the cycle associated to a closed subscheme. Recall that Z(s) ⊂ X denotes the zero scheme of a global section s of an invertible sheaf on a scheme X, see Divisors, Definition 26.9.18. Lemma 29.25.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L be an invertible OX -module. Let Z ⊂ X be a closed subscheme. Assume dimδ (Z) ≤ k + 1. Let s ∈ Γ(Z, L|Z ). Assume (1) dimδ (Z(s)) ≤ k, and (2) for every generic point ξ of an irreducible component of Z(s) of dimension k the multiplication by s induces an injection OZ,ξ → (L|Z )ξ . This holds for example if s is a regular section of L|Z . Then [Z(s)]k = c1 (L) ∩ [Z]k+1 in Ak (X). Proof. Write [Z]k+1 =

X

ni [Wi ]

where Wi ⊂ Z are the irreducible components of Z of δ-dimension k + 1 and ni > 0. By assumption the restriction si = s|Wi ∈ Γ(Wi , L|Wi ) is not zero, and hence is a regular section. By Lemma 29.24.2 we see that [Z(si )]k represents c1 (L|Wi ). Hence by definition X c1 (L) ∩ [Z]k+1 = ni [Z(si )]k In fact, the proof below will show that we have X (29.25.3.1) [Z(s)]k = ni [Z(si )]k

29.25. INTERSECTING WITH CARTIER DIVISORS

1695

as k-cycles on X. Let Z 0 ⊂ X be an integral closed subscheme of δ-dimension k. Let ξ 0 ∈ Z 0 be its coefficient n of [Z 0 ] in the expression P generic point. We want to compare the 0 ni [Z(si )]k with the coefficient m of [Z ] in the expression [Z(s)]k . Choose a generator sξ0 ∈ Lξ . Let I ⊂ OX be the ideal sheaf of Z. Write A = OX,ξ0 , L = Lξ0 and I = Iξ0 . Then L = Asξ0 and L/IL = (A/I)sξ0 = (L|Z )ξ0 . Write s = f sξ0 for some (unique) f ∈ A/I. Hypothesis (2) means that f : A/I → A/I is injective. Since dimδ (Z) ≤ k + 1 and dimδ (Z 0 ) = k we have dim(A/I) = 0 or 1. We have m = lengthA (A/(f, I)) which is finite in either case. If dim(A/I) = 0, then f : A/I → A/I being injective implies that f ∈ (A/I)∗ . Hence in this case m is zero. Moreover, the condition dim(A/I) = 0 means that ξ 0 does not lie on any irreducible component of δ-dimension k + 1, i.e., n = 0 as well. Now, let dim(A/I) = 1. Since A is a Noetherian local ring there are finitely many minimal primes q1 , . . . , qt ⊃ I over I. These correspond 1-1 with Wi passing through ξ 0 . Moreover ni = lengthAq ((A/I)qi ). Also, the multiplicity of [Z 0 ] in i [Z(si )]k is lengthA (A/(f, qi )). Hence the equation to prove in this case is X lengthA (A/(f, I)) = lengthAq ((A/I)qi )lengthA (A/(f, qi )) i

which follows from Lemma 29.5.6.



Lemma 29.25.4. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Let L be an invertible sheaf on Y . Let α be a k-cycle on Y . Then f ∗ (c1 (L) ∩ α) = c1 (f ∗ L) ∩ f ∗ α in Ak+r−1 (X). P Proof. Write α = ni [Wi ]. We claim it suffices to show that f ∗ (c1 (L) ∩ [Wi ]) = ∗ ∗ c1 (f L) ∩ f [Wi ] for each i. Proof of this claim is omitted. (Remarks: it is clear in the quasi-compact case. Something similar happened in the proof of Lemma 29.20.1, and one can copy the method used there here. Another possibility is to check the cycles and rational equivalences used for all Wi combined at each step form a locally finite collection). Let W ⊂ Y be an integral closed subscheme of δ-dimension k. We have to show that f ∗ (c1 (L) ∩ [W ]) = c1 (f ∗ L) ∩ f ∗ [W ]. Consider the following fibre product diagram W 0 = W ×Y X

/X

 W

 /Y

and let Wi0P⊂ W 0 be the irreducible components of δ-dimension P k +0 r. Write [W 0 ]k+r = ni [Wi0 ] with ni > 0 as per definition. So f ∗ [W ] = ni [Wi ]. Choose a nonzero meromorphic section s of L|W . Since each Wi0 → W is dominant we see that si = s|Wi0 is a nonzero meromorphic section for each i. We claim that we have the following equality of cycles X ni divL|Wi (si ) = f ∗ divL|W (s)

1696

29. CHOW HOMOLOGY AND CHERN CLASSES

in Zk+r−1 (X). Having formulated the problem as an equality of cycles we may work locally on Y . Hence we may assume Y and also W affine, and s = p/q for some nonzero sections p ∈ Γ(W, L) and q ∈ Γ(W, O). If we can show both X X ni divL|Wi (pi ) = f ∗ divL|W (p), and ni divO|Wi (qi ) = f ∗ divO|W (q) (with obvious notations) then we win by the additivity, see Lemma 29.24.3. Thus we may assume that s ∈ Γ(W, L|W ). In this case we may apply the equality (29.25.3.1) obtained in the proof of Lemma 29.25.3 to see that X ni divL|Wi (si ) = [Z(s0 )]k+r−1 where s0 ∈ f ∗ L|W 0 denotes the pullback of s to W 0 . On the other hand we have f ∗ divL|W (s) = f ∗ [Z(s)]k−1 = [f −1 (Z(s))]k+r−1 , by Lemmas 29.24.2 and 29.14.4. Since Z(s0 ) = f −1 (Z(s)) we win.



Lemma 29.25.5. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a proper morphism. Let L be an invertible sheaf on Y . Let s be a nonzero meromorphic section s of L on Y . Assume X, Y integral, f dominant, and dimδ (X) = dimδ (Y ). Then f∗ (divf ∗ L (f ∗ s)) = [R(X) : R(Y )]divL (s). In particular f∗ (c1 (f ∗ L) ∩ [X]) = c1 (L) ∩ f∗ [Y ]. Proof. The last equation follows from the first since f∗ [X] = [R(X) : R(Y )][Y ] by definition. It turns out that we can re-use Lemma 29.18.1 to prove this. Namely, since we are trying to prove an equality of cycles, we may work locally on Y . Hence we may assume that L = OY . In this case s corresponds to a rational function g ∈ R(Y ), and we are simply trying to prove f∗ (divX (g)) = [R(X) : R(Y )]divY (g). Comparing with the result of the aforementioned Lemma 29.18.1 we see this true since NmR(X)/R(Y ) (g) = g [R(X):R(Y )] as g ∈ R(Y )∗ .  Lemma 29.25.6. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let p : X → Y be a proper morphism. Let α ∈ Zk+1 (X). Let L be an invertible sheaf on Y . Then p∗ (c1 (p∗ L) ∩ α) = c1 (L) ∩ p∗ α in Ak (Y ). Proof. Suppose that p has the property that for every integral closed subscheme W ⊂ X the map p|W : W → Y is a closed immersion. Then, by definition of capping wiht c1 (L) the lemma holds. P We will use this remark to reduce to a special case. Namely, write α = ni [Wi ] with ni 6= 0 and Wi pairwise distinct. Let Wi0 ⊂ Y be the image of Wi (as an

29.26. CARTIER DIVISORS AND K-GROUPS

1697

integral closed subscheme). Consider the diagram ` X 0 = Wi q / X p0

Y0 =

p

 `

Wi0

q

0

 / Y.

Since {Wi } is locally finite on X, and p is proper we see that {Wi0 }P is locally finite on Y and that q, q 0 , p0 are also proper morphisms. We may think of ni [Wi ] also as a k-cycle α0 ∈ Zk (X 0 ). Clearly q∗ α0 = α. We have q∗ (c1 (q ∗ p∗ L)∩α0 ) = c1 (p∗ L)∩q∗ α0 and (q 0 )∗ (c1 ((q 0 )∗ L) ∩ p0∗ α0 ) = c1 (L) ∩ q∗0 p0∗ α0 by the initial remark of the P proof. Hence it suffices to prove the lemma for the morphism p0 and the cycle ni [Wi ]. Clearly, this means we may assume X, Y integral, f : X → Y dominant and α = [X]. In this case the result follows from Lemma 29.25.5.  29.26. Cartier divisors and K-groups In this section we describe how the intersection with the first chern class of an invertible sheaf L corresponds to tensoring with L − O in K-groups. Lemma 29.26.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L be an invertible OX -module. Let F be a coherent OX -module. Let s ∈ Γ(X, KX (L)) be a meromorphic section of L. Assume (1) dimδ (X) ≤ k + 1, (2) X has no embedded points, (3) F has no embedded associated points, (4) the support of F is X, and (5) the section s is regular meromorphic. In this situation let I ⊂ OX be the ideal of denominators of s, see Divisors, Definition 26.15.14. Then we have the following: (1) there are short exact sequences 0 0

→ IF → IF

1

− → s − →

F → Q1 F ⊗ OX L → Q 2

→ 0 → 0

(2) the coherent sheaves Q1 , Q2 are supported in δ-dimension ≤ k, (3) the section s restricts to a regular meromorphic section si on every irreducible componentPXi of X of δ-dimension k + 1, and (4) writing [F]k+1 = mi [Xi ] we have X [Q2 ]k − [Q1 ]k = mi (Xi → X)∗ divL|Xi (si ) in Zk (X), in particular [Q2 ]k − [Q1 ]k = c1 (L) ∩ [F]k+1 in Ak (X). Proof. Recall from Divisors, Lemma 26.15.15 the existence of injective maps 1 : IF → F and s : IF → F ⊗OX L whose cokernels are supported on a closed nowhere dense subsets T . Denote Qi there cokernels as in the lemma. We conclude that dimδ (Supp(Qi )) ≤ k. By Divisors, Lemmas 26.15.4 and 26.15.11 the pullbacks si are defined and are regular meromorphic sections for L|Xi . The equality of cycles

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in (4) implies the equality of cycle classes in (4). Hence the only remaining thing to show is that X [Q2 ]k − [Q1 ]k = mi (Xi → X)∗ divL|Xi (si ) holds in Zk (X). To see this, let Z ⊂ X be an integral closed subscheme of δdimension k. Let ξ ∈ Z be the generic point. Let A = OX,ξ and M = Fξ . Moreover, choose a generator sξ ∈ Lξ . Then we can write s = (a/b)sξ where a, b ∈ A are nonzerodivisors. In this case I = Iξ = {x ∈ A | x(a/b) ∈ A}. In this case the coefficient of [Z] in the left hand side is lengthA (M/(a/b)IM ) − lengthA (M/IM ) and the coefficient of [Z] in the right hand side is X lengthAq (Mqi )ordA/qi (a/b) i

where q1 , . . . , qt are the minimal primes of the 1-dimensional local ring A. Hence the result follows from Lemma 29.5.7.  Lemma 29.26.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L be an invertible OX -module. Let F be a coherent OX -module. Assume dimδ (Support(F)) ≤ k + 1. Then the element [F ⊗OX L] − [F] ∈ K0 (Coh≤k+1 (X)/Coh≤k−1 (X)) lies in the subgroup Bk (X) of Lemma 29.22.5 and maps to the element c1 (L)∩[F]k+1 via the map Bk (X) → Ak (X). Proof. Let 0 → K → F → F0 → 0 be the short exact sequence constructed in Divisors, Lemma 26.4.4. This in particular means that F 0 has no embedded associated points. Since the support of K is nowhere dense in the support of F we see that dimδ (Supp(K)) ≤ k. We may re-apply Divisors, Lemma 26.4.4 starting with K to get a short exact sequence 0 → K00 → K → K0 → 0 where now dimδ (Supp(K00 )) < k and K0 has no embedded associated points. Suppose we can prove the lemma for the coherent sheaves F 0 and K0 . Then we see from the equations [F]k+1 = [F 0 ]k+1 + [K0 ]k+1 + [K00 ]k+1 (use Lemma 29.10.4), [F ⊗OX L] − [F] = [F 0 ⊗OX L] − [F 0 ] + [K0 ⊗OX L] − [K0 ] + [K00 ⊗OX L] − [K00 ] (use the ⊗L is exact) and the trivial vanishing of [K00 ]k+1 and [K00 ⊗OX L] − [K00 ] in K0 (Coh≤k+1 (X)/Coh≤k−1 (X)) that the result holds for F. What this means is that we may assume that the sheaf F has no embedded associated points. Assume X, F as in the lemma, and assume in addition that F has no embedded associated points. Consider the sheaf of ideals I ⊂ OX , the corresponding closed subscheme i : Z → X and the coherent OZ -module G constructed in Divisors, Lemma 26.4.5. Recall that Z is a locally Noetherian scheme without embedded points, G is a coherent sheaf without embedded associated points, with Supp(G) = Z and such that i∗ G = F. Moreover, set N = L|Z .

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1699

By Divisors, Lemma 26.15.12 the invertible sheaf N has a regular meromorphic section s over Z. Let us denote J ⊂ OZ the sheaf of denominators of s. By Lemma 29.26.1 there exist short exact sequences 0 0

→ JG → JG

1

− → s − →

G G ⊗ OZ N

→ Q1 → Q2

→ 0 → 0

such that dimδ (Supp(Qi )) ≤ k and such that the cycle [Q2 ]k − [Q1 ]k is a representative of c1 (N ) ∩ [G]k+1 . We see (using the fact that i∗ (G ⊗ N ) = F ⊗ L by the projection formula, see Cohomology, Lemma 18.7.2) that [F ⊗OX L] − [F] = [i∗ Q2 ] − [i∗ Q1 ] in K0 (Coh≤k+1 (X)/Coh≤k−1 (X)). This already shows that [F ⊗OX L] − [F] is an element of Bk (X). Moreover we have [i∗ Q2 ]k − [i∗ Q1 ]k

=

i∗ ([Q2 ]k − [Q1 ]k )

=

i∗ (c1 (N ) ∩ [G]k+1 )

=

c1 (L) ∩ i∗ [G]k+1

=

c1 (L) ∩ [F]k+1

by the above and Lemmas 29.25.6 and 29.12.3. And this agree with the image of the element under Bk (X) → Ak (X) by definition. Hence the lemma is proved.  29.27. Blowing up lemmas In this section we prove some lemmas on representing Cartier divisors by suitable effective Cartier divisors on blow-ups. These lemmas can be found in [Ful98, Section 2.4]. We have adapted the formulation so they also work in the non-finite type setting. It may happen that the morphism b of Lemma 29.27.7 is a composition of infinitely many blow ups, but over any given quasi-compact open W ⊂ X one needs only finitely many blow-ups (and this is the result of loc. cit.). Lemma 29.27.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a proper morphism. Let D ⊂ Y be an effective Cartier divisor. Assume X, Y integral, n = dimδ (X) = dimδ (Y ) and f dominant. Then f∗ [f −1 (D)]n−1 = [R(X) : R(Y )][D]n−1 . In particular if f is birational then f∗ [f −1 (D)]n−1 = [D]n−1 . Proof. Immediate from Lemma 29.25.5 and the fact that D is the zero scheme of the canonical section 1D of OX (D).  Lemma 29.27.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral with dimδ (X) = n. Let L be an invertible OX module. Let s be a nonzero meromorphic section of L. Let U ⊂ X be the maximal open subscheme such that s corresponds to a section of L over U . There exists a projective morphism π : X 0 −→ X such that (1) X 0 is integral, (2) π|π−1 (U ) : π −1 (U ) → U is an isomorphism,

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29. CHOW HOMOLOGY AND CHERN CLASSES

(3) there exist effective Cartier divisors D, E ⊂ X 0 such that π ∗ L = OX 0 (D − E), (4) the meromorphic section s corresponds, via the isomorphism above, to the meromorphic section 1D ⊗ (1E )−1 (see Divisors, Definition 26.9.14), (5) we have π∗ ([D]n−1 − [E]n−1 ) = divL (s) in Zn−1 (X). Proof. Let I ⊂ OX be the quasi-coherent ideal sheaf of denominators of s. Namely, we declare a local section f of OX to be a local section of I if and only if f s is e for a local section of L. On any affine open U = Spec(A) of X write L|U = L some invertible A-module L. Then A is a Noetherian domain with fraction field K = R(X) and we may think of s|U as an element of L ⊗A K (see Divisors, Lemma 26.15.6). Let I = {x ∈ A | xs ∈ L}. Then we see that I|U = Ie (details omitted) and hence I is quasi-coherent. Consider the closed subscheme Z ⊂ X defined by I. It is clear that U = X \ Z. This suggests we should blow up Z. Let M  I n −→ X π : X 0 = ProjX n≥0 L be the blowing up of X along Z. The quasi-coherent sheaf of OX -algebras n≥0 I n is generated in degree 1 over OX . Moreover, the degree 1 part is a coherent OX module, in particular of finite type. Hence we see that π is projective and OX 0 (1) is relatively very ample. By Divisors, Lemma 26.17.7 we have X 0 is integral. By Divisors, Lemma 26.17.4 there exists an effective Cartier divisor E ⊂ X 0 such that π −1 I · OX 0 = IE . Also, by the same lemma we see that π −1 (U ) ∼ = U. Denote s0 the pullback of the meromorphic section s to a meromorphic section of L0 = π ∗ L over X 0 . It follows from the fact that Is ⊂ L that IE s0 ⊂ L0 . In other words, s0 gives rise to an OX 0 -linear map IE → L0 , or in other words a section t ∈ L0 ⊗ OX 0 (E). By Divisors, Lemma 26.9.20 we obtain a unique effective Cartier divisor D ⊂ X 0 such that L0 ⊗ OX 0 (E) ∼ = OX 0 (D) with t corresponding to 1D . Reversing this procedure we conclude that L0 = OX 0 (−E) ∼ = OX 0 (D) with s0 −1 corresponding to 1D ⊗ 1E as in (4). We still have to prove (5). By Lemma 29.25.5 we have π∗ (divL0 (s0 )) = divL (s). Hence it suffices to show that divL0 (s0 ) = [D]n−1 − [E]n−1 . This follows from the equality s0 = 1D ⊗ 1−1  E and additivity, see Lemma 29.24.3. Definition 29.27.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let D1 , D2 be two effective Cartier divisors in X. Let Z ⊂ X be an integral closed subscheme with dimδ (Z) = n − 1. The -invariant of this situation is Z (D1 , D2 ) = nZ · mZ where nZ , resp. mZ is the coefficient of Z in the (n−1)-cycle [D1 ]n−1 , resp. [D2 ]n−1 .

