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EMS Tracts in Mathematics 9
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EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 2 3 4 5 6 7 8
Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups Ralf Meyer, Local and Analytic Cyclic Homology Gohar Harutyunyan and B.-Wolfgang Schulze, Elliptic Mixed, Transmission and Singular Crack Problems Gennadiy Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups , Erich Novak and Henryk Wozniakowski, Tractability of Multivariate Problems. Volume I: Linear Information Hans Triebel, Function Spaces and Wavelets on Domains Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems
Boeckle_titelei
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Gebhard Böckle Richard Pink
Cohomological Theory of Crystals over Function Fields
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Authors: Gebhard Böckle Universität Duisburg-Essen Campus Essen 45117 Essen Germany E-mail:
[email protected] Richard Pink Departement Mathematik ETH Zürich Rämistrasse 101 8092 Zürich Switzerland E-mail:
[email protected] 2010 Mathematical Subject Classification: 11-02, 14-02 Key words: Characteristic p cohomologies, Drinfeld modules, A-motives, L-functions, Weil conjecture
ISBN 978-3-03719-074-6 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2009 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321
Preface
For global fields of positive characteristic p, Drinfeld and later Anderson defined objects, now called Drinfeld modules and A-motives, which bear many similarities to abelian varieties over number fields. The Tate modules of these objects provide interesting examples of strictly compatible systems of Galois representations. In many cases they also possess an analytic uniformization and a kind of Hodge theory, which altogether can be viewed as étale, Betti and de Rham realizations. A natural next step was taken by Goss, who assigned an L-function to any family of such objects over a variety of finite type over Fp , which is a power series in a variable T . By analogy to the L-function of a constructible `-adic sheaf in the case ` ¤ p, Goss conjectured that this L-function is, in fact, a rational function of T . In 1996 Taguchi and Wan proved this conjecture, using analytic methods à la Dwork. This development parallels that around the Weil conjecture, where Dwork’s analytic proof of rationality predated the algebraic proof of Grothendieck et al. based on `-adic étale cohomology. The present monograph performs some of the next logical steps in this direction. It provides a set of algebro-geometric and homological tools to give a purely algebraic proof of Goss’s rationality conjecture and – hopefully – much more. Specifically, we introduce a notion of crystals, which contains families of A-motives as special cases, and develop a complete cohomological theory for them. We also prove a Lefschetz trace formula that expresses the L-function in terms of explicitly computable cohomology groups. These cohomology groups contain finer information than the analytic rationality proof and will deserve further study. We also describe their relation to the cohomology of étale sheaves of Fp -modules over varieties in characteristic p. The book is intended for researchers and advanced graduate students interested in the arithmetic of function fields and/or cohomology theories for varieties in positive characteristic. A prerequisite to reading this book is a good working knowledge in algebraic geometry as well as familiarity with homological algebra and derived categories. Most of the necessary background can be found, for instance, in the books [26] on Algebraic Geometry by Hartshorne and [47] on Homological Algebra by Weibel. From that point on, we have tried to be largely self-contained. Acknowledgments. We thank David Goss for providing much interest and stimulus toward the present work. His motivation was invaluable during certain stages. We also thank Brian Conrad for helpful correspondence regarding some questions in algebraic geometry.
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
1
Introduction
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Categorical preparations 2.1 Categories . . . . . . . . . . . . . 2.2 Localization . . . . . . . . . . . . 2.3 Abelian categories . . . . . . . . . 2.4 Grothendieck categories . . . . . 2.5 Triangulated categories . . . . . . 2.6 Derived categories . . . . . . . . 2.7 Derived functors . . . . . . . . . 2.8 Construction of derived functors . 2.9 Comparison of derived categories
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13 13 17 20 23 26 27 30 32 36
Fundamental concepts 3.1 Conventions . . . 3.2 -sheaves . . . . 3.3 Nilpotence . . . . 3.4 A-crystals . . . . 3.5 Examples . . . .
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39 39 40 42 47 49
Functors 4.1 Inverse image . . . . . 4.2 Tensor product . . . . 4.3 Change of coefficients 4.4 Direct image . . . . . . 4.5 Extension by zero . . . 4.6 Constructibility . . . .
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52 52 58 60 61 64 69
Derived categories 5.1 The affine case: ind-acyclic T Œ-modules . . . . . 5.2 Ind-acyclic -sheaves . . . . . . . . . . . . . . . . 5.3 Derived categories of -sheaves and quasi-crystals . ˇ 5.4 Cech resolution . . . . . . . . . . . . . . . . . . .
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71 71 76 81 84
Derived functors 6.1 Inverse image . . . . . 6.2 Tensor product . . . . 6.3 Change of coefficients 6.4 Direct image I . . . . . 6.5 Direct image II . . . .
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88 88 90 93 93 97
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viii
Contents
6.6 6.7 7
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9
Extension by zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Direct image with compact support . . . . . . . . . . . . . . . . . . . 101
Flatness 7.1 Flatness of modules . . . . . . . . . . . 7.2 Basic properties . . . . . . . . . . . . . 7.3 Flatness of the canonical representative 7.4 Functoriality and constructibility . . . . 7.5 Representability . . . . . . . . . . . . . 7.6 Complexes of finite Tor-dimension . . . 7.7 Regular coefficient rings . . . . . . . .
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105 105 107 109 112 113 117 120
Naive L-functions 8.1 Basic properties . . . . . . . . 8.2 Duality . . . . . . . . . . . . 8.3 Anderson’s trace formula . . . 8.4 A cohomological trace formula 8.5 An extended example . . . . .
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Crystalline L-functions 9.1 Characteristic polynomials . . . . . . . . . . . 9.2 A primary decomposition for rational functions 9.3 The local L-factor . . . . . . . . . . . . . . . . 9.4 The global L-function . . . . . . . . . . . . . . 9.5 The L-function of a complex . . . . . . . . . . 9.6 Functoriality . . . . . . . . . . . . . . . . . . . 9.7 Arbitrary coefficients . . . . . . . . . . . . . . 9.8 Change of coefficients . . . . . . . . . . . . .
10 Étale cohomology 10.1 Basic Definitions . . . . . . . . 10.2 Functors . . . . . . . . . . . . . 10.3 Equivalence of categories . . . . 10.4 Derived categories and functors . 10.5 Flatness . . . . . . . . . . . . . 10.6 L-functions . . . . . . . . . . .
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138 139 143 145 148 150 151 157 160
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163 163 167 169 173 173 174
Bibliography
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List of notation
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Index
185
Chapter 1
Introduction
In the late 1940s Weil posed the challenge to create a cohomology theory for algebraic varieties X over an arbitrary field k with coefficients in Z. Weil had a deep arithmetic application in mind, namely to prove a conjecture of his concerning the -function attached to a variety over a finite field: Its first part asserts the rationality of this function, the second claims the existence of a functional equation, and the third predicts the locations of its zeros and poles. Assuming a well-developed cohomology theory with coefficients in Z, possessing in particular a Lefschetz trace formula, he indicated a proof of these conjectures, cf. [48]. As we now know, for fields k of positive characteristic one cannot hope to construct meaningful cohomology groups H i .X; Z/. Nevertheless Weil’s quest provided an important stimulus. Starting with Serre and then with the main driving force of Grothendieck, several good substitutes for such a theory have been constructed. Many later developments in arithmetic algebraic geometry hinge crucially on these theories. Among the most important cohomology theories for algebraic varieties over a field k are the following: If k is of characteristic zero, one may pass from k to the field of complex numbers C and consider the singular cohomology of the associated complex analytic space X.C/. Other invariants in the same situation are given by the algebraic de Rham cohomology of X . For general k and a prime ` different from the characteristic of k, there is the cohomology theory of étale `-adic sheaves due to Grothendieck et al. All these are examples of good cohomology theories in the sense of Grothendieck, in that they possess a full set of functors f , ˝, fŠ , f , Hom, f Š and a duality with certain properties. Moreover, there are important comparison isomorphisms between them. If k is of positive characteristic p and ` is equal to p, one also has cohomological theories of p-adic and mod p étale sheaves. But these are not good theories in Grothendieck’s sense, as they do not possess a duality and not all of the six functors above have a reasonable definition. For proper varieties over fields of characteristic p, a good theory for ` D p is crystalline cohomology. The first significant progress towards the Weil conjecture came, however, from another approach by Dwork, who resolved the first part and some cases of the second part of the Weil conjecture by p-adic analytic methods, cf. [16]. Only later Grothendieck et al. gave a cohomological proof of this along the lines proposed by Weil. Key ingredients in this proof are the cohomological theory of `-adic étale sheaves together with the Lefschetz trace formula. The latter yields an explicit formula for the L-function of an `-adic étale sheaf as the L-function of a complex representing the cohomology with compact support of this sheaf. It thereby provides finer information about the L-function than Dwork’s analytic proof. In 1974, Deligne gave an ingenious proof of the remaining parts of the Weil conjecture, again using extensively the cohomological method, cf. [13].
2
1 Introduction
Around the same time, Drinfeld initiated an arithmetic theory for objects over global fields of positive characteristic p, cf. [14]. For this one fixes a Dedekind domain A that is finitely generated over the field with p elements Fp and whose group of units is finite. The simplest example is the polynomial ring A D Fq Œt over a finite extension with q elements Fq . The ring A plays a role analogous to that of Z in classical arithmetic geometry. The first examples of these new objects were what are now called Drinfeld Amodules. Related but more algebro-geometric objects, also due to Drinfeld, are elliptic sheaves and more generally shtukas. Inspired by this, in the mid 1980s Anderson introduced the notion of t-motives for A D Fq Œt , which has a natural generalization to arbitrary A under the name of A-motives. The category of A-motives contains (via a contravariant embedding) that of Drinfeld A-modules, but is more flexible than the latter: While the only operation on Drinfeld A-modules is pullback along morphisms, on A-motives, in addition, one has all the standard operations from linear algebra such as direct sum and tensor product. These A-motives bear many analogies to abelian varieties. For any prime ` of Z one associates the `-adic Tate module to an abelian variety, and for any place v of A one associates the v-adic Tate module to an A-motive. The former is isomorphic to Z2g if ` the abelian variety has dimension g, and the latter is isomorphic to Arv if the A-motive has rank r. In either case the Tate module carries a continuous action of the absolute Galois group of the base field. If the base field is finite, the Galois representation is completely described by the action of the Frobenius automorphism. Its main invariant is therefore the dual characteristic polynomial of Frobenius, which is an element of 1 C t Z` Œt or 1 C tAv Œt . One proves that this polynomial is independent of the choice of ` or v and lies in 1 C tZŒt or 1 C tAŒt , respectively. More generally consider a family of abelian varieties or A-motives over a base scheme X of finite type over Fp . Following Weil, it is customary to attach an Lfunction to such a family by taking the product over all closed points x 2 X of the inverses of the above dual characteristic polynomials, stretched by the substitutions t 7! t deg.x/ . This L-function is a priori a power series in 1 C t ZŒŒt or 1 C tAŒŒt , respectively. The first part of the Weil conjecture asserts that the L-function of a family of abelian varieties is, in fact, a rational function of t . Inspired by the analogy with abelian varieties, Goss [22] conjectured that the Lfunction of a family of A-motives should be rational as well. This was proved in 1996 by Taguchi and Wan [44] for '-sheaves, which include A-motives as a special case. Their method was inspired by Dwork’s, with the field of p-adic numbers replaced by the field of Laurent series over Fp . Taguchi and Wan also proved a large portion of a conjecture by Goss on a certain analytic L-function similar to the -function of a scheme of finite type over Spec Z. We shall return to this point at the end of this introduction. With the paradigm of the development around the Weil conjecture, our main motivation for the present work was to develop a set of algebro-geometric and cohomological tools to give a purely algebraic proof of Goss’s rationality conjecture. With further
1 Introduction
3
applications in mind that are described at the end of this introduction, we develop our theory in great generality and much detail. For the same reasons as with p-adic étale cohomology in characteristic p, we do not obtain a good cohomology theory in the sense of Grothendieck. In particular we do not obtain an Rf Š or an internal Hom, and certainly there is no duality theory. But with the functors f , ˝L , Rf for proper f , and RfŠ for compactifiable f , as well as the trace formula we achieve a good half, enough for the application to L-functions. A trace formula for L-functions. One of our central results is a trace formula. Since it motivates much of the theory, we explain a basic version of it before giving more details on the individual sections of the book. From now on we abbreviate k ´ Fq , where q is a power of a prime p. Let A be a Dedekind domain that is finitely generated over k and whose group of units is finite, such as A D kŒt . Let X be a scheme of finite type over k and set C ´ Spec A. The Frobenius endomorphism of X relative to k, which on sections takes the form f 7! f q , is denoted X or simply . In what follows, schemes X , Y , etc., are thought of as base schemes for families of objects with coefficient ring A. We define these objects in terms of sheaves on X C , so that C plays the role of a Coefficient scheme. As a general convention, whenever tensor or fiber products are formed over k, the subscript k will be omitted. The basic objects of our theory are pairs F D .F ; /, where F is a coherent sheaf on X C and an OXC -linear homomorphism . id/ F ! F . Such pairs are called coherent -sheaves on X . For simplicity of exposition we assume that F is the pullback of a coherent sheaf F0 on X . To any such F we wish to assign an L-function as a product of pointwise L-factors. Let jXj denote the set of closed points of X . For any x 2 jX j let kx denote its residue field and dx its degree over k. Then the pullback Fx of F to x C inherits a homomorphism x W .x id/ Fx ! Fx ; hence it corresponds to a free kx ˝A-module of finite rank Mx together with a x ˝ idA -linear endomorphism x W Mx ! Mx . The iterate xdx of the latter is kx ˝ A-linear, and one knows that detkx ˝A .id t dx xdx jMx / D detA .id t x jMx /: This is therefore a polynomial in 1 C t dx AŒt dx . Since there are at most finitely many x 2 jXj with fixed dx , the following product makes sense: Definition 1.1. The naive L-function of F is Q Lnaive .X; F ; t / ´ detA .id t x jMx /1 2 1 C tAŒŒt : x2jXj
For the trace formula suppose first that X is proper over k. Then for every integer i the coherent cohomology group H i .X; F0 / is a finite dimensional vector space over k. Moreover, the equality F D pr 1 F0 yields a natural isomorphism H i .X C; F / Š H i .X; F0 / ˝ A. This is therefore a free A-module of finite rank. It also carries a natural endomorphism induced by ; hence we can consider it as a coherent -sheaf on
4
1 Introduction
Spec k, denoted by H i .X; F /. The first instance of the trace formula for L-functions then reads as follows: Q i Theorem 1.2. Lnaive .X; F ; t / D i2Z Lnaive .Spec k; H i .X; F /; t /.1/ . A standard procedure to extend this formula to non-proper X is via cohomology with compact support. For this we fix a dense open embedding j W X ,! Xx into a proper scheme of finite type over k. We want to extend the given F on X to a coherent -sheaf Fz on Xx without changing the L-function. Any extension whose z on Fzz is zero for all z 2 jXx n Xj has that property, and it is not hard to construct one. In fact, any coherent sheaf on Xx extending F , multiplied by a sufficiently high power of the ideal sheaf of Xx n X, does the job. However, there are many choices for this Fz , and none is functorial. Thus there is none that we can consider a natural extension by zero “jŠ F ” in the sense of -sheaves. Ignoring this for the moment, let us nevertheless provisionally regard H i .Xx ; Fz / as the cohomology with compact support Hci .X; F /. Then from Theorem 1.2 we obtain the following more general trace formula: Q i Theorem 1.3. Lnaive .X; F ; t / D i2Z Lnaive .Spec k; Hci .X; F /; t /.1/ . Since the factors on the right hand side are polynomials in 1 C tAŒt or inverses of such polynomials, the rationality of Lnaive .X; F ; t / is an immediate consequence. We hasten to add that the order of presentation of the above theorems is for expository purposes only. We actually first prove Theorem 1.3 when X is regular and affine over k and then generalize it to arbitrary X by devissage. The proof in the affine case is based on a trace formula by Anderson from [2]. While Anderson formulated it only for A D k, Taguchi and Wan [45] already noted that it holds whenever A is a field, and we extend it further. Also, the formula in [2] is really the Serre dual of the one in Theorem 1.3 and therefore avoids any mention of cohomology. The program in this book is to develop a full cohomological theory for -sheaves and to prove a relative trace formula for arbitrary compactifiable morphisms f W Y ! X . The formalism includes the inverse image functor f , the tensor product ˝, and the direct image under a proper morphism f . To define an extension by zero functor jŠ , we formally invert any morphism of coherent -sheaves on whose kernel and cokernel is nilpotent. The result is called a localization of the category of coherent -sheaves on X, with the same objects but other morphisms and more isomorphisms, which we call the category of crystals on X . In this category all the above Fz become naturally isomorphic, and any one of them represents the extension by zero jŠ F in the sense of crystals. This somewhat artificial construction is more than a cheap trick. It has several beneficial side effects; for example, it turns the pullback functor f , which is only right exact on coherent sheaves, into an exact functor on crystals. Thus crystals behave more like constructible sheaves than like coherent sheaves. The passage to crystals is perhaps the key innovation of this book compared to [2] or [45]. We construct a derived category of crystals and the corresponding derived functors f , ˝L and RfŠ for compactifiable f , and we calculate them in terms of the underlying -sheaves. We discuss flat crystals and more generally complexes of crystals
1 Introduction
5
of finite Tor-dimension, to which we can associate what we call crystalline L-functions. One of the main results is then the following relative trace formula generalizing Theorem 1.3. Theorem 1.4. Let f W Y ! X be a morphism of schemes of finite type over k. Suppose that A is an integral domain that is finitely generated over k. Then for any complex F of finite Tor-dimension of crystals on Y , one has
Lcrys .Y; F ; t / D Lcrys .X; RfŠ F ; t /: We now discuss in detail the individual chapters of the book. With the exception of Chapter 2 we follow their order of appearance. From now on X , Y , : : : will denote arbitrary noetherian schemes over k, and A will denote an arbitrary localization of a finitely generated k-algebra. In the examples A will typically be a finitely generated field, a Dedekind domain, or finite. Basic objects (Chapter 3). The starting point is the notion of a coherent -sheaf, which, due to its importance, we state again: Definition 1.5. A coherent -sheaf over A on X is a pair F ´ .F ; F / consisting of a coherent sheaf F on X C and an OXC -linear homomorphism F
. id/ F ! F : A homomorphism of coherent -sheaves F ! G on X is a homomorphism of the underlying sheaves ' W F ! G which is compatible with the action of . The category of coherent -sheaves over A on X is denoted by Coh .X; A/. It is an abelian A-linear category, and all constructions like kernel, cokernel, etc. are the usual ones on the underlying coherent sheaves, with the respective added by functoriality. Any '-sheaf on X in the sense of Taguchi and Wan is a coherent -sheaf whose underlying coherent sheaf is locally free, where A is a suitable Dedekind domain such as kŒt. In particular any Drinfeld A-module and any A-motive can be regarded as a coherent -sheaf. The next definition is a preparation for the concept of crystals: Definition 1.6. (a) A coherent -sheaf F is called nilpotent if the iterated homomorphism Fn W . n id/ F ! F vanishes for some n 0. (b) A homomorphism of coherent -sheaves is called a nil-isomorphism if both its kernel and cokernel are nilpotent. Nil-isomorphisms satisfy certain formal properties that make them a multiplicative system in the category Coh .X; A/. To any multiplicative system one associates a certain localized category in which all members of the multiplicative system become isomorphisms and which has a universal property, like the localization of a ring. Definition 1.7. The category Crys.X; A/ of A-crystals on X is the localization of Coh .X; A/ at the multiplicative system of nil-isomorphisms.
6
1 Introduction
The objects of Crys.X; A/ are those of Coh .X; A/, but the homomorphisms are different. Homomorphisms F ! G in Crys.X; A/ are diagrams F F 0 ! G in 0 Coh .X; A/ where the homomorphism F ! F is a nil-isomorphism, up to a certain equivalence relation. The result is again an A-linear abelian category, and a coherent -sheaf represents the zero crystal if and only if it is nilpotent. Functors (Chapter 4). Consider a morphism f W Y ! X and a k-algebra homomorphism A ! A0 . Then we have the following A-linear, resp. A-bilinear, functors: (i) inverse image: f W Coh .X; A/ ! Coh .Y; A/, (ii) tensor product:
˝ W Coh .X; A/ Coh .X; A/ ! Coh .X; A/,
(iii) change of coefficients:
˝A A0 W Coh .X; A/ ! Coh .X; A0 /,
(iv) direct image: f W Coh .Y; A/ ! Coh .X; A/ if f is proper. All these functors are defined by the corresponding operations on the underlying coherent sheaves, with .f id/ and .f id/ in place of f and f , and with the associated added by functoriality. All of them preserve nil-isomorphisms and therefore pass to functors between the corresponding categories of crystals. Next we consider an open embedding j W U ,! X with a closed complement i W Y ,! X. Then we can extend any coherent -sheaf F on U to a coherent -sheaf Fz on X such that i Fz is nilpotent (compare the discussion preceding Theorem 1.3). Any homomorphism between two such extensions, which is the identity on U , is a nil-isomorphism. Thus on crystals we obtain: Theorem 1.8. There is an exact A-linear functor (v) extension by zero: jŠ W Crys.U; A/ ! Crys.X; A/; uniquely characterized by the properties j jŠ D id and i jŠ D 0. Sheaf-theoretic properties (Chapter 4). Surprisingly, crystals behave more like constructible sheaves than like coherent sheaves. For instance: Theorem 1.9. For any morphism f W Y ! X the inverse image functor f on crystals is exact. In particular, let ix W x ,! X denote the natural embedding of a point of X . Then the stalk at x of a crystal F on X is defined as the crystal F x ´ ix F . The following result justifies this definition: Theorem 1.10. (a) A sequence of crystals is exact if and only if it is exact in all stalks. (b) The support of a crystal F , i.e., the set of points x 2 X for which ix F is non-zero (as a crystal!), is a constructible subset of X . Moreover crystals enjoy a rigidity property that is not shared by -sheaves, but well known for étale sheaves. We state it here in its simplest form:
1 Introduction
7
Theorem 1.11. Let f W Xred ,! X denote the canonical closed immersion of the induced reduced subscheme. Then the functors f and f are mutually quasi-inverse equivalences between Crys.X; A/ and Crys.Xred ; A/. Thus crystals extend in a unique way under infinitesimal extensions. In other words they grow in a prescribed way, as the name suggests. Derived categories and functors (Chapters 5 and 6). A major part of this book deals with the extension of the functors (i)–(v) to derived functors between derived categories of crystals. As usual, the construction of derived functors requires resolutions by acyclic objects, but these cannot be found within the categories of coherent -sheaves or crystals. We therefore consider the much larger categories of quasi-coherent sheaves and quasi-crystals, which are defined in an analogous way except that the ˇ underlying sheaves are only quasi-coherent. In these we dispose of Cech resolutions and resolutions by injectives. These categories and the functors between them are already studied in Chapters 3 and 4. For technical reasons we also need to consider the category of all filtered direct limits of coherent -sheaves, which we call ind-coherent -sheaves. This category is properly sandwiched between that of coherent -sheaves and quasi-coherent -sheaves, An analogous situation occurs for crystals and quasi-crystals. A large part of Chapter 5 is devoted to clarifying the relations between the derived categories of quasi-coherent, ind-coherent and coherent -sheaves and of the corresponding quasi-crystals and crystals. Once the comparison of derived categories is complete, it is relatively straightforward to construct the derived functors arising from (i)–(iv), namely derived pullback Lf , which equals f on the derived category of crystals, derived tensor product ˝L , and derived direct image under a proper morphism Rf . The derived direct image with compact support RfŠ for a compactifiable morphism f D fN B j is then defined as the composite of the exact functor jŠ from (v) with the derived direct image under a proper morphism RfN . We establish all compatibilities between these functors that could reasonably be expected, including the proper base change theorem and the projection formula. The proofs of these formulas make essential use of the universal properties of derived functors. We also show that the individual derived functors Li f and H i . ˝L / and Ri f for proper f can be computed in the ‘naive’ way by taking the corresponding derived functors of (quasi-) coherent sheaves with the respective added by functoriality. This is important for calculations, but would not serve as a very good definition of these functors, because one would still not know the true derived functors of crystals. Categorical preparations (Chapter 2). We also have to deal with a number of categorical issues in relation to derived categories. Let us give three examples: (i) There are at least two natural ways to go from the category of quasi-coherent -sheaves to the derived category of quasi-crystals. One can first localize at nil-isomorphisms and then derive the abelian category of quasi-crystals or, alternatively, one may first pass to the derived category of quasi-coherent -sheaves and then localize at nil-quasi-isomorphisms. We show that the two procedures agree.
8
1 Introduction
(ii) In our constructions of derived functors, we want to stay within the setting of quasi-coherent -sheaves and quasi-crystals. The reader familiar with the approach in Hartshorne [25] may remember that the construction of the derived inverse image and the derived tensor product uses categories of non-quasi-coherent sheaves of modules. We bypass this by giving another criterion for the existence of derived functors. (iii) The construction in [25] of a derived functor Rf of coherent sheaves for proper f depends on a comparison theorem between the derived category of coherent sheaves and the full triangulated subcategory of the derived category of quasi-coherent sheaves with coherent cohomology. The proof of this comparison theorem uses the fact that every quasi-coherent sheaf is a filtered direct limit of coherent sheaves. As mentioned above, this is not so for -sheaves and (quasi-) crystals. Nevertheless we prove the corresponding comparison theorem, although its proof is surprisingly intricate. Chapter2 also contains a brief review of categorical foundations and further preparations. Flatness (Chapter 7). Traces and characteristic polynomials and therefore L-functions can be defined only for locally free modules. The corresponding notion for crystals is flatness. Definition 1.12. A crystal F is flat if the functor F ˝
on crystals is exact.