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Lemma 29.27.4. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let D1 , D2 be two effective Cartier divisors in X. Let Z be an open and closed subscheme of the scheme D1 ∩ D2 . Assume dimδ (D1 ∩ D2 \ Z) ≤ n − 2. Then there exists a morphism b : X 0 → X, and Cartier divisors D10 , D20 , E on X 0 with the following properties (1) (2) (3) (4) (5) (6) (7)

X 0 is integral, b is projective, b is the blow up of X in the closed subscheme Z, E = b−1 (Z), b−1 (D1 ) = D10 + E, and b−1 D2 = D20 + E, dimδ (D10 ∩ D20 ) ≤ n − 2, and if Z = D1 ∩ D2 then D10 ∩ D20 = ∅, for every integral closed subscheme W 0 with dimδ (W 0 ) = n − 1 we have (a) if W 0 (D10 , E) > 0, then setting W = b(W 0 ) we have dimδ (W ) = n−1 and W 0 (D10 , E) < W (D1 , D2 ), (b) if W 0 (D20 , E) > 0, then setting W = b(W 0 ) we have dimδ (W ) = n−1 and W 0 (D20 , E) < W (D1 , D2 ),

Proof. Note that the quasi-coherent ideal sheaf I = ID1 + ID2 defines the scheme theoretic intersection D1 ∩ D2 ⊂ X. Since Z is a union of connected components of D1 ∩ D2 we see that for every z ∈ Z the kernel of OX,z → OZ,z is equal to Iz . Let b : X 0 → X be the blow up of X in Z. (So Zariski locally around Z it is the blow up of X in I.) Denote E = b−1 (Z) the corresponding effective Cartier divisor, see Divisors, Lemma 26.17.4. Since Z ⊂ D1 we have E ⊂ f −1 (D1 ) and hence D1 = D10 +E for some effective Cartier divisor D10 ⊂ X 0 , see Divisors, Lemma 26.9.8. Similarly D2 = D20 + E. This takes care of assertions (1) – (5). Note that if W 0 is as in (7) (a) or (7) (b), then the image W of W 0 is contained in D1 ∩ D2 . If W is not contained in Z, then b is an isomorphism at the generic point of W and we see that dimδ (W ) = dimδ (W 0 ) = n − 1 which contradicts the assumption that dimδ (D1 ∩ D2 \ Z) ≤ n − 2. Hence W ⊂ Z. This means that to prove (6) and (7) we may work locally around Z on X. Thus we may assume that X = Spec(A) with A a Noetherian domain, and D1 = Spec(A/a), D2 = Spec(A/b) and Z = D1 ∩ D2 . Set I = (a, b). Since A is a domain and a, b 6= 0 we can cover the blow up by two patches, namely U = Spec(A[s]/(as − b)) and V = Spec(A[t]/(bt − a)). These patches are glued using the isomorphism A[s, s−1 ]/(as − b) ∼ = A[t, t−1 ]/(bt − a) which maps s to t−1 . The effective Cartier divisor E is described by Spec(A[s]/(as − b, a)) ⊂ U and Spec(A[t]/(bt − a, b)) ⊂ V . The closed subscheme D10 corresponds to Spec(A[t]/(bt − a, t)) ⊂ U . The closed subscheme D20 corresponds to Spec(A[s]/(as − b, s)) ⊂ V . Since “ts = 1” we see that D10 ∩ D20 = ∅. Suppose we have a prime q ⊂ A[s]/(as − b) of height one with s, a ∈ q. Let p ⊂ A be the corresponding prime of A. Observe that a, b ∈ p. By the dimension formula we see that dim(Ap ) = 1 as well. The final assertion to be shown is that ordAp (a) ordAp (b) > ordBq (a) ordBq (s)

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29. CHOW HOMOLOGY AND CHERN CLASSES

where B = A[s]/(as−b). By Algebra, Lemma 7.116.1 we have ordAp (x) ≥ ordBq (x) for x = a, b. Since ordBq (s) > 0 we win by additivity of the ord function and the fact that as = b.  Definition 29.27.5. Let X be a scheme. Let {Di }i∈I be a locally finite collection of effective Cartier divisors on X. Suppose givenPa function I → Z≥0 , i 7→ ni . The sum of the effective Cartier divisors D = ni Di , is the unique effective CartierP divisor D ⊂ X such that on any quasi-compact open U ⊂ X we have D|U = Di ∩U 6=∅ ni Di |U is the sum as in Divisors, Definition 26.9.6. Lemma 29.27.6. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let {Di }i∈I be a locally finite collection of effective Cartier divisors on X. Suppose given ni ≥ 0 for i ∈ I. Then X [D]n−1 = ni [Di ]n−1 i

in Zn−1 (X). Proof. Since we are proving an equality of cycles we may work locally on X. Hence this reduces to a finite sum, and by induction to a sum of two effective Cartier divisors D = D1 + D2 . By Lemma 29.24.2 we see that D1 = divOX (D1 ) (1D1 ) where 1D1 denotes the canonical section of OX (D1 ). Of course we have the same statement for D2 and D. Since 1D = 1D1 ⊗ 1D2 via the identification OX (D) = OX (D1 ) ⊗ OX (D2 ) we win by Lemma 29.24.3.  Lemma 29.27.7. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = d. Let {Di }i∈I be a locally finite collection of effective Cartier divisors on X. Assume that for all {i, j, k} ⊂ I, #{i, j, k} = 3 we have Di ∩ Dj ∩ Dk = ∅. Then there exist (1) an open subscheme U ⊂ X with dimδ (X \ U ) ≤ d − 3, (2) a morphism b : U 0 → U , and (3) effective Cartier divisors {Dj0 }j∈J on U 0 with the following properties: (1) b is proper morphism b : U 0 → U , (2) U 0 is integral, (3) b is an isomorphism over the complement of the union of the pairwise intersections of the Di |U , (4) {Dj0 }j∈J is a locally finite collection of effective Cartier divisors on U 0 , (5) dimδ (Dj0 ∩ Dj0 0 ) ≤ d − 2 if j 6= j 0 , and P (6) b−1 (Di |U ) = nij Dj0 for certain nij ≥ 0. Moreover, if X is quasi-compact, then we may assume U = X in the above. Proof. Let us first prove this in the quasi-compact case, since it is perhaps the most interesting case. In this case we produce inductively a sequence of blowups b

b

0 1 X = X0 ←− X1 ←− X2 ← . . .

and finite sets of effective Cartier divisors {Dn,i }i∈In . At each stage these will have the property that any triple intersection Dn,i ∩ Dn,j ∩ Dn,k is empty. Moreover, for ` each n ≥ 0 we will have In+1 = In P (In ) where P (In ) denotes the set of pairs of elements of In . Finally, we will have X b−1 Dn+1,{i,i0 } n (Dn,i ) = Dn+1,i + 0 0 i ∈In ,i 6=i

29.27. BLOWING UP LEMMAS

1703

We conclude that for each n ≥ 0 we have (b0 ◦ . . . ◦ bn )−1 (Di ) is a nonnegative integer combination of the divisors Dn+1,j , j ∈ In+1 . To start the induction we set X0 = X and I0 = I and D0,i = Di . Given S (Xn , {Dn,i }i∈In ) let Xn+1 be the blow up of Xn in the closed subscheme Zn = {i,i0 }∈P (In ) Dn,i ∩ Dn,i0 . Note that the closed subschemes Dn,i ∩ Dn,i0 are pairwise disjoint ` by our assumption on triple intersections. In other words we may write Zn = {i,i0 }∈P (In ) Dn,i ∩ Dn,i0 . Moreover, in a Zariski neighbourhood of Dn,i ∩ Dn,i0 the morphism bn is equal to the blow up of the scheme Xn in the closed subscheme Dn,i ∩ Dn,i0 , and the results of Lemma 29.27.4 apply. Hence setting Dn+1,{i,i0 } = b−1 n (Di ∩ Di0 ) we get an effective Cartier divisor. The Cartier divisors Dn+1,{i,i0 } are pairwise disjoint. Clearly we have b−1 n (Dn,i ) ⊃ Dn+1,{i,i0 } for every i0 ∈ In , i0 6= i. Hence, applying Divisors, Lemma 26.9.8 we see that P indeed b−1 (Dn,i ) = Dn+1,i + i0 ∈In ,i0 6=i Dn+1,{i,i0 } for some effective Cartier divisor Dn+1,i on Xn+1 . In a neighbourhood of Dn+1,{i,i0 } these divisors Dn+1,i play the role of the primed divisors of Lemma 29.27.4. In particular we conclude that Dn+1,i ∩ Dn+1,i0 = ∅ if i 6= i0 , i, i0 ∈ In by part (6) of Lemma 29.27.4. This already implies that triple intersections of the divisors Dn+1,i are zero. OK, and at this point we can use the quasi-compactness of X to conclude that the invariant (29.27.7.1) (X, {Di }i∈I ) = max{Z (Di , Di0 ) | Z ⊂ X, dimδ (Z) = d − 1, {i, i0 } ∈ P (I)} is finite, since after all each Di has at most finitely many irreducible components. We claim that for some n the invariant (Xn , {Dn,i }i∈In ) is zero. Namely, if not then by Lemma 29.27.4 we have a strictly decreasing sequence (X, {Di }i∈I ) = (X0 , {D0,i }i∈I0 ) > (X1 , {D1,i }i∈I1 ) > . . . of positive integers which is a contradiction. Take n with invariant (Xn , {Dn,i }i∈In ) equal to zero. This means that there is no integral closed subscheme Z ⊂ Xn and no pair of indices i, i0 ∈ In such that Z (Dn,i , Dn,i0 ) > 0. In other words, dimδ (Dn,i , Dn,i0 ) ≤ d − 2 for all pairs {i, i0 } ∈ P (In ) as desired. Next, we come to the general case where we no longer assume that the scheme X is quasi-compact. The problem with the idea from the first part of the proof is that we may get and infinite sequence of blow ups with centers dominating a fixed point of X. In order to avoid this we cut out suitable closed subsets of codimension ≥ 3 at each stage. Namely, we will construct by induction a sequence of morphisms having the following shape X =O X0 j0

U0 o

b0

XO 1 j1

U1 o

b1

XO 2 j2

U2 o

b2

X3

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29. CHOW HOMOLOGY AND CHERN CLASSES

Each of the morphisms jn : Un → Xn will be an open immersion. Each of the morphisms bn : Xn+1 → Un will be a proper birational morphism of integral schemes. As in the quasi-compact case we will have effective Cartier divisors {Dn,i }i∈In on Xn . At each stage these will have the property that any triple intersection ` Dn,i ∩Dn,j ∩Dn,k is empty. Moreover, for each n ≥ 0 we will have In+1 = In P (In ) where P (In ) denotes the set of pairs of elements of In . Finally, we will arrange it so that X b−1 Dn+1,{i,i0 } n (Dn,i |Un ) = Dn+1,i + 0 0 i ∈In ,i 6=i

We start the induction by setting X0 = X, I0 = I and D0,i = Di . Given (Xn , {Dn,i }) we construct the open subscheme Un as follows. For each pair {i, i0 } ∈ P (In ) consider the closed subscheme Dn,i ∩ Dn,i0 . This has “good” irreducible components which have δ-dimension d−2 and “bad” irreducible components which have δ-dimension d − 1. Let us set [ Bad(i, i0 ) = W W ⊂Dn,i ∩Dn,i0 irred. comp. with dimδ (W )=d−1

and similarly Good(i, i0 ) =

[ W ⊂Dn,i ∩Dn,i0 irred. comp. with dimδ (W )=d−2

W.

Then Dn,i ∩ Dn,i0 = Bad(i, i0 ) ∪ Good(i, i0 ) and moreover we have dimδ (Bad(i, i0 ) ∩ Good(i, i0 )) ≤ d − 3. Here is our choice of Un : [ Un = Xn \ Bad(i, i0 ) ∩ Good(i, i0 ). 0 {i,i }∈P (In )

By our condition on triple intersections of the divisors Dn,i we see that the union is actually a disjoint union. Moreover, we see that (as a scheme) a Dn,i |Un ∩ Dn,i0 |Un = Zn,i,i0 Gn,i,i0 where Zn,i,i0 is δ-equidimension of dimension d − 1 and Gn,i,i0 is δ-equidimensional of dimension d − 2. (So toplogically Zn,i,i0 is the union of the bad components but throw out intersections with good components.) Finally we set [ a Zn = Zn,i,i0 = Zn,i,i0 , 0 0 {i,i }∈P (In )

{i,i }∈P (In )

and we let bn : Xn+1 → Xn be the blow up in Zn . Note that Lemma 29.27.4 applies to the morphism bn : Xn+1 → Xn locally around each of the loci Dn,i |Un ∩ Dn,i0 |Un . Hence, exactly as in the first part of the proof we obtain effective Cartier divisors Dn+1,{i,i0 } for {i, i0 } ∈ P (In ) and effective Cartier divisors Dn+1,i for i ∈ In such P that b−1 n (Dn,i |Un ) = Dn+1,i + i0 ∈In ,i0 6=i Dn+1,{i,i0 } . For each n denote πn : Xn → X the morphism obtained as the composition j0 ◦ . . . ◦ jn−1 ◦ bn−1 . Claim: given any quasi-compact open V ⊂ X for all sufficiently large n the maps −1 πn−1 (V ) ← πn+1 (V ) ← . . . −1 are all isomorphisms. Namely, if the map πn−1 (V ) ← πn+1 (V ) is not an isomor−1 0 phism, then Zn,i,i0 ∩ πn (V ) 6= ∅ for some {i, i } ∈ P (In ). Hence there exists an irreducible component W ⊂ Dn,i ∩ Dn,i0 with dimδ (W ) = d − 1. In particular we see that W (Dn,i , Dn,i0 ) > 0. Applying Lemma 29.27.4 repeatedly we see that

W (Dn,i , Dn,i0 ) < (V, {Di |V }) − n

29.28. INTERSECTING WITH EFFECTIVE CARTIER DIVISORS

1705

with (V, {Di |V }) as in (29.27.7.1). Since V is quasi-compact, we have (V, {Di |V }) < ∞ and taking n > (V, {Di |V }) we see the result. Note that by construction the difference Xn \ Un has dimδ (Xn \ Un ) ≤ d − 3. Let Tn = πn (Xn \ Un ) be its image in X. Traversing in the diagram of maps above using each bn is closed it follows that T0 ∪ . . . ∪ Tn is a closed subset of X for each n. Any t ∈ Tn satisfies δ(t) ≤ d − 3 by construction. Hence Tn ⊂ X is a closed subset with dimδ (Tn ) ≤ d − 3. By the claim above we see that for any quasi-compact open V ⊂ X we have Tn ∩ V 6= ∅ for at most finitely many n. Hence {Tn }n≥0 is a locally S finite collection of closed subsets, and we may set U = X \ Tn . This will be U as in the lemma. Note that Un ∩ πn−1 (U ) = πn−1 (U ) by construction of U . Hence all the morphisms −1 bn : πn+1 (U ) −→ πn−1 (U )

are proper. Moreover, by the claim they eventually become isomorphisms over each quasi-compact open of X. Hence we can define U 0 = limn πn−1 (U ). The induced morphism b : U 0 → U is proper since this is local S on U , and over each compact open the limit stabilizes. Similarly we set J = n≥0 In using the inclusions In → In+1 from the construction. For j ∈ J choose an n0 such that j corresponds to i ∈ In0 and define Dj0 = limn≥n0 Dn,i . Again this makes sense as locally over X the morphisms stabilize. The other claims of the lemma are verified as in the case of a quasi-compact X.  29.28. Intersecting with effective Cartier divisors To be able to prove the commutativity of intersection products we need a little more precision in terms of supports of the cycles. Here is the relevant notion. Definition 29.28.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let D be an effective Cartier divisor on X, and denote i : D → X the closed immersion. We define, for every integer k, a Gysin homomorphism i∗ : Zk+1 (X) → Ak (D). (1) Given a integral closed subscheme W ⊂ X with dimδ (W ) = k + 1 we define (a) if W 6⊂ D, then i∗ [W ] = [D ∩ W ]k as a k-cycle on D, and (b) if W ⊂ D, then i∗ [W ] = i0∗ (c1 (OX (D)|W ) ∩ [W ]), where i0 : W → D is the induced closed immersion. P (2) For a general (k + 1)-cycle α = nj [Wj ] we set X i∗ α = nj i∗ [Wj ] (3) We denote D · α = i∗ i∗ α the pushforward of the class to a class on X. In fact, as we will see later, this Gysin homomorphism i∗ can be viewed as an example of a non-flat pullback. Thus we will sometimes informally call the class i∗ α the pullback of the class α. Lemma 29.28.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let D be an effective Cartier divisor on X. Let α be a (k + 1)-cycle on X. Then D · α = c1 (OX (D)) ∩ α in Ak (X).

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29. CHOW HOMOLOGY AND CHERN CLASSES

P Proof. Write α = nj [Wj ] where ij : Wj → X are integral closed subschemes with dimδ (Wj ) = k. Since D is the zero scheme of the canonical section 1D of OX (D) we see that D ∩ Wj is the zero scheme of the restriction 1D |Wj . Hence for each j such that Wj 6⊂ D we have c1 (OX (D)) ∩ [Wj ] = [D ∩ Wj ]k by Lemma 29.25.3. So we have X X c1 (OX (D)) ∩ α = nj [D ∩ Wj ]k + nj ij,∗ (c1 (OX (D)|Wj ) ∩ [Wj ]) Wj 6⊂D

Wj ⊂D

in Ak (X) by Definition 29.25.1. The right hand side matches (termwise) the pushforward of the class i∗ α on D from Definition 29.28.1. Hence we win.  The following lemma will be superseded later. Lemma 29.28.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let D be an effective Cartier divisor on X. Let W ⊂ X be a closed subscheme such that D0 = W ∩ D is an effective Cartier divisor on W . D0

i0

i00

 D

i

/W  /X

For any (k + 1)-cycle on W we have i∗ α = (i00 )∗ (i0 )∗ α in Ak (D). Proof. Suppose α = [Z] for some integral closed subscheme Z ⊂ W . In case Z 6⊂ D we have Z ∩ D0 = Z ∩ D scheme theoretically. Hence the equality holds as cycles. In case Z ⊂ D we also have Z ⊂ D0 and the equality holds since OX (D)|Z ∼  = OW (D0 )|Z and the definition of i∗ and (i0 )∗ in these cases. Lemma 29.28.4. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let i : D → X be an effective Cartier divisor on X. (1) Let Z ⊂ X be a closed subscheme such that dimδ (Z) ≤ k +1 and such that D ∩ Z is an effective Cartier divisor on Z. Then i∗ [Z]k+1 = [D ∩ Z]k . (2) Let F be a coherent sheaf on X such that dimδ (Support(F)) ≤ k + 1 and 1D : F → F ⊗OX OX (D) is injective. Then i∗ [F]k+1 = [i∗ F]k in Ak (D). Proof. Assume Z ⊂ X as in (1). Then set F = OZ . The assumption that D ∩ Z is an effective Cartier divisor is equivalent to the assumption that 1D : F → F ⊗OX OX (D) is injective. Moreover [Z]k+1 = [F]k+1 ] and [D ∩ Z]k = [OD∩Z ]k = [i∗ F]k . See Lemma 29.10.3. Hence part (1) follows from part (2). P Write [F]k+1 = mj [Wj ] with mj > 0 and pairwise distinct integral closed subschemes Wj ⊂ X of δ-dimension k + 1. The assumption that 1D : F → F ⊗OX OX (D) is injective implies that Wj 6⊂ D for all j. By definition we see that X i∗ [F]k+1 = [D ∩ Wj ]k . We claim that X

[D ∩ Wj ]k = [i∗ F]k

as cycles. Let Z ⊂ D be an integral closed subscheme of δ-dimension k. Let ξ ∈ Z be its generic point. Let A = OX,ξ . Let M = Fξ . Let f ∈ A be an

29.28. INTERSECTING WITH EFFECTIVE CARTIER DIVISORS

1707

element generating the ideal of D, i.e., such that OD,ξ = A/f A. By assumption dim(M ) = 1, f : M → M is injective, and lengthA (M/f M ) < ∞. Moreover, lengthA (M/f M ) is the coefficient of [Z] in [i∗ F]k . On the other hand, let q1 , . . . , qt be the minimal primes in the support of M . Then X lengthAq (Mqi )ordA/qi (f ) i P is the coefficient of [Z] in [D ∩ Wj ]k . Hence we see the equality by Lemma 29.5.6.  Lemma 29.28.5. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let {ij : Dj → X}j∈J be a locally P finite collection of effective Cartier divisors on X. Let nj > 0, j ∈ J. Set D = j∈J nj Dj , and denote i : D → X the inclusion morphism. Let α ∈ Zk+1 (X). Then a p: Dj −→ D j∈J

is proper and i∗ α = p∗

X

nj i∗j α



in Ak (D). Proof. The proof of this lemma is made a bit longer than expected by a subtlety concerning infinite sums of rational equivalences. In the quasi-compact case the family Dj is finite and the result is altogether easy and a straightforward consequence of Lemmas 29.24.2 and 29.24.3 and the definitions. The P morphism p is proper since the family {Dj }j∈J is locally finite. Write α = a∈A ma [Wa ] with Wa ⊂ X an integral closed subscheme of δ-dimension k + 1. Denote ia : Wa → X the closed immersion. We assume that ma 6= 0 for all a ∈ A such that {Wa }a∈A is locally finite on X. P Observe that by Definition 29.28.1 the class i∗ α is the class of a cycle ma βa for certain βa ∈ Zk (Wa ∩ D). Namely, if Wa 6⊂ D then βa = [D ∩ Wa ]k and if Wa ⊂ D, then βa is a cycle representing c1 (OX (D)) ∩ [Wa ]. ` ` For each a ∈ A write J = Ja,1 Ja,2 Ja,3 where (1) j ∈ Ja,1 if and only if Wa ∩ Dj = ∅, (2) j ∈ Ja,2 if and only if Wa 6= Wa ∩ D1 6= ∅, and (3) j ∈ Ja,3 if and only if Wa ⊂ Dj . Since the family {Dj } is locally finite we see that Ja,3 is a finite set. For every a ∈ A and j ∈ J we choose a cycle βa,j ∈ Zk (Wa ∩ Dj ) as follows (1) if j ∈ Ja,1 we set βa,j = 0, (2) if j ∈ Ja,2 we set βa,j = [Dj ∩ Wa ]k , and (3) if j ∈ Ja,3 we choose βa,j ∈ Zk (Wa ) representing c1 (i∗a OX (Dj )) ∩ [Wj ]. We claim that βa ∼rat

X j∈J

nj βa,j

in Ak (Wa ∩ D). Case P I: Wa 6⊂ D. In this case Ja,3 = ∅. Thus it suffices to show that [D ∩ Wa ]k = nj [Dj ∩ Wa ]k as cycles. This is Lemma 29.27.6.

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29. CHOW HOMOLOGY AND CHERN CLASSES

∗ Case II: Wa ⊂ D. In this case βa is a cycle P representing c1 (ia OX (D)) ∩ [Wa ]. Write D = Da,1 + Da,2 + Da,3 with Da,s = j∈Ja,s nj Dj . By Lemma 29.24.3 we have

c1 (i∗a OX (D)) ∩ [Wa ]

=

c1 (i∗a OX (Da,1 )) ∩ [Wa ] + c1 (i∗a OX (Da,2 )) ∩ [Wa ] +c1 (i∗a OX (Da,3 )) ∩ [Wa ].

It is clear that the first term P of the sum is zero. Since Ja,3 is finite we see that the last term agrees with j∈Ja,3 nj c1 (i∗a Lj ) ∩ [Wa ], see Lemma 29.24.3. This is P represented by j∈Ja,3 nj βa,j . Finally, by Case I we see that the middle term is P P represented by the cycle j∈Ja,2 nj [Dj ∩ Wa ]k = j∈Ja,2 nj βa,j . Whence the claim in this case. At this point P we are ready to finish the proof of the lemma. Namely, P we have i∗ D ∼rat ma βa by our choice of βa . For each a we have βa ∼rat j βa,j with the rational equivalence taking place on D ∩ Wa . Since P the collection P of closed subschemes D∩Wa is locally finite on D, we see that also ma βa ∼rat a,j ma βa,j P on D! (See Remark 29.19.4.) Ok, and now it is clear that a ma βP a,j (viewed as P a cycle on Dj ) represents i∗j α and hence a,j ma βa,j represents p∗ j i∗j α and we win.  Lemma 29.28.6. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let D, D0 be effective Cartier divisors on X. Assume dimδ (D ∩ D0 ) = n − 2. Let i : D → X, resp. i0 : D0 → X be the corresponding closed immersions. Then (1) there exists a cycle α ∈ Zn−2 (D ∩ D0 ) whose pushforward to D represents i∗ [D0 ]n−1 ∈ An−2 (D) and whose pushforward to D0 represents (i0 )∗ [D]n−1 ∈ An−2 (D0 ), and (2) we have D · [D0 ]n−1 = D0 · [D]n−1 in An−2 (X). Proof. Part (3) is a trivial consequence of parts P (1) and (2). Because of symmetry P we only need to prove (1). Let us write [D]n−1 = na [Za ] and [D0 ]n−1 = mb [Zb ] with Za the irreducible components of D and [Zb ] the P irreducible components of 0 D to Definition 29.28.1, we have i∗ D0 = mb i∗ [Zb ] and (i0 )∗ D = P . According 0 ∗ na (i ) [Za ]. By assumption, none of the irreducible components Zb is contained in D, and hence i∗ [Zb ] = [Zb ∩ D]n−2 by definition. Similarly (i0 )∗ [Za ] = [Za ∩ D0 ]n−2 . Hence we are trying to prove the equality of cycles X X na [Za ∩ D0 ]n−2 = mb [Zb ∩ D]n−2 which are indeed supported on D ∩D0 . Let W ⊂ X be an integral closed subscheme with dimδ (W ) = n − 2. Let ξ ∈ W be its generic point. Set R = OX,ξ . It is a Noetherian local domain. Note that dim(R) = 2. Let f ∈ R, resp. f 0 ∈ R be an element defining the ideal of D, resp. D0 . By assumption dim(R/(f, f 0 )) = 0. Let q01 , . . . , q0t ⊂ R be the minimal primes over (f 0 ), let q1 , . . . , qs ⊂ R be the minimal primes over (f ). The equality above comes down to the equality X X lengthRq (Rqi /(f ))ordR/qi (f 0 ) = lengthRq (Rq0j /(f 0 ))ordR/qj (f ). i

i=1,...,s

j

j=1,...,t

29.28. INTERSECTING WITH EFFECTIVE CARTIER DIVISORS

1709

By Lemma 29.5.5 applied with M = R/(f ) the left hand side of this equation is equal to lengthR (R/(f, f 0 )) − lengthR (Ker(f 0 : R/(f ) → R/(f ))) OK, and now we note that Ker(f 0 : R/(f ) → R/(f )) is canonically isomorphic to ((f ) ∩ (f 0 ))/(f f 0 ) via the map x mod (f ) 7→ f 0 x mod (f f 0 ). Hence the left hand side is lengthR (R/(f, f 0 )) − lengthR ((f ) ∩ (f 0 )/(f f 0 )) Since this is symmetric in f and f 0 we win.



Lemma 29.28.7. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let {Dj }j∈J be a locally finite collection of effective Cartier X. P divisors on PLet nj , mj ≥ 0 be collections of nonnegative integers. Set D = nj Dj and D0 = mj Dj . Assume that dimδ (Dj ∩ Dj 0 ) = n − 2 for every j 6= j 0 . Then D · [D0 ]n−1 = D0 · [D]n−1 in An−2 (X). Proof. This lemma is a trivial consequence of Lemmas 29.27.6 and 29.28.6 in case the sums are finite, e.g., if X is quasi-compact. Hence we suggest the reader skip the proof. Here is the ` proof in the general case. Let ij : Dj → X be the closed immersions Let p : Dj → X denote coproduct of S the morphisms ij . Let {Za }a∈A be the collection of irreducible components of Dj . For each j we write [Dj ]n−1 =

X

dj,a [Za ].

By Lemma 29.27.6 we have [D]n−1 =

X

nj dj,a [Za ],

By Lemma 29.28.5 we have X  D · [D0 ]n−1 = p∗ nj i∗j [D0 ]n−1 ,

[D0 ]n−1 =

X

mj dj,a [Za ].

D0 · [D]n−1 = p∗

X

 mj 0 i∗j 0 [D]n−1 .

As in the definition of the Gysin homomorphisms (see Definition 29.28.1) we choose cycles βa,j on Dj ∩ Za representing i∗j [Za ]. (Note that in fact βa,j = [Dj ∩ Za ]n−2 if Za is not contained in Dj , i.e., there is no choice in that case.) Now since p is a closed immersion when restricted to each of the Dj we can (and we will) view βa,j as a cycle on X. Plugging in the formulas for [D]n−1 and [D0 ]n−1 obtained above we see that X X 0 0 dj 0 ,a βa,j , D · [D0 ]n−1 = n m D · [D] = mj 0 nj dj,a βa,j 0 . j j n−1 0 0 j,j ,a

j,j ,a

Moreover, with the same conventions we also have Dj · [Dj 0 ]n−1 =

X

dj 0 ,a βa,j .

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29. CHOW HOMOLOGY AND CHERN CLASSES

In these terms Lemma 29.28.6 (see also its proof) says that for j 6= j 0 the cycles P P dj 0 ,a βa,j and dj,a βa,j 0 are equal as cycles! Hence we see that X D · [D0 ]n−1 = nj mj 0 dj 0 ,a βa,j j,j 0 ,a X  X X = nj m j 0 dj 0 ,a βa,j + nj mj dj,a βa,j 0 j6=j a j,a   X X X = nj mj 0 dj,a βa,j 0 + nj mj dj,a βa,j 0 j6=j a j,a X = mj 0 nj dj,a βa,j 0 0 j,j ,a

=

D0 · [D]n−1

and we win.



Here is the key lemma of this chapter. A stronger version of this lemma asserts that D · [D0 ]n−1 = D0 · [D]n−1 holds in An−2 (D ∩ D0 ) for suitable representatives of the dot products involved. The first proof of the lemma together with Lemmas 29.28.5, 29.28.6, and 29.28.7 can be modified to show this (see [Ful98]). It is not so clear how to modify the second proof to prove the refined version. An application of the refined version is a proof that the Gysin homomorphism factors through rational equivalence. We will show this by another method later. Lemma 29.28.8. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let D, D0 be effective Cartier divisors on X. Then D · [D0 ]n−1 = D0 · [D]n−1 in An−2 (X). First proof of Lemma 29.28.8. First, let us prove this in case X is quasi-compact. In this case, apply Lemma 29.27.7 to X and the two element set {D, D0 } of effective Cartier divisors. Thus we get a proper morphism b : X 0 → X, a finite collection of effective Cartier divisors Dj0 ⊂ X 0 intersecting pairwise in codimension ≥ 2, with P P b−1 (D) = nj Dj0 , and b−1 (D0 ) = mj Dj0 . Note that b∗ [b−1 (D)]n−1 = [D]n−1 in 0 Zn−1 (X) and similarly for D , see Lemma 29.27.1. Hence, by Lemma 29.25.6 we have
D · [D0 ]n−1 = b∗ b−1 (D) · [b−1 (D0 )]n−1 in An−2 (X) and similarly for the other term. Hence the lemma follows from the equality b−1 (D) · [b−1 (D0 )]n−1 = b−1 (D0 ) · [b−1 (D)]n−1 in An−2 (X 0 ) of Lemma 29.28.7. Note that in the proof above, each referenced lemma works also in the general case (when X is not assumed quasi-compact). The only minor change in the general case is that the morphism b : U 0 → U we get from applying Lemma 29.27.7 has as its target an open U ⊂ X whose complement has codimension ≥ 3. Hence by Lemma 29.19.2 we see that An−2 (U ) = An−2 (X) and after replacing X by U the rest of the proof goes through unchanged.  Second proof of Lemma 29.28.8. Let I = OX (−D) and I 0 = OX (−D0 ) be the 0 invertible ideal sheaves of D and D0 . We denote ID0 = I ⊗OX OD0 and ID = 0 0 I ⊗OX OD . We can restrict the inclusion map I → OX to D to get a map ϕ : ID0 −→ OD0

29.29. COMMUTATIVITY

1711

and similarly 0 ψ : ID −→ OD

It is clear that Coker(ϕ) ∼ = OD∩D0 ∼ = Coker(ψ) and Ker(ϕ) ∼ =

I ∩ I0 ∼ = Ker(ψ). II 0

Hence we see that 0 γ = [ID0 ] − [OD0 ] = [ID ] − [OD ] in K0 (Coh≤n−1 (X)). On the other hand it is clear that 0 [ID ]n−1 = [D]n−1 ,

[ID0 ]n−1 = [D0 ]n−1 .

and that 0 = OD , OX (D) ⊗ ID0 = OD0 . OX (D0 ) ⊗ ID By Lemma 29.26.2 (applied two times) this means that the element γ is an element of Bn−2 (X), and maps to both c1 (OX (D0 )) ∩ [D]n−1 and to c1 (OX (D)) ∩ [D0 ]n−1 and we win (since the map Bn−2 (X) → An−2 (X) is well defined – which is the key to this proof). 

29.29. Commutativity At this point we can start using the material above and start proving more interesting results. Lemma 29.29.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X integral and dimδ (X) = n. Let L, N be invertible on X. Choose a nonzero meromorphic section s of L and a nonzero meromorphic section t of N . Set α = divL (s) and β = divN (t). Then c1 (N ) ∩ α = c1 (L) ∩ β in An−2 (X). Proof. By Lemma 29.27.2 (applied twice) there exists a proper morphism π : X 0 → X and effective Cartier divisors D1 , E1 , D2 , E2 on X 0 such that b∗ L = OX 0 (D1 − E1 ),

b∗ N = OX 0 (D2 − E2 ),

and such that α = π∗ ([D1 ]n−1 − [E1 ]n−1 ),

β = π∗ ([D2 ]n−1 − [E2 ]n−1 ).

By the projection formula of Lemma 29.25.6 and the additivity of Lemma 29.25.2 it is enough to show the equality c1 (OX 0 (D1 )) ∩ [D2 ]n−1 = c1 (OX 0 (D2 )) ∩ [D1 ]n−1 and three other similar equalities involving Di and Ej . By Lemma 29.28.2 this is the same as showing that D1 · [D2 ]n−1 = D2 · [D1 ]n−1 and so on. Thus the result follows from Lemma 29.28.8.  Lemma 29.29.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L be invertible on X. The operation α 7→ c1 (L) ∩ α factors through rational equivalence to give an operation c1 (L) ∩ − : Ak+1 (X) → Ak (X)

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29. CHOW HOMOLOGY AND CHERN CLASSES

Proof. Let α ∈ Zk+1 (X), and α ∼rat 0. We have to show that c1 (L) ∩ α as defined in Definition 29.25.1 is zero. By Definition 29.19.1 there exists a locally finite family {Wj } of integral closed subschemes with dimδ (Wj ) = k + 2 and rational functions fj ∈ R(Wj )∗ such that X α= (ij )∗ divWj (fj ) ` Note that ` p : Wj → X is a proper morphism, and hence α = p∗ α0 where 0 α ∈ Zk+1 ( Wj ) is the sum of the principal divisors divWj (fj ). By the projection formula (Lemma 29.25.6) we have c1 (L) ∩ α = p∗ (c1 (p∗ L) ∩ α0 ). Hence it suffices to show that each c1 (L|Wj ) ∩ divWj (fj ) is zero. In other words we may assume that X is integral and α = divX (f ) for some f ∈ R(X)∗ . Assume X is integral and α = divX (f ) for some f ∈ R(X)∗ . We can think of f as a regular meromorphic section of the invertible sheaf N = OX . Choose a meromorphic section s of L and denote β = divL (s). By Lemma 29.29.1 we conclude that c1 (L) ∩ α = c1 (OX ) ∩ β. However, by Lemma 29.25.2 we see that the right hand side is zero in Ak (X) as desired.  For any integer s ≥ 0 we will denote c1 (L)s ∩ − : Ak+s (X) → Ak (X) the s-fold iterate of the operation c1 (L) ∩ −. This makes sense by the lemma above. Lemma 29.29.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let L, N be invertible on X. For any α ∈ Ak+2 (X) we have c1 (L) ∩ c1 (N ) ∩ α = c1 (N ) ∩ c1 (L) ∩ α as elements of Ak (X). P Proof. Write α = mj [Zj ] for some lolcally finite collection of integral closed subschemes Z ⊂ X with j ` P dimδ (Zj ) = k + 2. Consider ` the proper morphism p : Zj → X. Set α0 = mj [Zj ] as a (k + 2)-cycle on Zj . By several applications of Lemma 29.25.6 we see that c1 (L) ∩ c1 (N ) ∩ α = p∗ (c1 (p∗ L) ∩ c1 (p∗ N ) ∩ α0 ) and c1 (N ) ∩ c1 (L) ∩ α = p∗ (c1 (p∗ N ) ∩ c1 (p∗ L) ∩ α0 ). Hence it suffices to prove the formula in case X is integral and α = [X]. In this case the result follows from Lemma 29.29.1 and the definitions.  29.30. Gysin homomorphisms We want to show the Gysin homomorphisms factor through rational equivalence. One method (see [Ful98]) is to prove a more precise version of the key Lemma 29.28.8 keeping track of supports. Having obtained this one can find anlogues of the lemmas of Section 29.29 for the Gysin homomorphism and get the result. We will use another method. Lemma 29.30.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let X be integral and n = dimδ (X). Let a ∈ Γ(X, OX ) be a nonzero function. Let i : D = Z(a) → X be the closed immersion of the zero scheme of a. Let f ∈ R(X)∗ . In this case i∗ divX (f ) = 0 in An−2 (D).