Clearly any crystal whose underlying coherent sheaf is locally free is flat. More importantly, any crystal on X whose underlying sheaf is a pullback of a coherent sheaf under the projection pr 1 W X C ! X is flat. Some basic properties on functorialities and on stalks are: Theorem 1.13. (a) Flatness of crystals is preserved under inverse image, tensor product, change of coefficients and extension by zero. (b) A crystal is flat if and only if all its stalks are flat.
(c) If f is compactifiable and F is a bounded complex of flat crystals on Y , then RfŠ F is represented by a bounded complex of flat crystals. Note that by (a) the analog of assertion (c) for inverse image, tensor product and change of coefficients also holds. We have observed that any crystal represented by a coherent -sheaf whose underlying sheaf is locally free is flat. For the definition of an L-function we would wish to go the other way and represent every flat crystal by such a -sheaf. It is a highly non-trivial matter to decide when, precisely, that is possible. As the L-function is a product of Euler factors attached to points x 2 X with finite residue field, it suffices to address this question over a single such point. But even then the question does not always have a positive answer. We have, however, the following important special case: Theorem 1.14. Suppose that x D Spec kx for a finite field extension kx of k and that A is artinian. Then for any A-crystal F on x we have:
1 Introduction
9
(a) The crystal F can be represented by a coherent -sheaf F 0 whose homomorphism W . id/ F 0 ! F 0 is an isomorphism. (b) The representative in (a) is unique up to unique isomorphism. We denote it by F ss and call it the semisimple representative of F . (c) The assignment F 7! F ss is functorial. (d) The crystal F is flat if and only if the coherent sheaf underlying F ss is locally free. (Crystalline) L-functions (Chapters 8 and 9). In view of Theorem 1.14 (d) – and its failure for general A – we first assume that A is artinian. Consider a flat crystal F over a scheme X of finite type over k. As before let jX j denote the set of closed points of X . For any x 2 jX j let dx denote its degree over k and ix W x ,! X its natural embedding. Then the stalk F x ´ ix F is again flat by Theorem 1.13, and so the coherent sheaf Fx;ss underlying its semisimple representative F x;ss is locally free. Choose a locally free coherent sheaf G on xC such that Fx;ss ˚G is free, and turn it into a -sheaf G by setting G ´ 0. Then the local factor at x of the crystalline L-function of F is defined as Lcrys .x; F x ; t / ´ Lnaive .x; F x;ss ˚ G ; t / 2 1 C t dx AŒŒt dx with Lnaive from Definition 1.1. One easily shows that this is independent of the choice of G . As there are at most finitely many x 2 jX j with fixed dx , the following product makes sense: Q crys Lcrys .X; F ; t / ´ L .x; F x ; t / 2 1 C tAŒŒt : x2jXj
This definition is extended to any bounded complex F of flat A-crystals on X by defining its crystalline L-function as Q crys i Lcrys .X; F ; t / ´ L .x; F i ; t /.1/ 2 1 C tAŒŒt : i2Z
This definition is invariant under quasi-isomorphisms and thus naturally extends to any complex of crystals of bounded Tor-dimension. Since the L-function of a crystal must be independent of the representing -sheaf, there is essentially no other sensible definition. In particular, before taking the characteristic polynomial one must purge any -subsheaf or quotient on which is nilpotent, because any such -sheaf represents the zero crystal. This has the somewhat unfortunate consequence that the naive L-function of a coherent -sheaf whose underlying coherent sheaf is free may differ from the crystalline L-function of the associated crystal. For example, if on F is nilpotent, the naive L-function can be any polynomial with constant term 1 and nilpotent higher coefficients, while the crystalline L-function is necessarily 1. The problem disappears only when A is reduced. It was also the reason for the name ‘naive L-function’.
10
1 Introduction
When A is reduced, the crystalline L-function satisfies all the usual cohomological formulas (except duality). For arbitrary artinian A, finite or not, these formulas hold only up to ‘unipotent factors’. In some sense these factors correspond to missing nilpotent -sheaves; hence this defect is an unavoidable consequence of the very definition of crystals. For arbitrary artinian A let nA denote its nilradical, i.e., the ideal consisting of all nilpotent elements of A. Any polynomial in 1 C t nA Œt will be called unipotent. We regard two power series P , Q 2 1 C tAŒŒt as equivalent and write P Q if the quotient P =Q is a unipotent polynomial. One easily checks that this defines an equivalence relation on 1 C tAŒŒt . Based on Theorem 1.13 (c), we can now state a main result of Chapter 9: Theorem 1.15. Let f W Y ! X be a morphism of schemes of finite type over k. Then for any complex F of crystals on Y of bounded Tor-dimension, we have
Lcrys .Y; F ; t / Lcrys .X; RfŠ F ; t /: If A is reduced, then equality holds. If we apply this to the structure morphism X ! Spec k, the right hand side becomes a finite alternating product of polynomials. We therefore deduce: Corollary 1.16. The crystalline L-function of a complex of crystals of bounded Tordimension on a scheme of finite type over k is a rational function of t . We now revoke the assumption that A is artinian. Let QA denote the direct sum of the localizations of A at all minimal prime ideals. Since A is noetherian, this QA is artinian. Thus for any complex F of A-crystals of bounded Tor-dimension on a scheme X of finite type over k, we can consider the L-function of the associated complex of QA -crystals
Lcrys .X; F ˝AL QA ; t / 2 1 C tQA ŒŒt : On the other hand, any good theory of L-functions should be invariant under change of coefficients. We therefore attempt to extract a crystalline L-function of F from L that of F ˝A QA . An obvious necessary condition for this is that the canonical homomorphism A ! QA is injective. But additional conditions are needed to guarantee that the coefficients actually lie in A. For this we introduce a notion of good coefficient rings. Particular examples of these are normal integral domains and artinian rings. For any good coefficient ring A we show that Lcrys .X; F ˝AL QA ; t / lies in 1 C tAŒŒt , crys and so we take it as the definition of L .X; F ; t /. All formulas for crystalline L-functions, such as the trace formula in Theorem 1.15, extend directly from artinian rings to good coefficient rings. Moreover, for any ring homomorphism W A ! A0 denote the induced group homomorphism 1 C tAŒŒt ! 1 C tA0 ŒŒt again by . Then the change of coefficients formula for crystalline Lfunctions states:
1 Introduction
11
Theorem 1.17. For any homomorphism between good coefficient rings W A ! A0 and any complex F of A-crystals of bounded Tor-dimension on a scheme X of finite type over k, we have .Lcrys .X; F ; t // Lcrys .X; F ˝AL A0 ; t /:
The case of finite A. Our original motivation to study -sheaves and crystals stemmed from the theory of Drinfeld modules and A-motives. However, in relation with Artin– Schreier theory, the case of finite A was discussed in the literature already quite some time ago, cf. [29, § 4] or [SGA4 12 , § 3]. Let A be finite, and let F be a coherent -sheaf over A on X . To its underlying coherent sheaf one can canonically assign a sheaf FeK t on the small étale site over X . By functoriality, it inherits an endomorphism from F . The subsheaf ".F / of FeK t of -invariant sections is an étale sheaf of A-modules, and the assignment F 7! ".F / is functorial. Since " preserves the property of being noetherian, it takes its image in the category Étc .X; A/ of constructible étale sheaves of A-modules on X . Moreover " is left exact and zero on nilpotent -sheaves. Therefore it maps nil-isomorphisms to isomorphisms, and so passes to a functor "N W Crys.X; A/ ! Étc .X; A/;
F 7! "N.F /:
The following is the main result of Chapter 10. In an important special case its first part is due to Katz, cf. [29, Theorem 4.1]. Theorem 1.18. Suppose that A is finite. Then the functor "N is an equivalence of categories. It commutes with all functors and derived functors mentioned above, and it preserves flatness. Finally, the crystalline L-function of a flat crystal agrees with the L-function of the associated étale sheaf. Theorem 1.18 indicates that our theory for general A may be regarded as a global counterpart to the theory of étale p-torsion sheaves. To be more specific, let A be a Dedekind domain that is finitely generated over k and whose group of units is finite. Then for any maximal ideal p of A, the change of coefficients functor for A ! A=p composed with the functor "N above induces a functor Š Crys.X; A/ ! Crys.X; A=p/ ! Étc .X; A=p/:
Combining all of the above results, it follows that this functor has all compatibilities that one could hope for. Relations to other work. For smooth schemes X , a theory similar to ours was developed independently by Emerton and Kisin in part I of [18]. The relation between the two theories is basically given by duality in the derived category of bounded complexes of coherent sheaves as described in [25]. This duality transforms id-linear homomorphisms into 1 id-linear homomorphisms and vice versa. The precise correspondence is presently worked out in detail by Blickle and the first-named author.
12
1 Introduction
Emerton and Kisin go beyond our setting when, in part II of their work, they extend their theory to p-adic coefficients and prove a trace formula in that context. The present theory is a natural extension of the theory of A-motives. Our main motivation was to prove a trace formula for L-functions of families of A-motives. The motivation of Emerton and Kisin was to prove a Riemann–Hilbert correspondence, which bears much resemblance to our last Chapter 10, and a trace formula in the padic context, cf. [17]. Their theory has applications to p-adic unit root crystals and local cohomology in characteristic p. Motivated by our theory, they also developed a full-fledged formalism of coefficients similar to ours and a trace formula in that context. Further applications. The L-functions in the present book are analogues of the Lfunctions of p-adic and `-adic sheaves on varieties in characteristic p. In [22] Goss also defined analytic L-functions of families of A-motives which are analogues of the L-functions of `-adic sheaves on schemes of finite type over Spec Z. He conjectured that these L-functions extend to entire resp. meromorphic functions on a domain in characteristic p which replaces the usual complex plane, and gave first evidence for this in [22]. In the case A D Fq Œt this conjecture was proved by Taguchi and Wan in [44] and [45]. The special values at negative integers of Goss’s analytic L-functions are rational functions and turn out to have an interpretation in terms of the cohomology theory developed here. In [4] explicit cohomological expressions are used to prove that the degrees of these special values at negative integers n grow at most logarithmically in n. From this, in [4] and, independently, in [23], the remaining conjectures of Goss on meromorphy and entireness were deduced. In an interesting recent preprint [31], V. Lafforgue uses and extends parts of the present theory to study special values of Goss’s L-functions at critical values. In [5] the theory of the present book was applied to Drinfeld modular forms. The moduli scheme of Drinfeld A-modules of rank 2 with a full level n-structure carries a canonical locally free -sheaf F n of rank 2, which corresponds to the universal Drinfeld module. Following the classical case of elliptic modular forms, one is naturally led to studying the cohomology of the k-th symmetric power Symk F n for any k 0. In [5] the ‘analytic realization’ of this cohomology is shown to be dual to the space of cuspidal Drinfeld modular forms of weight k C 2 and level n. This isomorphism is equivariant for naturally defined Hecke operators. On the other hand, the ‘étale realization’ of this cohomology yields abelian Av -adic Galois representation for any place v of A. Combining both realizations, any cuspidal Drinfeld Hecke eigenform yields a onedimensional v-adic Galois representation for any v, where the correspondence is given by an Eichler–Shimura type relation. This is analogous to constructions by Eichler– Shimura and Deligne attaching `-adic Galois representation to cuspidal elliptic modular forms. For a survey on both developments we refer to [6].
Chapter 2
Categorical preparations
In commutative algebra the word localization refers to the process of constructing new rings or modules from old ones, such that certain previously specified ring elements become invertible. The same term is used when, given a category and a suitable collection of morphisms in it, a new category is constructed such that the selected morphisms correspond to isomorphisms there. This construction plays an essential role in the definition of derived categories. Another special case is the localization of an abelian category such that all objects of a given subcategory become zero. We will need to clarify the interplay between these notions and to compare the derived categories of the original abelian category and its localization. In Sections 2.2–2.7 we discuss localization in various instances; most of this material stems from the existing literature such as [20], [27], [46], [47]. For better readability we briefly review the relevant definitions and results concerning abstract, additive, abelian, and derived categories and derived functors in Sections 2.1–2.7; these are all very well known and described in standard textbooks, e.g. [34], [47, Appendix A]. The remaining two sections of this chapter are devoted to different problems. In Section 2.8 we generalize a well-known criterion [25, Chapter I, Theorem 5.1] on the existence of derived functors. In Section 2.9 we study the natural functor D .B/ ! D .A/ associated to an embedding of abelian categories B A. Its image is contained .A/ consisting of those objects all of whose in the full triangulated subcategory DB cohomology lies in B. If the induced functor D .B/ ! DB .A/ is an equivalence of categories, many constructions on D .B/ can be carried out using the greater freedom of D .A/. It is therefore important to dispose of useful sufficient conditions for this equivalence. Again the known conditions, e.g. from [25, Chapter I], were not enough for all our purposes and so, in Section 2.9, we derive some further results in this direction.
2.1 Categories A category C consists of a class of objects of C, denoted Ob.C/ or just C, a set HomC .M; N / for any two objects M , N 2 C, whose elements are called morphisms from M to N , and a map HomC .M; N / HomC .N; L/ ! HomC .M; L/;
.f; g/ 7! g B f;
for any three objects M , N , L 2 C, called composition, which is associative and possesses (unique) two-sided identity elements idN 2 HomC .N; N /. The last conditions
14
2 Categorical preparations
mean that h B .g B f / D .h B g/ B f and idN B f D f and g D g B idN whenever these relations make sense. For ease of notation we often abbreviate gf ´ g B f . A morphism f 2 HomC .M; N / is usually denoted by a solid arrow f W M ! N . It is called an isomorphism if it possesses a two-sided inverse, i.e., if there exists a morphism g W N ! M satisfying gf D idM and fg D idN . In that case we write f W M ! N and M Š N and call M and N isomorphic. This defines an equivalence relation on the class of objects of C. A morphism f W M ! N is called a monomorphism if the map HomC .L; M / ! HomC .L; N /, g 7! fg is injective for every object L 2 C. Monomorphisms are usually depicted by hooked arrows M ,! N . Two monomorphisms f W M ,! N and f 0 W M 0 ,! N are called equivalent if there exists an isomorphism u W M ! M0 0 such that f D f u. An equivalence class of monomorphisms is called a subobject of N , but the distinction between the two notions is usually somewhat blurred. Dually f W M ! N is called an epimorphism if the map HomC .N; L/ ! HomC .M; L/, g 7! gf is injective for every object L 2 C. Epimorphisms are usually depicted by two-headed arrows M N . Two epimorphisms f W M N and f 0 W M N 0 are called equivalent if there exists an isomorphism u W N ! N0 0 such that f D uf . An equivalence class of epimorphisms is called a quotient object of M ; though again the distinction is blurred. A category is called small if the class of its objects is a set. It is called essentially small if the class of its objects possesses a set of representatives under isomorphism. Many standard categories are not small, but many categories with finiteness conditions are essentially small. For example, the category of coherent sheaves of OX -modules on a noetherian scheme X is essentially small, but not small. A category C0 is called a subcategory of a category C if Ob.C0 / Ob.C/ and HomC0 .M; N / HomC .M; N / for any two objects M , N 2 C0 . The subcategory is called full if HomC0 .M; N / D HomC .M; N / for all M , N 2 C0 . Any subclass of Ob.C/ is the class of objects of a unique full subcategory of C. The opposite Copp of a category C is the same category with all arrows reversed, i.e., with HomCopp .M; N / D HomC .N; M / and the only natural notion of composition in Copp . A functor of categories F W C ! D consists of a map Ob.C/ ! Ob.D/, M 7! FM , and maps HomC .M; N / ! HomD .FM; F N /, f 7! F .f /, for any two objects M , N 2 C, such that F .gf / D F .g/F .f / for any two composable morphisms f , g, and F .idN / D idF N for any N 2 C. A functor F W C ! D is also called a covariant functor from C to D, and a functor Copp ! D or C ! Dopp is called a contravariant functor from C to D. We will follow the standard convention and mean by a contravariant functor from C to D a functor Copp ! D. Two functors F
G
!D ! E possess an evident composite GF W C ! E. C Given two functors F , F 0 W C ! D, a natural transformation or morphism of functors ˛ W F ! F 0 consists of a map ˛.M / 2 HomD .FM; F 0 M / for any object M 2 C such that F 0 .f /˛.M / D ˛.N /F .f / for all morphisms f 2 HomC .M; N /. An isomorphism of functors or a functorial isomorphism is a morphism of functors
2.1 Categories
15
˛ W F ! F 0 such that ˛.M / is an isomorphism for every M 2 C. It then possesses a unique two-sided inverse F 0 ! F . A functor F W C ! D is called an equivalence of categories if it possesses a two-sided quasi-inverse, i.e., a functor G W D ! C such that GF W C ! C and F G W D ! D are isomorphic to the respective identity functors. A functor F W C ! D is called faithful if HomC .M; N / ! HomD .FM; F N /, f 7! F .f / is injective for any two objects M , N 2 C. It is called fully faithful if this map is bijective for any two objects M , N 2 C. The essential image of F is the class of all objects in D that are isomorphic to FM for some M 2 C. If the essential image contains all objects of D, we call F essentially surjective. A functor is an equivalence of categories if and only if it is fully faithful and essentially surjective. Two (covariant) functors F W C ! D and G W D ! C are called adjoint, and F is called left adjoint to G and G right adjoint to F , if there is a bijection HomD .FM; N / ! HomC .M; GN /;
u 7! ˛.u/;
for any objects M 2 C and N 2 D which is functorial in M and N , i.e., which satisfies f
u
g
˛.guF .f // D G.g/˛.u/f for all morphisms M 0 ! M in C and FM ! N ! N 0 in D. The bijection ˛ is tacitly understood in any given situation. The maps ˛.idF M / 2 HomC .M; GFM / for all M 2 C constitute a morphism of functors idC ! GF called adjunction morphism. Dually, the maps ˛ 1 .idGN / 2 HomD .F GN; N / for all N 2 D constitute a morphism of functors F G ! idD , also called adjunction morphism. On replacing C by Copp one obtains the analogous concepts for contravariant functors. Adjunction of functors and the associated adjunction morphisms play an important role in many categorical constructions. A commutative diagram in a category C is a functor from a small category I to C. Finite commutative diagrams are often depicted by labeled nodes and arrows in the plane, where identity morphisms and composites of other morphisms are mostly left out. The adjective commutative then means that for any two sequences of arrows beginning in a node M and ending in a node N , the two resulting composite morphisms M ! N are equal. A commutative diagram I ! C is also called a direct system in C. It is often written with indices as a collection of objects Mi and morphisms 'ij W Mi ! Mj in C. A morphism from the direct system .Mi ; 'ij / to an object N 2 C is a collection of morphisms i W Mi ! N satisfying j 'ij D i for every morphism 'ij . A direct limit of the system is an object M 2 C together with a morphism . i / from .Mi ; 'ij / to M , such that any morphism from .Mi ; 'ij / to any object N 2 C is the composite of . i / with a unique morphism M ! N . If a direct limit exists, the universal property implies that it is unique up to unique isomorphism. It is denoted by lim Mi , where the !i index category I and the morphisms 'ij and i are tacitly understood. The direct limit is functorial in .Mi ; 'ij / to the extent that it exists. Thus if all direct limits of diagrams I ! C exist, any morphism of diagrams (i.e., of functors) .Mi ; 'ij / ! .Mi0 ; 'ij0 / induces a unique morphism lim Mi ! lim Mi0 , defining a functor from the category !i !i of such diagrams to C that is unique up to unique isomorphism.
16
2 Categorical preparations
Some special cases of direct limits are the following. When the system contains no transition morphisms besides the identity morphisms idMi , then lim Mi is called the !i coproduct of the objects Mi . The coproduct of an empty collection of objects is called an initial object of C. The direct limit M of a system of the form M1 M0 ! M2 is called the pushout of M1 and M2 over M0 , and the commutative diagram / M1 M0 M2
/M
is then called a pushout diagram. Along another vein, a direct system .Mi ; 'ij / is called filtered if for any indices i and j there exist an index k and morphisms 'ik W Mi ! Mk and 'j k W Mj ! Mk in the system, and for any two morphisms in the system with the same source and target 'ij , 'ij0 W Mi ! Mj there exist an index k and a morphism in the system 'j k W Mj ! Mk such that 'j k 'ij D 'j k 'ij0 . The direct limit of a filtered direct system is called a filtered direct limit. Filtered direct limits become interesting when the index category is infinite. Applying the above concepts to the opposite category Copp , i.e., with all arrows reversed, one obtains the corresponding dual notions of inverse system, inverse limit, product, final object, pullback (diagram), filtered inverse system, and filtered inverse limit. For any objects Mi 2 C there is a natural morphism from their coproduct to their product, provided these exist. If this morphism is an isomorphism, we call the (co)product the biproduct of the Mi. Equivalently, with Q it is an object M of C together functorial isomorphisms Hom M; N Š N; M Š Hom .M ; N / and Hom i C C C i Q Hom .N; M / for all objects N 2 C. The biproduct of an empty collection of i C i objects of C is by definition simultaneously an initial and a final object of C, if that exists. It is called a zero object of C and denoted by 0. An additive category A is a category whose morphism sets HomA .M; N / are endowed with the structure of an abelian group, written additively, such that composition is bilinear and arbitrary finite biproducts exist. In particular an additive category contains a zero object. A morphism in an additive category is often called a homomorphism. L A coproduct in an additive category is also called a direct sum and denoted by i Mi . The opposite Aopp of an additive category A is in an evident way again an additive category. A functor between two additive categories F W A ! B is called additive if the map HomA .M; N / ! HomB .FM; F N /, f 7! F .f / is a group homomorphism for all M , N 2 A. An additive functor automatically commutes with finite direct sums and maps zero objects to zero objects. An additive subcategory A0 of an additive category A is a subcategory whose inclusion maps HomA0 .M 0 ; N 0 / ,! HomA .M 0 ; N 0 / are additive for all objects M 0 , N 0 2 A0 . An additive subcategory is called closed under taking direct summands if for any object M 0 2 A0 and any decomposition M 0 D M10 ˚ M20 in A, the direct summands M10 , M20 again lie in A0 .
2.2 Localization
17
2.2 Localization Consider a category C. In the following will denote a collection of morphisms in C. To distinguish morphisms in from arbitrary morphisms in C we will denote them by double arrows H). Definition 2.2.1. A collection of morphisms in a category C is called a multiplicative system if it satisfies the following three axioms. (a) is closed under composition and contains the identity morphism for every object of C. (b) For any t W N 0 ) N in and any f W M ! N in C, there exist s W M 0 ) M in and f 0 W M 0 ! N 0 in C such that the following diagram commutes: M0
f0
s
M
/ N0 t
f
/ N.
The same statement with all arrows reversed is also required. (c) For any pair of morphisms f; g W M ! N the following are equivalent: (i) There exists s 2 such that sf D sg. (ii) There exists t 2 such that f t D gt . From any such multiplicative system one constructs a new category 1 C as follows. First, its objects are the same as those of C. Next any diagram f
s
M ! L ( HN in C with s 2 is called a right fraction from M to N . Two right fractions fi
si
M ! Li (H N are called equivalent if they occur in a commutative diagram = L1 \d AAA AAAAs1 || | AAAA | | | AAA || f s / s k MB LO N BB }}}} BB } } } B }}}}s f2 BB! z }}} 2 L2 f1
in C with s 2 . Using the axioms 2.2.1 one easily shows that this defines an equivalence relation on the class of all right fractions from M to N . One wants the equivalence classes to correspond to the morphisms in 1 C. Here one encounters a set-theoretical
18
2 Categorical preparations
problem, because Hom 1 C .M; N / must be a set, while the class of right fractions from M to N may be too large. The construction is permitted if and only if the following condition holds: Definition 2.2.2. is called essentially locally small if for all M and N the class of right fractions from M to N possesses a set of representatives. Due to the symmetry in the definition of multiplicative systems one can also work with left fractions M ( L ! N . The axioms 2.2.1 imply that every equivalence class of left fractions corresponds to a unique equivalence class of right fractions, and vice versa, so that one obtains the same result. Using axiom 2.2.1 (b) one defines the composition of right fractions. Thus under the condition 2.2.2 one obtains a welldefined category 1 C, called the localization of C by . There is also a natural localization functor q W C ! 1 C mapping any object f
to itself and any morphism M ! N to the equivalence class of the right fraction f
id
M !N ( H N . To distinguish morphisms in 1 C from those in C we often denote them by dotted arrows K. If a morphism in 1 C is the image of a morphism in C, by abuse of notation we also denote it by a solid arrow. We will often abbreviate x ´ 1 C when is clear from the context. C Theorem 2.2.3 ([25, Chapter I, §3] or [47, Theorem 10.3.7]). Let be an essentially locally small multiplicative system in a category C. x (a) For every s 2 the morphism q.s/ is an isomorphism in C. (b) For any category D and any functor F W C ! D such that F .s/ is an isox ! D such that morphism for all s 2 , there exists a unique functor Fx W C x F D F q. x and q is an additive functor. In this (c) If C is an additive category, then so is C case, if the functor F in (b) is additive, then so is Fx. x ! D, the natural map xW C (d) For any category D and any two functors Fx, G x ! Hom.Fxq; Gq/ x Hom.Fx; G/ is bijective. x and q up to equivalence of categories. The properties 2.2.3 (a)–(b) characterize C x One could use these properties to define C, but we prefer to work with explicit objects, so we use the direct construction above. Proposition 2.2.4. Let C and C0 be categories with essentially locally small multiplicative systems and 0 , respectively. Consider a functor F W C0 ! C which maps x0 ! C x such that the following diagram 0 to . Then there exists a unique functor Fx W C
2.2 Localization
19
commutes: /C
F
C0
q
q
x0 C
/C x.