29.30. GYSIN HOMOMORPHISMS

1713

P Proof. Write divX (f ) = nj [Zj ] for some integral closed subschemes Zj ⊂ X of δ-dimension n − 1. We may assume S that the family {Zj }j∈J is locally finite and ∗ that f ∈ Γ(U, OU ) where U = X \ Zj (see Lemma 29.16.3 and its proof). ` Write J = J1 J2 where J1 = {j ∈ J | Zj ⊂ D}. Note that OX (D) ∼ = OX because a−1 is a trivializing global section. Hence by Definition 29.28.1 of i∗ we see that i∗ divX (f ) is represented by X nj [D ∩ Zj ]n−2 . j∈J2

Namely, the terms involving c1 (OX (D)|Zj ) ∩ Zj may be dropped since c1 (O) ∩ − is the zero operation anyway (see Lemma 29.25.2). For each j let ξj ∈ Zj be its generic point. Let Bj = OX,ξj , which has residue field κj = κ(ξj ) = R(Zj ). For j ∈ J1 , let fj = dBj (f, a) be the tame symbol, see Definition 29.4.5. We claim that we have the following equality of cycles X X nj [D ∩ Zj ]n−2 = (Zj → D)∗ divZj (fj ) j∈J2

j∈J1

on D. Indeed, note that [D∩Zj ]n−2 = divZj (a). Hence nj [D∩Zj ]n−2 = divZj (anj ). Since nj = ordBj (f ) we see that in fact also nj [D ∩ Zj ]n−2 = divZj (dBj (a, f )), as a is a unit in Bj see Lemma 29.4.6. Note that dBj (f, a) = dBj (a, f )−1 , see Lemma 29.4.4. Hence altogether we are trying to show that X (Zj → D)∗ divZj (dBj (a, f )) = 0 j∈J

as an (n − 2)-cycle. Consider any codimension 2 integral closed subscheme W ⊂ X with generic point ζ ∈ X. Set A = OX,ζ . Applying Lemma 29.6.1 to (A, a, f ) we see that the coefficient of [W ] in the expression above is zero as desired.  Lemma 29.30.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let X be integral and n = dimδ (X). Let i : D → X be an effective Cartier divisor. Let f ∈ R(X)∗ . In this case i∗ divX (f ) = 0 in An−2 (D). Proof. This proof is a repeat of the proof of Lemma 29.30.1. So make sure you’ve read that one first. P Write divX (f ) = nj [Zj ] for some integral closed subschemes Zj ⊂ X of δdimension n − 1. We may assume S that the family {Zj }j∈J is locally finite and that ∗ f ∈ Γ(U, OU ) where U = X \ Zj (see Lemma 29.16.3 and its proof). ` Write J = J1 J2 where J1 = {j ∈ J | Zj ⊂ D}. For each j let ξj ∈ Zj be its generic point. Let us write L = OX (D). Choose s˜j ∈ Lξj a generator. Denote sj ∈ Lξj ⊗ κ(ξj ) the corresponding nonzero meromorphic section of L|Zj . Then by Definition 29.28.1 of i∗ we see that i∗ divX (f ) is represented by the cycle X X nj [D ∩ Zj ]n−2 + nj divL|Zj (sj ) j∈J2

j∈J2

on D. Our goal is to show that this is rationally equivalent to zero on D. Let Bj = OX,ξj , which has residue field κj = κ(ξj ) = R(Zj ). Write s = aj s˜j for some aj ∈ Bj . For j ∈ J1 let fj = dBj (f, aj ) ∈ κ∗j = R(Zj )∗

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29. CHOW HOMOLOGY AND CHERN CLASSES

be the tame symbol, see Definition 29.4.5. We claim that we have the following equality of cycles X X X (Zj → D)∗ divZj (fj ) nj divL|Zj (sj ) = nj [D ∩ Zj ]n−2 + j∈J1

j∈J2

j∈J2

on D. This will clearly prove the lemma. Note that for j ∈ J2 we have [D ∩ Zj ]n−2 = divL|Zj (s|Zj ). Since s|Zj = aj |Zj sj we see that [D ∩ Zj ]n−2 = divL|Zj (sj ) + divZj (aj |Zj ). Hence, still for j ∈ J2 , we have nj [D ∩ Zj ]n−2 = nj divL|Zj (sj ) + divZj ((aj |Zj )nj ) Since nj = ordBj (f ) we see that divZj ((aj |Zj )nj ) = divZj (dBj (aj , f )), as aj is a unit in Bj (since j ∈ J2 ), see Lemma 29.4.6. Note that dBj (f, aj ) = dBj (aj , f )−1 , see Lemma 29.4.4. Hence altogether we are trying to show that X X (29.30.2.1) nj divL|Zj (sj ) = (Zj → D)∗ divZj (dBj (aj , f )) j∈J

j∈J

as an (n − 2)-cycle. Consider any codimension 2 integral closed subscheme W ⊂ X with generic point ζ ∈ X. Set A = OX,ζ . Choose a generator sζ ∈ Lζ . For those j such that ζ ∈ Zj we may write s˜j = bj sζ with bj ∈ Bj∗ . We may also write s = aζ sζ for some aζ ∈ A. Then we see that aj = bj aζ . The coefficient of [W ] on the right hand side of Equation (29.30.2.1) is X nj ordA/qj (bj ). ζ∈Zj

where qj ⊂ A is the height one prime corresponding to Zj . Note that Bj = Aqj in this case. The coefficient of [W ] on the left hand side of Equation (29.30.2.1) is X ordA/qj (dAqj (bj aζ , f )). ζ∈Zj

nj

Since bj is a unit, and nj = ordAqj (f ) we see that dAqj (bj aζ , f ) = bj dAqj (aζ , f ) by Lemmas 29.4.4 and 29.4.6. By additivity of ord we see that it suffices to prove X 0= ordA/qj (dAqj (aζ , f )) ζ∈Zj

which is Lemma 29.6.1.



Lemma 29.30.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let i : D → X be an effective Cartier divisor on X. The Gysin homomorphism factors through rational equivalence to give a map i∗ : Ak+1 (X) → Ak (D). Proof. Let α ∈ Zk+1 (X) and assume that α ∼rat 0. This means there exists a locally finite collection of integral P closed subschemes Wj ⊂ X of δ-dimension k + 2 and fj ∈ R(Wj )∗ such that α = ij,∗ divWj (fj ). By construction of the map i∗ P ∗ ∗ we see that i α = i ij,∗ divWj (fj ) where each cycle i∗ ij,∗ divWj (fj ) is supported on D ∩ Wj . If we can show that each i∗ ij,∗ divWj (fj ) is rationally equivalent on Wj ∩ D, then we see that i∗ α ∼rat 0 (this is clear if the sum is finite, in general see Remark 29.19.4). Pick an index j. If Wj ⊂ D, then we see that i∗ ij,∗ divWj (fj ) is simply equal to i0j,∗ c1 (OX (D)|Wj ) ∩ divWj (fj )

29.32. AFFINE BUNDLES

1715

where i0j : Wj → D is the inclusion map. This is rationally equivalent to zero by Lemma 29.29.2. If Wj 6⊂ D, then we see that i∗ ij,∗ divWj (fj ) is simply equal to (i0 )∗ divWj (fj ) where i0 : D∩Wj → Wj is the corresponding closed immersion (see Lemma 29.28.3). Hence in this case Lemma 29.30.2 applies, and we win.  29.31. Relative effective Cartier divisors Lemma 29.31.1. Let A → B be a ring map. Let f ∈ B. Assume that (1) A → B is flat, (2) f is a nonzerodivisor, and (3) A → B/f B is flat. Then for every ideal I ⊂ A the map f : B/IB → B/IB is injective. Proof. Note that IB = I ⊗A B and I(B/f B) = I ⊗A B/f B by the flatness of B and B/f B over A. In particular IB/f IB ∼ = I ⊗A B/f B maps injectively into B/f B. Hence the result follows from the snake lemma applied to the diagram 0

/ I ⊗A B f

0



/B f

/ I ⊗A B

 /B

/ B/IB

/0

f

 / B/IB

with exact rows.

/0 

Lemma 29.31.2. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let p : X → Y be a flat morphism of relative dimension r. Let i : D → X be an effective Cartier divisor with the property that p|D : D → Y is flat of relative dimension r − 1. Let L = OX (D). For any α ∈ Ak+1 (Y ) we have i∗ p∗ α = (p|D )∗ α in Ak+r (D) and c1 (L) ∩ p∗ α = i∗ ((p|D )∗ α) in Ak+r (X). Proof. Let W ⊂ Y be an integral closed subvariety of δ-dimension k + 1. By Lemma 29.31.1 we see that D ∩ p−1 W is an effective Cartier divisor on p−1 W . By Lemma 29.28.4 we see that i∗ [p−1 W ]k+r+1 = [D∩W ]k+r = [(p|D )−1 (W )]k+r . Since ∗ −1 by definition p∗ [W ] = [p−1 W ]k+r+1 and k+r we see we P (p|D ) [W ] = [(p|D ) ∗(W )]P have equality of cycles. Hence if α = m [W ], then we get i α = mj i∗ [Wj ] = j j P ∗ mj (p|D ) [Wj ] as cycles. This proves then first equality. To deduce the second from the first apply Lemma 29.28.2.  29.32. Affine bundles Lemma 29.32.1. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let f : X → Y be a flat morphism of relative dimension r. Assume that for every y ∈ Y , there exists an open neighbourhood U ⊂ Y such that f |f −1 (U ) : f −1 (U ) → U is identified with the morphism U × Ar → U . Then f ∗ : Ak (Y ) → Ak+r (X) is surjective for all k ∈ Z.

1716

29. CHOW HOMOLOGY AND CHERN CLASSES

P Proof. Let α ∈ Ak+r (X). Write α = mj [Wj ] with mj 6= 0 and Wj pairwise distinct integral closed subschemes of δ-dimension k + r. Then the family {Wj } is locally finite in X. For any quasi-compact open V ⊂ Y we see that f −1 (V ) ∩ Wj is nonempty only for finitely many j. Hence the collection Zj = f (Wj ) of closures of images is a locally finite collection of integral closed subschemes of Y . Consider the fibre product diagrams f −1 (Zj ) fj

 Zj

/X f

 /Y

∗ Suppose that [Wj ] ∈ Zk+r (f −1 P(Zj )) is rationally equivalent to fj∗ βj for P some kmj βj will be a k-cycle on Y and f β = mj fj∗ βj cycle βj ∈ Ak (Zj ). Then β = will be rationally equivalent to α (see Remark 29.19.4). This reduces us to the case Y integral, and α = [W ] for some integral closed subscheme of X dominating Y . In particular we may assume that d = dimδ (Y ) < ∞.

Hence we can use induction on d = dimδ (Y ). If d < k, then Ak+r (X) = 0 and the lemma holds. By assumption there exists a dense open V ⊂ Y such that f −1 (V ) ∼ = V ×Ar as schemes over V . Suppose that we can show that α|f −1 (V ) = f ∗ β for some β ∈ Zk (V ). By Lemma 29.14.2 we see that β = β 0 |V for some β 0 ∈ Zk (Y ). By the exact sequence Ak (f −1 (Y \ V )) → Ak (X) → Ak (f −1 (V )) of Lemma 29.19.2 we see that α − f ∗ β 0 comes from a cycle α0 ∈ Ak+r (f −1 (Y \ V )). Since dimδ (Y \ V ) < d we win by induction on d. Thus we may assume that X = Y × Ar . In this case we can factor f as X = Y × Ar → Y × Ar−1 → . . . → Y × A1 → Y. Hence it suffices to do the case r = 1. By the argument in the second paragraph of the proof we are reduced to the case α = [W ], Y integral, and W → Y dominant. Again we can do induction on d = dimδ (Y ). If W = Y × A1 , then [W ] = f ∗ [Y ]. Lastly, W ⊂ Y × A1 is a proper inclusion, then W → Y induces a finite field extension R(Y ) ⊂ R(W ). Let P (T ) ∈ R(Y )[T ] be the monic irreducible polynomial such that the generic fibre of W → Y is cut out by P in A1R(Y ) . Let V ⊂ Y be a nonempty open such that P ∈ Γ(V, OY )[T ], and such that W ∩ f −1 (V ) is still cut out by P . Then we see that α|f −1 (V ) ∼rat 0 and hence α ∼rat α0 for some cycle α0 on (Y \ V ) × A1 . By induction on the dimension we win.  Remark 29.32.2. We will see later (Lemma 29.33.3) that if X is a vectorbundle over Y then the pullback map Ak (Y ) → Ak+r (X) is an isomorphism. Is this true in general? 29.33. Projective space bundle formula Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Consider a finite locally free OX -module E of rank r. Our convention is that the projective bundle associated to E is the morphism P(E) = ProjX (Sym∗ (E))

π

/X

29.33. PROJECTIVE SPACE BUNDLE FORMULA

1717

over X with OP(E) (1) normalized so that π∗ (OP(E) (1)) = E. In particular there is a surjection π ∗ E → OP(E) (1). We will say informally “let (π : P → X, OP (1)) be the projective bundle associated to E” to denote the situation where P = P(E) and OP (1) = OP(E) (1). Lemma 29.33.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free OX -module E of rank r. Let (π : P → X, OP (1)) be the projective bundle associated to E. For any α ∈ Ak (X) we the element π∗ (c1 (OP (1))s ∩ π ∗ α) ∈ Ak+r−1−s (X) is 0 if s < r − 1 and is equal to α when s = r − 1. Proof. Let Z ⊂ X be an integral closed subscheme of δ-dimension k. Note that π ∗ [Z] = [π −1 (Z)] as π −1 (Z) is integral of δ-dimension r − 1. If s < r − 1, then by construction c1 (OP (1))s ∩ π ∗ [Z] is represented by a (k + r − 1 − s)-cycle supported on π −1 (Z). Hence the pushforward of this cycle is zero for dimension reasons. Let s = r−1. By the argument given above we see that π∗ (c1 (OP (1))s ∩π ∗ α) = n[Z] for some n ∈ Z. We want to show that n = 1. For the same dimension reasons as above it suffices to prove this result after replacing X by X \ T where T ⊂ Z is a proper closed subset. Let ξ be the generic point of Z. We can choose elements e1 , . . . , er−1 ∈ Eξ which form part of a basis of Eξ . These give rational sections s1 , . . . , sr−1 of OP (1)|π−1 (Z) whose common zero set is the closure of the image a rational section of P(E|Z ) → Z union a closed subset whose support maps to a proper closed subset T of Z. After removing T from X (and correspondingly π −1 (T ) from P ), we see that s1 , . . . , sn form a sequence of global sections si ∈ Γ(π −1 (Z), Oπ−1 (Z) (1)) whose common zero set is the image of a section Z → π −1 (Z). Hence we see succesively that π ∗ [Z]

=

[π −1 (Z)]

∗

=

[Z(s1 )]

2

∗

=

[Z(s1 ) ∩ Z(s2 )]

r−1

∗

c1 (OP (1)) ∩ π [Z] c1 (OP (1)) ∩ π [Z] ... c1 (OP (1))

∩ π [Z]

= ... =

[Z(s1 ) ∩ . . . ∩ Z(sr−1 )]

by repeated applications of Lemma 29.25.3. Since the pushforward by π of the image of a section of π over Z is clearly [Z] we see the result when α = [Z]. We omit P the verification that these arguments imply the result for a general cycle α= nj [Zj ].  Lemma 29.33.2 (Projective space bundle formula). Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free OX module E of rank r. Let (π : P → X, OP (1)) be the projective bundle associated to E. The map Mr−1 Ak+i (X) −→ Ak+r−1 (P ), i=0

(α0 , . . . , αr−1 ) 7−→ π ∗ α0 + c1 (OP (1)) ∩ π ∗ α1 + . . . + c1 (OP (1))r−1 ∩ π ∗ αr−1 is an isomorphism.

1718

29. CHOW HOMOLOGY AND CHERN CLASSES

Proof. Fix k ∈ Z. We first show the map is injective. Suppose that (α0 , . . . , αr−1 ) is an element of the left hand side that maps to zero. By Lemma 29.33.1 we see that 0 = π∗ (π ∗ α0 + c1 (OP (1)) ∩ π ∗ α1 + . . . + c1 (OP (1))r−1 ∩ π ∗ αr−1 ) = αr−1 Next, we see that 0 = π∗ (c1 (OP (1))∩(π ∗ α0 +c1 (OP (1))∩π ∗ α1 +. . .+c1 (OP (1))r−2 ∩π ∗ αr−2 )) = αr−2 and so on. Hence the map is injective. It remains to show the map is surjective. Let Xi , i ∈ I be the irreducible components of X. Then Pi = P(E|Xi ), i ∈ I are the irreducible components of P . If the map is surjective for each of the morphisms Pi → Xi , then the map is surjective for π : P → X. Details omitted. Hence we may assume X is irreducible. Thus dimδ (X) < ∞ and in particular we may use induction on dimδ (X). The result is clear if dimδ (X) L < k. Let α ∈ Ak+r−1 (P ). For any locally closed subscheme T ⊂ X denote γT : Ak+i (T ) → Ak+r−1 (π −1 (T )) the map γT (α0 , . . . , αr−1 ) = π ∗ α0 + . . . + c1 (Oπ−1 (T ) (1))r−1 ∩ π ∗ αr−1 . Suppose for some nonempty open U ⊂ X we have α|π−1 (U ) = γU (α0 , . . . , αr−1 ). 0 Then we may choose lifts αi0 ∈ Ak+i (X) and we see that α − γX (α00 , . . . , αr−1 ) is by Lemma 29.19.2 rationally equivalent to a k-cycle on PY = P(E|Y ) where Y = X \U as a reduced closed subscheme. Note that dimδ (Y ) < dimδ (X). By induction the result holds for PY → Y and hence the result holds for α. Hence we may replace X by any nonempty open of X. ⊕r . In this case P(E) = X × Pr−1 . Let us In particular we may assume that E ∼ = OX use the stratification a a a Pr−1 = Ar−1 Ar−2 ... A0

The closure of each stratum is a Pr−1−i which is a representative of c1 (O(1))i ∩ [Pr−1 ]. Hence P has a similar stratification a a a P = U r−1 U r−2 ... U0 Let P i be the closure of U i . Let π i : P i → X be the restriction of π to P i . Let α ∈ Ak+r−1 (P ). By Lemma 29.32.1 we can write α|U r−1 = π ∗ α0 |U r−1 for some α0 ∈ Ak (X). Hence the difference α − π ∗ α0 is the image of some α0 ∈ Ak+r−1 (P r−2 ). By Lemma 29.32.1 again we can write α0 |U r−2 = (π r−2 )∗ α1 |U r−2 for some α1 ∈ Ak+1 (X). By Lemma 29.31.2 we see that the image of (π r−2 )∗ α1 represents c1 (OP (1)) ∩ π ∗ α1 . We also see that α − π ∗ α0 − c1 (OP (1)) ∩ π ∗ α1 is the image of some α00 ∈ Ak+r−1 (P r−3 ). And so on.  Lemma 29.33.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let p : E = Spec(Sym∗ (E)) −→ X be the associated vector bundle over X. Then p∗ : Ak (X) → Ak+r (E) is an isomorphism for all k.

29.34. THE CHERN CLASSES OF A VECTOR BUNDLE

1719

Proof. For surjectivity see Lemma 29.32.1. Let (π : P → X, OP (1)) be the projective space bundle associated to the finite locally free sheaf E ⊕ OX . Let s ∈ Γ(P, OP (1)) correspond to the global section (0, 1) ∈ Γ(X, E ⊕ OX ). Let D = Z(s) ⊂ P . Note that (π|D : D → X, OP (1)|D ) is the projective space bundle associated to E. We denote πD = π|D and OD (1) = OP (1)|D . Moreover, D is an effective Cartier divisor on P . Hence OP (D) = OP (1) (see Divisors, Lemma 26.9.20). Also there is an isomorphism E ∼ = P \ D. Denote j : E → P the corresponding open immersion. For injectivity we use that the kernel of j ∗ : Ak+r (P ) −→ Ak+r (E) are the cycles supported in the effective Cartier divisor D, see Lemma 29.19.2. So if p∗ α = 0, then π ∗ α = i∗ β for some β ∈ Ak+r (D). By Lemma 29.33.2 we may write ∗ ∗ β = πD β0 + . . . + c1 (OD (1))r−1 ∩ πD βr−1 . for some βi ∈ Ak+i (X). By Lemmas 29.31.2 and 29.25.6 this implies π ∗ α = i∗ β = c1 (OP (1)) ∩ π ∗ β0 + . . . + c1 (OD (1))r ∩ π ∗ βr−1 . Since the rank of E ⊕ OX is r + 1 this contradicts Lemma 29.25.6 unless all α and all βi are zero.  29.34. The Chern classes of a vector bundle We can use the projective space bundle formula to define the chern classes of a rank r vector bundle in terms of the expansion of c1 (O(1))r in terms of the lower powers, see formula (29.34.1.1). The reason for the signs will be explained later. Definition 29.34.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let E be a finite locally free sheaf of rank r on X. Let (π : P → X, OP (1)) be the projective space bundle associated to E. (1) By Lemma 29.33.2 there are elements ci ∈ An−i (X), i = 0, . . . , r such that c0 = [X], and Xr (29.34.1.1) (−1)i c1 (OP (1))i ∩ π ∗ cr−i = 0. i=0

(2) With notation as above we set ci (E) ∩ [X] = ci as an element of An−i (X). We call these the chern classes of E on X. (3) The total chern class of E on X is the combination c(E) ∩ [X] = c0 (E) ∩ [X] + c1 (E) ∩ [X] + . . . + cr (E) ∩ [X] L which is an element of A∗ (X) = k∈Z Ak (X). Let us check that this does not give a new notion in case the vector bundle has rank 1. Lemma 29.34.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Assume X is integral and n = dimδ (X). Let L be an invertible OX module. The first chern class of L on X of Definition 29.34.1 is equal to the Weil divisor associated to L by Definition 29.24.1.

1720

29. CHOW HOMOLOGY AND CHERN CLASSES

Proof. In this proof we use c1 (L) ∩ [X] to denote the construction of Definition 29.24.1. Since L has rank 1 we have P(L) = X and OP(L) (1) = L by our normalizations. Hence (29.34.1.1) reads (−1)1 c1 (L) ∩ c0 + (−1)0 c1 = 0 Since c0 = [X], we conclude c1 = c1 (L) ∩ [X] as desired.



Remark 29.34.3. We could also rewrite equation 29.34.1.1 as Xr (29.34.3.1) c1 (OP (−1))i ∩ π ∗ cr−i = 0. i=0

but we find it easier to work with the tautological quotient sheaf OP (1) instead of its dual. 29.35. Intersecting with chern classes Definition 29.35.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. We define, for every integer k and any 0 ≤ j ≤ r, an operation cj (E) ∩ − : Zk (X) → Ak−j (X) called intersection with the jth chern class of E. (1) Given an integral closed subscheme i : W → X of δ-dimension k we define cj (E) ∩ [W ] = i∗ (cj (i∗ E) ∩ [W ]) ∈ Ak−j (X) where cj (i∗ E) ∩ [W ] is as defined in Definition 29.34.1. P (2) For a general k-cycle α = ni [Wi ] we set X cj (E) ∩ α = ni cj (E) ∩ [Wi ] Again, if E has rank 1 then this agrees with our previous definition. Lemma 29.35.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let (π : P → X, OP (1)) be the projective bundle associated to E. For α ∈ Zk (X) the elements cj (E) ∩ α are the unique elements αj of Ak−j (X) such that α0 = α and Xr (−1)i c1 (OP (1))i ∩ π ∗ (αr−i ) = 0 i=0

holds in the Chow group of P . Proof. The uniqueness of α0 , . . . , αr such that α0 = α and such that the displayed equation holds follows from the projective space bundle formula Lemma 29.33.2. The identity holds by definition for α = [X]. For a general k-cycle α on X write P α = na [Wa ] with na 6= 0, and ia : Wa → X pairwise distinct integral closed subschemes. Then the family {Wa } is locally finite on X. Set Pa = π −1 (Wa ) = P(E|Wa ). Denote i0a : Pa → P the corresponding closed immersions. Consider the fibre product diagram ` /P P0 Pa 0 ia

π

0

 X0

πa

` Wa

π ia

 /X

29.35. INTERSECTING WITH CHERN CLASSES

1721

The morphism p : X 0 → X ` is proper. Moreover π 0 : P 0 → X 0 together with the invertible sheaf OP 0 (1) = OPa (1) which is also the pullback of OP (1) is the projective bundle associated to E 0 = p∗ E. By definition X cj (E) ∩ [α] = ia,∗ (cj (E|Wa ) ∩ [Wa ]). Write βa,j = cj (E|Wa ) ∩ [Wa ] which is an element of Ak−j (Wa ). We have Xr (−1)i c1 (OPa (1))i ∩ πa∗ (βa,r−i ) = 0 i=0

for each a by definition. Thus clearly we have Xr (−1)i c1 (OP 0 (1))i ∩ (π 0 )∗ (βr−i ) = 0 i=0 P ` 0 with βj = na βa,j ∈ Ak−j (X 0 ). Denote p0 : P 0 → P the morphism ia . We have π ∗ p∗ βj = p0∗ (π 0 )∗ βj by Lemma 29.15.1. By the projection formula of Lemma 29.25.6 we conclude that Xr (−1)i c1 (OP (1))i ∩ π ∗ (p∗ βj ) = 0 i=0

Since p∗ βj is a representative of cj (E) ∩ α we win.



This characterization of chern classes allows us to prove many more properties. Lemma 29.35.3. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. If α ∼rat β are rationally equivalent k-cycles on X then cj (E) ∩ α = cj (E) ∩ β in Ak−j (X). Proof. By Lemma 29.35.2 the elements αj = cj (E) ∩ α, j ≥ 1 and βj = cj (E) ∩ β, j ≥ 1 are uniquely determined by the same equation in the chow group of the projective bundle associated to E. (This of course relies on the fact that flat pullback is compatible with rational equivalence, see Lemma 29.20.1.) Hence they are equal.  In other words capping with chern classes of finite locally free sheaves factors through rational equivalence to give maps cj (E) ∩ − : Ak (X) → Ak−j (X). Lemma 29.35.4. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let E be a finite locally free sheaf of rank r on Y . Let f : X → Y be a flat morphism of relative dimension r. Let α be a k-cycle on Y . Then f ∗ (cj (E) ∩ α) = cj (f ∗ E) ∩ f ∗ α Proof. Write αj = cj (E) ∩ α, so α0 = α. By Lemma 29.35.2 we have Xr (−1)i c1 (OP (1))i ∩ π ∗ (αr−i ) = 0 i=0

in the chow group of the projective bundle (π : P → Y, OP (1)) associated to E. Consider the fibre product diagram PX = P(f ∗ E) πX

 X

f

fP

/P  /Y

π

1722

29. CHOW HOMOLOGY AND CHERN CLASSES

Note that OPX (1) = fP∗ OP (1). By Lemmas 29.25.4 and 29.14.3 we see that Xr ∗ (−1)i c1 (OPX (1))i ∩ πX (f ∗ αr−i ) = 0 i=0

holds in the chow group of PX . Since f ∗ α0 = f ∗ α the lemma follows from the uniqueness in Lemma 29.35.2.  Lemma 29.35.5. Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let p : X → Y be a proper morphism. Let α be a k-cycle on X. Let E be a finite locally free sheaf on Y . Then p∗ (cj (p∗ E) ∩ α) = cj (E) ∩ p∗ α Proof. Write αj = cj (p∗ E) ∩ α, so α0 = α. By Lemma 29.35.2 we have Xr i=0

∗ (−1)i c1 (OP (1))i ∩ πX (αr−i ) = 0

in the chow group of the projective bundle (πX : PX → X, OPX (1)) associated to p∗ E. Let (π : P → Y, OP (1)) be the projective bundle associated to E. Consider the fibre product diagram PX = P(p∗ E)

pP

πX

 X

p

/P  /Y

π

Note that OPX (1) = p∗P OP (1). Pushing the displayed equality above to P and using Lemmas 29.15.1, 29.25.6 and 29.14.3 we see that Xr (−1)i c1 (OP (1))i ∩ π ∗ (p∗ αr−i ) = 0 i=0

holds in the chow group of P . Since p∗ α0 = p∗ α the lemma follows from the uniqueness in Lemma 29.35.2.  Lemma 29.35.6. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E, F be finite locally free sheaves on X of ranks r and s. For any α ∈ Ak (X) we have ci (E) ∩ cj (F) ∩ α = cj (F) ∩ ci (E) ∩ α as elements of Ak−i−j (X). Proof. Consider π : P(E) ×X P(F) −→ X with invertible sheaves L = pr∗1 OP(E) (1) and N = pr∗2 OP(F ) (1). Write αi,j for the left hand side and βi,j for the right hand side. Also write αj = cj (F) ∩ α and βi = ci (E) ∩ α. In particular this means that α0 = α = β0 , and α0,j = αj = β0,j , αi,0 = βi = βi,0 . From Lemma 29.35.2 (pulled back to the space above using

29.36. POLYNOMIAL RELATIONS AMONG CHERN CLASSES

1723

Lemma 29.25.4 for the first two) we see that X 0 = (−1)j c1 (N )j ∩ π ∗ αs−j j=0,...,s X 0 = (−1)i c1 (L)i ∩ π ∗ βr−i i=0,...,r X 0 = (−1)i c1 (L)i ∩ π ∗ αr−i,s−j i=0,...,r X 0 = (−1)j c1 (N )j ∩ π ∗ βr−i,s−j j=0,...,s

We can combine the first and the third of these to get (−1)r+s c1 (L)r ∩ c1 (N )s ∩ π ∗ α X = (−1)r+j−1 c1 (L)r ∩ c1 (N )j ∩ π ∗ αs−j j=1,...,s X (−1)j−1+r c1 (N )j ∩ c1 (L)r ∩ π ∗ α0,s−j = j=1,...,s Xs Xr = (−1)i+j c1 (N )j ∩ c1 (L)i ∩ π ∗ αr−i,s−j j=1

i=1

using that capping with c1 (L) commutes with capping with c1 (N ). In exactly the same way one shows that Xs Xr (−1)r+s c1 (L)r ∩ c1 (N )s ∩ π ∗ α = (−1)i+j c1 (N )j ∩ c1 (L)i ∩ π ∗ βr−i,s−j j=1

i=1

By the projective space bundle formula Lemma 29.33.2 applied twice these representations are unique. Whence the result.  29.36. Polynomial relations among chern classes Definition 29.36.1. Let P (xi,j ) ∈ Z[xi,j ] be a polynomial. We write P as a finite sum X X aI xi1 ,j1 . . . xis ,js . s

I=((i1 ,j1 ),(i2 ,j2 ),...,(is ,js ))

Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let Ei be a finite collection of finite locally free sheaves on X. We say that P is a polynomial relation among the chern classes and we write P (cj (Ei )) = 0 if for any morphism f : Y → X of an integral scheme locally of finite type over S the cycle X X aI cj1 (f ∗ Ei1 ) ∩ . . . ∩ cjs (f ∗ Eis ) ∩ [Y ] s

I=((i1 ,j1 ),(i2 ,j2 ),...,(is ,js ))

is zero in A∗ (Y ). This is not an elegant definition but it will do for now. It makes sense because we showed in Lemma 29.35.6 that capping with chern classes of vector bundles is commutative. By our definitions and results above this is equivalent with requiring all the operations X X aI cj1 (f ∗ Ei1 ) ∩ . . . ∩ cjs (f ∗ Eis ) ∩ − : A∗ (Y ) → A∗ (Y ) s

I

to be zero for all morphisms f : Y → X which are locally of finite type. An example of such a relation is the relation c1 (L ⊗OX N ) = c1 (L) + c1 (N ) proved in Lemma 29.25.2. More generally, here is what happens when we tensor an arbitrary locally free sheaf by an invertible sheaf.

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Lemma 29.36.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free sheaf of rank r on X. Let L be an invertible sheaf on X. Then Xi  r − i + j  (29.36.2.1) ci (E ⊗ L) = ci−j (E)c1 (L)j j=0 j is a valid polynomial relation in the sense described above. Proof. This should hold for any triple (X, E, L). In particular it should hold when X is integral, and in fact by definition of a polynomial relation it is enough to prove it holds when capping with [X] for such X. Thus assume that X is integral. Let (π : P → X, OP (1)), resp. (π 0 : P 0 → X, OP 0 (1)) be the projective space bundle associated to E, resp. E ⊗ L. Consider the canonical morphsm P

/ P0

g π

X

~

π0

see Constructions, Lemma 22.20.1. It has the property that g ∗ OP 0 (1) = OP (1) ⊗ π ∗ L. This means that we have Xr (−1)i (ξ + x)i ∩ π ∗ (cr−i (E ⊗ L) ∩ [X]) = 0 i=0

in A∗ (P ), where ξ represents c1 (OP (1)) and x represents c1 (π ∗ L). By simple algebra this is equivalent to X    Xr r j j−i (−1)i ξ i (−1)j−i x ∩ π ∗ (cr−j (E ⊗ L) ∩ [X]) = 0 i=0 j=i i Comparing with Equation (29.34.1.1) it follows from this that Xr j  cr−i (E) ∩ [X] = (−c1 (L))j−i ∩ cr−j (E ⊗ L) ∩ [X] j=i i Reworking this (getting rid of minus signs, and renumbering) we get the desired relation.  Some example cases of (29.36.2.1) are c1 (E ⊗ L) = c1 (E) + rc1 (L)   r c1 (L)2 2     r−1 r c3 (E ⊗ L) = c3 (E) + (r − 2)c2 (E)c1 (L) + c1 (E)c1 (L)2 + c1 (L)3 2 3 c2 (E ⊗ L) = c2 (E) + (r − 1)c1 (E)c1 (L) +

29.37. Additivity of chern classes All of the preliminary lemmas follow trivially from the final result. Lemma 29.37.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E, F be finite locally free sheaves on X of ranks r, r − 1 which fit into a short exact sequence 0 → OX → E → F → 0

29.37. ADDITIVITY OF CHERN CLASSES

1725

Then cr (E) = 0, cj (E) = cj (F), j = 0, . . . , r − 1 are valid polynomial relations among chern classes. Proof. By Definition 29.36.1 it suffices to show that if X is integral then cj (E) ∩ [X] = cj (F) ∩ [X]. Let (π : P → X, OP (1)), resp. (π 0 : P 0 → X, OP 0 (1)) denote the projective space bundle associated to E, resp. F. The surjection E → F gives rise to a closed immersion i : P 0 −→ P over X. Moreover, the element 1 ∈ Γ(X, OX ) ⊂ Γ(X, E) gives rise to a global section s ∈ Γ(P, OP (1)) whose zero set is exactly P 0 . Hence P 0 is an effective Cartier divisor on P such that OP (P 0 ) ∼ = OP (1). Hence we see that c1 (OP (1)) ∩ π ∗ α = i∗ ((π 0 )∗ α) for any cycle class α on X by Lemma 29.31.2. By Lemma 29.35.2 we see that αj = cj (F) ∩ [X], j = 0, . . . , r − 1 satisfy Xr−1 (−1)j c1 (OP 0 (1))j ∩ (π 0 )∗ αj = 0 j=0

Pushing this to P and using the remark above as well as Lemma 29.25.6 we get Xr−1 (−1)j c1 (OP (1))j+1 ∩ π ∗ αj = 0 j=0

By the uniqueness of Lemma 29.35.2 we conclude that cr (E) ∩ [X] = 0 and cj (E) ∩ [X] = αj = cj (F) ∩ [X] for j = 0, . . . , r − 1. Hence the lemma holds.  Lemma 29.37.2. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E, F be finite locally free sheaves on X of ranks r, r − 1 which fit into a short exact sequence 0→L→E →F →0 where L is an invertible sheaf Then c(E) = c(L)c(F) is a valid polynomial relation among chern classes. Proof. This relation really just says that ci (E) = ci (F)+c1 (L)ci−1 (F). By Lemma 29.37.1 we have cj (E ⊗L⊗−1 ) = cj (E ⊗L⊗−1 ) for j = 0, . . . , r (were we set cr (F) = 0 by convention). Applying Lemma 29.36.2 we deduce   i  i  X X r−i+j r−1−i+j j j (−1) ci−j (E)c1 (L) = (−1)j ci−j (F)c1 (L)j j j j=0 j=0 Setting ci (E) = ci (F) + c1 (L)ci−1 (F) gives a “solution” of this equation. The lemma follows if we show that this is the only possible solution. We omit the verification.  Lemma 29.37.3. Let (S, δ) be as in Situation 29.7.1. Let X be a scheme locally of finite type over S. Suppose that E sits in an exact sequence 0 → E1 → E → E2 → 0 of finite locally free sheaves Ei of rank ri . Then c(E) = c(E1 )c(E2 )

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29. CHOW HOMOLOGY AND CHERN CLASSES

is a polynomial relation among chern classes. Proof. We may assume that X is integral and we have to show the identity when capping against [X]. By induction on r1 . The case r1 = 1 is Lemma 29.37.2. Assuem r1 > 1. Let (π : P → X, OP (1)) denote the projective space bundle associated to E1 . Note that (1) π ∗ : A∗ (X) → A∗ (P ) is injective, and (2) π ∗ E1 sits in a short exact sequence 0 → F → π ∗ E1 → L → 0 where L is invertible. The first assertion follows from the projective space bundle formula and the second follows from the definition of a projective space bundle. (In fact L = OP (1).) Let Q = π ∗ E/F, which sits in an exact sequence 0 → L → Q → π ∗ E2 → 0. By induction we have c(π ∗ E) ∩ [P ]

= c(F) ∩ c(π ∗ E/F) ∩ [P ] = c(F) ∩ c(L) ∩ c(π ∗ E2 ) ∩ [P ] = c(π ∗ E1 ) ∩ c(π ∗ E2 ) ∩ [P ]

Since [P ] = π ∗ [X] we win by Lemma 29.35.4.



Lemma 29.37.4. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let Li , i = 1, . . . , r be invertible OX -modules on X. Let E be a finite locally free rank r OX -module endowed with a filtration 0 = E0 ⊂ E1 ⊂ E2 ⊂ . . . ⊂ Er = E such that Ei /Ei−1 ∼ = Li . Set c1 (Li ) = xi . Then Yr c(E) = (1 + xi ) i=1

is a valid polynomial relation among chern classes in the sense of Definition 29.36.1. Proof. Apply Lemma 29.37.2 and induction.