Fx
Proof. This is a direct consequence of Theorem 2.2.3 (b). Proposition 2.2.5. Let C and C0 be categories with essentially locally small multiF
G
!C ! C0 . plicative systems and 0 , respectively, and consider functors C0 x
x
F G x x 0 as in 2.2.4, and F is left adjoint to x0 !C !C (a) If F and G induce functors C x x G, then F is left adjoint to G. H
(b) Consider a third functor C ! C such that F , H , and GH induce functors between the localized categories, as in the commutative diagram C0
F
/C
H
q
q
x0 C
Fx
/C x
x H
/C
G
/ C0
q
q
/C x
x 0. 7C
GH
Then any functorial isomorphism HomC .F .M 0 /; H.N // Š HomC0 .M 0 ; GH.N // induces a functorial isomorphism x .q.N /// Š Hom x 0 .q.M 0 /; GH.q.N ///: HomCx .Fx.q.M 0 //; H C Proof. To prove (b) observe that the given functorial isomorphism is a natural transformation between the two functors HomC .F . /; H. //;
HomC0 . ; GH. // W .C0 /opp C ! Sets:
Applying Theorem 2.2.3 (d) twice to these functors, the isomorphism corresponds to a unique isomorphism between the functors x . //; HomCN .Fx. /; H
N 0 /opp C N ! Sets: HomCN 0 . ; GH. // W .C
This proves (b), and (a) is the special case of (b) with H D id. Definition 2.2.6. A multiplicative system is called saturated if, in addition to 2.2.1 (a)–(c), it also satisfies the condition
20
2 Categorical preparations
(d) For any morphism f W M ! N in C, if there exist g W N ! N 0 and h W M 0 ! M such that gf and f h are in , then f is in . f
g
If if saturated, one easily shows that for any morphisms L !M ! N , if two of f , g, and gf are in , then so is the third. This property is useful in simplifying arguments. Definition 2.2.7 (Compare [25, Chapter I, § 5]). Let C be a category with an essentially locally small multiplicative system . A full subcategory C0 of C is called localizing (with respect to ) if the following conditions hold: (a) 0 ´ \ C0 is a multiplicative system in C0 . (b) 0 is essentially locally small. x0 ! C x as in 2.2.4 is fully faithful. (c) The induced functor C x 0 can be proved by calculating inside C. x The condition (c) means that identities in C 0 Note also that, if C is localizing, the functor in (c) is an equivalence of categories if and only if it is essentially surjective. We will use the following criteria for a subcategory to be localizing: Proposition 2.2.8. Let C be a category with an essentially locally small multiplicative system , and let C0 be a full subcategory of C such that 0 ´ \ C0 is a multiplicative system in C0 . (a) Assume that for any s W M 0 ) M in with M 0 2 C0 there exists h W M ! N 0 in C with N 0 2 C0 and hs 2 . Then C0 is localizing. (a0 ) Same as (a) with all arrows reversed. (b) Assume that for any M in C there exists a morphism M ) M 0 in with M 0 2 C0 . x0 ! C x is an equivalence of categories. Then C0 is localizing and the functor C (b0 ) Same as (b) with all arrows reversed. Proof. These are easy exercises and are omitted, as in [25, Chapter I, Proposition 3.3].
2.3 Abelian categories Consider an additive category A and a morphism f 2 HomA .M; N /. A kernel of f is an object K 2 A together with a morphism i 2 HomA .K; M /, such that for any object L and any morphism u 2 HomA .L; M / with f u D 0 there exists a unique morphism v 2 HomA .L; K/ satisfying u D iv. If such a pair .K; i / exists, it is determined up
2.3 Abelian categories
21
to unique isomorphism. Usually the object K is denoted by ker.f / and the morphism i is tacitly understood. The same definition with all arrows reversed yields the cokernel of f . This is an object coker.f / 2 A together with a morphism q 2 HomA .N; coker.f //, such that for any object L and any morphism u 2 HomA .N; L/ with uf D 0 there exists a unique morphism v 2 HomA .coker.f /; L/ satisfying u D vq. The kernel of the natural morphism N ! coker.f / is called the coimage of f and denoted by coim.f / if it exists. Dually, the cokernel of the natural morphism ker.f / ! M is called the image of f and denoted by im.f / if it exists. By construction these objects come with natural morphisms q W M coim.f / and i W im.f / ,! N . The universal properties imply that there is a unique morphism W coim.f / ! im.f / such that iq D f . An abelian category is an additive category in which every morphism f possesses a kernel and cokernel, and consequently also a coimage and an image, such that the canonical morphism coim.f / ! im.f / is an isomorphism. An additive category is abelian if and only if its opposite category is abelian, because the notions of kernel and cokernel, resp. of image and coimage, are exchanged under A $ Aopp . Let A be an abelian category. A morphism in A is a monomorphism if and only if its kernel is zero, i.e., a zero object of A. Dually, a morphism in A is an epimorphism if and only if its cokernel is zero. A morphism in A is an isomorphism if and only if both its kernel and cokernel are zero. f
g
A sequence of morphisms L ! M ! N in A with im.f / D ker.g/ is called exact. A longer bounded or unbounded sequence ! Mi1 ! Mi ! MiC1 ! is exact if every 3-term subsequence is exact. An exact sequence of the form 0 ! M 0 ! M ! M 00 ! 0 is called a short exact sequence. In such a short exact sequence M 0 ,! M is a monomorphism and M M 00 is an epimorphism, and the object M is called an extension of M 00 by M 0 . Every abelian category satisfies the usual diagram lemmas for abelian groups, in particular the snake lemma, the 5-lemma, and the 33-lemma. Assertions in an abelian category that involve diagram chasing can be proved as if the objects were composed of elements, and one freely uses terminology based on this idea. For instance the pullback of two monomorphisms M1 ,! M and M2 ,! M is also called the intersection of the corresponding subobjects. The cokernel of a monomorphism M 0 ,! M is also called the quotient of M by M 0 and denoted by M=M 0 . A covariant additive functor between two abelian categories F W A ! B is called left exact if it maps every exact sequence of the form 0 ! M 0 ! M ! M 00 in A to an exact sequence 0 ! FM 0 ! FM ! FM 00 in B. It is called right exact if it maps every exact sequence of the form M 0 ! M ! M 00 ! 0 in A to an exact sequence FM 0 ! FM ! FM 00 ! 0 in B. It is called exact if it is both left and right exact, or equivalently if it maps every exact sequence M 0 ! M ! M 00 in A to an exact sequence FM 0 ! FM ! FM 00 in B. A contravariant additive functor F W A ! B between abelian categories is left
22
2 Categorical preparations
exact, right exact, resp. exact if the corresponding covariant functor Aopp ! B has the same property. For instance, a contravariant functor F W A ! B is left exact if it maps every exact sequence of the form M 0 ! M ! M 00 ! 0 in A to an exact sequence 0 ! FM 00 ! FM ! FM 0 in B. Definition 2.3.1. A full subcategory of an abelian category which is closed under taking subobjects, quotients, extensions, and isomorphisms, is called a Serre subcategory. Proposition 2.3.2 ([47, Examples 10.3.2]). For any abelian category A there is a bijection between the class of saturated multiplicative systems and the class of Serre subcategories B. Explicitly, given B one defines as the class of those morphisms whose kernel and cokernel are in B. Conversely, given one defines B as the full subcategory consisting of those objects M of A such that 0 ! M is in . Definition 2.3.3. An abelian category is called locally small if for every object the equivalence classes of subobjects form a set. Proposition 2.3.4. If A is locally small, the multiplicative system associated to any Serre subcategory B of A is essentially locally small. Moreover the localized category x ´ 1 A is abelian and the functor q W A ! A x is exact. A x by Proof. See [42, Theorem 4.3.3] or [47, Examples 10.3.2]. The construction of A localization is equivalent to that of the quotient category A=B in [20]. Proposition 2.3.5. Let A, B, and be as in Proposition 2.3.4, and consider a morphism f W M ! N in A. (a)
x is zero if and only if M 2 B. (i) The object q.M / 2 A (ii) The morphism q.f / is zero if and only if im f 2 B. (iii) The morphism q.f / is a monomorphism if and only if ker f 2 B. (iv) The morphism q.f / is an epimorphism if and only if coker f 2 B. (v) The morphism q.f / is an isomorphism if and only if both ker f , coker f 2 B.
(b) If q.f / is an isomorphism, then f can be factored as f D gh where h is an epimorphism with kernel in B and h is a monomorphism with cokernel in B. x is isomorphic to the image of a short exact (c) Every short exact sequence in A sequence in A. x is isomorphic to the image of a complex in A. (d) Every complex in A Proof. Straightforward. See also [20, Chapter III, Corollary 1]. Next recall that an object M 2 A is noetherian if every increasing sequence of subobjects becomes stationary. x is noetherian. Proposition 2.3.6. If M 2 A is noetherian, then q.M / 2 A
2.4 Grothendieck categories
23
f
Proof. First consider any left fraction N ( Nz ! M representing a subobject of x Then Nz ! M represents the same subobject, and since ker.f / 2 B, so q.M / in A. does im.f / M . Thus every subobject of q.M / can be represented by a subobject of M . Next consider two subobjects M 0 , M 00 M such that q.M 0 / q.M 00 /. Then x so its image lies in B. Thus M 0 CM 00 the morphism M 0 ! M=M 00 becomes zero in A, 00 represents the same subobject of q.M / as M . Now assume that M is noetherian, i.e., every increasing sequence of subobjects becomes stationary. By the above remarks every increasing sequence of subobjects of q.M / can be represented by a sequence of subobjects MP i of M . Moreover the inclusions q.Mj / q.Mi / for all j i imply that MiC ´ j i Mj represents the same subobject of q.M / as Mi . Since M is noetherian, the sequence MiC becomes stationary; hence so does the sequence q.MiC / D q.Mi /, as desired. Proposition 2.3.7. Let A and A0 be abelian categories with essentially locally small x and A x 0 . Let multiplicative systems and 0 and corresponding localized categories A 0 0 F W A ! A be an additive functor mapping to . Then F induces an additive x 0 ! A. x Moreover if F is left exact, right exact, or exact, then so is Fx. functor Fx W A Proof. The first statement follows from Theorem 2.2.3 (c). The others follow from the exactness of the localization functors q from Proposition 2.3.4 and the fact that the exactness of Fx can be tested on exact sequences coming from A by Proposition 2.3.5 (c). Proposition 2.3.8. Let A, B and be as in Proposition 2.3.4, and let A0 be a Serre subcategory of A containing B. Then A0 is localizing with respect to , the resulting x0 ! A x multiplicative system 0 ´ \ A0 is saturated, and the natural functor A x 0 with a Serre subcategory of A. x In particular a sequence in A x 0 is exact if identifies A x is exact. and only if its image in A Proof. The assumptions imply that for any morphism M ) N in , if one of M , N is in A0 , then so is the other. This together with Proposition 2.2.8 (a) implies everything. Compare also [42, § 4.3, Example 6].
2.4 Grothendieck categories An object I of an abelian category A is called injective if for any monomorphism i W M 0 ,! M and any morphism f 0 W M 0 ! I there exists a morphism f W M ! I such that f i D f 0 . The category A is said to possess enough injectives if every object M of A possesses a monomorphism into some injective object M ,! I . These concepts are important in the construction of right derived functors: see Section 2.8. The same concepts with all arrows reversed yield the notions of projective objects and enough projectives, which are used in the construction of left derived functors.
24
2 Categorical preparations
An object U of a category C is called a generator of C if for every non-zero morphism f W M ! N in C there exists a morphism g W U ! M with fg 6D 0. In an abelian category with arbitrary direct sums this is L equivalent to saying that for every object M there exists a set I and an epimorphism i2I U M . Definition 2.4.1. An abelian category with exact filtered direct limits and a generator is called a Grothendieck category. Proposition 2.4.2 ([42, Theorem 3.10.10]). Any Grothendieck category possesses enough injectives. Proposition 2.4.3. Any Grothendieck category is locally small. Proof. If U is a generator, a subobject M 0 of an object M is determined completely by the associated subset Hom.U; M 0 / Hom.U; M /. Thus the equivalence classes of all subobjects of M form a set of cardinality bounded by the cardinality of the set of subsets of Hom.U; M /. Obviously we can deduce that every full subcategory of a Grothendieck category is also locally small. Next we formalize some useful assumptions: Assumption 2.4.4. A is a Grothendieck category and B is a Serre subcategory which is closed under filtered direct limits. x We will deduce from this several properties relating A with B and with A. Proposition 2.4.5. In the situation of 2.4.4 B is a Grothendieck category. Proof. We have to prove only that B has a generator. For this fix any generator U of A. Since A is locally small, the equivalence classes of quotients of U form a set. In particular there exists a set of representatives fU˛ g of the quotients of U that lie in B. We claim that their direct sum U 0 is a generator of B. To show this consider any non-zero morphism f W M 0 ! N 0 in B. Since U is a generator of A, there exists a morphism h W U ! M 0 such that f h 6D 0. But as B is a Serre subcategory, the image h.U / is an object in B, so it is isomorphic to some U˛ . Letting h0 denote the composite morphism U 0 U˛ Š h.U / ,! M 0 , we deduce that f h0 6D 0, proving that U 0 is a generator of B. Proposition 2.4.6. In the situation of 2.4.4 the inclusion functor i W B ,! A possesses a right adjoint r W A ! B whose restriction to B is the identity. Proof. The defining property of r asserts HomB .N; r.M // D HomA .N; M / for all M in A and N in B. Thus r.M / must be the largest subobject of M that lies in B. Clearly that can be constructed as lim M 0 where M 0 runs through all subobjects of M ! that lie in B. Note that set-theoretic problems are avoided by Proposition 2.4.3. Since the construction of lim M 0 is functorial in M , this shows the existence of r. !
2.4 Grothendieck categories
25
Proposition 2.4.7. The functor r is left exact and maps injectives to injectives. Moreover, the category B possesses enough injectives of the form r.I / with I injective in A. Proof. The first two assertions follow directly from the fact that r is the right adjoint of an exact functor. For the last statement consider any M 2 B and choose an embedding M ,! I with I injective in A. We then have M D r.M / ,! r.I /, with r.I / injective in B, as desired. x posProposition 2.4.8. In the situation of 2.4.4 the localization functor q W A ! A x sesses a right adjoint s W A ! A such that the adjunction morphism qs ! id is an isomorphism and the adjunction morphism id ! sq consists of morphisms in . Proof. See [42, Proposition 4.6.3 and § 4.4]. One can also construct s explicitly as follows. For any object M 2 C consider the filtered direct system of all morphisms y ´ lim M 0 exists. For this note that M ) M 0 in . We first show that its direct limit M ! composing any morphism M ) M 0 in with the projection M 0 M 0 =r.M 0 / yields another morphism in ; hence the morphisms M ) M 0 with r.M 0 / D 0 form a cofinal subsystem. These morphisms factor through a monomorphism M=r.M / ) M 0 . Next fix any embedding M=r.M / ,! I with I injective, using Proposition 2.4.2. Then for any M=r.M / ) M 0 we can map M 0 to I and replace it by its image. Thus the morphisms M ) M 0 with r.M 0 / D 0 and M 0 a subobject of I form another cofinal subsystem. Now by Proposition 2.4.3 the isomorphism classes of these M 0 form a set; hence the whole system possesses a cofinal subsystem which is indexed by a set. Thus the direct limit exists by the definition of Grothendieck category. y . By the exactness of filtered direct Next consider the natural morphism M ! M limits its kernel and cokernel are the direct limits of the kernels and cokernels of the morphisms M ) M 0 . By assumption these lie in B, and B is closed under filtered y lie in B. Thus this morphism direct limits; hence the kernel and cokernel of M ! M is an element of . In particular, this morphism is itself part of the system of all M ) M 0 ; so it alone forms a cofinal subsystem. This implies that for any other object N 2 A we have y / Š lim HomA .N; M 0 / §2.2 D HomA HomA .N; M x .q.N /; q.M //: ! 0 M )M
y . The rest But this implies that q possesses a right adjoint s such that s.q.M // D M follows from the construction. Proposition 2.4.9. In the situation of 2.4.4 we have: x is a Grothendieck category. (a) A x commutes with filtered direct limits. (b) The localization functor q W A ! A x are up to isomorphy the objects of the form q.I / with I 2 A (c) The injectives of A injective such that r.I / D 0.
26
2 Categorical preparations
x 0 its localization (d) Let A0 be another Serre subcategory of A containing B, and A 0 x0 from Proposition 2.3.8. If A is closed under filtered direct limits in A, then A x and itself a Grothendieck category. is closed under filtered direct limits in A Proof. Assertion (a) is [42, Corollary 4.6.2]. Assertions (b) and (c) are [42, Theorem 1.5.4 (1)] and [42, Proposition 4.5.3 (2)] together with Proposition 2.4.8. They can x By Si / in A. also be shown directly. For one part consider any filtered direct system .M Si // the functoriality of the splitting s it is the image of the filtered direct system .s.M Si // has the universal property of the direct limit in A. One easily shows that q.lim s.M ! Si /. This together with Proposition 2.4.5 also shows assertion (d). of .M
2.5 Triangulated categories Let C be an additive category endowed with an automorphism t W C ! C, called translation functor. Any sequence of morphisms L ! M ! N ! t.L/ in C is called a triangle, and any commutative diagram L
/M
/N
/ t.L/
L0
/ M0
/ N0
/ t.L0 /
is called a morphism of triangles. A triangulated category is an additive category endowed with a translation functor and a collection of triangles, called distinguished triangles, satisfying a number of axioms. We will almost never use these axioms directly; thus for their formulation and explanation and for more background on triangulated categories we refer the reader to [20], [27], [46], and [47]. An exact functor between two triangulated categories is an additive functor that commutes with translation and preserves distinguished triangles. Definition 2.5.1. A multiplicative system in a triangulated category C is called compatible with the triangulation if, in addition to 2.2.1 (a)–(c), it also satisfies the following conditions: (e) The system is invariant under the translation functor t W C ! C. (f) Given two distinguished triangles L ! M ! N ! t.L/
and L0 ! M 0 ! N 0 ! t.L0 /
and morphisms L ) L0 and M ) M 0 in such that the square formed by L, M , L0 , M 0 commutes, there is a morphism N ) N 0 in inducing a morphism
2.6 Derived categories
27
of distinguished triangles L
/M
/N
/ t.L/
L0
/ M0
/ N0
/ t.L0 /.
In the following a multiplicative system in a triangulated category will always be assumed compatible with the triangulation. Proposition 2.5.2. For any triangulated category C, there is a bijection between the class of saturated multiplicative systems and the class of strictly full triangulated subcategories N of C which are closed under taking direct summands. Explicitly, given N one defines as the set of those morphisms f W L ! M which are part of f
a distinguished triangle L ! M ! N ! t.L/ such that N lies in N. Conversely, given one defines N as the full triangulated subcategory consisting of those objects N of C which appear in a distinguished triangle as above where f is in . Proof. [27, § XI.6] or [46, Corollary II.2.2.11]. Proposition 2.5.3. Let be an essentially locally small multiplicative system in a x carries a natural triangulated category C. Then the associated localized category C x structure of triangulated category, and the functor q W C ! C is exact. x is induced by that in C, and a triangle in Proof. Explicitly, the translation functor in C x C is distinguished if and only if it is isomorphic to the image of a distinguished triangle in C. See [46, Theorem II.2.2.6]. Proposition 2.5.4. Let and 0 be essentially locally small multiplicative systems in triangulated categories C and C0 . Then any exact functor F W C0 ! C that maps 0 to x x 0 ! C. induces an exact functor of triangulated categories C Proof. Direct consequence of Propositions 2.2.4 and 2.5.3.
2.6 Derived categories Let A be an additive category. An (infinite) complex M in A is a sequence of objects and morphisms i 2 dM
i 1 dM
i dM
i C1 dM
! M i1 ! M i ! M iC1 ! ; iC1 i indexed by i 2 Z, such that dM dM D 0 for all i . A morphism of complexes i f W M ! N is a system of morphisms f i W M i ! N i satisfying f iC1 dM D dNi f i for all i . The category of complexes in A is denoted by C .A/.
28
2 Categorical preparations
A complex M is called bounded below if M i D 0 for all i 0, bounded above if M i D 0 for all i 0, and bounded on both sides or just bounded if M i D 0 whenever ji j 0. For any symbol 2 fC; ; bg we let C .A/ denote the full additive subcategory of complexes that are bounded below if D C, bounded above if D , resp. bounded on both sides if D b. A homotopy between two morphisms of complexes f1 , f2 W M ! N is a system i C dNi1 hi D f1i f2i of morphisms hi W M iC1 ! N i for all i such that hiC1 dM for all i . The existence of a homotopy defines an equivalence relation on the set HomC .A/ .M ; N /, and the equivalence class of a morphism f is called the homotopy class of f . The homotopy class of a composite morphism gf depends only on the homotopy classes of g and f . Thus for any symbol 2 f¿; C; ; bg one can define the homotopy category K .A/, whose objects are the same as those of C .A/ and whose morphisms are homotopy classes of morphisms with the evident composition. It is again an additive category, because the homotopy class of a sum of morphisms depends only on the homotopy classes of the summands. It also possesses a natural structure of triangulated category. Namely, the translation of a complex t.L / ´ L Œ1 is the left shift defined by .L Œ1/i D LiC1 and dLi Œ1 D dLiC1 . The cone of a morphism f W L ! M is the complex Cone.f / defined by Cone.f /i ´ M i ˚ LiC1 together with the morphisms M i ˚ LiC1 ! M iC1 ˚ LiC2 ;
i .m; `/ 7! .dM .m/ f iC1 .`/; dLiC1 .`//:
The inclusion morphism and minus the projection morphism M i ,! M i ˚ LiC1 LiC1 define morphisms of complexes M ! Cone.f / ! L Œ1. The corresponding morphisms in K .A/ form a triangle
Œf
L ! M ! Cone.f / ! L Œ1: Any triangle isomorphic to one like this is called a distinguished triangle in K .A/. It is tedious but standard to verify that with this definition the axioms of triangulated categories are satisfied (see [46, Proposition II.1.3.2]). In the following we assume that A is an abelian category. Then the category of complexes C .A/ is again an abelian category, though the homotopy category K .A/ is usually not. For any integer i one defines the i -th cohomology of a complex M as i i1 H i .M / ´ ker.dM /= im.dM /. If its cohomology vanishes for all i, the complex M is called acyclic. This is equivalent to M being exact. Any morphism of complexes f W M ! N induces a natural morphism H i .f / W H i .M / ! H i .N / on cohomology. This morphism depends only on the homotopy class of f , so that H i induces an additive functor K .A/ ! A. It satisfies H i .L Œ1/ D H iC1 .L /, and any distinguished triangle L ! M ! N ! L Œ1 induces a long exact cohomology sequence
! H i .L / ! H i .M / ! H i .N / ! H iC1 .L / ! H iC1 .M / ! : A morphism of complexes f is called a quasi-isomorphism if H i .f / is an isomorphism for all i . Again this notion depends only on the homotopy class of f . Let
2.6 Derived categories
29
qi denote the collection of all quasi-isomorphisms in K .A/. It is a saturated multiplicative system in the sense of Definitions 2.2.1 and 2.2.6, and compatible with the triangulation in the sense of Definition 2.5.1. Proposition 2.6.1. If A is a Grothendieck category, the multiplicative system qi is essentially locally small and the localization exists. Proof. See [47, Proposition 10.4.4, Remark 10.4.5], whose proof applies to each of the four choices of . The localized category D .A/ ´ qi1 K .A/ is called the derived category of A, if it exists. It is again a triangulated category, and the localization functor W K .A/ ! D .A/ is exact. The universal property 2.2.3 (b) of localization implies that H i factors through a unique additive functor D .A/ ! A for any i . Using truncations it is easy to show that D C .A/ is equivalent to the full triangulated subcategory of D.A/ consisting of those complexes M whose cohomology H i .M / vanishes for all i 0. The analogous remarks apply to D and D b (see [47, Example 10.3.15]). More generally, let A0 be a full additive subcategory of A which is closed under taking direct summands. Then K .A0 / is a full triangulated subcategory of K .A/ and by Proposition 2.5.2 the collection qi induces a multiplicative system in K .A0 /. If this multiplicative system is essentially locally small, the associated localization is called the derived category of A0 and denoted D .A0 /. This is again a triangulated category, but it is by no means clear how it relates to D .A/. It is important not to confuse morphisms in D .A0 / with their images in D .A/. .A/ Now let B be a Serre subcategory of A. Then by CB .A/ C .A/ and KB K .A/ and DB .A/ D .A/ we denote the full subcategories consisting of those .A/ is closed under quasi-isomorcomplexes whose cohomology lies in B. Since KB phisms in K .A/, its localization is simply DB .A/. The properties of B show that DB .A/ is a strictly full triangulated subcategory of D .A/ which is closed under taking direct summands. All derived categories considered in the present monograph are obtained in this way. Note that while there is a natural functor D .B/ ! DB .A/, it is non-trivial to decide whether this functor is an equivalence of categories. We will take up this problem in Section 2.9. In the rest of this section we discuss the relation between the derived categories x where A x denotes the localization of A at B from Proposition 2.3.4. D .A/ and D .A/, Assume that both derived categories exist. Then via the equivalence in Proposition 2.5.2 the subcategory DB .A/ corresponds to a saturated multiplicative system in D .A/ x induces an exact functor on which we denote by Bqi . The functor q W A ! A x derived categories D .A/ ! D .A/ which clearly maps all morphisms in Bqi to isomorphisms. It therefore induces an exact functor 1 x D .A/ ! D .A/: Bqi
30
2 Categorical preparations
Theorem 2.6.2. In the situation of 2.4.4 consider any 2 f¿; C; ; bg such that x exist. Then 1 D .A/ exists and the above functor is an equivD .A/ and D .A/ Bqi alence of categories. Proof. A proof under much weaker assumptions can be found in [38], where the case D ¿ was left out but can be handled in the same fashion. In the situation of 2.4.4 x ! A from Proposition 2.4.8 to give a short proof, as we can use the splitting s W A follows. First we observe that the fact that Bqi is essentially locally small follows once we have proved that the functor induces a bijection on morphisms. Thus as long as we do not perform any set theoretic constructions with collections of morphisms in 1 Bqi D .A/, we may already use that this localization exists in the weaker sense where 1 morphisms form only classes instead of sets. With this in mind observe that Bqi D .A/ can be obtained from K .A/ in one step by localizing at the multiplicative system of morphisms whose cones have cohomology in B. Denoting this multiplicative system again by Bqi , we consider the diagram K .A/
u
s 1 Bqi K .A/
s q
s q
/ K .A/ x
/ 1 K .A/D x x D .A/:
x Here the functors q and s in the upper row are induced by the functors q W A ! A x and its splitting s W A ! A, which induce the functors q and s in the lower row by Proposition 2.5.4 and the definition of the respective multiplicative systems. Now since x by functoriality the same follows for the corresponding morphisms in qs Š id on A, the above diagram. Similarly the adjunction morphism id ! sq on A consists of 1 morphisms in , which by construction become isomorphisms in Bqi K .A/. Thus q and s in the lower row are quasi-inverses; hence q is an equivalence of categories, as desired.