29.38. The splitting principle In our setting it is not so easy to say what the splitting principle exactly says/is. Here is a possible formulation. Lemma 29.38.1. Let (S, δ) be as in Situation 29.7.1. Let X be locally of finite type over S. Let E be a finite locally free sheaf E on X of rank r. There exists a projective flat morphism of relative dimension d π : P → X such that (1) for any morphism f : Y → X the map πY∗ : A∗ (Y ) → A∗+r (Y ×X P ) is injective, and (2) π ∗ E has a filtration with succesive quotients L1 , . . . , Lr for some invertible OP -modules Li . Proof. Omitted. Hint: Use a composition of projective space bundles, i.e., a flag variety over X.  The splitting principle refers to the practice of symbolically writing Y c(E) = (1 + xi )

29.39. CHERN CLASSES AND TENSOR PRODUCT

1727

with xi = c1 (Li ). The expressions xi are then called the Chern roots of E. In order to prove polynomial relations amoing chern classes of vector bundles it is permissible to do calculations using the chern roots. For example, let us calculate the chern classes of the dual vector bundle E ∧ . Note that if E has a filtration with subquotients invertible sheaves Li then E ∧ has a filtration with subquotients the invertible sheaves L−1 i . Hence if xi are the chern roots of E, then the −xi are the chern roots of E ∧ . It follows that cj (E ∧ ) = (−1)j cj (E) is a valid polynomial relation among chern classes. In the same vain, let us compute the chern classes of a tensor product of vector bundles. Namely, suppose that E, F are finite locally free of ranks r, s. Write Yr Ys c(E) = (1 + xi ), c(E) = (1 + yj ) i=1

j=1

where xi , yj are the chern roots of E, F. Then we see that Y c(E ⊗OX F) = (1 + xi + yj ) i,j

Here are some examples of what this means in terms of chern classes c1 (E ⊗ F) = rc1 (F) + sc1 (E) c2 (E ⊗ F) = r2 c2 (F) + rsc1 (F)c1 (E) + s2 c2 (E) 29.39. Chern classes and tensor product We define the Chern character of a finite locally free sheaf of rank r to be the formal expression Xr ch(E) := exi i=1

if the xi are the chern roots of E. Writing this in terms of chern classes ci = ci (E) we see that 1 1 1 ch(E) = r+c1 + (c21 −2c2 )+ (c31 −3c1 c2 +3c3 )+ (c41 −4c21 c2 +4c1 c3 +2c22 −4c4 )+. . . 2 6 24 What does it mean that the coefficients are rational numbers? Well this simply means that we think of these as operations chj (E) ∩ − : Ak (X) −→ Ak−j (X) ⊗Z Q and we think of polynomial relations among them as relations between these operations with values in the groups Ak−j (Y ) ⊗Z Q for varying Y . By the above we have in case of an exact sequence 0 → E1 → E → E2 → 0 that ch(E) = ch(E1 ) + ch(E2 ) Using the Chern character we can express the compatibility of the chern classes and tensor product as follows: ch(E1 ⊗OX E2 ) = ch(E1 )ch(E2 ) This follows directly from the discussion of the chern roots of the tensor product in the previous section.

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29.40. Todd classes A final class associated to a vector bundle E of rank r is its Todd class T odd(E). In terms of the chern roots x1 , . . . , xr it is defined as Yr xi T odd(E) = i=1 1 − e−xi In terms of the chern classes ci = ci (E) we have 1 1 1 1 T odd(E) = 1 + c1 + (c21 + c2 ) + c1 c2 + (−c41 + 4c21 c2 + 3c22 + c1 c3 − c4 ) + . . . 2 12 24 720 We have made the appropriate remaks about denominators in the previous section. It is the case that given an exact sequence 0 → E1 → E → E2 → 0 we have T odd(E) = T odd(E1 )T odd(E2 ). 29.41. Grothendieck-Riemann-Roch Let (S, δ) be as in Situation 29.7.1. Let X, Y be locally of finite type over S. Let E be a finite locally free sheaf E on X of rank r. Let f : X → Y be a proper smooth morphism. Assume that Ri f∗ E are locally free sheaves on Y of finite rank (for example if Y is a point). The Grothendieck-Riemann-Roch theorem implies that in this case we have X f∗ (T odd(TX/Y )ch(E)) = (−1)i ch(Ri f∗ E) Here TX/Y = Hom OX (ΩX/Y , OX ) is the relative tangent bundle of X over Y . The theorem is more general and becomes easier to prove when formulated in correct generality. We will return to this elsewhere (insert future reference here). 29.42. Other chapters (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)

Introduction Conventions Set Theory Categories Topology Sheaves on Spaces Commutative Algebra Brauer Groups Sites and Sheaves Homological Algebra Derived Categories More on Algebra Smoothing Ring Maps Simplicial Methods Sheaves of Modules Modules on Sites Injectives

(18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)

Cohomology of Sheaves Cohomology on Sites Hypercoverings Schemes Constructions of Schemes Properties of Schemes Morphisms of Schemes Cohomology of Schemes Divisors Limits of Schemes Varieties Chow Homology Topologies on Schemes Descent Adequate Modules More on Morphisms More on Flatness

29.42. OTHER CHAPTERS

(35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52) (53) (54)

Groupoid Schemes More on Groupoid Schemes ´ Etale Morphisms of Schemes ´ Etale Cohomology Crystalline Cohomology Algebraic Spaces Properties of Algebraic Spaces Morphisms of Algebraic Spaces Decent Algebraic Spaces Cohomology of Algebraic Spaces Limits of Algebraic Spaces Topologies on Algebraic Spaces Descent and Algebraic Spaces More on Morphisms of Spaces Quot and Hilbert Spaces Spaces over Fields Stacks Formal Deformation Theory Groupoids in Algebraic Spaces More on Groupoids in Spaces

(55) (56) (57) (58) (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73)

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Bootstrap Examples of Stacks Quotients of Groupoids Algebraic Stacks Sheaves on Algebraic Stacks Criteria for Representability Artin’s Axioms Properties of Algebraic Stacks Morphisms of Algebraic Stacks Cohomology of Algebraic Stacks Introducing Algebraic Stacks Examples Exercises Guide to Literature Desirables Coding Style Obsolete GNU Free Documentation License Auto Generated Index

CHAPTER 30

Topologies on Schemes 30.1. Introduction In this document we explain what the different topologies on the category of schemes are. Some references are [Gro71] and [BLR90]. Before doing so we would like to point out that there are many different choices of sites (as defined in Sites, Definition 9.6.2) which give rise to the same notion of sheaf on the underlying category. Hence our choices may be slightly different from those in the references but ultimately lead to the same cohomology groups, etc.

30.2. The general procedure In this section we explain a general procedure for producing the sites we will be working with. Suppose we want to study sheaves over schemes with respect to some topology τ . In order to get a site, as in Sites, Definition 9.6.2, of schemes with that topology we have to do some work. Namely, we cannot simply say “consider all schemes with the Zariski topology” since that would give a “big” category. Instead, in each section of this chapter we will proceed as follows: (1) We define a class Covτ of coverings of schemes satisfying the axioms of Sites, Definition 9.6.2. It will always be the case that a Zariski open covering of a scheme is a covering for τ . (2) We single out a notion of standard τ -covering within the category of affine schemes. (3) We define what is an “absolute” big τ -site Schτ . These are the sites one gets by appropriately choosing a set of schemes and a set of coverings. (4) For any object S of Schτ we define the big τ -site (Sch/S)τ and for suitable τ the small1 τ -site Sτ . (5) In addition there is a site (Aff/S)τ using the notion of standard τ -covering of affines whose category of sheaves is equivalent to the category of sheaves on (Sch/S)τ . The above is a little clumsy in that we do not end up with a canonical choice for the big τ -site of a scheme, or even the small τ -site of a scheme. If you are willing to ignore set theoretic difficulties, then you can work with classes and end up with canonical big and small sites...

1The words big and small here do not relate to bigness/smallness of the corresponding categories. 1731

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30.3. The Zariski topology Definition 30.3.1. Let T be a scheme. A Zariski covering of T is a family of morphisms {fi S : Ti → T }i∈I of schemes such that each fi is an open immersion and such that T = fi (Ti ). This defines a (proper) class of coverings. Next, we show that this notion satisfies the conditions of Sites, Definition 9.6.2. Lemma 30.3.2. Let T be a scheme. (1) If T 0 → T is an isomorphism then {T 0 → T } is a Zariski covering of T . (2) If {Ti → T }i∈I is a Zariski covering and for each i we have a Zariski covering {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is a Zariski covering. (3) If {Ti → T }i∈I is a Zariski covering and T 0 → T is a morphism of schemes then {T 0 ×T Ti → T 0 }i∈I is a Zariski covering. Proof. Omitted.



Lemma 30.3.3. Let T be an affine scheme. Let {Ti → T }i∈I be a Zariski covering of T . Then there exists a Zariski covering {Uj → T }j=1,...,m which is a refinement of {Ti → T }i∈I such that each Uj is a standard open of T , see Schemes, Definition 21.5.2. Moreover, we may choose each Uj to be an open of one of the Ti . Proof. Follows as T is quasi-compact and standard opens form a basis for its topology. This is also proved in Schemes, Lemma 21.5.1.  Thus we define the corresponding standard coverings of affines as follows. Definition 30.3.4. Compare Schemes, Definition 21.5.2. Let T be an affine scheme. A standard Zariski covering of T is a a Zariski covering {Uj → T }j=1,...,m with each Uj → T inducing an isomorphism with a standard affine open of T . Definition 30.3.5. A big Zariski site is any site SchZar as in Sites, Definition 9.6.2 constructed as follows: (1) Choose any set of schemes S0 , and any set of Zariski coverings Cov0 among these schemes. (2) As underlying category of SchZar take any category Schα constructed as in Sets, Lemma 3.9.2 starting with the set S0 . (3) As coverings of SchZar choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category Schα and the class of Zariski coverings, and the set Cov0 chosen above. It is shown in Sites, Lemma 9.8.6 that, after having chosen the category Schα , the category of sheaves on Schα does not depend on the choice of coverings chosen in (3) above. In other words, the topos Sh(SchZar ) only depends on the choice of the category Schα . It is shown in Sets, Lemma 3.9.9 that these categories are closed under many constructions of algebraic geometry, e.g., fibre products and taking open and closed subschemes. We can also show that the exact choice of Schα does not matter too much, see Section 30.10. Another approach would be to assume the existence of a strongly inaccessible cardinal and to define SchZar to be the category of schemes contained in a chosen universe with set of coverings the Zariski coverings contained in that same universe.

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Before we continue with the introduction of the big Zariski site of a scheme S, let us point out that the topology on a big Zariski site SchZar is in some sense induced from the Zariski topology on the category of all schemes. Lemma 30.3.6. Let SchZar be a big Zariski site as in Definition 30.3.5. Let T ∈ Ob(SchZar ). Let {Ti → T }i∈I be an arbitrary Zariski covering of T . There exists a covering {Uj → T }j∈J of T in the site SchZar which is tautologically equivalent (see Sites, Definition 9.8.2) to {Ti → T }i∈I Proof. Since each Ti → T is an open immersion, we see by Sets, Lemma 3.9.9 that each Ti is isomorphic to an object Vi of SchZar . The covering {Vi → T }i∈I is tautologically equivalent to {Ti → T }i∈I (using the identity map on I both ways). Moreover, {Vi → T }i∈I is combinatorially equivalent to a covering {Uj → T }j∈J of T in the site SchZar by Sets, Lemma 3.11.1.  Definition 30.3.7. Let S be a scheme. Let SchZar be a big Zariski site containing S. (1) The big Zariski site of S, denoted (Sch/S)Zar , is the site SchZar /S introduced in Sites, Section 9.21. (2) The small Zariski site of S, which we denote SZar , is the full subcategory of (Sch/S)Zar whose objects are those U/S such that U → S is an open immersion. A covering of SZar is any covering {Ui → U } of (Sch/S)Zar with U ∈ Ob(SZar ). (3) The big affine Zariski site of S, denoted (Aff/S)Zar , is the full subcategory of (Sch/S)Zar whose objects are affine U/S. A covering of (Aff/S)Zar is any covering {Ui → U } of (Sch/S)Zar which is a standard Zariski covering. It is not completely clear that the small Zariski site and the big affine Zariski site are sites. We check this now. Lemma 30.3.8. Let S be a scheme. Let SchZar be a big Zariski site containing S. Both SZar and (Aff/S)Zar are sites. Proof. Let us show that SZar is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 9.6.2. Since (Sch/S)Zar is a site, it suffices to prove that given any covering {Ui → U } of (Sch/S)Zar with U ∈ Ob(SZar ) we also have Ui ∈ Ob(SZar ). This follows from the definitions as the composition of open immersions is an open immersion. Let us show that (Aff/S)Zar is a site. Reasoning as above, it suffices to show that the collection of standard Zariski coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 9.6.2. Let R be a ring. Let f1 , . . . , fn ∈ R generate the unit ideal. For each i ∈ {1, . . . , n} let gi1 , . . . , gini ∈ Rfi be elements generating the unit e ideal of Rfi . Write gij = fij /fi ij which is possible. After replacing fij by fi fij if necessary, we have that D(fij ) ⊂ D(fi ) ∼ = Spec(Rfi ) is equal to D(gij ) ⊂ Spec(Rfi ). Hence we see that the family of morphisms {D(gij ) → Spec(R)} is a standard Zariski covering. From these considerations it follows that (2) holds for standard Zariski coverings. We omit the verification of (1) and (3).  Lemma 30.3.9. Let S be a scheme. Let SchZar be a big Zariski site containing S. The underlying categories of the sites SchZar , (Sch/S)Zar , SZar , and (Aff/S)Zar have fibre products. In each case the obvious functor into the category Sch of all

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30. TOPOLOGIES ON SCHEMES

schemes commutes with taking fibre products. The categories (Sch/S)Zar , and SZar both have a final object, namely S/S. Proof. For SchZar it is true by construction, see Sets, Lemma 3.9.9. Suppose we have U → S, V → U , W → U morphisms of schemes with U, V, W ∈ Ob(SchZar ). The fibre product V ×U W in SchZar is a fibre product in Sch and is the fibre product of V /S with W/S over U/S in the category of all schemes over S, and hence also a fibre product in (Sch/S)Zar . This proves the result for (Sch/S)Zar . If U → S, V → U and W → U are open immersions then so is V ×U W → S and hence we get the result for SZar . If U, V, W are affine, so is V ×U W and hence the result for (Aff/S)Zar .  Next, we check that the big affine site defines the same topos as the big site. Lemma 30.3.10. Let S be a scheme. Let SchZar be a big Zariski site containing S. The functor (Aff/S)Zar → (Sch/S)Zar is a special cocontinuous functor. Hence it induces an equivalence of topoi from Sh((Aff/S)Zar ) to Sh((Sch/S)Zar ). Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 9.25.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 9.25.1. Denote the inclusion functor u : (Aff/S)Zar → (Sch/S)Zar . Being cocontinuous just means that any Zariski covering of T /S, T affine, can be refined by a standard Zariski covering of T . This is the content of Lemma 30.3.3. Hence (1) holds. We see u is continuous simply because a standard Zariski covering is a Zariski covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that u is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering.  Let us check that the notion of a sheaf on the small Zariski site corresponds to notion of a sheaf on S. Lemma 30.3.11. The category of sheaves on SZar is equivalent to the category of sheaves on the underlying topological space of S. Proof. We will use repeatedly that for any object U/S of SZar the morphism U → S is an isomorphism onto an open subscheme. Let F be a sheaf on S. Then we define a sheaf on SZar by the rule F 0 (U/S) = F(Im(U → S)). For the converse, we choose for every open subscheme U ⊂ S an object U 0 /S ∈ Ob(SZar ) with Im(U 0 → S) = U (here you have to use Sets, Lemma 3.9.9). Given a sheaf G 0 0 on SZar we define a sheaf on S by setting G(U S ) = G(U /S). To see that G is a sheaf we use that for any open covering U = i∈I Ui the covering {Ui → U }i∈I is combinatorially equivalent to a covering {Uj0 → U 0 }j∈J in SZar by Sets, Lemma 3.11.1, and we use Sites, Lemma 9.8.4. Details omitted.  From now on we will not make any distinction between a sheaf on SZar or a sheaf on S. We will always use the procedures of the proof of the lemma to go between the two notions. Next, we esthablish some relationships between the topoi associated to these sites. Lemma 30.3.12. Let SchZar be a big Zariski site. Let f : T → S be a morphism in SchZar . The functor TZar → (Sch/S)Zar is cocontinuous and induces a morphism of topoi if : Sh(TZar ) −→ Sh((Sch/S)Zar )

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For a sheaf G on (Sch/S)Zar we have the formula (i−1 f G)(U/T ) = G(U/S). The −1 functor if also has a left adjoint if,! which commutes with fibre products and equalizers. Proof. Denote the functor u : TZar → (Sch/S)Zar . In other words, given and open immersion j : U → T corresponding to an object of TZar we set u(U → T ) = (f ◦ j : U → S). This functor commutes with fibre products, see Lemma 30.3.9. Moreover, TZar has equalizers (as any two morphisms with the same source and target are the same) and u commutes with them. It is clearly cocontinuous. It is also continuous as u transforms coverings to coverings and commutes with fibre products. Hence the lemma follows from Sites, Lemmas 9.19.5 and 9.19.6.  Lemma 30.3.13. Let S be a scheme. Let SchZar be a big Zariski site containing S. The inclusion functor SZar → (Sch/S)Zar satisfies the hypotheses of Sites, Lemma 9.19.8 and hence induces a morphism of sites πS : (Sch/S)Zar −→ SZar and a morphism of topoi iS : Sh(SZar ) −→ Sh((Sch/S)Zar ) such that πS ◦ iS = id. Moreover, iS = iidS with iidS as in Lemma 30.3.12. In −1 particular the functor i−1 S = πS,∗ is described by the rule iS (G)(U/S) = G(U/S). Proof. In this case the functor u : SZar → (Sch/S)Zar , in addition to the properties seen in the proof of Lemma 30.3.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows.  Definition 30.3.14. In the situation of Lemma 30.3.13 the functor i−1 S = πS,∗ is often called the restriction to the small Zariski site, and for a sheaf F on the big Zariski site we denote F|SZar this restriction. With this notation in place we have for a sheaf F on the big site and a sheaf G on the big site that MorSh(SZar ) (F|SZar , G) = MorSh((Sch/S)Zar ) (F, iS,∗ G) MorSh(SZar ) (G, F|SZar ) = MorSh((Sch/S)Zar ) (πS−1 G, F) Moreover, we have (iS,∗ G)|SZar = G and we have (πS−1 G)|SZar = G. Lemma 30.3.15. Let SchZar be a big Zariski site. Let f : T → S be a morphism in SchZar . The functor u : (Sch/T )Zar −→ (Sch/S)Zar ,

V /T 7−→ V /S

is cocontinuous, and has a continuous right adjoint v : (Sch/S)Zar −→ (Sch/T )Zar ,

(U → S) 7−→ (U ×S T → T ).

They induce the same morphism of topoi fbig : Sh((Sch/T )Zar ) −→ Sh((Sch/S)Zar ) −1 fbig (G)(U/T )

We have = G(U/S). We have fbig,∗ (F)(U/S) = F(U ×S T /T ). Also, −1 fbig has a left adjoint fbig! which commutes with fibre products and equalizers.

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Proof. The functor u is cocontinuous, continuous, and commutes with fibre products and equalizers (details omitted; compare with proof of Lemma 30.3.12). Hence −1 Sites, Lemmas 9.19.5 and 9.19.6 apply and we deduce the formula for fbig and the existence of fbig! . Moreover, the functor v is a right adjoint because given U/T and V /S we have MorS (u(U ), V ) = MorT (U, V ×S T ) as desired. Thus we may apply Sites, Lemmas 9.20.1 and 9.20.2 to get the formula for fbig,∗ .  Lemma 30.3.16. Let SchZar be a big Zariski site. Let f : T → S be a morphism in SchZar . (1) We have if = fbig ◦ iT with if as in Lemma 30.3.12 and iT as in Lemma 30.3.13. (2) The functor SZar → TZar , (U → S) 7→ (U ×S T → T ) is continuous and induces a morphism of topoi fsmall : Sh(TZar ) −→ Sh(SZar ). −1 fsmall

The functors and fsmall,∗ agree with the usual notions f −1 and f∗ is we identify sheaves on TZar , resp. SZar with sheaves on T , resp. S via Lemma 30.3.11. (3) We have a commutative diagram of morphisms of sites TZar o fsmall



SZar o

πT

(Sch/T )Zar fbig

πS

 (Sch/S)Zar

so that fsmall ◦ πT = πS ◦ fbig as morphisms of topoi. (4) We have fsmall = πS ◦ fbig ◦ iT = πS ◦ if . −1 −1 Proof. The equality if = fbig ◦ iT follows from the equality i−1 f = iT ◦ fbig which is clear from the descriptions of these functors above. Thus we see (1).