2.7 Derived functors Let F W A ! B be an additive functor of additive categories. Applied to each term in a complex it induces an exact functor of triangulated categories F W K .A/ ! K .B/. In general, the composite functor F
! K .B/ ! D .B/ K .A/
D .A/. Instead one defines does not factor through the canonical functor K .A/ ! a (total) right derived functor of F as an exact functor of triangulated categories
2.7 Derived functors
31
RF W D .A/ ! D .B/ together with a natural transformation W F ! .RF / such that for every exact functor G W D .A/ ! D .B/ and every natural transformation W F ! G there exists a unique natural transformation W RF ! G such that D . If a right derived functor exists, it is unique up to unique isomorphism. If F is a left exact functor of abelian categories and A possesses enough injectives, one can construct a right derived functor RF W D C .A/ ! D C .B/ as follows. First, for every object M 2 A there exists an infinite exact sequence 0 ! M ! I 0 ! I 1 ! with all I i injective, called an injective resolution of M . Using this fact for each term in a complex M 2 K C .A/, one can construct a quasi-isomorphism M ! I in K C .A/ with all I i injective. Any two such injective resolutions of M are naturally isomorphic in K C .A/, turning M 7! I into an exact functor D C .A/ ! K C .A/, and one shows that RF .M / ´ F .I / defines a right derived functor of F . Identify an object M 2 A with the complex having M in degree 0 and the zero object in all other degrees. The (individual) derived functors Ri F W A ! B are defined as Ri F .M / ´ H i .RF .M //. They vanish for i < 0 and satisfy R0 F Š F . Moreover, every short exact sequence 0 ! L ! M ! N ! 0 induces a long exact cohomology sequence ! Ri F .L/ ! Ri F .M / ! Ri F .N / ! RiC1 F .L/ ! RiC1 F .M / ! : An object M with Ri F .M / D 0 for all i > 0 is called F -acyclic. One can repeat the whole construction with F -acyclic instead of injective objects and obtain the same derived functor RF . The cohomological dimension of F is the largest integer d such that Rd F 6D 0 if that exists, and is 1 otherwise. If the cohomological dimension d is finite, one can repeat the above construction with F -acyclic resolutions of length d and thereby construct a derived functor RF W D .A/ ! D .B/ for every symbol 2 f¿; C; ; bg. For a complex M there is a procedure to calculate the cohomology of RF .M / from Ri F .M j / for all i, j , called a spectral sequence with E1 -term E1ij D Ri F .M j / H) H iCj .RF .M //:
The complete details are too involved to be sketched here; see [47, Chapter 5, § 5.6, 5.7.9, Corollary 10.5.7]. An important special case occurs if E1ij D 0 whenever i 6D 0, that is, if all M j are F -acyclic, where H i .RF .M // is the i -th cohomology of the complex F .M /. F
G
!B ! C. Then Now consider two left exact functors of abelian categories A GF is again left exact. Assume that A and B possess enough injectives, so that the derived functors RF , RG, and R.GF / exist. Then we have R.GF / D RG B RF if and only if F maps all injectives in A to G-acyclic objects in B. If that is the case, one can calculate the individual derived functors of GF from those of G and F by means of a spectral sequence with E2 -term E2ij D Ri G.Rj F .M // H) RiCj .GF /.M /:
32
2 Categorical preparations
See [47, Corollary 10.8.3] for the details. An important special case occurs if M is F -acyclic, so that E2ij D 0 whenever j 6D 0, in which case Ri .GF /.M / D Ri G.FM / for all i . Another special case occurs if all Rj F .M / are G-acyclic, so that E2ij D 0 whenever i 6D 0 and Ri .GF /.M / D G.Ri F .M // for all i. All the above facts have direct analogues for left derived functors and projective resolutions. The following result relates derived functors with localization. Theorem 2.7.1. In the situation of Theorem 2.6.2 consider another abelian category x ! C. Assume that the composite functor F ´ C and a left exact functor Fx W A Fxq W A ! C possesses a right derived functor RF W D .A/ ! D .C/ which factors x ! D .C/. Then RF is the right derived functor of through a functor RF W D .A/ x F . The corresponding statement is true for the left derived functor of a right exact functor Fx. Proof. By symmetry it suffices to deal with the case of right derived functors. By assumption we have a diagram F
K .A/
q
D .A/
/ K .A/ x
Fx
q
/ D .A/ x
+ / K .C/
RF
/ D .C/, 3
RF
where everything except the right hand square commutes. That RF is a right derived functor of F means that there is a natural transformation F ! RF which is universal among all natural transformations from F to exact functors K .A/ ! D .C/ that factor through D .A/. Now by Theorem 2.2.3 (d) the natural transformation Fxq D F ! RF D RF q comes from a unique natural transformation Fx ! RF . Similarly for every exact x ! D .C/ the composition with q yields a bijection between functor G W D .A/ natural transformations Fx ! G and Fxq ! Gq D Gq, and composition with and q yields bijections between natural transformations RF ! G and RF ! G and RF q ! Gq. Using this one easily sees that RF satisfies the universal property for the right derived functor of Fx, as desired.
2.8 Construction of derived functors The aim of this section is to prove the following result on the construction of derived functors:
2.8 Construction of derived functors
33
Theorem 2.8.1. Let A and B be abelian categories and let 2 fb; C; ; ¿g be such that D .A/ and D .B/ exist. Let F W A ! B be a right exact functor. Furthermore, let A0 be a full additive subcategory of A, let C W K .A/ ! K .A/ be an exact functor, and let v W id ) C be a quasi-isomorphism of functors such that the following conditions hold: (a) C maps K .A0 / to itself. (b) For every M 2 K .A/ there exists M 0 2 K .A0 / and a quasi-isomorphism M 0 ) CM . (c) F .v/ W F ! F C is a quasi-isomorphism of functors K .A/ ! K .B/. (d) F maps quasi-isomorphisms in K .A0 / to quasi-isomorphisms. Then F possesses a left derived functor LF W D .A/ ! D .B/ which on K .A0 / coincides with F . Clearly Theorem 2.8.1 is equivalent to its dual version obtained by inverting all arrows and asserting the existence of a right derived functor of a left exact functor. The same remark applies to the special case C D id, which is well known by [25, Chapter I, Theorem 5.1]: Theorem 2.8.2. Let A and B be abelian categories and let 2 fb; C; ; ¿g be such that D .A/ and D .B/ exist. Let F W A ! B be a right exact functor. Furthermore, let A0 be a full additive subcategory of A such that the following conditions hold: (a) For every M 2 K .A/ there exists M 0 2 K .A0 / and a quasi-isomorphism M 0 ) M . (b) F maps quasi-isomorphisms in K .A0 / to quasi-isomorphisms. Then F possesses a left derived functor LF W D .A/ ! D .B/ which on K .A0 / coincides with F . The rest of this section is devoted to proving Theorem 2.8.1. For simplicity we drop the upper index . / in our notation for complexes. We first show: Proposition 2.8.3. In the situation of Theorem 2.8.1 the derived category D .A0 / exists and the natural functor i W D .A0 / ! D .A/ is an equivalence of categories. Proof. As in the proof of Theorem 2.6.2 the existence of D .A0 / follows from the bijectivity on morphisms. We use left fractions instead of right fractions, which is more vM
efficient. First, for every M 2 K .A/ the diagram M H) CM ( M 0 furnished by 2.8.1 (b) shows that the functor i is essentially surjective. Next, consider any morphism in D .A/ represented by a left fraction M 0 ( L ! N 0 in K .A/ with
34
2 Categorical preparations
M 0 , N 0 2 K .A0 /. Using the functoriality of C and v and applying 2.8.1 (b) to L we obtain a commutative diagram M 0 ks
/ N0
L
vM 0
vN 0
vL
/ CN 0 CL CM 0_g Fks S K < FFFF xx FFFF xx FFFF x x FFF xx 0 L
with L0 2 K .A0 /. From 2.8.1 (a) we deduce that all the arrows around the left, right, and lower edges lie in K .A0 /. Since double arrows induce isomorphisms in D .A0 /, we find that the path from M 0 to N 0 along the lower edge defines a morphism in D .A0 / whose image in D .A/ is the given one. This shows that the functor i is surjective on morphisms. Thirdly, to prove injectivity on morphisms consider a morphism in D .A0 / represented by a left fraction M 0 ( L0 ! N 0 in K .A0 / whose image in D .A/ is zero. Then there exists a quasi-isomorphism L ) L0 in K .A/ such that the composite L ) L0 ! N 0 is zero. Using the functoriality of C and v and applying 2.8.1 (b) to L we obtain a commutative diagram M 0 ks vM 0
CM 0 ks
L0 ld PPP PPP P
vL0
L
/ 0 mm6 N m m mmm 0
vN 0
vL / CN 0 CLX` 09dl P 6 999 PPPP mm B mmmm 9999 0 9999 CL 9999 KS 0 9999 99 L00
with L00 2 K .A0 / and where the indicated morphisms are zero. Removing L and CL yields a commutative diagram purely within K .A0 /, which shows that the given morphism vanishes in D .A0 /, as desired. Knowing already that i W D .A0 / ! D .A/ possesses a quasi-inverse T , we can construct one easily as follows. Using 2.8.1 (b) for every M 2 K .A/ we choose a diagram vM
˛M
M HH) CM (HH TM
(2.8.4)
in K .A/ with TM 2 K .A0 /. If M already lies in K .A0 /, we choose TM ´ M and ˛M ´ vM , so that (2.8.4) represents the identity morphism in D .A0 /. Since this prescription defines a quasi-inverse on objects, it can be extended uniquely to a full quasi-inverse functor T W D .A/ ! D .A0 /. The next lemma tells us what this functor does on morphisms:
2.8 Construction of derived functors
35
Lemma 2.8.5. For every morphism ' W M ! N in K .A/ there exists a commutative diagram vM +3 CM ks ˛M TM M ck PPP PP 0 ' C' oo L o o vN woo +3 CN ks ˛N T N N with L0 2 K .A0 /. Furthermore, if T ' denotes the morphism in D .A0 / defined by the left fraction TM ( L0 ! T N , then i.T .'// D '. Proof. Everything in the diagram apart from L0 results from the functoriality of C and v and our choices for T . Applying the axiom 2.2.1 (b) to the right fraction TM ! CN ( T N yields the desired completion, with an object L 2 K .A/ in place of L0 . But by the proof of the surjectivity on morphisms in 2.8.3 the resulting left fraction TM ( L ! T N is dominated by a left fraction in K .A0 /. This proves the first assertion, and the rest follows from the construction. Now we consider the functor F . The assumption 2.8.1 (d) implies that the induced functor F W K .A0 / ! D .B/ factors through a unique exact functor Fx W D .A0 / ! D .B/. Setting LF ´ FxT we obtain the following diagram:
C
K .A/ II II F II II II $ # / D .B/ D .A/ LF u: O uu uu T u i uu x uu F 0 D .A /.
By construction the lower triangle commutes. To analyze the upper triangle we apply F to the morphisms in (2.8.4), yielding morphisms F .vM /
F .˛M /
FM ! F CM F TM in D .B/. Here the morphism on the left is an isomorphism by 2.8.1 (c); hence we obtain a composite morphism FM F TM . Applying F to the commutative diagram in Lemma 2.8.5 shows that this defines a natural transformation u LF D FxT ! F:
To prove Theorem 2.8.1 it remains to show that LF and u satisfy the universal property for the left derived functor of F . For this consider any exact functor G W D .A/ ! w ! F . Applying w to the morphisms D .B/ together with a natural transformation G
36
2 Categorical preparations
in (2.8.4) yields a commutative diagram GM
GvM
wM
FM
/ GCM o
G˛M
wCM F vM
/ F CM o
GTM wTM
˛M
F TM .
Here the arrows in the first row are isomorphisms because G is a functor on D .A/, and the arrow on the lower left is an isomorphism by the assumption 2.8.1 (c). The composite arrow GTM ! F TM defines a natural transformation w x W G ! FxT D LF such that uw x D w. It is in fact the only such natural transformation, because its restriction to K .A0 / is determined by the fact that u is an isomorphism there, and these objects represent all isomorphism classes of objects of D .A/ by 2.8.1 (b). This finishes the proof of Theorem 2.8.1.
2.9 Comparison of derived categories In this section we discuss the relation between the derived categories D .B/ and DB .A/. The first result concerns passage to direct limits: Theorem 2.9.1. Let A be an abelian category and B a Serre subcategory such that (a) every object in B is noetherian, and (b) every object in A is a filtered direct limit of objects in B. Let 2 fb; g and assume that D .A/ exists. Then D .B/ exists and the natural functor D .B/ ! DB .A/ is an equivalence of categories. Proof. By Proposition 2.2.8 (b0 ) it suffices to prove that every complex M 2 CB .A/ possesses a subcomplex N 2 C .B/ whose inclusion N ,! M is a quasiisomorphism. di
To show this consider any complex ! M i ! M iC1 ! in CB .A/. Take an integer i such that M j already lies in B for all j > i. By assumption we can write M i as a filtered direct limit of objects N˛i in B. For simplicity we may, and do, assume that N˛i M i . Now as M iC1 is noetherian, we have d i .N˛i / D d i .M i / for all sufficiently large ˛. Furthermore, as H i .M / is noetherian, the morphism N˛i \ker.d i / ! H i .M / is an epimorphism for all sufficiently large ˛. Set N i ´ N˛i and N i1 ´ .d i /1 .N˛i /, and N j ´ M j for all j 6D i , i 1. If ˛ is sufficiently large, this defines a quasi-isomorphism N ,! M such that N j lies in B for all j i . By assumption M is bounded above, so we can begin this construction for some i 0 and repeat it with the resulting complexes by downward induction on i . In the limit we obtain a subcomplex N whose embedding is a quasi-isomorphism and whose components all lie in B. Clearly N is bounded below if M is, finishing the proof.
2.9 Comparison of derived categories
37
In the remainder of this section, we return to the situation of 2.4.4. Recall from Proposition 2.4.6 that the inclusion B ,! A possesses a right adjoint splitting r W A ! B. As A has enough injectives, the right derived functors Ri r W A ! B exist. As usual, an object M in A is called r-acyclic if Ri r.M / D 0 for all i 1. Lemma 2.9.2. In the situation of 2.4.4 assume that r has cohomological dimension
n, that is, that Ri r D 0 for all i > n. Then for any exact sequence M 0 ! M 1 ! ! M n ! 0 in A, if M 0 ; : : : ; M n1 are r-acyclic, so is M n . Proof. Set N ´ ker.M 0 ! M 1 /. Using long exact sequences and induction on n, one finds a natural isomorphism Ri r.M n / Š RiCn r.N / for all i 1. The latter vanishes by assumption; hence M n is r-acyclic, as desired. Proposition 2.9.3. In the situation of 2.4.4 consider any symbol 2 fb; C; ; ¿g. If 6D C, assume that r has finite cohomological dimension. (a) Any complex M 2 C .A/ possesses a quasi-isomorphic embedding into a complex N 2 C .A/ consisting of r-acyclic objects. (b) The total right derived functor Rr W D .A/ ! D .B/ exists and is calculated by Rr.M / D r.N / for any N as in (a). Proof. (b) follows from (a) by the dual version of Theorem 2.8.2. Part (a) for 2 fC; ¿g is the assertion of [25, Chapter I, Lemma 4.6], where Lemma 2.9.2 is taken into account when D ¿. For 2 fb; g suppose that M i D 0 for i > i0 and assume that r has cohomological dimension n. We first select M ,! N which solves the problem within C C .A/ or C .A/, respectively. Let d i W N i ! N iC1 denote the differential in N . Then the sequence N i0 ! N i0 C1 ! : : : ! N i0 Cn1 ! ker d i0 Cn ! 0 is exact and all terms but its last one are r-acyclic. Thus by Lemma 2.9.2 the last term is r-acyclic, too. We may therefore replace N by its right truncation ! N i0 Cn1 ! ker d i0 Cn ! 0, which lies in C .A/, as desired. Theorem 2.9.4. In the situation of 2.4.4 assume that r has finite cohomological dimension and that every object of B is r-acyclic. Then for any symbol 2 fb; C; ; ¿g the natural functors i / DB .A/ D .B/ o Rr
are mutually quasi-inverse equivalences of categories. Proof. For M 2 C .B/ we know by the second assumption and Proposition 2.9.3 (b) that Rr.i.M // D r.i.M // D M . Thus Rr i Š id. On the other hand consider M 2 CB .A/ and choose M ,! N as in Proposition 2.9.3 (a). Then N is also in
38
2 Categorical preparations
CB .A/. We claim that the inclusion r.N / ,! N is a quasi-isomorphism. To prove this note that by the first assumption there is a spectral sequence E2ij D Ri r.H j .N // H) H iCj .Rr.N //:
By the second assumption all H j .N / are r-acyclic; hence E2ij D 0 whenever i 6D 0. The spectral sequence thus degenerates to an isomorphism
H j .N / D r.H j .N // Š H j .Rr.N // for every j . As N consists of r-acyclic objects, we have Rr.N / D r.N /. It follows that r.N / ,! N is an isomorphism on cohomology, i.e., a quasi-isomorphism, as desired. Finally, this implies that i.Rr.M // Š i.r.N // ! N ŠM
in DB .A/; hence i Rr Š id, as desired.
Chapter 3
Fundamental concepts In this chapter we introduce the basic objects that are being studied in this book. The main concept is that of an A-crystal, which is a coherent sheaf together with a semilinear endomorphism, viewed modulo a certain equivalence relation. This equivalence relation annihilates precisely those objects whose semi-linear endomorphism is nilpotent. Roughly speaking, an A-crystal is like a system of homogeneous linear differential equations, where the derivative is replaced by a partial Frobenius map. In other words, we are looking at systems of ‘Frobenius equations’ modulo degenerate ones. The prefix ‘A-’ indicates the ring of coefficients of these objects and is often dropped for brevity. Although our main interest concerns coherent objects, we are forced to relax this finiteness condition in order to define derived functors. In the interest of brevity we allow this greater generality right from the start, working with (A-)quasi-crystals. But the gentle reader is advised to ignore all such technicalities on first reading and to restrict his or her attention to coherent objects. In the last section of this chapter we give a few examples which relate our concepts with existing ones, such as families of Drinfeld modules or t -motives.
3.1 Conventions From now on we fix a finite field k with q elements. All schemes are assumed to be noetherian and separated over k. All morphisms and fiber products, and all tensor products of modules and algebras, are taken over k, unless otherwise specified. By a (quasi)-coherentsheaf on a scheme X we will always mean a (quasi)-coherent sheaf of OX -modules. Any homomorphism of such sheaves is assumed OX -linear, and any tensor product of such sheaves is taken over OX . The Frobenius morphism on X over k, which acts on functions by x 7! x q , is denoted W X ! X . Throughout most of the monograph we fix a scheme C which is assumed to be a localization of a scheme of finite type over k. The notation is intended to reflect the role of C as a C oefficient system for the crystals in the present work. To guarantee the existence of sufficiently many functions we assume that C is affine; thus C D Spec A, where A is a localization of a finitely generated k-algebra. Interesting special cases are that of an affine curve of finite type over k, or the generic point thereof, or of a finite local Artin ring. The assumptions on C imply that X C is again noetherian for every noetherian scheme X over k. This is useful in dealing with coherent sheaves on X C . As a general rule, sheaves on X will be distinguished from those on X C by an index . /0 . Throughout we let pr 1 W X C ! X denote the projection to the first factor. For any coherent sheaf of ideals 0 OX we abbreviate 0 F ´ .pr 1 1 0 /F .
40
3 Fundamental concepts
3.2 -sheaves The basic objects in this book are the following: Definition 3.2.1. A -sheaf over A on X is a pair F ´ .F ; F / consisting of a quasi-coherent sheaf F on X C and an OXC -linear homomorphism . id/ F
F
/F.
As A remains fixed for the most part, we usually speak of -sheaves on X . A homomorphism of -sheaves F ! G on X is a homomorphism of the underlying sheaves ' W F ! G for which the following diagram commutes: . id/ F
F
. id/ '
. id/ G
/F '
G
/ G.