Statement (2): See Sites, Example 9.14.2. Part (3) follows because πS and πT are given by the inclusion functors and fsmall and fbig by the base change functor U 7→ U ×S T . Statement (4) follows from (3) by precomposing with iT .



In the situation of the lemma, using the terminology of Definition 30.3.14 we have: for F a sheaf on the big Zariski site of T (fbig,∗ F)|SZar = fsmall,∗ (F|TZar ), This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small Zariski site of T , resp. S is given by πT,∗ , resp. πS,∗ . A similar formula involving pullbacks and restrictions is false. Lemma 30.3.17. Given schemes X, Y , Y in (Sch/S)Zar and morphisms f : X → Y , g : Y → Z we have gbig ◦ fbig = (g ◦ f )big and gsmall ◦ fsmall = (g ◦ f )small . Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 30.3.15. For the functors on the small sites this is Sheaves, Lemma 6.21.2 via the identification of Lemma 30.3.11. 

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We can think about a sheaf on the big Zariski site of S as a collection of “usual” sheaves on all schemes over S. Lemma 30.3.18. Let S be a scheme contained in a big Zariski site SchZar . A sheaf F on the big Zariski site (Sch/S)Zar is given by the following data: (1) for every T /S ∈ Ob((Sch/S)Zar ) a sheaf FT on T , (2) for every f : T 0 → T in (Sch/S)Zar a map cf : f −1 FT → FT 0 . These data are subject to the following conditions: (i) given any f : T 0 → T and g : T 00 → T 0 in (Sch/S)Zar the composition g −1 cf ◦ cg is equal to cf ◦g , and (ii) if f : T 0 → T in (Sch/S)Zar is an open immersion then cf is an isomorphism. Proof. Given a sheaf F on Sh((Sch/S)Zar ) we set FT = i−1 p F where p : T → S is the structure morphism. Note that FT (U ) = F(U 0 /S) for any open U ⊂ T , and U 0 → T an open immersion in (Sch/T )Zar with image U , see Lemmas 30.3.11 and 30.3.12. Hence given f : T 0 → T over S and U, U 0 → T we get a canonical map FT (U ) = F(U 0 /S) → F(U 0 ×T T 0 /S) = FT 0 (f −1 (U )) where the middle is the restriction map of F with respect to the morphism U 0 ×T T 0 → U 0 over S. The collection of these maps are compatible with restrictions, and hence define an f -map cf from FT to FT 0 , see Sheaves, Definition 6.21.7 and the discussion surrounding it. It is clear that cf ◦g is the composition of cf and cg , since composition of restriction maps of F gives restriction maps. Conversely, given a system (FT , cf ) as in the lemma we may define a presheaf F on Sh((Sch/S)Zar ) by simply setting F(T /S) = FT (T ). As restriction mapping, given f : T 0 → T we set for s ∈ F(T ) the pullback f ∗ (s) equal to cf (s) (where we think of cf as an f -map again). The condition on the cf garantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.  30.4. The ´ etale topology Let S be a scheme. We would like to define the ´etale-topology on the category of schemes over S. According to our general principle we first introduce the notion of an ´etale covering. Definition 30.4.1. Let T be a scheme. An ´etale covering of T is a family of morphisms {fi : Ti → T }i∈I of schemes such that each fi is ´etale and such that S T = fi (Ti ). Lemma 30.4.2. Any Zariski covering is an ´etale covering. Proof. This is clear from the definitions and the fact that an open immersion is an ´etale morphism, see Morphisms, Lemma 24.37.9.  Next, we show that this notion satisfies the conditions of Sites, Definition 9.6.2. Lemma 30.4.3. Let T be a scheme. (1) If T 0 → T is an isomorphism then {T 0 → T } is an ´etale covering of T . (2) If {Ti → T }i∈I is an ´etale covering and for each i we have an ´etale covering {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is an ´etale covering.

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(3) If {Ti → T }i∈I is an ´etale covering and T 0 → T is a morphism of schemes then {T 0 ×T Ti → T 0 }i∈I is an ´etale covering. Proof. Omitted.



Lemma 30.4.4. Let T be an affine scheme. Let {Ti → T }i∈I be an ´etale covering of T . Then there exists an ´etale covering {Uj → T }j=1,...,m which is a refinement of {Ti → T }i∈I such that each Uj is an affine scheme. Moreover, we may choose each Uj to be open affine in one of the Ti . Proof. Omitted.



Thus we define the corresponding standard coverings of affines as follows. Definition 30.4.5. Let T be an affine scheme. A standard ´etale covering of T is a S family {fj : Uj → T }j=1,...,m with each Uj is affine and ´etale over T and T = fj (Uj ). In the definition above we do not assume the morphisms fj are standard ´etale. The reason is that if we did then the standard ´etale coverings would not define a site on Aff/S, for example because of Algebra, Lemma 7.133.14 part (4). On the other hand, an ´etale morphism of affines is automatically standard smooth, see Algebra, Lemma 7.133.2. Hence a standard ´etale covering is a standard smooth covering and a standard syntomic covering. Definition 30.4.6. A big ´etale site is any site Sche´tale as in Sites, Definition 9.6.2 constructed as follows: (1) Choose any set of schemes S0 , and any set of ´etale coverings Cov0 among these schemes. (2) As underlying category take any category Schα constructed as in Sets, Lemma 3.9.2 starting with the set S0 . (3) Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category Schα and the class of ´etale coverings, and the set Cov0 chosen above. See the remarks following Definition 30.3.5 for motivation and explanation regarding the definition of big sites. Before we continue with the introduction of the big ´etale site of a scheme S, let us point out that the topology on a big ´etale site Sche´tale is in some sense induced from the ´etale topology on the category of all schemes. Lemma 30.4.7. Let Sche´tale be a big ´etale site as in Definition 30.4.6. Let T ∈ Ob(Sche´tale ). Let {Ti → T }i∈I be an arbitrary ´etale covering of T . (1) There exists a covering {Uj → T }j∈J of T in the site Sche´tale which refines {Ti → T }i∈I . (2) If {Ti → T }i∈I is a standard ´etale covering, then it is tautologically equivalent to a covering in Sche´tale . (3) If {Ti → T }i∈I is a Zariski covering, then it is tautologically equivalent to a covering in Sche´tale . S Proof. For each i choose an affine open covering Ti = j∈Ji Tij such that each Tij maps into an affine open subscheme of T . By Lemma 30.4.3 the refinement {Tij → T }i∈I,j∈Ji is an ´etale covering of T as well. Hence we may assume each

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Ti is affine, and maps into an affine open Wi of T . Applying Sets, Lemma 3.9.9 we see that Wi is isomorphic to an object of SchZar . But then Ti as a finite type scheme over Wi is isomorphic to an object Vi of SchZar by a second application of Sets, Lemma 3.9.9. The covering {Vi → T }i∈I refines {Ti → T }i∈I (because they are isomorphic). Moreover, {Vi → T }i∈I is combinatorially equivalent to a covering {Uj → T }j∈J of T in the site SchZar by Sets, Lemma 3.9.9. The covering {Uj → T }j∈J is a refinement as in (1). In the situation of (2), (3) each of the schemes Ti is isomorphic to an object of Sche´tale by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want.  Definition 30.4.8. Let S be a scheme. Let Sche´tale be a big ´etale site containing S. (1) The big ´etale site of S, denoted (Sch/S)e´tale , is the site Sche´tale /S introduced in Sites, Section 9.21. (2) The small ´etale site of S, which we denote Se´tale , is the full subcategory of (Sch/S)e´tale whose objects are those U/S such that U → S is ´etale. A covering of Se´tale is any covering {Ui → U } of (Sch/S)e´tale with U ∈ Ob(Se´tale ). (3) The big affine ´etale site of S, denoted (Aff/S)e´tale , is the full subcategory of (Sch/S)e´tale whose objects are affine U/S. A covering of (Aff/S)e´tale is any covering {Ui → U } of (Sch/S)e´tale which is a standard ´etale covering. It is not completely clear that the big affine ´etale site or the small ´etale site are sites. We check this now. Lemma 30.4.9. Let S be a scheme. Let Sche´tale be a big ´etale site containing S. Both Se´tale and (Aff/S)e´tale are sites. Proof. Let us show that Se´tale is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 9.6.2. Since (Sch/S)e´tale is a site, it suffices to prove that given any covering {Ui → U } of (Sch/S)Zar with U ∈ Ob(Se´tale ) we also have Ui ∈ Ob(Se´tale ). This follows from the definitions as the composition of ´etale morphisms is an ´etale morphism. Let us show that (Aff/S)e´tale is a site. Reasoning as above, it suffices to show that the collection of standard ´etale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 9.6.2. This is clear since for example, given a standard ´etale covering {Ti → T }i∈I and for each i we have a standard ´etale covering S {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is a standard ´etale covering because i∈I Ji is finite and each Tij is affine.  Lemma 30.4.10. Let S be a scheme. Let Sche´tale be a big ´etale site containing S. The underlying categories of the sites Sche´tale , (Sch/S)e´tale , Se´tale , and (Aff/S)e´tale have fibre products. In each case the obvious functor into the category Sch of all schemes commutes with taking fibre products. The categories (Sch/S)e´tale , and Se´tale both have a final object, namely S/S. Proof. For Sche´tale it is true by construction, see Sets, Lemma 3.9.9. Suppose we have U → S, V → U , W → U morphisms of schemes with U, V, W ∈ Ob(Sche´tale ). The fibre product V ×U W in Sche´tale is a fibre product in Sch and is the fibre product of V /S with W/S over U/S in the category of all schemes over S, and

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hence also a fibre product in (Sch/S)e´tale . This proves the result for (Sch/S)e´tale . If U → S, V → U and W → U are ´etale then so is V ×U W → S and hence we get the result for Se´tale . If U, V, W are affine, so is V ×U W and hence the result for (Aff/S)e´tale .  Next, we check that the big affine site defines the same topos as the big site. Lemma 30.4.11. Let S be a scheme. Let Sche´tale be a big ´etale site containing S. The functor (Aff/S)e´tale → (Sch/S)e´tale is special cocontinuous and induces an equivalence of topoi from Sh((Aff/S)e´tale ) to Sh((Sch/S)e´tale ). Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 9.25.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 9.25.1. Denote the inclusion functor u : (Aff/S)e´tale → (Sch/S)e´tale . Being cocontinuous just means that any ´etale covering of T /S, T affine, can be refined by a standard ´etale covering of T . This is the content of Lemma 30.4.4. Hence (1) holds. We see u is continuous simply because a standard ´etale covering is a ´etale covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that u is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering.  Next, we esthablish some relationships between the topoi associated to these sites. Lemma 30.4.12. Let Sche´tale be a big ´etale site. Let f : T → S be a morphism in Sche´tale . The functor Te´tale → (Sch/S)e´tale is cocontinuous and induces a morphism of topoi if : Sh(Te´tale ) −→ Sh((Sch/S)e´tale ) For a sheaf G on (Sch/S)e´tale we have the formula (i−1 f G)(U/T ) = G(U/S). The −1 functor if also has a left adjoint if,! which commutes with fibre products and equalizers. Proof. Denote the functor u : Te´tale → (Sch/S)e´tale . In other words, given an ´etale morphism j : U → T corresponding to an object of Te´tale we set u(U → T ) = (f ◦ j : U → S). This functor commutes with fibre products, see Lemma 30.4.10. Let a, b : U → V be two morphisms in Te´tale . In this case the equalizer of a and b (in the category of schemes) is V ×∆V /T ,V ×T V,(a,b) U ×T U which is a fibre product of schemes ´etale over T , hence ´etale over T . Thus Te´tale has equalizers and u commutes with them. It is clearly cocontinuous. It is also continuous as u transforms coverings to coverings and commutes with fibre products. Hence the Lemma follows from Sites, Lemmas 9.19.5 and 9.19.6.  Lemma 30.4.13. Let S be a scheme. Let Sche´tale be a big ´etale site containing S. The inclusion functor Se´tale → (Sch/S)e´tale satisfies the hypotheses of Sites, Lemma 9.19.8 and hence induces a morphism of sites πS : (Sch/S)e´tale −→ Se´tale and a morphism of topoi iS : Sh(Se´tale ) −→ Sh((Sch/S)e´tale ) such that πS ◦ iS = id. Moreover, iS = iidS with iidS as in Lemma 30.4.12. In −1 particular the functor i−1 S = πS,∗ is described by the rule iS (G)(U/S) = G(U/S).

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Proof. In this case the functor u : Se´tale → (Sch/S)e´tale , in addition to the properties seen in the proof of Lemma 30.4.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 9.19.8.  Definition 30.4.14. In the situation of Lemma 30.4.13 the functor i−1 S = πS,∗ is often called the restriction to the small ´etale site, and for a sheaf F on the big ´etale site we denote F|Se´tale this restriction. With this notation in place we have for a sheaf F on the big site and a sheaf G on the big site that MorSh(Se´tale ) (F|Se´tale , G) = MorSh((Sch/S)e´tale ) (F, iS,∗ G) MorSh(Se´tale ) (G, F|Se´tale ) = MorSh((Sch/S)e´tale ) (πS−1 G, F) Moreover, we have (iS,∗ G)|Se´tale = G and we have (πS−1 G)|Se´tale = G. Lemma 30.4.15. Let Sche´tale be a big ´etale site. Let f : T → S be a morphism in Sche´tale . The functor u : (Sch/T )e´tale −→ (Sch/S)e´tale ,

V /T 7−→ V /S

is cocontinuous, and has a continuous right adjoint v : (Sch/S)e´tale −→ (Sch/T )e´tale ,

(U → S) 7−→ (U ×S T → T ).

They induce the same morphism of topoi fbig : Sh((Sch/T )e´tale ) −→ Sh((Sch/S)e´tale ) −1 fbig (G)(U/T )

We have = G(U/S). We have fbig,∗ (F)(U/S) = F(U ×S T /T ). Also, −1 fbig has a left adjoint fbig! which commutes with fibre products and equalizers. Proof. The functor u is cocontinuous, continuous and commutes with fibre products and equalizers (details omitted; compare with the proof of Lemma 30.4.12). −1 Hence Sites, Lemmas 9.19.5 and 9.19.6 apply and we deduce the formula for fbig and the existence of fbig! . Moreover, the functor v is a right adjoint because given U/T and V /S we have MorS (u(U ), V ) = MorT (U, V ×S T ) as desired. Thus we may apply Sites, Lemmas 9.20.1 and 9.20.2 to get the formula for fbig,∗ .  Lemma 30.4.16. Let Sche´tale be a big ´etale site. Let f : T → S be a morphism in Sche´tale . (1) We have if = fbig ◦ iT with if as in Lemma 30.4.12 and iT as in Lemma 30.4.13. (2) The functor Se´tale → Te´tale , (U → S) 7→ (U ×S T → T ) is continuous and induces a morphism of topoi fsmall : Sh(Te´tale ) −→ Sh(Se´tale ). We have fsmall,∗ (F)(U/S) = F(U ×S T /T ). (3) We have a commutative diagram of morphisms of sites Te´tale o fsmall



Se´tale o

πT

(Sch/T )e´tale fbig

πS

 (Sch/S)e´tale

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so that fsmall ◦ πT = πS ◦ fbig as morphisms of topoi. (4) We have fsmall = πS ◦ fbig ◦ iT = πS ◦ if . −1 −1 Proof. The equality if = fbig ◦ iT follows from the equality i−1 f = iT ◦ fbig which is clear from the descriptions of these functors above. Thus we see (1).

The functor u : Se´tale → Te´tale , u(U → S) = (U ×S T → T ) transforms coverings into coverings and commutes with fibre products, see Lemma 30.4.3 (3) and 30.4.10. Moreover, both Se´tale , Te´tale have final objects, namely S/S and T /T and u(S/S) = T /T . Hence by Sites, Proposition 9.14.6 the functor u corresponds to a morphism of sites Te´tale → Se´tale . This in turn gives rise to the morphism of topoi, see Sites, Lemma 9.15.3. The description of the pushforward is clear from these references. Part (3) follows because πS and πT are given by the inclusion functors and fsmall and fbig by the base change functors U 7→ U ×S T . Statement (4) follows from (3) by precomposing with iT .



In the situation of the lemma, using the terminology of Definition 30.4.14 we have: for F a sheaf on the big ´etale site of T (fbig,∗ F)|Se´tale = fsmall,∗ (F|Te´tale ), This equality is clear from the commutativity of the diagram of sites of the lemma, since restriction to the small ´etale site of T , resp. S is given by πT,∗ , resp. πS,∗ . A similar formula involving pullbacks and restrictions is false. Lemma 30.4.17. Given schemes X, Y , Y in Sche´tale and morphisms f : X → Y , g : Y → Z we have gbig ◦ fbig = (g ◦ f )big and gsmall ◦ fsmall = (g ◦ f )small . Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 30.4.15. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 30.4.16.  We can think about a sheaf on the big ´etale site of S as a collection of “usual” sheaves on all schemes over S. Lemma 30.4.18. Let S be a scheme contained in a big ´etale site Sche´tale . A sheaf F on the big ´etale site (Sch/S)e´tale is given by the following data: (1) for every T /S ∈ Ob((Sch/S)e´tale ) a sheaf FT on Te´tale , −1 (2) for every f : T 0 → T in (Sch/S)e´tale a map cf : fsmall FT → FT 0 . These data are subject to the following conditions: (i) given any f : T 0 → T and g : T 00 → T 0 in (Sch/S)e´tale the composition −1 gsmall cf ◦ cg is equal to cf ◦g , and (ii) if f : T 0 → T in (Sch/S)e´tale is ´etale then cf is an isomorphism. Proof. Given a sheaf F on Sh((Sch/S)e´tale ) we set FT = i−1 p F where p : T → S is the structure morphism. Note that FT (U ) = F(U/S) for any U → T in Te´tale see Lemma 30.4.12. Hence given f : T 0 → T over S and U → T we get a canonical map FT (U ) = F(U/S) → F(U ×T T 0 /S) = FT 0 (U ×T T 0 ) where the middle is the restriction map of F with respect to the morphism U ×T T 0 → U over S. The collection of these maps are compatible with restrictions, and hence define a map c0f : FT → fsmall,∗ FT 0 where u : Te´tale → Te´0tale is the base change functor −1 associated to f . By adjunction of fsmall,∗ (see Sites, Section 9.13) with fsmall this

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−1 is the same as a map cf : fsmall FT → FT 0 . It is clear that c0f ◦g is the composition 0 0 of cf and fsmall,∗ cg , since composition of restriction maps of F gives restriction maps, and this gives the desired relationship among cf , cg and cf ◦g .