The sheaf underlying a -sheaf F will always be denoted F , unless otherwise specified. To avoid cumbersome notation we will often abbreviate D F if the underlying sheaf is clear from the context. On any affine chart Spec R X a -sheaf over A corresponds to an R ˝ A-module M together with a ˝ id-linear homomorphism W M ! M . In other words, it corresponds to a left module over the non-commutative polynomial ring .R ˝ A/Œ , which is defined by the commutation rule .u ˝ a/ ´ .uq ˝ a/ for all u 2 R and a 2 A. Definition 3.2.2. (a) A -sheaf F is called coherent if its underlying sheaf is coherent. (b) A -sheaf F is called ind-coherent if it is a union of coherent -subsheaves. Remark 3.2.3. There exist ind-coherent -sheaves which are not coherent, for instance any infinite direct sum of non-zero coherent -sheaves. There also exist -sheaves which are not ind-coherent, as the following example shows. Take R D kŒx and X ´ Spec R and A D k, and let F be the -sheaf on X corresponding to the Rmodule M D kŒx; x 1 =kŒx with .m/ ´ mq for all m 2 M . Then F contains no non-zero coherent -subsheaf. Definition 3.2.4. The category formed by all -sheaves over A on X and with the above homomorphisms is denoted QCoh .X; A/. The full subcategory of all coherent -sheaves is denoted Coh .X; A/, that of all ind-coherent -sheaves IndCoh .X; A/. Clearly these are abelian A-linear categories, and all constructions like kernel, cokernel, etc. are the usual ones on the underlying quasi-coherent sheaves, with the respective added by functoriality. In particular, the formation of kernel, cokernel, image and coimage is preserved under the inclusions Coh .X; A/ IndCoh .X; A/ QCoh .X; A/. But more is true:
3.2 -sheaves
41
Proposition 3.2.5. Both Coh .X; A/ and IndCoh .X; A/ are Serre subcategories of QCoh .X; A/. To prove this we will need the following lemma, which in turn depends on a construction whose natural place is in Section 4.5. We permit ourselves this forward reference, because the careful reader can readily verify that those results depend neither on the lemma nor on any other intervening theory, and should in fact find it easy to simplify the relevant construction for the case at hand. Lemma 3.2.6. Consider a short exact sequence of -sheaves 0 ! F 0 ! F ! F 00 ! 0 where F 00 is coherent and F 0 is ind-coherent. Then there exists a coherent -sheaf G F which maps epimorphically to F 00 . Proof. Assume first that X is affine, and choose global sections m1 ; : : : ; mr of F whose images generate F 00 as a coherent P sheaf. Then there exist global sections ˛i;j of OXC such that all ni ´ .mi / j ˛i;j mj lie in F 0 . By assumption there exists a coherent -subsheaf G 0 F 0 containing all these sections. By construction the coherent subsheaf G F generated by G 0 together with all sections mi underlies a coherent -subsheaf G F . Clearly G has the desired properties. In the general case let fUi g be a finite affine open cover of X and denote the respective open embeddings by ji W Ui ,! X . We will use the concept of inverse image from Definition 4.1.1. By the above special case there exist coherent -subsheaves G i of ji F that map epimorphically to ji F 00 . By Propositions 4.5.1 and 4.5.2 there exist coherent -sheaves Gz i on X with ji Gz i Š G i such that the inclusions ji Gz i Š G i ,! ji F extend to homomorphisms Gz i ! F . Let Gz F be the sum of their images. By construction this is a coherent -subsheaf, and the induced homomorphism Gz ! F 00 is locally epimorphic; hence it is itself epimorphic, as desired. Proof of Proposition 3.2.5. The assertion on Coh .X; A/ follows immediately from the corresponding property of coherent sheaves on a noetherian scheme, inside the category of all quasi-coherent sheaves. For the category IndCoh .X; A/ only the invariance under extensions is non-trivial. So consider a short exact sequence 0 ! F 0 ! F ! F 00 ! 0 in QCoh .X; A/ with F 0 , F 00 2 IndCoh .X; A/. By Lemma 3.2.6 every coherent -subsheaf G 00 F 00 is the image of a coherent -subsheaf of F . By varying G 00 we deduce that F is ind-coherent, as desired. Theorem 3.2.7. The category QCoh .X; A/ is a Grothendieck category. Its subcategory IndCoh .X; A/ is closed under filtered direct limits in QCoh .X; A/ and is itself a Grothendieck category. Proof. We first show all assertions concerning filtered direct limits. Since the category of quasi-coherent sheaves has exact filtered direct limits, so does the category QCoh .X; A/. Next consider any filtered direct system .F i / in IndCoh .X; A/ and let F be its limit in QCoh .X; A/. Then F is the union of the images of all F i ! F , which are ind-coherent; hence F itself is ind-coherent. This shows that IndCoh .X; A/
42
3 Fundamental concepts
is closed under filtered direct limits in QCoh .X; A/, which are then also exact filtered direct limits in IndCoh .X; A/. It remains to construct generators. For this let us call a -sheaf !-coherent if its underlying quasi-coherent sheaf is locally countably generated. We claim that the isomorphism classes of !-coherent -sheaves form a set. To show this let F be any quasi-coherent sheaf on X C which is locally countably generated. Then its restriction to any sufficiently small open affine is a quotient of the direct sum of countably many copies of the structure sheaf. The isomorphism classes of such quotients form a set, and for any such quotients on two open affines, the possible isomorphisms on the intersection again form a set. Thus the gluing data for F lies in a set, and so the isomorphism classes of F form a set. Now for each such F the possible homomorphisms F W . id/ F ! F form a set; hence so do the isomorphism classes of all possible -sheaves F with underlying sheaf F . Varying F we deduce that the isomorphism classes of all !-coherent -sheaves form a set, as desired. We claim that the direct sum U of a set of representatives is a generator of QCoh .X; A/. To show this consider any -sheaf F on X . Its underlying sheaf F is the union of coherent subsheaves G ; hence F is the union of the -subsheaves X n .. n id/ G /; n0
which by construction are !-coherent. Thus F is a union of quotients of U; hence U is a generator, as desired. Finally, we can obtain a generator of IndCoh .X; A/ either as in the proof of Proposition 2.4.5, or directly as the direct sum of a set of representatives of coherent -sheaves on X .
3.3 Nilpotence For any -sheaf F we define the iterates Fn of F by setting inductively F0 ´ id and FnC1 ´ F B . id/ Fn . Thus they are OXC -linear homomorphisms . n id/ F ! F : Each . n id/ F ´ . n id/ F ; . n id/ F is a -sheaf in its own right, and Fn is a homomorphism of -sheaves . n id/ F ! F . In particular, its image is a -subsheaf of F . Definition 3.3.1. (a) A -sheaf F is called nilpotent if Fn vanishes for some, or equivalently all, n 0. (b) A -sheaf F is called locally nilpotent if it is a union of nilpotent -subsheaves. (c) A -sheaf F is called locally nil-coherent if there is a short exact sequence of -sheaves 0 ! G ! F ! N ! 0 where G is coherent and N locally nilpotent.
3.3 Nilpotence
43
Definition 3.3.2. The full subcategories of QCoh .X; A/ formed by all nilpotent coherent, locally nilpotent, resp. locally nil-coherent -sheaves are denoted Nil .X; A/ LNil .X; A/ LNilCoh .X; A/. Proposition 3.3.3. We have the following inclusions of categories: / LNil .X; A/ LNil .X; LNil .X; Nil .X; _ A/ _ _ A/ _ A/ Coh .X; A/
/ LNilCoh .X; A/
/ IndCoh .X; A/
/ QCoh .X; A/,
where furthermore Nil .X; A/ D Coh .X; A/ \ LNil .X; A/. Proof. For every -sheaf F the underlying sheaf F is the union of coherent subsheaves G ; hence F is the union of the -subsheaves X n .. n id/ G /: n0
If F is locally nilpotent, this sum is really finite for every G , so it defines a coherent -subsheaf of F . It follows that F is ind-coherent, which proves the inclusion LNil .X; A/ IndCoh .X; A/. Together with the fact that IndCoh .X; A/ is closed under extensions this implies the inclusion LNilCoh .X; A/ IndCoh .X; A/. The remaining assertions are obvious. Proposition 3.3.4. The category LNil .X; A/ is closed under filtered direct limits. Proof. Consider a filtered direct system .F i / in LNil .X; A/ and let F be its limit in QCoh .X; A/. Then F is the union of the images of all F i ! F , which are again locally nilpotent; hence F itself is locally nilpotent, as desired. Proposition 3.3.5. All the categories in 3.3.3 are Serre subcategories of QCoh .X; A/. Proof. By Proposition 3.2.5 it remains to prove the assertion for the subcategories (a) Nil .X; A/, (b) LNil .X; A/, (c) LNilCoh .X; A/. In each case the non-trivial part is to prove the invariance under extensions. So consider a short exact sequence 0 ! F 0 ! F ! F 00 ! 0 in QCoh .X; A/ with F 0 and F 00 in the respective subcategories. In case (a), there exist n; m 2 N such that n . id/ F F 0
and
m . id/ .F 0 / D 0:
The nilpotence of F now follows from FnCm D 0. In case (b), Proposition 3.2.5 shows that F is ind-coherent, and thus a union of coherent -subsheaves G . By (a) applied to the extension 0 ! G \ F 0 ! G ! .G C F 0 /=F 0 ! 0; we deduce the nilpotence of G . Because F is the union of such G , it is locally nilpotent.
44
3 Fundamental concepts
Finally, we come to (c). In light of (b) and Proposition 3.2.5, the important case to consider is that in which F 0 is locally nilpotent and F 00 is coherent. Here Lemma 3.2.6 yields a coherent -subsheaf G 0 in F which surjects onto F 00 . But then F is the extension of the locally nilpotent -sheaf F =G 0 by the coherent -sheaf G 0 , and therefore locally nil-coherent. Remark 3.3.6. The above proof shows that any extension of a locally nilpotent subsheaf by a coherent quotient- -sheaf can be written as an extension in reverse order, i.e., of a coherent -subsheaf by a locally nilpotent quotient- -sheaf. The converse is not true, as the following example shows. S n Take R D kŒx and A D k, and let M D n kŒx q with the -action m 7! mq . Then M 0 ´ R M is of finite type and M=M 0 is locally nilpotent, so M corresponds to a locally nil-coherent -sheaf on A1k which is, of course, not coherent. But W M ! M is bijective; hence M possesses no non-zero locally nilpotent -subsheaf. By Proposition 2.3.2, the Serre subcategory LNil .X; A/ defines a corresponding multiplicative system: Definition 3.3.7. A homomorphism of -sheaves is called a nil-isomorphism if both its kernel and cokernel are locally nilpotent. Of course, by Proposition 3.3.3 a homomorphism of coherent -sheaves is a nilisomorphism if and only if its kernel and cokernel are nilpotent. Proposition 3.3.8. Any nil-isomorphism ' W F ! G of ind-coherent -sheaves is the filtered direct limit of nil-isomorphisms of coherent -sheaves. Proof. For any coherent -subsheaf G 0 G we have a commutative diagram with exact rows 0
/ ker '
/F
k
[
0
/ ker '
/ ' 1 .G 0 /
'
/G '0
/ coker '
[
[
/ G0
/ coker ' 0
/0 / 0,
where ' 0 is again a nil-isomorphism. Since G 0 is coherent, so is its -subsheaf im ' 0 . Thus by Lemma 3.2.6, applied to the left part of the lower exact sequence, there exists a coherent -subsheaf F 0 ' 1 .G 0 / with the same image in G 0 . Moreover, we may assume that F 0 contains any given coherent -subsheaf H 0 ker ', because we may still replace F 0 by F 0 C H 0 . In any case the construction implies that the homomorphism F 0 ! G 0 induced by ' is a nil-isomorphism. Finally the union of all G 0 is G , and as G 0 and H 0 vary, the union of all F 0 is F . The lemma follows. The following characterization of nil-isomorphisms will be useful. Note that inverse image by n id always preserves coherence.
3.3 Nilpotence
45
Proposition 3.3.9. A homomorphism of -sheaves ' W F ! G is a nil-isomorphism if there exist n 0 and a homomorphism of -sheaves ˛ making the following diagram commute: n / . n id/ F 9F tt t ˛ tt ' t . n id/ ' tt t t n / G. . n id/ G If F and G are coherent, the converse is also true. Proof. During this proof, let us abbreviate H .n/ ´ . n id/ H for any -sheaf H . If the desired ˛ exists, we have a commutative diagram 0
/ .ker '/.n/
0
/ ker '
n
/ F .n/
/ G.n/ z ˛ zz n zzz n z |zz ' /F /G
/ .coker '/.n/
/0
n
/ coker '
/ 0.
Here the bottom row is exact and, since inverse image is a right exact functor on quasicoherent sheaves, the top row is a complex whose right half is exact. It is straightforward to deduce that the outer vertical homomorphisms vanish. This shows that ker ' and coker ' are nilpotent; hence ' is a nil-isomorphism, as desired. Conversely, suppose that F and G are coherent and ' is a nil-isomorphism. Then ker ' and coker ' are coherent and locally nilpotent; hence they are nilpotent. Choose m so that m vanishes on both of them, and consider the commutative diagram .ker '/.m/ 0
ker '
/ F .m/
/ / .im '/.m/ / G.m/ t u ˛1 t ˛2 u u m m m t u t zt zu /F / / im ' /G
/ .coker '/.m/ 0
/ coker ',
where the dashed arrows are not yet defined. Since its rows are exact at F .m/ and at G , one finds unique diagonal homomorphisms ˛1 and ˛2 making everything commute. Defining ˛ as the composite G.2m/
. m id/ ˛2
/ .im '/.m/
˛1
/F,
the desired assertion follows with n D 2m. Corollary 3.3.10. For every -sheaf F and for every n 0 the homomorphism Fn W . n id/ F ! F is a nil-isomorphism. Proof. Apply Proposition 3.3.9 to ' D Fn and ˛ D idF .
46
3 Fundamental concepts
The preceding results possess dual versions, involving direct image instead of inverse image by n id. For any -sheaf F and any n 0, the sheaf . n id/ F together with the composite homomorphism / . n id/ F i4 i i ii i i i base change iiii n iiii . id/ F . n id/ . id/ F
. id/ . n id/ F
is a -sheaf . n id/ F (compare Definition 4.4.1), and the adjoint of Fn defines a natural homomorphism F ! . n id/ F . If the morphism W X ! X is finite, this operation maps Coh .X; A/ to itself, but in general this is not so. Note that this assumption is guaranteed whenever X is a localization of a scheme of finite type over k. The dual version of Proposition 3.3.9 is obtained simply by adjunction with respect to . n id/ and . n id/ . The dual version of Corollary 3.3.10 reads: Corollary 3.3.11. For every -sheaf and for every n 0 the natural homomorphism F ! . n id/ F is a nil-isomorphism. Passing to the direct limit over the above natural homomorphisms, we can associate to any F the -sheaf Fy ´ lim . n id/ F : (3.3.12) ! n
Proposition 3.3.13. (a) The assignment F 7! Fy defines an exact functor QCoh .X; A/ ! QCoh .X; A/ which maps nil-isomorphisms to isomorphisms. (b) The induced natural transformation id ! y is a nil-isomorphism. (c) The subcategories LNilCoh .X; A/ and IndCoh .X; A/ are preserved under the functor F 7! Fy . In particular Fy is a canonical representative of the nil-isomorphism class of F . Note that for F 2 Coh .X; A/ the -sheaf Fy is usually only in LNilCoh .X; A/. Proof. Clearly F 7! Fy defines an additive functor, which is exact by the exactness of filtered direct limits in QCoh .X; A/. If F is nilpotent, say annihilated by m , the homomorphisms . n id/ F ! . nCm id/ F in the direct system vanish for all n; hence Fy D 0. As F 7! Fy commutes with filtered direct limits, the same follows when F is locally nilpotent. By exactness this implies that the functor maps nil-isomorphisms to isomorphisms, proving (a). Next observe that by the exactness of filtered direct limits and Proposition 3.3.4, the class of nil-isomorphisms is preserved under filtered direct limits. Since the natural homomorphism F ! Fy is such a limit, it is a nil-isomorphism, proving (b). Finally, both LNilCoh .X; A/ and IndCoh .X; A/ are Serre subcategories of QCoh .X; A/ containing LNil .X; A/; hence they are closed under nil-isomorphisms. Thus (c) follows from (b).
3.4 A-crystals
47
3.4 A-crystals By Proposition 3.3.5 the categories in the upper row of 3.3.3 are Serre subcategories of the categories in the lower row. Proposition 2.3.2 identifies the corresponding saturated multiplicative systems with the respective classes of nil-isomorphisms. In this section we will study basic properties of the associated localized categories. Our first concern is existence. Recall first that QCoh .X; A/ is a Grothendieck category by Theorem 3.2.7. Thus by Proposition 2.4.3 it is locally small, and the same follows for any full subcategory. From Proposition 2.3.4 we deduce that in each case the localization with respect to nil-isomorphisms yields a well defined abelian category. Definition 3.4.1. In the following commutative diagram the lower row is obtained from the upper row by localization with respect to nil-isomorphisms, the vertical arrows are the respective localization functors, and the lower horizontal arrows are obtained from the upper horizontal arrows by the universal property of localization: Coh .X; A/ q
Crys.X; A/
/ LNilCoh .X; A/ q
/ LNilCrys.X; A/
/ IndCoh .X; A/ q
/ IndCrys.X; A/
/ QCoh .X; A/ q
/ QCrys.X; A/.
We refer to the objects of Crys.X; A/ as A-crystals on X , and to the objects of QCrys.X; A/ as A-quasi-crystals on X. As A remains fixed throughout most of this book, we mostly speak only of (quasi)crystals on X. Proposition 3.4.2. (a) The horizontal functors in the bottom row of the diagram in 3.4.1 are fully faithful. (b) LNilCrys.X; A/ and IndCrys.X; A/ are Serre subcategories of the category QCrys.X; A/. (c) The functor Crys.X; A/ ! LNilCrys.X; A/ is an equivalence of categories. Proof. For the right and middle horizontal functors this follows from Proposition 2.3.8, for the left horizontal functor from Proposition 2.2.8 (b0 ) and the definition of LNilCoh .X; A/. Remark 3.4.3. By Proposition 3.4.2 (a) we can, and do, view all the stated categories as full abelian subcategories of QCrys.X; A/. However, we must be a little careful with Crys.X; A/. The definition of LNilCoh .X; A/ and the fact that it is closed under nil-isomorphisms shows that LNilCrys.X; A/ is the closure of Crys.X; A/ under isomorphisms in QCrys.X; A/. But the example in Remark 3.3.6 shows that Crys.X; A/ 6D LNilCrys.X; A/, so Crys.X; A/ is not a Serre subcategory of QCrys.X; A/. Whenever an object of QCrys.X; A/ can be replaced by an isomorphic one, this is not a problem.
48
3 Fundamental concepts
Remark 3.4.4. By our convention from Section 2.2 a localized category possesses the same class of objects as the original one, only with other homomorphism sets. To avoid confusion one should keep in mind that the categorical properties of an object depend solely on the homomorphisms. Thus in the interest of clarity we will always speak of the (quasi)-crystal associated to a -sheaf, and of the -sheaf underlying a (quasi)-crystal, although these objects are really ‘the same’. This rule will help to clarify whether a property refers to the category QCoh .X; A/ or to QCrys.X; A/. For example we can paraphrase Proposition 2.3.5 (a) (i) as saying that a quasi-crystal is zero if and only if its underlying -sheaf is locally nilpotent. We also observe that the quasi-crystal associated to a -sheaf is isomorphic to a crystal if and only if the -sheaf is locally nil-coherent. As before we use solid arrows ! to denote homomorphisms in QCoh .X; A/, double arrows H) to denote nil-isomorphisms, and dotted arrows K to emphasize homomorphisms in QCrys.X; A/. We retain solid arrows for functors and natural transformations, even if their target is a category of (quasi)-crystals. In any case, the rule regarding dotted arrows will be relaxed to some extent in the later chapters. Theorem 3.4.5. The category QCrys.X; A/ is a Grothendieck category. Its subcategory IndCrys.X; A/ is closed under filtered direct limits and itself a Grothendieck category. Furthermore every object of IndCrys.X; A/ is a union of objects in Crys.X; A/. Proof. It follows from Theorem 3.2.7 and Propositions 3.3.5 and 3.3.4 that the categories LNil .X; A/ QCoh .X; A/ satisfy Assumption 2.4.4, so the first assertion follows from Proposition 2.4.9 (a). The second assertion follows from Theorem 3.2.7 and Proposition 2.4.9 (d). Finally the last assertion is immediate from the definition of IndCrys.X; A/ and Proposition 2.4.9 (b). As with nil-isomorphisms between coherent -sheaves, there is a standard way to represent homomorphisms of crystals. Proposition 3.4.6. Any homomorphism ' W F K G in Crys.X; A/ can be represented for suitable n by a diagram n
F (H . n id/ F ! G : If the morphism W X ! X is finite, then ' can be represented by a diagram n
F ! . n id/ G (H G : s
Proof. From Section 2.2 recall that ' can be represented by a diagram F (H F 0 ! G in Coh .X; A/ where s is a nil-isomorphism. Applying Proposition 3.3.9 to s, we obtain a commutative diagram s / F skia JJ F0 t: G O JJJJJ tt t JJJJ tt JJJJ JJJ ˛ tttt ˛ n JJ t . n id/ F ,
3.5 Examples
49
where n is again a nil-isomorphism by Corollary 3.3.10. Thus ' is also represented by the lower edge of this diagram, which proves the first assertion. The second assertion is proved in the same way, using the dual versions of 3.3.9 and 3.3.10. Finally, we continue the discussion of the functor F 7! Fy defined in the previous section. By Proposition 2.4.8 the localization functor q W QCoh .X; A/ ! QCrys.X; A/ has a right adjoint splitting snil W QCrys.X; A/ ! QCoh .X; A/:
(3.4.7)
Proposition 3.4.8. (a) The functor F 7! Fy coincides with snil q. (b) The functor snil is exact. (c) snil maps the subcategories LNilCrys.X; A/ IndCrys.X; A/ to the subcategories LNilCoh .X; A/ IndCoh .X; A/. Proof. By the proof of Proposition 2.4.8 we have snil .q.F // Š lim F 0 , where the ! limit is taken over all nil-isomorphisms F ) F 0 . By Proposition 3.3.13 any such nil-isomorphism yields a commutative diagram of nil-isomorphisms +3 F 0 F > >>> > >>> >>>> >>" / Fy Fy 0 ,
where the lower horizontal arrow is an isomorphism. Thus the single nil-isomorphism F ) Fy is cofinal in the whole direct system. It follows that lim F 0 is functorially ! isomorphic to Fy , proving (a). Assertion (b) follows from the exactness of F 7! Fy and Proposition 2.3.7. Assertion (c) follows from the fact that LNilCoh .X; A/ and IndCoh .X; A/ are closed under nil-isomorphisms. Corollary 3.4.9. Any homomorphism F K G in QCrys.X; A/ is represented by a unique homomorphism Fy ! Gy .
3.5 Examples Throughout this section, we assume that C D Spec A is an irreducible smooth curve over k whose smooth compactification Cx is obtained by adjoining precisely one closed point. For any non-zero element a 2 A we set deg.a/ ´ logq ŒA=Aa. By a line bundle L on X we mean a group scheme over X which is Zariski locally isomorphic to the additive group scheme Ga X . Its endomorphism ring Endk .L/ consists of all k-linear endomorphisms as a group scheme. If X D Spec R is affine and L is free, one can identify Endk .L/ with the non-commutative polynomial ring RŒ defined by the commutation rule u ´ uq for all u 2 R.
50
3 Fundamental concepts
Definition 3.5.1. A Drinfeld A-module of rank r > 0 on X consists of a line bundle L on X and a ring homomorphism ' W A ! Endk .L/; a 7! 'a , such that for all points x 2 X with residue field kx the induced map 'x W A ! Endk .Ljx/ Š kx Œ ;
a 7!
1 X
ui .a/ i
iD0
has coefficients ui .a/ D 0 for i > r deg.a/ and ur deg.a/ .a/ 2 kx for a 2 A non-zero. A homomorphism .L; '/ ! .L0 ; ' 0 / of Drinfeld A-modules over X is a homomorphism of line bundles L ! L0 that is equivariant with respect to the actions ' and ' 0 . The characteristic of .L; '/ is the morphism of schemes ch' W X ! C corresponding to the ring homomorphism d' W A ! EndOX .Lie.L// Š .X; OX /. Example 3.5.2. In the special case where X D Spec R is affine and L is free over X , any Drinfeld A-module is isomorphic to one in the standard form X
r deg.a/
' W A ! Rfg;
a 7!
ui .a/ i ;
iD0
where ur deg.a/ .a/ is a unit in R for all non-zero a 2 A. The characteristic of ' is then the morphism corresponding to the ring homomorphism A ! R, a 7! u0 .a/. We now give a construction due to Drinfeld which attaches a coherent -sheaf to any Drinfeld A-module .L; '/ of rank r on X . Note first that U 7! Homk LjU; Ga U defines a quasi-coherent sheaf of OX -modules on X . By right composition with 'a it becomes a sheaf of OX ˝ A-modules. We let M.'/ be the corresponding quasicoherent sheaf of OXC -modules on X C , which is easily seen to be locally free of rank r (compare [3]). Let now 2 Endk .Ga X / denote the Frobenius endomorphism relative to X. Then left composition with defines an OXC -linear homomorphism M.'/ ! . id/ M.'/, and thus via adjunction an OXC -linear homomorphism W . id/ M.'/ ! M.'/: We denote the resulting -sheaf by M.'/. Clearly this construction is functorial in .L; '/. Also it is easy to see that coker. / is supported on the graph of ch' and locally free of rank 1 over X. Example 3.5.3. In Example 3.5.2 the module underlying M.'/ is M.'/ ´ RŒ . Here R and act by left multiplication, and a 2 A by right multiplication with 'a . It is easy to see that M.'/ is finitely generated over R ˝ A. In the special case A D kŒt it is free over R ˝ A Š RŒt with basis f1; ; 2 ; : : : ; r1 g.
3.5 Examples
51
The above properties of M.'/ are part of the motivation for the following definition, essentially due to Anderson. We fix a morphism ch W X ! C . Definition 3.5.4. A family of A-motives over X, or short an A-motive on X , of rank r and of characteristic ch is a coherent -sheaf M on X such that (a) the underlying sheaf M is locally free of rank r, and (b) the set-theoretic support of coker. / is a subset of the graph of ch. When X is the spectrum of a field K and the module corresponding to M is finitely generated over KŒ, the above M is a Drinfeld-Anderson A-motive in the sense of [43, Definition 5.1]. If furthermore A D kŒt , it is a t -motive in the sense of Anderson [1]. The category of Drinfeld A-modules does not permit the formation of direct sums or tensor products or related operations from linear algebra. The passage to Anderson’s t -motives, and more generally the A-motives above, adds this missing flexibility. In the following chapters we will see that A-crystals are even more flexible in that they form an abelian category with tensor product, which possesses a cohomology theory with compact supports with most of the usual properties. There are further generalization of A-motives and Drinfeld modules, such as elliptic sheaves [3], or F -sheaves and shtukas [15], [30]. These are systems of coherent sheaves on X Cx with a certain kind of semi-linear operation , where the coefficient scheme is made projective at the expense of having to deal with a diagram of objects. The theory in the following chapters is not directly applicable to these more general objects, so we do not pursue this direction.
Chapter 4
Functors
In this chapter we define various functors of A-crystals and A-quasi-crystals and study their basic properties. We will see that, in spite of the seemingly artificial definition of the category Crys.X; A/, its objects behave very much like constructible sheaves of Amodules. For instance, although the inverse image functor on coherent sheaves is only right exact, the functor f on A-crystals is exact (see 4.1.12). Besides, many properties can be tested by passing to ‘stalks’(cf. 4.1.7 and 4.1.11). We also discuss tensor product and direct image. Certain results on finite radicial morphisms (4.4.6 and 4.4.7) exhibit a close analogy with étale sheaves. Furthermore, for every open embedding there is an ‘extension by zero’ functor jŠ , not present for coherent or quasicoherent sheaves. All these functors satisfy the usual relations between them, and all except non-proper direct image preserve coherence. The last section in this chapter provides additional evidence for the analogy with constructible sheaves. Throughout this chapter we fix a morphism f W Y ! X .
4.1 Inverse image Definition 4.1.1. For any -sheaf F on X we let f F denote the -sheaf on Y consisting of .f id/ F and the composite homomorphism f F
/ .f id/ F 4 j j j jj j j j jjjj jjjj .f id/ F .f id/ . id/ F .
. id/ .f id/ F
For any homomorphism ' W F ! F 0 we abbreviate f ' ´ .f id/ '. This defines an A-linear functor f W QCoh .X; A/ ! QCoh .Y; A/ which is clearly right exact. When f is flat, it is exact. In general, its exactness properties are governed by the associated Tor-objects: Definition 4.1.2. For any -sheaf F on X and any integer i we let Li f F denote the -sheaf on Y consisting of the quasi-coherent sheaf 1 O XC
id/ Tor .f i
.f id/1 F ; OY C
4.1 Inverse image
53
together with its own natural Li f F arising from F . Any homomorphism ' W F ! F 0 induces a homomorphism Li f ' W Li f F ! Li f F 0 in the obvious way. Proposition 4.1.3. (a) If F is locally nilpotent, then so is Li f F . (b) If ' is a nil-isomorphism, then so is Li f '. (c) The functor Li f on -sheaves induces a unique A-linear functor Li f W QCrys.X; A/ ! QCrys.Y; A/: (d) These functors form a sequence of ı-functors. (e) These functors commute with filtered direct limits. (f) The functor f ´ L0 f on quasi-crystals is right exact. When f is flat, it is exact. (g) The functor f induces functors f W IndCrys.X; A/ ! IndCrys.Y; A/; f W Crys.X; A/ ! Crys.Y; A/: Proof. If Fn vanishes, then so does Lni f F , proving (a) in the case that F is nilpotent. In the general case one writes F as a union of nilpotent -sheaves and uses the fact that .f id/ and its derived functors on quasi-coherent sheaves commute with filtered direct limits. Next consider a short exact sequence '
0 ! F 0 ! F ! F 00 ! 0 in QCoh .X; A/. Then we have a long exact sequence in QCoh .Y; A/, : : : ! L1 f F 00 ! f F 0 ! f F ! f F 00 ! 0: To prove (b) we may reduce ourselves to mono- and epimorphisms by Proposition 2.3.5 (b). Thus if F 0 or F 00 in the above short exact sequence is locally nilpotent, the long exact sequence shows that Li f , respectively Li f ', is a nil-isomorphism for all i , proving (b). Next, (b) implies that the composite functor QCoh .X; A/ ! QCoh .Y; A/ ! QCrys.Y; A/ maps nil-isomorphisms to isomorphisms; hence by Proposition 2.3.7 it factors through a unique functor on quasi-crystals, proving (c). Assertion (d) follows from Proposition 2.3.5 (c) and the long exact sequence above. Assertions (e) and (f) follow from the corresponding property of tensor product and Tor. Finally, assertion (g) follows from (e) and the fact that .f id/ on quasi-coherent sheaves preserves coherence. Remark 4.1.4. The notation Li f above suggests that these functors are the derived functors of f on quasi-crystals. In Proposition 6.1.4 we will see that this is actually so, but until then one should view them simply as an ad hoc sequence of ı-functors.
54
4 Functors g
f
Proposition 4.1.5. For any two morphisms Z ! Y ! X there is a natural isomorphism of functors .fg/ Š g f . Proof. Obvious from the construction. The following result shows that quasi-crystals behave like sheaves: S Proposition 4.1.6. Suppose that X D i Ui is an open covering with embeddings ji W Ui ,! X. (a) A quasi-crystal F on X is zero if and only if ji F is zero in QCrys.Ui ; A/ for all i . (b) A homomorphism ' in QCrys.X; A/ is a monomorphism, an epimorphism, an isomorphism, resp. zero, if and only if its inverse image ji ' has that property for all i . Proof. The ‘only if’ part of (a) is contained in Proposition 4.1.3 (a). The ‘if’ part means that a -sheaf F on X is locally nilpotent if ji F is locally nilpotent for all i. To prove this consider any coherent subsheaf G F . Its restriction to Ui C is then a coherent subsheaf of .ji id/ F , so it is annihilated by some n . Since X is noetherian, the open covering possesses a finite subcovering, for which one choice of n is enough. Thus the sum X n .. n id/ G / n0
is finite and defines a nilpotent -subsheaf of F . Varying G we deduce that F is a union of nilpotent -subsheaves, proving (a). By applying (a) to kernels, cokernels, and images, and using the exactness of ji , one deduces (b). Next consider any point x 2 X . Let kx denote its residue field and ix W x Š Spec kx ,! X its natural embedding. The object ix F can be viewed as the stalk of F at x. The next few results concern its properties. First we show that the vanishing of a crystal can be detected on stalks. For coherent -sheaves this means that nilpotence can be detected stalkwise; this is an analogue of the well-known fact that an element of a commutative ring is nilpotent if and only if it is contained in every prime ideal. A stronger version of this result will be proved in Section 4.6. Theorem 4.1.7. (a) A coherent -sheaf F on X is nilpotent if and only if the -sheaf ix F is nilpotent for every x 2 X . (b) An object F in IndCrys.X; A/ is zero if and only if ix F is zero in IndCrys.x; A/ for every x 2 X . Proof. The ‘only if’ part of (a) is a special case of Proposition 4.1.3 (a). For the ‘if’ part note first that the images of Fn form a decreasing sequence of coherent subsheaves of F . Thus their supports form a decreasing sequence of closed subschemes Zn X C .
4.1 Inverse image
55
As X C is noetherian, we deduce that Z1 ´ Zn is independent of n whenever n 0. If Z1 D ¿ we are done. Otherwise let be a generic point of Z1 and set x ´ pr 1 . /. We claim that ix F is not nilpotent. To prove this we may replace X by its localization at x, after which X D Spec R for a noetherian local ring R, and x corresponds to the maximal ideal m R. Let M be the .R ˝ A/-module corresponding to F and Mn the submodule generated by the image of n . By construction the support of Mn is non-empty and contained in x C whenever n 0. Since M is of finite type over the noetherian ring R ˝ A, so is Mn ; hence mr Mn D 0 for any r 0. If ix F is nilpotent, we have s M mM for some s 1. For every i 0 this implies i iCs M i .mM / mq M: Using the Artin–Rees lemma [36, Theorem 8.5] we deduce iCsCn M iCs Mn Mn \ iCs M i
Mn \ mq M i D mq c Mn \ mc M mq
i c
Mn
for some constant c and all i 0. But the last submodule is zero whenever q i c r; hence MiCsCn D 0, contradicting our assumption on Z1 . This proves (a). Assertion (b) for crystals follows directly from (a) and Proposition 2.3.5 (a). The general case is a consequence of Proposition 4.1.3 (e). Remark 4.1.8. The analogous statement for quasi-crystals is false. For example take X ´ A1k D Spec kŒx and A D k. Then -sheaves on X correspond to kŒx-modules together with a -linear endomorphism . Set M ´ kŒx; x 1 =kŒx with m D mq . Then M is supported only at the point x D 0, but M=xM D 0. On the other hand is not locally nilpotent on M . In the remainder of this section we study exactness properties. It will turn out that the functor f on IndCrys is exact. This phenomenon is related to the fact that the Frobenius morphism W X ! X is strongly contracting in an algebraic sense, because X its derivative vanishes everywhere. We will see that the higher Tor O of coherent i sheaves inherit a similar contractive property; this implies nilpotence and therefore the desired exactness. We begin with the following technical lemma. Lemma 4.1.9. If X is affine, every coherent -sheaf on X is the quotient of a coherent -sheaf whose underlying coherent sheaf is free. Proof. Suppose that X D Spec R and let M be the .R ˝ A/Œ -module corresponding to a coherent -sheaf on X. Then M is of finite type over R ˝ A, so we may write
56
4 Functors
it as a quotient of a free module of finite type N ´ .R ˝ A/r . Since N is free, the semi-linear endomorphism of M can be lifted to a semi-linear endomorphism of N . Proposition 4.1.10. (a) For any coherent -sheaf F on X , any point x 2 X , and any i 1, the -sheaf Li ix F is nilpotent. (b) The functor Li ix W IndCrys.X; A/ ! IndCrys.x; A/ is zero for any x 2 X and i 1. (c) The functor ix W IndCrys.X; A/ ! IndCrys.x; A/ is exact for any x 2 X . Proof. To prove (a) we may replace X by its localization at x, so we may assume that X D Spec R for a noetherian local ring with maximal ideal m R. Let M be the .R ˝ A/Œ-module corresponding to F . Using Lemma 4.1.9 we may write it as the quotient of an .R ˝ A/Œ -module P which is free of finite type over R ˝ A. Consider the resulting short exact sequence 0 ! K ! P ! M ! 0: The long exact Tor-sequence yields isomorphisms Tor R˝A ! Tor R˝A K; .R=m/ ˝ A iC1 M; .R=m/ ˝ A i for all i 1. Thus (a) follows by induction on i , if it is proved for i D 1. The long exact Tor-sequence also yields the exact sequence 0 ! Tor R˝A M; .R=m/ ˝ A ! K=mK ! P =mP ! M=mM ! 0 1 and hence an isomorphism
Tor R˝A M; .R=m/ ˝ A Š .K \ mP /=mK: 1
By the Artin–Rees lemma there exists j0 such that for all j j0 we have K \ mj P D mj j0 .K \ mj0 P /: For any ` with q ` > j0 we then have `
` .K \ mP / K \ mq P D mq
` j
0
.K \ mj0 P / mK:
Thus the endomorphism of .K \ mP /=mK induced by is nilpotent, as desired. Assertion (b) for crystals follows directly from (a) and Proposition 2.3.5 (a), and the general case is a consequence of Proposition 4.1.3 (e). Finally, (c) results from (b) and Proposition 4.1.3 (d). Proposition 4.1.11. (a) A homomorphism ' in IndCrys.X; A/ is a monomorphism, an epimorphism, an isomorphism, resp. zero, if and only if its inverse image ix ' has that property for every point x 2 X. '
(b) A sequence F K F 0 K F 00 in IndCrys.X; A/ is a complex, resp. exact in the middle, if and only if the induced sequence ix F 0 K ix F K ix F 00 in IndCrys.x; A/ has that property for every point x 2 X .
4.1 Inverse image
57
Proof. The exactness 4.1.10 (c) shows that ix commutes with the formation of kernels, cokernels, and images, i.e., that ix .ker '/ D ker.ix '/, and so on. This together with Theorem 4.1.7 (b) implies (a). Applying (a) to the composite homomorphism in (b) proves the assertion about being a complex, and applying it to the inclusion im ' / ker shows the rest. Theorem 4.1.12. (a) For any coherent -sheaf F on X and any i 1, the -sheaf Li f F is nilpotent. (b) The functor Li f W IndCrys.X; A/ ! IndCrys.Y; A/ is zero for any i 1. (c) The functor f W IndCrys.X; A/ ! IndCrys.Y; A/ is exact. Proof. For any y 2 Y set x ´ f .y/ and consider the commutative diagram y fy
iy
/Y f
x
ix
/ X.
Consider the associated commutative diagram of inverse image functors Coh .y; A/ o O
iy
Coh .Y; A/ O
fy
Coh .x; A/ o
f ix
Coh .X; A/.
From the underlying coherent sheaves we have spectral sequences in Coh .y; A/ Eij2 D Li iy .Lj f F / H) LiCj .f iy / F 0 2 Eij
D Li fy .Lj ix F / H) LiCj .ix fy / F .
Proposition 4.1.10 (a) implies that Eij2 is nilpotent whenever i 6D 0 and that 0Eij2 is nilpotent whenever j 6D 0. Thus the corresponding spectral sequences in Crys.y; A/ degenerate to an isomorphism iy .Lj f F / Š Lj fy .ix F / in Crys.y; A/ for every j . On the other hand, for j 6D 0 the latter object vanishes already in Coh .y; A/, because fy is flat. This proves that iy .Lj f F / is nilpotent as a -sheaf. Since y was arbitrary, Theorem 4.1.7 implies that Lj f F is nilpotent, proving (a). The rest follows from (a) just as in the proof of Proposition 4.1.10. Proposition 4.1.13. If f is surjective, the functor f W IndCrys.X; A/ ! IndCrys.Y; A/ is faithful.
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4 Functors
Proof. Since f is exact and commutes with filtered direct limits, it suffices to show that for any crystal F on X , if f F is zero, then so is F . Using Theorem 4.1.7 both on X and on Y we can reduce ourselves to the case that X D Spec K and Y D Spec L where K L are fields. Then the K ˝ A-module M underlying F embeds into the .L ˝ A/-module L ˝K M underlying f F . Thus if is nilpotent on the latter, it is so on the former, as desired.
4.2 Tensor product Definition 4.2.1. For any -sheaves F and G on X over A, we define F ˝ G as the -sheaf consisting of F ˝ G and the composite homomorphism . id/ F ˝ G
F ˝G
/F ˝G jj4 j j j jjj jjjj. /˝ j j j F G jjj . id/ F ˝ . id/ G .
With the usual tensor product of homomorphisms this defines an A-bilinear bifunctor ˝ W QCoh .X; A/ QCoh .X; A/ ! QCoh .X; A/; which is right exact in both variables. Its exactness properties are governed by the associated Tor-objects: Definition 4.2.2. For any F and G as above and any integer i we let Tor i .F ; G / X C denote the -sheaf consisting of the quasi-coherent sheaf Tor O .F ; G / and the i natural homomorphism Tor i .F ; G /. Proposition 4.2.3. (a) If F or G is locally nilpotent, then so are F ˝G and Tor i .F ; G / for all i . (b) If ' and are nil-isomorphisms, then so are ' ˝ and Tor i .'; / for all i . (c) The above bi-functors on -sheaves induce unique A-linear bi-functors ˝; Tor i W QCrys.X; A/ QCrys.X; A/ ! QCrys.X; A/: (d) These bi-functors form a sequence of ı-functors in each variable. (e) These bi-functors commute with filtered direct limits. (f) The bi-functor ˝ on quasi-crystals is right exact in each variable. (g) The bi-functors ˝ induces bi-functors ˝ W IndCrys.X; A/ IndCrys.X; A/ ! IndCrys.X; A/; ˝ W Crys.X; A/ Crys.X; A/ ! Crys.X; A/:
4.2 Tensor product
59
Proof. Entirely analogous to that of Proposition 4.1.3. Example 4.2.4. Let 1lX;A denote the crystal on X consisting of the structure sheaf OXC and the natural isomorphism . id/ OXC ! OXC : The canonical isomorphism F ˝ 1lX;A Š F allows us to view it as a unit object for the tensor product. Proposition 4.2.5. There are functorial isomorphisms for all F , G , H in QCrys.X; A/: (a) F ˝ G Š G ˝ F . (b) F ˝ .G ˝ H / Š .F ˝ G / ˝ H . (c) f .F ˝ G / Š f F ˝ f G . Proof. Obvious from the construction. Of course, the tensor product of quasi-crystals is not exact in general: compare Remark 6.2.8. Thus its acyclic objects become important: Definition 4.2.6. A quasi-crystal F on X is called flat if Tor i .F ; G / D 0 for all i 1 and all G 2 IndCrys.X; A/. It is called very flat if Tor i .F ; G / D 0 for all i 1 and all G 2 QCrys.X; A/. Since we are speaking of quasi-crystals, the vanishing of Tor i .F ; G / is meant in QCrys.X; A/. The corresponding full subcategories are denoted QCrysvflat .X; A/ QCrysflat .X; A/ QCrys.X; A/: The notion ‘flat’ is the one we are really interested in; we will study it in detail in Chapter 7. The notion ‘very flat’ is needed only for technical reasons. Clearly ‘very flat’ implies ‘flat’, but the converse is not true, as the following example shows: Example 4.2.7. Let X ´ Spec kŒx and A ´ k. As in Remark 3.2.3 define M by the short exact sequence 0 ! kŒx ! kŒx; x 1 ! M ! 0: Set N ´ kŒx=.x/ and let act on all these modules by .m/ ´ mq . Then the long exact Tor-sequence yields a -equivariant isomorphism Tor kŒx 1 .M; N / Š kŒx ˝kŒx N Š N: Thus if F is the quasi-crystal associated to M and G the crystal associated to N , we deduce that Tor 1 .F ; G / is a non-zero quasi-crystal. Therefore G is not very flat. But it is flat, for instance by Corollary 7.3.6.
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4 Functors
Proposition 4.2.8. (a) If F is very flat, then Li f F D 0 in QCrys.Y; A/ for all i 1. (b) If F is very flat, then f F is very flat. (c) If F and G are very flat, then F ˝ G is very flat. Proof. All assertions may be verified locally on X and Y by Proposition 4.1.6 (a), so without loss of generality we may assume that X D Spec R and Y D Spec S. Let M be the .R ˝ A/Œ -module corresponding to F . Then S ˝ A with the action .u ˝ a/ ´ uq ˝ a can be viewed either as an .S ˝A/Œ -module or as an .R ˝A/Œ .M; S ˝A/ corresponds to the -sheaf module. In the former sense the module Tor R˝A i Li f F , and in the later sense it corresponds to a -sheaf of the form Tor i .F ; G /. By assumption, the latter is locally nilpotent for i 1; hence so is Li f F , proving (a). For (b) let N be the .S ˝ A/Œ -module corresponding to a -sheaf G on Y . Then Tor i .f F ; G / corresponds to the module Tor S˝A .M ˝.R˝A/ .S ˝ A/; N /: i 2 This is the term Ei0 of the spectral sequence R˝A Eij2 D Tor S˝A .M; S ˝ A/; N / ) Tor R˝A Torj i iCj .M; N /:
We have already seen that TorjR˝A .M; S ˝ A/ is locally nilpotent for all j 1; hence the terms Eij2 are locally nilpotent for all j 1 by Proposition 4.2.3 (a). Again since F is very flat, the limit Tor R˝A iCj .M; N / is locally nilpotent for all i C j 1. It follows 2 that Ei0 is locally nilpotent for all i 1, hence f F is very flat, proving (b). The proof of (c) along the same lines is left to the reader.
4.3 Change of coefficients Consider an algebra homomorphism A ! A0 , where A0 is again a localization of a finitely generated k-algebra. Set C 0 ´ Spec A0 and let h W C 0 ! C denote the corresponding morphism of schemes. Definition 4.3.1. For any -sheaf F on X over A we let F ˝A A0 denote the -sheaf on X over A0 consisting of .id h/ F and the composite homomorphism F ˝ A0 A
/ .id h/ F j4 j j j jjjj jjj.idh/ j j j F jj .id h/ . id/ F .
. id/ .id h/ F
For any homomorphism ' W F ! F 0 we write ' ˝ id ´ .id h/ '.
4.4 Direct image
61
This defines a right exact A-linear functor ˝A A0 W QCoh .X; A/ ! QCoh .X; A0 /: Our notation is meant to emphasize the role of A and A0 as rings of coefficients. Again we will also consider the associated Tor-objects: 0 Definition 4.3.2. For any F as above and any integer i we let Tor A i .F ; A / denote the -sheaf consisting of 1 O XC Tor .idh/ .id h/1 F ; OXC 0 i
together with its own natural . As in the preceding sections one proves: Proposition 4.3.3. (a) If F is locally nilpotent, then so is F ˝A A0 . (b) If ' is a nil-isomorphism, then so is ' ˝ id. (c) The above functor on -sheaves induces a unique A-linear functor ˝A A0 W QCrys.X; A/ ! QCrys.X; A0 /: (d) This functor commutes with filtered direct limits. (e) This functor is right exact. (f) This functor induces functors ˝A A0 W IndCrys.X; A/ ! IndCrys.X; A0 /; ˝A A0 W Crys.X; A/ ! Crys.X; A0 /: Proposition 4.3.4. For any two homomorphisms of finitely generated algebras A ! A0 ! A00 there are functorial isomorphisms: (a) f . (b) .
˝A A0 / Š f
˝A A0 .
˝A A0 / ˝A0 A00 Š
˝A A00 .
4.4 Direct image Definition 4.4.1. For any -sheaf G on Y we let f G denote the -sheaf on X consisting of .f id/ G and the composite homomorphism f G
/ .f id/ G j4 j j j jjjj base change jjjj.f id/ G j j j j .f id/ . id/ G .
. id/ .f id/ G
For any homomorphism
W G ! G 0 we abbreviate f
´ .f id/ . /.
62
4 Functors
This defines an A-linear functor f W QCoh .Y; A/ ! QCoh .X; A/ which is clearly left exact. When f is affine, it is exact. More generally: Definition 4.4.2. For any -sheaf G on Y and any integer i we let Ri f G denote the -sheaf on X consisting of the quasi-coherent sheaf Ri .f id/ G together with its own natural Ri f G arising from G . Proposition 4.4.3. (a) If G is locally nilpotent, then so is Ri f G . (b) If
is a nil-isomorphism, then so is Ri f .
(c) The functor Ri f on -sheaves induces a unique A-linear functor Ri f W QCrys.Y; A/ ! QCrys.X; A/: (d) These functors form a sequence of ı-functors. (e) These functors commute with filtered direct limits. (f) The functor f D R0 f on quasi-crystals is left exact. (g) When f is affine, then Ri f is zero for all i 1, and f is exact. (h) When f is proper, the functor Ri f induces functors Ri f W IndCrys.Y; A/ ! IndCrys.X; A/; Ri f W Crys.Y; A/ ! Crys.X; A/: Proof. Entirely analogous to that of Proposition 4.1.3. g
f
Proposition 4.4.4. For any two morphisms Z ! Y ! X there is a functorial isomorphism .fg/ Š f g . Proof. Obvious from the construction. Proposition 4.4.5. The functor f on quasi-crystals is right adjoint to the functor f on quasi-crystals from Section 4.1. Proof. By Proposition 2.2.5 (a) it suffices to prove this for the categories QCoh .: : : /. So consider -sheaves F on X and G on Y , and homomorphisms of quasi-coherent '
sheaves F ! f G and f F ! G which correspond to each other by usual adjunction. One must then show that ' is compatible with F and G if and only if is compatible with them. This is straightforward and left to the reader. For the next results recall that f is radicial if and only if it is injective on points and the residue field extensions are purely inseparable (see [EGA1, § 3.7]). For instance, any closed embedding is finite and radicial.
4.4 Direct image
63
Theorem 4.4.6. If f is finite and radicial, the adjunction homomorphism f f ! id is an isomorphism in QCrys.Y; A/. Proof. The assertion is local on X by Proposition 4.1.6 (a), so without loss of generality we may assume that X is affine. Then Y is affine, say X D Spec R and Y D Spec S. Let N be the .S ˝ A/Œ -module corresponding to a -sheaf G on Y , and consider S ˝ A with the action .u ˝ a/ ´ uq ˝ a. Then f f G corresponds to the module .S ˝A/˝.R˝A/ N with the induced action, and the adjunction homomorphism corresponds to the multiplication map .S ˝ A/ ˝.R˝A/ N ! N . Since .S ˝ A/ ˝.R˝A/ N Š .S ˝ A/ ˝.R˝A/ .S ˝ A/ ˝.S˝A/ N; and tensor product preserves nil-isomorphisms, it suffices to show that the multiplication map .S ˝ A/ ˝.R˝A/ .S ˝ A/ ! S ˝ A is a nil-isomorphism. Clearly this reduces to the case A D k, i.e., to the same assertion for the multiplication map W S ˝R S ! S . Now the morphism of schemes corresponding to is the diagonal embedding Y ,! Y X Y , which is surjective by the equivalent characterization [EGA1, Proposition 3.7.1] of radicial morphisms. Thus its defining ideal ker consists of nilpotent elements. But un D 0 implies n n .u/ D uq D 0; hence ker is locally nilpotent under . Since coker D 0, it follows that is a nil-isomorphism, as desired. Theorem 4.4.7. If f is finite radicial and surjective, the adjunction homomorphism id ! f f is an isomorphism in QCrys.X; A/ and the functors QCrys.X; A/ o
f
/
QCrys.Y; A/
f
are mutually quasi-inverse equivalences of categories. Proof. Let 1lX;A be the unit crystal on X from Example 4.2.4 and 1l Y;A the analogue on Y . Then f defines a natural homomorphism ˛ W 1lX;A ! f 1l Y;A , and the composite homomorphism F Š F ˝ 1lX;A ! F ˝ f 1l Y;A Š f f F ˛˝id
is the adjunction homomorphism in question. As tensor product preserves isomorphisms, it suffices to prove that ˛ is an isomorphism in QCrys.X; A/. Now the composite homomorphism 1l Y;A Š f 1lX;A
f ˛
/ f f 1l Y;A
adj
/ 1l Y;A
is the identity, and the adjunction homomorphism on the right hand side is an isomorphism in QCrys.Y; A/ by Theorem 4.4.6. Thus f ˛ is an isomorphism in QCrys.Y; A/.