Conversely, given a system (FT , cf ) as in the lemma we may define a presheaf F on Sh((Sch/S)e´tale ) by simply setting F(T /S) = FT (T ). As restriction mapping, given f : T 0 → T we set for s ∈ F(T ) the pullback f ∗ (s) equal to cf (s) where we think of cf as a map FT → fsmall,∗ FT 0 again. The condition on the cf garantees that pullbacks satisfy the required functoriality property. We omit the verification that this is a sheaf. It is clear that the constructions so defined are mutually inverse.  30.5. The smooth topology In this section we define the smooth topology. This is a bit pointless as it will turn out later (see More on Morphisms, Section 33.27) that this topology defines the same topos as the ´etale topology. But still it makes sense and it is used occasionally. Definition 30.5.1. Let T be a scheme. An smooth covering of T is a family of morphisms {fi : Ti → T }i∈I of schemes such that each fi is smooth and such that S T = fi (Ti ). Lemma 30.5.2. Any ´etale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering. Proof. This is clear from the definitions, the fact that an ´etale morphism is smooth see Morphisms, Definition 24.37.1 and Lemma 30.4.2.  Next, we show that this notion satisfies the conditions of Sites, Definition 9.6.2. Lemma 30.5.3. Let T be a scheme. (1) If T 0 → T is an isomorphism then {T 0 → T } is an smooth covering of T . (2) If {Ti → T }i∈I is a smooth covering and for each i we have a smooth covering {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is a smooth covering. (3) If {Ti → T }i∈I is a smooth covering and T 0 → T is a morphism of schemes then {T 0 ×T Ti → T 0 }i∈I is a smooth covering. Proof. Omitted.



Lemma 30.5.4. Let T be an affine scheme. Let {Ti → T }i∈I be a smooth covering of T . Then there exists a smooth covering {Uj → T }j=1,...,m which is a refinement of {Ti → T }i∈I such that each Uj is an affine scheme, and such that each morphism Uj → T is standard smooth, see Morphisms, Definition 24.35.1. Moreover, we may choose each Uj to be open affine in one of the Ti . Proof. Omitted, but see Algebra, Lemma 7.127.10.



Thus we define the corresponding standard coverings of affines as follows. Definition 30.5.5. Let T be an affine scheme. A standard smooth covering of T is a family S {fj : Uj → T }j=1,...,m with each Uj is affine, Uj → T standard smooth and T = fj (Uj ). Definition 30.5.6. A big smooth site is any site Schsmooth as in Sites, Definition 9.6.2 constructed as follows:

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(1) Choose any set of schemes S0 , and any set of smooth coverings Cov0 among these schemes. (2) As underlying category take any category Schα constructed as in Sets, Lemma 3.9.2 starting with the set S0 . (3) Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category Schα and the class of smooth coverings, and the set Cov0 chosen above. See the remarks following Definition 30.3.5 for motivation and explanation regarding the definition of big sites. Before we continue with the introduction of the big smooth site of a scheme S, let us point out that the topology on a big smooth site Schsmooth is in some sense induced from the smooth topology on the category of all schemes. Lemma 30.5.7. Let Schsmooth be a big smooth site as in Definition 30.5.6. Let T ∈ Ob(Schsmooth ). Let {Ti → T }i∈I be an arbitrary smooth covering of T . (1) There exists a covering {Uj → T }j∈J of T in the site Schsmooth which refines {Ti → T }i∈I . (2) If {Ti → T }i∈I is a standard smooth covering, then it is tautologically equivalent to a covering of Schsmooth . (3) If {Ti → T }i∈I is a Zariski covering, then it is tautologically equivalent to a covering of Schsmooth . S Proof. For each i choose an affine open covering Ti = j∈Ji Tij such that each Tij maps into an affine open subscheme of T . By Lemma 30.5.3 the refinement {Tij → T }i∈I,j∈Ji is an smooth covering of T as well. Hence we may assume each Ti is affine, and maps into an affine open Wi of T . Applying Sets, Lemma 3.9.9 we see that Wi is isomorphic to an object of SchZar . But then Ti as a finite type scheme over Wi is isomorphic to an object Vi of SchZar by a second application of Sets, Lemma 3.9.9. The covering {Vi → T }i∈I refines {Ti → T }i∈I (because they are isomorphic). Moreover, {Vi → T }i∈I is combinatorially equivalent to a covering {Uj → T }j∈J of T in the site SchZar by Sets, Lemma 3.9.9. The covering {Uj → T }j∈J is a refinement as in (1). In the situation of (2), (3) each of the schemes Ti is isomorphic to an object of Schsmooth by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want.  Definition 30.5.8. Let S be a scheme. Let Schsmooth be a big smooth site containing S. (1) The big smooth site of S, denoted (Sch/S)smooth , is the site Schsmooth /S introduced in Sites, Section 9.21. (2) The big affine smooth site of S, denoted (Aff/S)smooth , is the full subcategory of (Sch/S)smooth whose objects are affine U/S. A covering of (Aff/S)smooth is any covering {Ui → U } of (Sch/S)smooth which is a standard smooth covering. Next, we check that the big affine site defines the same topos as the big site. Lemma 30.5.9. Let S be a scheme. Let Sche´tale be a big smooth site containing S. The functor (Aff/S)smooth → (Sch/S)smooth is special cocontinuous and induces an equivalence of topoi from Sh((Aff/S)smooth ) to Sh((Sch/S)smooth ).

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Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 9.25.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 9.25.1. Denote the inclusion functor u : (Aff/S)smooth → (Sch/S)smooth . Being cocontinuous just means that any smooth covering of T /S, T affine, can be refined by a standard smooth covering of T . This is the content of Lemma 30.5.4. Hence (1) holds. We see u is continuous simply because a standard smooth covering is a smooth covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that u is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering.  To be continued... Lemma 30.5.10. Let Schsmooth be a big smooth site. Let f : T → S be a morphism in Schsmooth . The functor u : (Sch/T )smooth −→ (Sch/S)smooth ,

V /T 7−→ V /S

is cocontinuous, and has a continuous right adjoint v : (Sch/S)smooth −→ (Sch/T )smooth ,

(U → S) 7−→ (U ×S T → T ).

They induce the same morphism of topoi fbig : Sh((Sch/T )smooth ) −→ Sh((Sch/S)smooth ) −1 We have fbig (G)(U/T ) = G(U/S). We have fbig,∗ (F)(U/S) = F(U ×S T /T ). Also, −1 fbig has a left adjoint fbig! which commutes with fibre products and equalizers.

Proof. The functor u is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 9.19.5 and 9.19.6 apply and we deduce −1 the formula for fbig and the existence of fbig! . Moreover, the functor v is a right adjoint because given U/T and V /S we have MorS (u(U ), V ) = MorT (U, V ×S T ) as desired. Thus we may apply Sites, Lemmas 9.20.1 and 9.20.2 to get the formula for fbig,∗ .  30.6. The syntomic topology In this section we define the syntomic topology. This topology is quite interesting in that it often has the same cohomology groups as the fppf topology but is technically easier to deal with. Definition 30.6.1. Let T be a scheme. An syntomic covering of T is a family of morphisms {fi : Ti → T }i∈I of schemes such that each fi is syntomic and such that S T = fi (Ti ). Lemma 30.6.2. Any smooth covering is a syntomic covering, and a fortiori, any ´etale or Zariski covering is a syntomic covering. Proof. This is clear from the definitions and the fact that a smooth morphism is syntomic, see Morphisms, Lemma 24.35.7 and Lemma 30.5.2.  Next, we show that this notion satisfies the conditions of Sites, Definition 9.6.2. Lemma 30.6.3. Let T be a scheme. (1) If T 0 → T is an isomorphism then {T 0 → T } is an syntomic covering of T.

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(2) If {Ti → T }i∈I is a syntomic covering and for each i we have a syntomic covering {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is a syntomic covering. (3) If {Ti → T }i∈I is a syntomic covering and T 0 → T is a morphism of schemes then {T 0 ×T Ti → T 0 }i∈I is a syntomic covering. Proof. Omitted.



Lemma 30.6.4. Let T be an affine scheme. Let {Ti → T }i∈I be a syntomic covering of T . Then there exists a syntomic covering {Uj → T }j=1,...,m which is a refinement of {Ti → T }i∈I such that each Uj is an affine scheme, and such that each morphism Uj → T is standard syntomic, see Morphisms, Definition 24.32.1. Moreover, we may choose each Uj to be open affine in one of the Ti . Proof. Omitted, but see Algebra, Lemma 7.126.16.



Thus we define the corresponding standard coverings of affines as follows. Definition 30.6.5. Let T be an affine scheme. A standard syntomic covering of T is a familyS{fj : Uj → T }j=1,...,m with each Uj is affine, Uj → T standard syntomic and T = fj (Uj ). Definition 30.6.6. A big syntomic site is any site Schsyntomic as in Sites, Definition 9.6.2 constructed as follows: (1) Choose any set of schemes S0 , and any set of syntomic coverings Cov0 among these schemes. (2) As underlying category take any category Schα constructed as in Sets, Lemma 3.9.2 starting with the set S0 . (3) Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category Schα and the class of syntomic coverings, and the set Cov0 chosen above. See the remarks following Definition 30.3.5 for motivation and explanation regarding the definition of big sites. Before we continue with the introduction of the big syntomic site of a scheme S, let us point out that the topology on a big syntomic site Schsyntomic is in some sense induced from the syntomic topology on the category of all schemes. Lemma 30.6.7. Let Schsyntomic be a big syntomic site as in Definition 30.6.6. Let T ∈ Ob(Schsyntomic ). Let {Ti → T }i∈I be an arbitrary syntomic covering of T . (1) There exists a covering {Uj → T }j∈J of T in the site Schsyntomic which refines {Ti → T }i∈I . (2) If {Ti → T }i∈I is a standard syntomic covering, then it is tautologically equivalent to a covering in Schsyntomic . (3) If {Ti → T }i∈I is a Zariski covering, then it is tautologically equivalent to a covering in Schsyntomic . S Proof. For each i choose an affine open covering Ti = j∈Ji Tij such that each Tij maps into an affine open subscheme of T . By Lemma 30.6.3 the refinement {Tij → T }i∈I,j∈Ji is an syntomic covering of T as well. Hence we may assume each Ti is affine, and maps into an affine open Wi of T . Applying Sets, Lemma 3.9.9 we see that Wi is isomorphic to an object of SchZar . But then Ti as a finite type scheme over Wi is isomorphic to an object Vi of SchZar by a second application

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of Sets, Lemma 3.9.9. The covering {Vi → T }i∈I refines {Ti → T }i∈I (because they are isomorphic). Moreover, {Vi → T }i∈I is combinatorially equivalent to a covering {Uj → T }j∈J of T in the site SchZar by Sets, Lemma 3.9.9. The covering {Uj → T }j∈J is a covering as in (1). In the situation of (2), (3) each of the schemes Ti is isomorphic to an object of SchZar by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want.  Definition 30.6.8. Let S be a scheme. Let Schsyntomic be a big syntomic site containing S. (1) The big syntomic site of S, denoted (Sch/S)syntomic , is the site Schsyntomic /S introduced in Sites, Section 9.21. (2) The big affine syntomic site of S, denoted (Aff/S)syntomic , is the full subcategory of (Sch/S)syntomic whose objects are affine U/S. A covering of (Aff/S)syntomic is any covering {Ui → U } of (Sch/S)syntomic which is a standard syntomic covering. Next, we check that the big affine site defines the same topos as the big site. Lemma 30.6.9. Let S be a scheme. Let Schsyntomic be a big syntomic site containing S. The functor (Aff/S)syntomic → (Sch/S)syntomic is special cocontinuous and induces an equivalence of topoi from Sh((Aff/S)syntomic ) to Sh((Sch/S)syntomic ). Proof. The notion of a special cocontinuous functor is introduced in Sites, Definition 9.25.2. Thus we have to verify assumptions (1) – (5) of Sites, Lemma 9.25.1. Denote the inclusion functor u : (Aff/S)syntomic → (Sch/S)syntomic . Being cocontinuous just means that any syntomic covering of T /S, T affine, can be refined by a standard syntomic covering of T . This is the content of Lemma 30.6.4. Hence (1) holds. We see u is continuous simply because a standard syntomic covering is a syntomic covering. Hence (2) holds. Parts (3) and (4) follow immediately from the fact that u is fully faithful. And finally condition (5) follows from the fact that every scheme has an affine open covering.  To be continued... Lemma 30.6.10. Let Schsyntomic be a big syntomic site. Let f : T → S be a morphism in Schsyntomic . The functor u : (Sch/T )syntomic −→ (Sch/S)syntomic ,

V /T 7−→ V /S

is cocontinuous, and has a continuous right adjoint v : (Sch/S)syntomic −→ (Sch/T )syntomic ,

(U → S) 7−→ (U ×S T → T ).

They induce the same morphism of topoi fbig : Sh((Sch/T )syntomic ) −→ Sh((Sch/S)syntomic ) −1 fbig (G)(U/T )

We have = G(U/S). We have fbig,∗ (F)(U/S) = F(U ×S T /T ). Also, −1 fbig has a left adjoint fbig! which commutes with fibre products and equalizers. Proof. The functor u is cocontinuous, continuous, and commutes with fibre products and equalizers. Hence Sites, Lemmas 9.19.5 and 9.19.6 apply and we deduce −1 the formula for fbig and the existence of fbig! . Moreover, the functor v is a right adjoint because given U/T and V /S we have MorS (u(U ), V ) = MorT (U, V ×S T ) as desired. Thus we may apply Sites, Lemmas 9.20.1 and 9.20.2 to get the formula for fbig,∗ . 

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30.7. The fppf topology Let S be a scheme. We would like to define the fppf-topology2 on the category of schemes over S. According to our general principle we first introduce the notion of an fppf-covering. Definition 30.7.1. Let T be a scheme. An fppf covering of T is a family of morphisms {fi : Ti → T }i∈I of S schemes such that each fi is flat, locally of finite presentation and such that T = fi (Ti ). Lemma 30.7.2. Any syntomic covering is an fppf covering, and a fortiori, any smooth, ´etale, or Zariski covering is an fppf covering. Proof. This is clear from the definitions, the fact that a synomtic morphism is flat and locally of finite presentation, see Morphisms, Lemmas 24.32.6 and 24.32.7, and Lemma 30.6.2.  Next, we show that this notion satisfies the conditions of Sites, Definition 9.6.2. Lemma 30.7.3. Let T be a scheme. (1) If T 0 → T is an isomorphism then {T 0 → T } is an fppf covering of T . (2) If {Ti → T }i∈I is an fppf covering and for each i we have an fppf covering {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is an fppf covering. (3) If {Ti → T }i∈I is an fppf covering and T 0 → T is a morphism of schemes then {T 0 ×T Ti → T 0 }i∈I is an fppf covering. Proof. The first assertion is clear. The second follows as the composition of flat morphisms is flat (see Morphisms, Lemma 24.26.5) and the composition of morphisms of finite presentation is of finite presentation (see Morphisms, Lemma 24.22.3). The third follows as the base change of a flat morphism is flat (see Morphisms, Lemma 24.26.7) and the base change of a morphism of finite presentation is of finite presentation (see Morphisms, Lemma 24.22.4). Moreover, the base change of a surjective family of morphisms is surjective (proof omitted).  Lemma 30.7.4. Let T be an affine scheme. Let {Ti → T }i∈I be an fppf covering of T . Then there exists an fppf covering {Uj → T }j=1,...,m which is a refinement of {Ti → T }i∈I such that each Uj is an affine scheme. Moreover, we may choose each Uj to be open affine in one of the Ti . Proof. This follows directly from the definitions using that a morphism which is flat and locally of finite presentation is open, see Morphisms, Lemma 24.26.9.  Thus we define the corresponding standard coverings of affines as follows. Definition 30.7.5. Let T be an affine scheme. A standard fppf covering of T is a family {fj : Uj → S T }j=1,...,m with each Uj is affine, flat and of finite presentation over T and T = fj (Uj ). Definition 30.7.6. A big fppf site is any site Schf ppf as in Sites, Definition 9.6.2 constructed as follows: (1) Choose any set of schemes S0 , and any set of fppf coverings Cov0 among these schemes. 2 The letters fppf stand for “fid` element plat de pr´ esentation finie”.

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(2) As underlying category take any category Schα constructed as in Sets, Lemma 3.9.2 starting with the set S0 . (3) Choose any set of coverings as in Sets, Lemma 3.11.1 starting with the category Schα and the class of fppf coverings, and the set Cov0 chosen above. See the remarks following Definition 30.3.5 for motivation and explanation regarding the definition of big sites. Before we continue with the introduction of the big fppf site of a scheme S, let us point out that the topology on a big fppf site Schf ppf is in some sense induced from the fppf topology on the category of all schemes. Lemma 30.7.7. Let Schf ppf be a big fppf site as in Definition 30.7.6. Let T ∈ Ob(Schf ppf ). Let {Ti → T }i∈I be an arbitrary fppf covering of T . (1) There exists a covering {Uj → T }j∈J of T in the site Schf ppf which refines {Ti → T }i∈I . (2) If {Ti → T }i∈I is a standard fppf covering, then it is tautologically equivalent to a covering of Schf ppf . (3) If {Ti → T }i∈I is a Zariski covering, then it is tautologically equivalent to a covering of Schf ppf . S Proof. For each i choose an affine open covering Ti = j∈Ji Tij such that each Tij maps into an affine open subscheme of T . By Lemma 30.7.3 the refinement {Tij → T }i∈I,j∈Ji is an fppf covering of T as well. Hence we may assume each Ti is affine, and maps into an affine open Wi of T . Applying Sets, Lemma 3.9.9 we see that Wi is isomorphic to an object of SchZar . But then Ti as a finite type scheme over Wi is isomorphic to an object Vi of SchZar by a second application of Sets, Lemma 3.9.9. The covering {Vi → T }i∈I refines {Ti → T }i∈I (because they are isomorphic). Moreover, {Vi → T }i∈I is combinatorially equivalent to a covering {Uj → T }j∈J of T in the site SchZar by Sets, Lemma 3.9.9. The covering {Uj → T }j∈J is a refinement as in (1). In the situation of (2), (3) each of the schemes Ti is isomorphic to an object of Schf ppf by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want.  Definition 30.7.8. Let S be a scheme. Let Schf ppf be a big fppf site containing S. (1) The big fppf site of S, denoted (Sch/S)f ppf , is the site Schf ppf /S introduced in Sites, Section 9.21. (2) The big affine fppf site of S, denoted (Aff/S)f ppf , is the full subcategory of (Sch/S)f ppf whose objects are affine U/S. A covering of (Aff/S)f ppf is any covering {Ui → U } of (Sch/S)f ppf which is a standard fppf covering. It is not completely clear that the big affine fppf site is a site. We check this now. Lemma 30.7.9. Let S be a scheme. Let Schf ppf be a big fppf site containing S. Then (Aff/S)f ppf is a site. Proof. Let us show that (Aff/S)f ppf is a site. Reasoning as in the proof of Lemma 30.4.9 it suffices to show that the collection of standard fppf coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 9.6.2. This is clear since for example, given a standard fppf covering {Ti → T }i∈I and for each i we have a

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standard fppf covering {Tij → Ti }j∈Ji , then {Tij → T }i∈I,j∈Ji is a standard fppf S covering because i∈I Ji is finite and each Tij is affine.  Lemma 30.7.10. Let S be a scheme. Let Schf ppf be a big fppf site containing S. The underlying categories of the sites Schf ppf , (Sch/S)f ppf , and (Aff/S)f ppf have fibre products. In each case