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Note that both 1lX;A and f 1l Y;A are actually crystals, because f is finite. Since f is exact on Crys.X; A/ by Theorem 4.1.12 (c), we deduce that f .ker ˛/ D ker.f ˛/ and f .coker ˛/ D coker.f ˛/ both vanish. As f is furthermore faithful on Crys.X; A/ by Proposition 4.1.13, both ker ˛ and coker ˛ vanish, proving that ˛ is an isomorphism. This shows the first assertion. The second assertion results from the first and Theorem 4.4.6. For later use we include a technical result on the property ‘very flat’: Proposition 4.4.8. If f is affine and flat, and G is very flat, then f G is very flat. Proof. The construction of Tor i .f G ; F / is local on X , and so is its vanishing by Proposition 4.1.6 (a). Thus without loss of generality we may assume that X is affine. Then Y is affine, say X D Spec R and Y D Spec S . Let N be the .S ˝ A/Œ -module corresponding to G , and M the .R ˝ A/Œ -module corresponding to any -sheaf F on X. Then Tor i .f G ; F / corresponds to the module Tor R˝A .N; M /. Since S is flat i over R, this module is equal to .N; M ˝.R˝A/ .S ˝ A//; Tor S˝A i which represents the quasi-crystal Tor i .G ; f F / on Y . By assumption this quasicrystal vanishes for i 1; hence so does Tor i .f G ; F /, as desired.
4.5 Extension by zero In this section we study the relations between crystals on X and on an open subscheme U X.To do this we choose a closed subscheme Y X whose underlying set of points is the complement of U . Let 0 OX denote its ideal sheaf. Although 0 is far from unique, this will cause no difficulties. We denote the respective inclusion morphisms by j / o i ?_ X Y. U Proposition 4.5.1. For any coherent -sheaf F on U there exists a coherent -sheaf Fz on X such that j Fz Š F and i Fz is nilpotent. Proof. To begin with, we can choose a coherent sheaf Fz on X C whose restriction to U C is F (see [12, no 1, Corollaire 2] or [26, Chapter II, Exercise 5.15]). Without loss of generality we may assume Fz .j id/ F . The homomorphism F then yields a homomorphism W . id/ Fz ! .j id/ F ; which we would like to factor through Fz . Consider the image of . id/ Fz in the (quasi-coherent) quotient sheaf .j id/ F =Fz . Being the image of a coherent sheaf,
4.5 Extension by zero
65
it is itself coherent. Since it also vanishes on U C , it is annihilated by 0n for some integer n 0. In other words, we have n . id/ Fz Fz : 0
Select an integer m with .q 1/m > n. Since 0 0q , we can calculate . id/ .0m Fz / 0q m . id/ Fz 0q mn Fz 0 0m Fz : Thus after replacing Fz by 0m Fz the homomorphism F extends to a homomorphism . id/ Fz ! 0 Fz . Let Fz be the corresponding crystal on X . Then the first condition holds by construction, and the second follows from the fact that i Fz vanishes. Proposition 4.5.2. Let F and Fz be as in Proposition 4.5.1. Then for any -sheaf Gz on X and any homomorphism ' W F ! j Gz there exists an integer n 0 such that ' extends to a homomorphism of -sheaves 'Q W 0n Fz ! Gz. Proof. First we disregard the desired compatibility with . Since any quasi-coherent sheaf on a noetherian scheme is the union of its coherent subsheaves (see [26, Chapter II, Exercise 5.15]), there exists a coherent subsheaf Gz0 Gz such that ' factors through .j id/ Gz0 . Let us abbreviate Gzj0 ´ .j id/ .j id/ Gz0 ; then ' corresponds to a homomorphism .j id/ F ! Gzj0 . Let Gx0 denote the image of Gz0 in Gzj0 and consider the diagram of quasi-coherent sheaves 0 SSSS)
0 Gztor SSS )
z w F w {w Gz0 SSSS) k5 Gx0 kkk 0 'Q
GG ' kkk5 0 GG z0 =Gx0 G # 5 j z0 kk G 5 k j k kk SSS S) 0,
where the oblique lines are exact and the dashed arrows do not yet exist. First we look at the image of Fz in the quotient sheaf Gzj0 =Gx0 . This is a coherent subsheaf which vanishes on U C , so it is annihilated by 0n for some n 0. Thus after replacing Fz by 0n Fz we may assume that this image is zero. Then we already have the vertical dotted arrow in the diagram above. 0 To lift this homomorphism to Gz0 we look at its extension class. The sheaf Gztor n is coherent and vanishes on U C , so it is annihilated by 0 for some n 0. 0 Thus 0n annihilates the local Ext-sheaf Ext 1 Fz ; Gztor , and after replacing Fz by n z 0 F the extension vanishes locally. It then corresponds to a cohomology class in 0 H 1 X C; Hom Fz ; Gztor . After replacing Fz again by 0n Fz this cohomology class vanishes, so there exists a homomorphism of quasi-coherent sheaves 'Q W Fz ! Gz0 Gz which lifts '.
66
4 Functors
It remains to make this extension compatible with . Consider the discrepancy 'Q B Fz Gz B . id/ 'Q 2 Hom . id/ Fz ; Gz : By construction it vanishes on U C . Since Fz is coherent, this homomorphism is annihilated locally by 0n for some n 0. Thus after replacing Fz by 0n Fz one last time the discrepancy vanishes, and 'Q is a homomorphism of -sheaves, as desired. Passing to the associated crystals we deduce: Proposition 4.5.3. Let F and Fz be as in Proposition 4.5.1. Then for any quasi-crystal Gz on X the inverse image under j induces a bijection j W HomQCrys .Fz ; Gz/ ! HomQCrys .F ; j Gz/: Proof. First consider a homomorphism of -sheaves F ! j Gz. By Proposition 4.5.2 it extends to a homomorphism 0n Fz ! Gz for some integer n 0. By assumption i i Fz Š Fz =0 Fz is nilpotent; hence so is Fz =0n Fz . Thus the inclusion 0n Fz ,! Fz is a nil-isomorphism, and the left fraction Fz ( 0n Fz ! Gz defines the desired extension. Next let F 0 ) F be a nil-isomorphism of coherent -sheaves on U , and Fz 0 any extension of F 0 as in Proposition 4.5.1. By applying the above remarks to Fz 0 the nil-isomorphism extends to a homomorphism Fz 0 K Fz in QCrys.X; A/, and hence in Crys.X; A/. By construction its restriction to U is an isomorphism of crystals, and since i Fz 0 D i Fz D 0 in Crys.Y; A/, so is its restriction to Y . Thus its restriction to all points on X is an isomorphism, hence it is itself an isomorphism in Crys.X; A/ by Proposition 4.1.11 (a). Therefore F 0 ) F extends to an isomorphism Fz 0 Š Fz in Crys.X; A/. Now consider an arbitrary homomorphism F K j Gz in QCrys.X; A/. It can be represented by a left fraction F ( F 0 ! j Gz in QCoh .X; A/. Here F 0 is locally nil-coherent, so it is nil-isomorphic to a coherent -subsheaf. After replacing F 0 by that -subsheaf we may assume that it is itself coherent. Let Fz 0 be any extension of F 0 as in Proposition 4.5.1. Then by the above remarks the left fraction extends to a diagram Fz Š Fz 0 K Gz in QCrys.X; A/. This proves the surjectivity of the restriction map. To prove the injectivity consider a homomorphism ı W Fz K Gz in QCrys.X; A/ with j ı D 0. Then im.ı/ is a quotient of Fz and hence a crystal. By construction j .im.ı// D 0, and since i Fz vanishes, so does its quotient i .im ı/. Thus Proposition 4.1.13 implies that im.ı/ and hence ı vanishes in Crys.X; A/, proving the injectivity. Proposition 4.5.4. (a) For any crystal F on U there exists a crystal Fz on X such that j Fz Š F , and i Fz D 0 in Crys.Y; A/. (b) The pair in (a) consisting of Fz and the isomorphism j Fz Š F is unique up to unique isomorphism, and it depends functorially on F .
4.5 Extension by zero
67
Proof. Let F and Fz be as in Proposition 4.5.1, then the associated crystals satisfy (a). The functoriality follows from Proposition 4.5.3, which in turn implies the uniqueness, proving (b). Theorem 4.5.5. (a) For every F 2 IndCrys.U; A/ there exists Fz 2 IndCrys.X; A/ such that j Fz Š F , and i Fz D 0 in IndCrys.Y; A/. (b) The pair in (a) consisting of Fz and the isomorphism j Fz Š F is unique up to unique isomorphism, and it depends functorially on F . (c) For any F and Fz as in (a) and any quasi-crystal Gz on X the inverse image under j induces a bijection j W HomQCrys .Fz ; Gz/ ! HomQCrys .F ; j Gz/: Proof. If F is a crystal this is a restatement of Propositions 4.5.3 and 4.5.4. In general we write F as a filtered union of crystals F i . By functoriality their extensions Fzi form a filtered direct system in Crys.X; A/, whose limit we denote by Fz . Property (a) then follows from the fact that j and i commute with filtered direct limits. Assertion (c) follows from the universal property of direct limits. Finally, (b) follows from (c). For every F 2 IndCrys.U; A/ we choose Fz as in Theorem 4.5.5 and denote it by jŠ F . This defines an A-linear functor extension by zero jŠ W IndCrys.U; A/ ! IndCrys.X; A/;
(4.5.6)
which is unique up to unique isomorphism. One should be aware that jŠ is not induced from a functor of coherent -sheaves, because in general a homomorphism jŠ F K jŠ G in IndCrys.X; A/ lifts to a homomorphism in IndCoh .X; A/ only after F or G is replaced by a nil-isomorphic -sheaf. Proposition 4.5.7. The following assertions hold within IndCrys.: : : /: (a) The functor jŠ is left adjoint to j . (b) The adjunction homomorphism id ! j jŠ is an isomorphism. (c) The composite i jŠ is zero. (d) The functor jŠ is exact. (e) There is a natural exact sequence of functors 0 ! jŠ j ! id ! i i ! 0: (f) The functor jŠ induces an exact functor jŠ W Crys.U; A/ ! Crys.X; A/:
68
4 Functors
Proof. Assertions (a) through (c) follow from Theorem 4.5.5, and (f) from Proposition 4.5.4. Assertions (d) and (e) may be checked pointwise by Proposition 4.1.11 (b), where they hold by construction. j0
j
Proposition 4.5.8. For any two open embeddings U 0 ,! U ,! X there are functorial isomorphisms: (a) .jj 0 /Š Š jŠ jŠ0 . (b) jŠ .
˝ j / Š jŠ
(c) jŠ .
˝
(d) jŠ .
˝A A0 / Š jŠ
/ Š jŠ
˝
.
˝ jŠ . ˝A A0 .
Proof. In each case one must show that the right hand side satisfies the characterization 4.5.5 (a) for the left hand side. The requirement on the open part is obviously true. The vanishing on the closed complement is seen by applying i and using various foregoing compatibilities, such as Proposition 4.2.5 (c). The details are left to the reader. Proposition 4.5.9. For any cartesian diagram U0
j0
g0
U
/ X0 g
j
/X
with open embeddings j , j 0 , the base change homomorphism jŠ0 g 0 ! g jŠ is an isomorphism. Proof. Verify the isomorphy separately on U 0 and on X 0 n U 0 . Finally we note the following technical result: Proposition 4.5.10. Every crystal on X is the quotient of a crystal whose underlying coherent sheaf on X C is a pullback from the first factor. S Proof. Let F be a crystal on X and X D i Ui a finite open affine covering with embeddings ji W Ui ,! X . Then by Lemma 4.1.9 every ji F is the quotient of a crystal L Gi whose underlying coherent sheaf is free. The natural homomorphism G ´ i jiŠ Gi K F obtained by adjunction is then surjective, because it is so over ˚di every Ui . Now suppose that Gi Š OUi C . Then the proof of Proposition 4.5.1 shows n
˚d
i that jiŠ Gi can be represented by a -sheaf with underlying coherent sheaf 0 i OXC ni ˚di for some ni 0. This sheaf is the pullback of the coherent sheaf .0 / on X ; hence the coherent sheaf underlying G is a pullback, as desired.
4.6 Constructibility
69
4.6 Constructibility While the preceding sections have established that crystals and quasi-crystals behave very much like sheaves, in this section we provide some evidence for the more precise analogy between Crys.X; A/ and categories of constructible sheaves of A-modules. Lemma 4.6.1. Any closedSsubset T X C satisfying . id/.T / T or T . id/.T / has the form riD1 Yi Di for closed subsets Yi X and Di C . n Proof. If . id/.T / T , set Tn ´ . id/ .T / for all n 0; otherwise set n 1 Tn ´ . id/ .T /. These are closed subsets of X C , because id is a homeomorphism on the underlying topological space. By assumption they form a decreasing sequence, so as X C is noetherian, we have TnC1 D Tn for all n 0. Since id is bijective, this implies that actually . id/.T / D T . In particular id permutes the irreducible components of T . Consider any irreducible component T0 T . Then T0 is mapped to itself by some power . id/n ; hence we may replace T by the union of the conjugates . id/i .T0 /. For both projections we then have pr i .T / D pr i .T0 /, which is irreducible. After red red red replacing X by pr 1 .T / and C by pr 2 .T / and T by its image in pr 1 .T / red pr 2 .T / , both X and C are integral and the projections pr 1 W T ! X and pr 2 W T ! C are dominant. It suffices to prove that this implies T D X C . For this we may localize on X, so we may assume that X D Spec R for an integral domain R. Let I I0 R ˝ A denote the ideals of T red and T0red . We must show that I D 0. Note first that I0 is a prime ideal containing no element of the form u ˝ 1 or 1 ˝ a for 0 6D u 2 R or 0 6D a 2 A. Thus I0 andP hence I does not contain u ˝ a. Suppose that I contains a non-zero element f D riD1 ui ˝ ai . Among all such expressions with 0 6D f 2 I we select one for which r is minimal. Then the ui and the ai are k-linearly independent, P and we must have r > 1. The . id/-invariance of T implies . ˝ id/.f / D riD1 uqi ˝ ai 2 I . Therefore
. ˝ id/.f /
uq1 1
f D
r X
.uqi uq1 1 ui / ˝ ai 2 I;
iD2
which is a shorter expression of the same kind. By minimality, its value must be zero. As the ai are linearly independent, we deduce that uqi uq1 1 ui D 0. Since k is a finite Q q q1 field of order q, we have ui u1 ui D ˛2k .ui ˛u1 /. As R is an integral domain, one of these factors must vanish. Thus the ui are k-linearly dependent, contradicting the minimality of r. This proves that I D 0, as desired. Theorem 4.6.2. For any crystal F on X , the set of points x 2 X with ix F 6Š 0 is constructible. Proof. We may prove the theorem separately for the restriction to a closed subscheme and to its open complement, using Proposition 4.1.5. Thus by noetherian induction we may assume the theorem on every proper closed subscheme. After replacing X by a
70
4 Functors
neighborhood of a generic point we may therefore assume that X is irreducible. We may also assume that X is reduced. Let F be a coherent -sheaf on X , and for all n 0 let Tn denote the set-theoretic support of im.Fn /. These are closed subsets of X C satisfying Tn Tn1 and . id/.Tn / Tn1 . Since X C is noetherian, there exists n0 such that Tn D Tn0 for Sr all n n0 and . id/.Tn0 / Tn0 . By Lemma 4.6.1 it follows that Tn0 D iD1 Yi Di for closed subsets Yi X and Di C . The terms with Yi 6D X can be discarded by localizing further on X . Afterwards we have Tn0 D X D for a closed subset D C . By Corollary 3.3.10 we may also replace F by im.Fn0 / without changing the respective crystals. Thereafter the set-theoretic support of F and of im.Fn / for every n 0 is X D. Next, observe that the length of im.Fn / at any generic point of X D cannot increase with n. Thus after replacing F by im.Fn1 / for suitable n1 we may assume that this length remains constant. This means that the homomorphism F W . id/ F ! F is surjective at every generic point of X D. If D is empty, then F D 0 and we are done. Otherwise let Z be the set theoretic support of coker F , which by construction is a nowhere dense closed subset of X D. Let D be the localization of a scheme D 0 of finite type over k, and let Z 0 be the closure of Z in X D 0 . Then W 0 ´ .X D 0 / n Z 0 is an open dense subset of X D 0 . Let V be the set of points x 2 X for which W 0 \ .x D 0 / is dense in x D 0 . As the projection pr 1 W X D 0 ! X is a morphism of finite type, [EGA4, Proposition 9.5.3] shows that V is constructible. By assumption it contains the generic point of X; hence V contains an open dense subset U X . Finally, by construction for every x 2 U the homomorphism F 0 is surjective at every generic point of xD. This means that the homomorphism on ix F is surjective at every generic point of x D. Since x D is the support of ix F , it follows that is not nilpotent on ix F , so the associated crystal is non-zero. By noetherian induction this implies the theorem. As a consequence we obtain the following variant of Theorem 4.1.7 (b): Corollary 4.6.3. If X is of finite type over a field, then a crystal F on X is zero if and only if the crystal ix F is zero for every closed point x 2 X . Proof. Since X is of finite type over a field, any non-empty constructible subset contains at least one closed point of X . Thus the subset in Theorem 4.6.2 is non-empty if and only if it contains a closed point of X . The assertion now follows from Theorem 4.1.7 (b).
Chapter 5
Derived categories
In this chapter we prove a number of basic results on derived categories of A-(quasi)crystals. Our main object of interest is the category D .Crys.X; A// for 2 fb; g. Unfortunately, this category by itself is too small to construct the total derived functor of the direct image functor f and to prove that it has all the desired properties. One problem is that Crys.X; A/ does not possess enough injectives. This can be solved by passing to the category IndCrys.X; A/, but that category is still too small to allow the ˇ calculation of Rf via Cech cohomology. Thus we are forced to work in the derived category of all quasi-crystals D .QCrys.X; A//. The main problem is then to show that the natural functor induces an equivalence of categories D .Crys.X; A// ! Dcrys .QCrys.X; A//;
where the target is the full subcategory of D .QCrys.X; A// formed by all complexes with cohomology in Crys.X; A/. The proof of this equivalence turns out to be surprisingly intricate. It is achieved by showing the equivalence separately for the functor D .Crys.X; A// ! Dcrys .IndCrys.X; A//;
which is the easier part, and then for the functor D .IndCrys.X; A// ! Dindcrys .QCrys.X; A//;
whose target is the full subcategory of D .QCrys.X; A// formed by all complexes with cohomology in IndCrys.X; A/. The latter part is achieved in Section 5.2, after some preparations for affine X in Section 5.1, and the remaining steps are given in Section 5.3. In order to construct and study derived functors, we need a sufficient supply of ˇ acyclic resolutions. In Section 5.4 they are constructed by combining the Cech resolution associated to an affine open covering with free resolutions on open affines. In particular we show that every complex in D .QCrys.X; A// is quasi-isomorphic to a complex of very flat quasi-crystals.
5.1 The affine case: ind-acyclic T Œ-modules Consider a noetherian commutative ring T with identity, endowed with an endomorphism x 7! x. Let T Œ denote the non-commutative polynomial ring in one variable over T subject to the commutation rule x D x for all x 2 T . All T Œ -modules below will be left modules. Giving such a module M is equivalent to giving M as a
72
5 Derived categories
T -module together with a -linear endomorphism W M ! M . Let Mod .T / denote the category of such modules, let Modft .T / be the full subcategory of modules which are of finite type over T , and Modind .T / the full subcategory of modules which are unions of modules in Modft .T /. For X D Spec R and T D R ˝ A with the action .u ˝ a/ ´ uq ˝ a the categories Modft .T / Modind .T / Mod .T / are equivalent to the categories of -sheaves Coh .X; A/ IndCoh .X; A/ QCoh .X; A/. We will return to these in the next section. Note that these inclusions are in general proper by Remark 3.2.3. Remark 5.1.1. The ring T Œ is not left noetherian in general, making the theory of its modules somewhat complicated. For example, take T D kŒ"=."2 / with k a finite field of order q, and .x C y"/ D .x C y"/q D x. Then " T Œ D " kŒ is a left ideal annihilated by both " and but not finitely generated. Proposition 5.1.2. Mod .T / is a Grothendieck category with generator T Œ . Both ind Modft .T / Modind .T / are Serre subcategories, and Mod .T / is closed under filtered direct limits. Proof. Every module category has exact filtered direct limits, and the free module of rank 1 is a generator. This proves the first assertion. The second assertion is shown as in the proof of Proposition 3.2.5, and the last one holds by construction. Proposition 5.1.2 together with Proposition 2.4.6 implies that the inclusion functor Modind .T / ,! Mod .T / possesses a right adjoint, mapping any module M to its unique largest submodule lying in Modind .T /. We denote this functor by ind W M 7! ind.M /. It is left exact, and the main aim of this section is to determine its right derived functors. This is based on the following construction. Let S denote the set of monic polynomials in T Œ , i.e., of non-zero elements whose highest non-zero coefficient with respect to is 1. It has the following properties: Proposition 5.1.3. (a) (Multiplicativity) 1 2 S , 0 62 S , and S S D S. (b) (Left Ore condition) For all f 2 S and u 2 T Œ there exist g 2 S and v 2 T Œ such that gu D vf . (c) (No right zero divisors) For all f 2 S and u 2 T Œ we have uf D 0 if and only if u D 0. (d) (Left saturation) For all f 2 S and u 2 T Œ we have uf 2 S if and only if u 2 S. (e) (Filtered system) For any two f1 , f2 2 S there exist g1 , g2 2 S so that g1 f1 D g2 f2 . Proof. Assertion (a) is obvious. For the rest suppose that f is monic of degree n in . To prove (b) we note that T Œ =T Œ f is a free T -module of finite rank n. By assumption T is noetherian, so for any u 2 T Œ the submodule .T Œ u C T Œ f /=T Œ f is finitely generated over T . Being generated by the residue classes of i u for all i 0, we deduce that the residue class of some m u is a T -linear combination of the residue
5.1 The affine case: ind-acyclic T Π-modules
73
classes of i u for all i < m. But this means that the residue class of gu vanishes for some g 2 S of degree m. Therefore gu 2 T Œ f ; hence gu D vf for some v 2 T Œ , proving (b). The ‘if’ parts of (c) and (d) are trivial. For the ‘only if’ parts suppose that u 6D 0 and let um m be its highest non-zero term. As the highest non-zero term of f is n , we deduce that uf D um mCn C .lower terms/. This shows that uf 6D 0, and that uf 2 S only if u 2 S , proving (c) and (d). Finally, in (e) one can find g1 2 S and g2 2 T Œ with g1 f1 D g2 f2 by (b). From (d) one then deduces that g2 2 S. For any T Œ-module M one defines its left Ore localization S 1 M as follows. (Compare [32, § 10A] where the dual construction of right localization is described.) Two pairs .f1 ; m1 /, .f2 ; m2 / 2 S M are called equivalent if there exist gi 2 S such that .g1 f1 ; g1 m1 / D .g2 f2 ; g2 m2 /. Using the properties 5.1.3 (a) and (e) one easily shows that this defines an equivalence relation on S M . The set of equivalence classes is denoted by S 1 M , and the equivalence class of a pair .f; m/ is called the left fraction f 1 m. Next, using 5.1.3 (e) again one finds that any two left fractions can be expanded to have the same denominator. Thus the sum of two elements of S 1 M can be defined by f 1 m1 C f 1 m2 ´ f 1 .m1 C m2 /. Moreover, for any u 2 T Œ one defines u f 1 m ´ g 1 .vm/ for g, v as in 5.1.3 (b). One easily shows that both operations are well defined and that they turn S 1 M into a T Œ -module. Moreover, any homomorphism of T Œ-modules ' W M ! N induces a natural homomorphism S 1 ' W S 1 M ! S 1 N , f 1 m 7! f 1 '.m/. Thus the construction defines a covariant functor S 1 from Mod .T / into itself. Furthermore we have a natural transformation ` W id ! S 1 defined by `M W M ! S 1 M , m 7! 11 m. Proposition 5.1.4. (a) All elements of S act bijectively on S 1 M . (b) (Universal property) Let N be any T Œ-module on which all elements of S act bijectively, and let ' W M ! N be any T Œ-linear map. Then there exists a unique T Œ-linear map W S 1 M ! N with ' D `M . (c) The functor M 7! S 1 M is exact. Proof. This is easy to prove directly. Part (c) can be deduced from the fact that S 1 M f
g
is the limit of the filtered direct system M ! M ! M ! with f , g; : : : 2 S. Compare also [32, § 10, Exercise 18 and Corollary 10.11] Remark 5.1.5. One can show that the functor S 1 is isomorphic to sq where s is the splitting from Proposition 2.4.8 for the inclusion Modind .T / Mod .T /. Proposition 5.1.6. For any injective I 2 Mod .T / the map `I W I ! S 1 I is surjective. Proof. We must show that every element f 1 x 2 S 1 I is equal to 11 y for some y 2 I . Clearly it suffices to show that x D f y, i.e., that multiplication by f induces
74
5 Derived categories
a surjection I ! I . To find y observe that the submodule T Œ f T Œ is free of rank 1 by Proposition 5.1.3 (c). Thus there exists a homomorphism T Œ f ! I which sends f to x. As I is injective, this map extends to a homomorphism ' W T Œ ! I . It follows that x D '.f / D f '.1/ with '.1/ 2 I , as desired. Now observe that the category Mod .T / possesses enough injectives by Propositions 5.1.2 and 2.4.2, or directly by [10, Chapter I, §§ 2–3]. Since the functor ind W Mod ! Modind .T / is left exact, its right derived functors Ri ind exist. Theorem 5.1.7. For every T Œ-module M we have: (a) ind.M / D ker `M W M ! S 1 M . (b) R1 ind.M / Š coker `M W M ! S 1 M . (c) Ri ind.M / D 0 for all i 2. i (d) If M 2 Modind .T /, then R ind.M / D 0 for all i 1.
Proof. An element m 2 M lies in ind.M / if and only if it is contained in a T Œ submodule M 0 which is of finite type over T . Since T is noetherian, the submodule T Œm M 0 is then itself of finite type over T . Clearly T Œ m is finitely generated over T if and only if some element n m can be expressed as a T -linear combination of the elements i m for all i < n. Thus m 2 ind.M / if and only if f m D 0 for some f 2 S. By the construction of S 1 M this is equivalent to `M .m/ D 0, proving (a). For (b) and (c) observe that the functor mapping M to the complex M ! S 1 M is exact by Proposition 5.1.4 (c). Its cohomology therefore defines a sequence of ıfunctors T i . Part (a) says that T 0 D ind, and Proposition 5.1.6 means that T i .I / D 0 for all i 1 if I is injective. It follows that T i D Ri ind for all i , which implies (b) and (c). Moreover, if M 2 Modind .T /, then `M .M / D `M .ind.M // D 0 by (a). Thus 1 f m D f 1 `M .m/ D 0 for all m 2 M and f 2 S. By construction this implies that S 1 M D 0; whence (d). Combining Theorems 5.1.7 and 2.9.4 yields: Theorem 5.1.8. For 2 fb; C; ; ¿g the natural functors D Modind .T / o
i R ind
/ D Mod .T / Modind
are mutually quasi-inverse equivalences of categories. For use in the next section we include a few more technical results.
5.1 The affine case: ind-acyclic T Π-modules
75
Definition 5.1.9. A T Œ-module M is called strongly ind-acyclic if multiplication by every f 2 S induces a surjection M ! M . Proposition 5.1.10. (a) If M is strongly ind-acyclic, then Ri ind.M / D 0 for all i 1, that is, M is ind-acyclic. (b) M is strongly ind-acyclic if and only if ind.M / is strongly ind-acyclic and `M W M ! S 1 M is surjective. Proof. If M is strongly ind-acyclic, as in the proof of Proposition 5.1.6 we find that `M is surjective. Thus (a) follows from Theorem 5.1.7. Moreover, for (b) we may assume that `M is surjective and must prove that M is strongly ind-acyclic if and only if ind.M / is strongly ind-acyclic. To do this take any f 2 S and consider the commutative diagram 0
/ ind.M / f
0
/ ind.M /
/M f
/M
/ S 1 M
/0
f
/ S 1 M
/ 0,
whose rows are exact by Theorem 5.1.7 (a). By Proposition 5.1.4 (a) the right vertical arrow is an isomorphism. Thus the snake lemma shows that the left vertical arrow is surjective if and only if the middle one is surjective, proving (b). Proposition 5.1.11. Every injective object in Mod .T / or in Modind .T / is strongly ind-acyclic. Proof. For I injective in Mod .T / this was shown in the proof of Proposition 5.1.6. For J injective in Modind .T / choose an embedding J I with I injective in Mod .T /. Then J ind.I /, and this embedding splits because J is injective in Modind .T /. Thus it suffices to show that ind.I / is strongly ind-acyclic. But this follows from the first part and Proposition 5.1.10 (b). We must also discuss the behavior of S 1 M under suitable localizations. For this we fix an element t 2 T such that .t / D t q for some integer q > 1. Then the multiplicative system t N ´ ft n j n 0g is mapped to itself under ; so induces an endomorphism of the localized ring T t ´ .t N /1 T . Let S t denote the multiplicative system of monic polynomials in T t Œ . Lemma 5.1.12. For every f 2 S t there exist integers j and k and an element g 2 S such that f D t j gt k in T t Œ . P ri Proof. Write f D n C n1 xi i with xi 2 T . Then for every integer k we iD0 t have n1 X n n i t k.q q / t ri xi i : t kq f t k D n C iD0
76
5 Derived categories
Since q > 1 by assumption, the new factors cancel out the denominators t ri whenever P n k.q n q i /ri k 0. Then g ´ n C n1 xi i lies in S and satisfies f D t kq gt k iD0 t in T t Œ, as desired. Now consider any T Œ-module M . Its localization M t ´ .t N /1 M becomes a T t Œ-module in the obvious way. Consider the diagram w ww ww w w {w 1 St Mt `M t
Mt I II II.`M / t II II $ .S 1 M / t ,
where the map on the right is deduced from `M W M ! S 1 M by localization. Proposition 5.1.13. There exists a unique isomorphism of T t Œ -modules S t1 M t Š .S 1 M / t making the above diagram commute. Proof. Each module is obtained from M by localizing twice: For S t1 M t one first localizes at t N and then at S t , for .S 1 M / t one first localizes at S and then at t N . Each of these localization steps has a universal property, as in Proposition 5.1.4 (b). Thus to prove the proposition it suffices to show that both modules satisfy the same total universal property. By construction S t1 M t is universal for all homomorphisms from M into T t Œ modules on which S t acts bijectively. Since t is invertible in T t , by Lemma 5.1.12 the latter condition is equivalent to S acting bijectively. Moreover a T t Œ -module is the same as a T Œ-module on which t acts bijectively. Thus S t1 M t is universal for all homomorphisms from M into T Œ-modules on which S and t act bijectively. Unraveling this in the reverse order we find that this is precisely the universal property of .S 1 M / t , as desired.
5.2 Ind-acyclic -sheaves Returning to -sheaves, we will now transfer the results of the preceding section to an arbitrary base X . Recall that IndCoh .X; A/ QCoh .X; A/ satisfy Assumption 2.4.4. Thus by Proposition 2.4.6 the inclusion functor possesses a right adjoint, mapping any -sheaf F to its unique largest -subsheaf lying in IndCoh .X; A/. We denote this functor by ind W F 7! ind.F /. It is left exact, and as in the preceding section we want to calculate its right derived functors. We first show that ind commutes with localization. Proposition 5.2.1. For any open embedding j W U ,! X and any -sheaf F 2 QCoh .X; A/ we have j .ind.F // D ind.j F /.
5.2 Ind-acyclic -sheaves
77
Proof. Since j preserves coherence and commutes with filtered direct limits, it also preserves ind-coherence. This shows the inclusion j .ind.F // ind.j F /. For the reverse inclusion consider any coherent -subsheaf G ind.j F /. By Propositions 4.5.1 and 4.5.2 there exists a coherent -sheaf Gz on X with j Gz Š G , such that the embedding G ,! j F extends to a homomorphism ' W Gz ! F . But then '.Gz/ ind.F /, and hence G j .ind.F //. The desired equality follows by taking the limit over all coherent G . For use in Chapter 10 we also include the following generalization: Proposition 5.2.2. For any étale morphism u W U ! X and any -sheaf F contained in QCoh .X; A/, we have u .ind.F // D ind.u F /. Proof. The proof proceeds by induction on ˚ d ´ deg.u/ ´ max dimkx .OU ˝OX kx / j x 2 X : Thus for fixed d , we may assume the assertion for all u and F with deg.u/ < d . To prove the assertion for deg.u/ D d , observe that by Proposition 5.2.1 the desired equality holds whenever u is an open embedding. Factoring a general étale morphism u into a surjective étale morphism composed with an open embedding, we can reduce ourselves to the case that u is surjective étale, and so in particular faithfully flat. For such u we consider the diagram U X U
v1 v2
//
U
u
/ X.
Since u is étale, the diagonal morphism U ! U X U is an open and closed embedding. Let V denote the complement of its image. Then deg.vi jV / < deg.vi / D deg.u/ D d for i D 1, 2, so by the induction hypothesis the proposition holds for the restrictions vi jV . As it trivially holds for the restriction of vi to the diagonal, it holds for vi itself. In particular we deduce that v1 .ind.u F // D ind.v1 u F / D ind.v2 u F / D v2 .ind.u F //: This means that the subsheaf ind.u F / u F possesses a descent datum relative to u. Thus by [SGA1, Exposé VIII, § 1] there exists a quasi-coherent subsheaf F 0 F such that .u id/ F 0 is the sheaf underlying ind.u F /. Moreover, since ind.u F / is invariant under the homomorphism u F , the subsheaf F 0 is invariant under the homomorphism F . We have therefore found a -subsheaf F 0 F satisfying u F 0 D ind.u F /. Now we have u .ind.F // ind.u F / D u F 0 ; where the inclusion follows as in the proof of Proposition 5.2.1. By descent this implies that ind.F / F 0 . To finish the proof it suffices to show the opposite inclusion, i.e., that F 0 is ind-coherent.
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5 Derived categories
For this, let G be an arbitrary coherent subsheaf of F 0 , and define X n . n id/ G : Gz ´ n0
By flatness of u the functor .u id/ is exact; hence it commutes with the formation of images. We deduce that X n . n id/ .u id/ G : .u id/ Gz D n0
Here .u id/ G is a coherent subsheaf of .u id/ F 0 , the sheaf underlying the indcoherent -sheaf u F 0 D ind.u F /. Therefore .u id/ Gz is coherent. Since descent preserves coherence by [SGA1, Exposé VIII, 1.10], we find that Gz defines a coherent -subsheaf of F 0 containing G . As G was arbitrary, we conclude that F 0 is the union of its coherent -subsheaves, so it is ind-coherent, as desired. This finishes the proof. Next we globalize the construction of S 1 M from the preceding section. Proposition 5.1.13 implies: Proposition 5.2.3. For every X there is a functor S W QCoh .X; A/ ! QCoh .X; A/ and a natural transformation ` W id ! S with the following properties: (a) If X D Spec R is affine, then S and ` are the sheaf theoretic equivalents of the functor M 7! S 1 M and the natural transformation ` of the preceding section, applied to T ´ R ˝ A with the action .u ˝ a/ ´ uq ˝ a. (b) S and ` commute with inverse image under open embeddings. Remark 5.2.4. One can show that the functor S is isomorphic to sq where s is the splitting from Proposition 2.4.8 for the inclusion IndCoh .X; A/ QCoh .X; A/. Proposition 5.2.5. The functor S is exact. Proof. Immediate from Propositions 5.2.3 and 5.1.4 (c). For the next few technical results a -sheaf on an affine X D Spec R will be called strongly ind-acyclic if its corresponding .R ˝ A/Œ -module is strongly ind-acyclic in the sense of Definition 5.1.9. Proposition 5.2.6. Consider a morphism f W Y ! X, where both X and Y are affine, and a -sheaf F 2 QCoh .Y; A/. (a) If F is strongly ind-acyclic, then so is f F . (b) If `F W F ! S.F / is an isomorphism, then `f F W f F ! S.f F / is an isomorphism.
5.2 Ind-acyclic -sheaves
79
Proof. Write X D Spec R and Y D Spec R0 and set T ´ R ˝ A and T 0 ´ R0 ˝ A, both with the action .u ˝ a/ ´ uq ˝ a. Then the functor f corresponds to the functor Mod .T 0 / ! Mod .T / mapping any T 0 Œ -module to itself, viewed as a T Œ module via the evident homomorphism T Œ ! T 0 Œ . Clearly this homomorphism maps the set of monic polynomials S T Œ to the set of monic polynomials S 0 T 0 Œ. Let M 0 be the T 0 Œ -module corresponding to F . If F is strongly ind-acyclic, by definition all elements of S 0 induce surjections M 0 ! M 0 . The same then follows for all elements of S; hence f F is strongly ind-acyclic, proving (a). For (b) assume that `F is an isomorphism. Then M 0 Š S 01 M 0 , which implies that all elements of S 0 act bijectively on M 0 . Since S is mapped to S 0 , the same follows for all elements of S . By the construction of S 1 M 0 this implies that M 0 ! S 1 M 0 is an isomorphism, proving (b). Proposition 5.2.7. Consider an open embedding j W U ,! X. If F is an injective in IndCoh .X; A/, then j F is an injective in IndCoh .U; A/. Proof. Consider ind-coherent -sheaves G 0 G on U . We must prove that any homomorphism G 0 ! j F extends to a homomorphism G ! j F . Using Zorn’s lemma this statement easily reduces to the case that G 0 and G are coherent. By Proposition 4.5.1 we may choose coherent -sheaves Gz0 , Gz on X such that j Gz0 Š G 0 and j Gz Š G . By Proposition 4.5.2 the inclusion G 0 ,! G extends to a homomorphism 0n Gz0 ! Gz for some n 0. After replacing Gz0 by its image we may assume that Gz0 Gz. Again by Proposition 4.5.2 the homomorphism G 0 ! j F extends to a homomorphism 0m Gz0 ! F for some m 0. Since F is injective, this homomorphism extends to a homomorphism 0m Gz ! F . Its restriction to U is the desired homomorphism G ! j F . Proposition 5.2.8. Consider an open embedding j W U ,! X where U is affine. If in QCoh .X; A/ is injective, then j is strongly ind-acyclic. Proof. By Theorem 5.1.7 we have a short exact sequence 0 ! ind. / ! ! S. / ! 0: Since ind and S commute with j by Propositions 5.2.1 and 5.2.3, we deduce from this a short exact sequence 0 ! ind.j / D j .ind. // ! j ! S.j / ! 0: Here ind. / is injective in IndCoh .X; A/ by Proposition 2.4.7. Thus Propositions 5.2.7 and 5.1.11 imply that j .ind. // is injective in IndCoh .U; A/ and therefore strongly ind-acyclic. Using this and the above exact sequence, Proposition 5.1.10 (b) implies that j is strongly ind-acyclic, as desired.
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5 Derived categories
Proposition 5.2.9. For any injective 2 QCoh .X; A/ the homomorphism ` W ! S. / is surjective. Proof. It suffices to verify this on any open affine U X . By Proposition 5.2.8 the restriction to U is strongly ind-acyclic; hence ` is surjective over U by Proposition 5.1.10 (b), as desired. Now observe that the category QCoh .X; A/ possesses enough injectives by Theorem 3.2.7 and Proposition 2.4.2. Since the functor ind W QCoh .X; A/ ! IndCoh .X; A/ is left exact, its right derived functors Ri ind exist. Theorem 5.2.10. For every -sheaf F 2 QCoh .X; A/ we have: (a) ind.F / D ker `F W F ! S.F / . (b) R1 ind.F / Š coker `F W F ! S.F / . (c) Ri ind.F / D 0 for all i 2. (d) If F 2 IndCoh .X; A/, then Ri ind.F / D 0 for all i 1. (e) If F is strongly ind-acyclic, then Ri ind.F / D 0 for all i 1, that is, F is ind-acyclic. Proof. The left side in (a) commutes with localization by Proposition 5.2.1, and the right side does so by construction. Thus (a) follows from Proposition 5.2.3 and Theorem 5.1.7 (a). For (b) and (c) observe that the functor mapping F to the complex F ! S.F / is exact by Proposition 5.2.5. Its cohomology therefore defines a sequence of ı-functors T i . Part (a) says that T 0 D ind, and Proposition 5.2.9 means that T i . / D 0 for all i 1 if is injective. It follows that T i D Ri ind for all i , which implies (b) and (c). Moreover, the first three assertions imply that Ri ind commutes with localization for every i . Thus assertions (d) and (e) follow from Theorem 5.1.7 (d) and Proposition 5.2.9, respectively. Finally, combining Theorems 5.2.10 and 2.9.4 yields: Theorem 5.2.11. For 2 fb; C; ; ¿g the natural functors D IndCoh .X; A/ o
i R ind
/
Dindcoh QCoh .X; A/
are mutually quasi-inverse equivalences of categories.
5.3 Derived categories of -sheaves and quasi-crystals
81
5.3 Derived categories of -sheaves and quasi-crystals In this section we discuss the relation between various derived categories of -sheaves and of crystals. We use the following notation for the strictly full triangulated subcategories defined by the indicated conditions on cohomology. For brevity we leave out .X; A/ in this section. subcategory Dcoh .IndCoh /
ambient category
condition on cohomology
D .IndCoh /
in Coh
Dcoh .QCoh /
D .QCoh /
in Coh
Dindcoh .QCoh /
D .QCoh /
in IndCoh
Dcrys .IndCrys/
D .IndCrys/
in the essential image of Crys
Dcrys .QCrys/
D .QCrys/
in the essential image of Crys
Dindcrys .QCrys/
D .QCrys/
in IndCrys
Recall from Remark 3.4.3 that there is a slight difference between Crys and its essential image in QCrys, which we determined to be LNilCrys. We are forced to define Dcrys .: : : / in terms of the latter, because the condition on cohomology must be invariant under isomorphisms. Thus a complex in QCoh represents an object of Dcrys .QCrys/ if and only if its cohomology is in LNilCoh . Next a morphism in a derived category of -sheaves will be called a nil-quasiisomorphism if it induces a nil-isomorphism on the cohomology in every degree. Clearly the nil-quasi-isomorphisms form a saturated multiplicative system that is compatible with the triangulation; we denote it by nilqi . The following theorem summarizes our comparison results: Theorem 5.3.1. Consider the following derived categories with the natural functors, all for some fixed .X; A/:
/ D .Crys/
1 / nilqi D .IndCoh /
/ D .IndCrys/
1 / nilqi D .QCoh /
/ D .QCrys/.
D .Coh /
1 / nilqi D .Coh /
D .IndCoh / D .QCoh /
(a) The categories in the middle and the lower row exist for any symbol 2 fb; C; ; ¿g, the categories in the upper row exist for any 2 fb; g.
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5 Derived categories
(b) All horizontal arrows on the right hand side are equivalences of categories. (c) All vertical arrows are fully faithful. More specifically, for any symbol 2 fb; C; ; ¿g they induce equivalences of categories ! Dindcoh .QCoh /; D .IndCoh / D .IndCrys/ ! Dindcrys .QCrys/;
and for any 2 fb; g they induce equivalences of categories ! Dcoh .IndCoh / ! Dcoh .QCoh /; D .Coh / D .Crys/ ! Dcrys .IndCrys/ ! Dcrys .QCrys/:
Proof. We start with the lower half of the commutative diagram for any 2 fb; C; ; ¿g. Since QCoh , IndCoh , QCrys, and IndCrys are Grothendieck categories by Theorems 3.2.7 and 3.4.5, their derived categories exist by Proposition 2.6.1. Next by applying Theorem 2.6.2 to the pairs of categories LNil QCoh and LNil IndCoh 1 1 we find that nilqi D .QCoh / and nilqi D .IndCoh / exist and their horizontal ar rows induces equivalences with D .QCrys/ and D .IndCrys/, respectively. This proves (a) and (b) for the lower half of the diagram. To prove (c) we enlarge this part as follows: D .IndCoh /
1 / nilqi D .IndCoh /
Dindcoh .QCoh / _
1 / nilqi Dindcoh .QCoh / _
D .QCoh /
1 / nilqi D .QCoh /
/ D .IndCrys/ / Dindcrys .QCrys/ _
/ D .QCrys/.
Here the upper left vertical arrow is an equivalence of categories by Theorem 5.2.11. This implies that the upper middle vertical arrow is an equivalence of categories as well. .QCoh / is closed under nil-quasi-isomorphisms Now since the subcategory Dindcoh within D .QCoh /, it is localizing for the multiplicative system nilqi . Thus the lower middle vertical arrow is fully faithful. Since by definition Dindcrys .QCrys/ is the 1 essential image of nilqi Dindcoh .QCoh / in D .QCrys/, we deduce from the horizontal equivalences that the upper right vertical arrow is an equivalence of categories. This proves everything in the lower half of the diagram. For the upper half we take 2 fb; g. Note that every object of Coh is noetherian, because X C is noetherian. Thus Proposition 2.3.6 implies that every object of Crys is noetherian. In view of Theorem 3.4.5 we may therefore apply Theorem 2.9.1 to the pairs of categories Coh IndCoh and Crys IndCrys, proving that D .Coh / and D .Crys/ exist and that the functors D .Coh / ! Dcoh .IndCoh / and D .Crys/ ! Dcrys .IndCrys/ are equivalences of categories. Together with the
5.3 Derived categories of -sheaves and quasi-crystals
83
first two equivalences in (c) this implies the last two, finishing the proof of (c). It remains 1 1 to show that nilqi D .Coh / exists and the functor nilqi D .Coh / ! D .Crys/ is an equivalence of categories. To do this we enlarge the upper half of the diagram as follows: 1 / nilqi D .Coh /
/ D .Crys/
Dcoh .IndCoh / _
1 / nilqi Dcoh .IndCoh / _
/ Dcrys .IndCrys/ _
D .IndCoh /
1 / nilqi D .IndCoh /
D .Coh / o
o
/ D .IndCrys/.
Here all the categories at the left, the right, or the bottom already exist, with the indicated equivalences. For the middle we use a variant of the argument in 2.9.1. Note that every complex in D .IndCoh / which is nil-quasi-isomorphic to a complex in Dcoh .IndCoh / has cohomology in LNilCoh . Conversely we have: Lemma 5.3.2. For any 2 fb; g and any complex F 2 C .IndCoh / with cohomology in LNilCoh there exists a subcomplex G in C .Coh / whose embedding into F is a nil-quasi-isomorphism.
di
Proof. Let ! F i ! F iC1 ! be a complex in C .IndCoh / and consider an index i such that F j is already coherent for all j > i. Write F i as a union of coherent -subsheaves G˛i . As F iC1 is coherent, we have d i .G˛i / D d i .F i / for all sufficiently large ˛. As H i .F / is nil-coherent, it possesses a coherent -subsheaf H i such that H i .F /=H i is locally nilpotent. Since H i is coherent, it is contained in the image of the homomorphism G˛i \ker.d i / ! H i .F / whenever ˛ is sufficiently large. i i i1 i 1 i j ´ .d / .G˛ /, and G ´ F j for all j 6D i , i 1. If ˛ is Set G ´ G˛ and G sufficiently large, the embedding G ,! F induces an isomorphism on cohomology in all degrees 6D i and a nil-isomorphism in degree i, and G j is coherent for all j i . As the given complex F is bounded above, we can begin this construction for some i 0 and repeat it with the resulting complexes by downward induction on i . The desired subcomplex is obtained by passing to the limit. Lemma 5.3.2 and Proposition 2.2.8 (a0 ) imply that D .Coh / is a localizing subcategory for the multiplicative system nilqi in D .IndCoh /. Thus the localiza1 1 D .Coh / ! nilqi D .IndCoh / is fully faithtion exists and the functor nilqi ful. Moreover, again by Lemma 5.3.2 its essential image consists of all complexes with cohomology in LNilCoh . Since these are precisely the complexes whose im .IndCrys/, we deduce an equivalence of categories ages in D .IndCrys/ lie in Dcrys 1 nilqi D .Coh / ! Dcrys .IndCrys/. This implies that the upper right horizontal arrow is an equivalence of categories, finishing the proof of Theorem 5.3.1.
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5 Derived categories
The relation between D .IndCrys/ and Dindcrys .QCrys/ can be clarified further as follows.
Proposition 5.3.3. (a) The functor ind W QCoh ! IndCoh induces by localization a left exact functor ind W QCrys ! IndCrys which is right adjoint to the inclusion functor IndCrys ,! QCrys. (b) For any 2 fb; C; ; ¿g the right derived functor R ind W D .QCoh / ! D .IndCoh / induces the right derived functor R ind W D .QCrys/ ! D .IndCrys/. .QCrys/ is a quasi-inverse of the natural (c) The restriction of R ind to Dindcrys functor D .IndCrys/ ! Dindcrys .QCrys/. Proof. By definition and Theorem 5.2.10 (d) the functor ind and its derivatives map LNil to LNil and hence nil-isomorphisms to nil-isomorphisms. Thus by the universal property 2.3.7 of localization it induces a left exact functor ind W QCrys ! IndCrys. Using Proposition 2.2.5 (a) we find that ind is precisely the right adjoint to the inclusion functor afforded by Proposition 2.4.6. Similarly R ind W D .QCoh / ! D .IndCoh / preserves nil-quasi-isomorphisms, so it induces an exact functor R ind W D .QCrys/ ! D .IndCrys/. Applying Theorem 2.7.1 to the composite functor D .QCoh / ! D .IndCrys/ we deduce that R ind is indeed a right derived functor of ind. This proves (a) and (b). Finally, (c) follows by localization from the corresponding property of R ind from Theorem 5.2.11.
ˇ 5.4 Cech resolution ˇ The Cech resolution is a general functorial construction that serves to reduce a calculation with quasi-coherent sheaves to an affine base. We discuss this construction and its properties when applied to -sheaves and quasi-crystals. Let U D fUi j i D 1; : : : ; mg be a finite affine T open covering of X . For any non-empty subset I f1; : : : ; mg we set UI ´ i2I Ui and denote its embedding into X by jI . For any -sheaf F on X we set 8 M ˆ jI jI F if p 0, < p L CU .F / ´ jI jDpC1 ˆ : 0 if p < 0. For any I D fi0 ; : : : ; ipC1 g with 1 i0 < < ipC1 m, and I 0 D I n fik g with 0 k p C 1, we have a relative embedding jI;I 0 W UI ,! UI 0 . Let dLI 0 ;I W jI 0 jI0 F ! jI jI F be .1/k times the natural homomorphism arising from the adjunction homomor 0 phism id ! jI;I 0 jI;I 0 . Summing up over all I and I we obtain a homomorphism
ˇ 5.4 Cech resolution
85
CL Up .F / ! CL UpC1 .F /. Furthermore, we let ˛U W F ! CL U0 .F / be the homomorphism corresponding to the adjunction homomorphism id ! jI jI in each term. It is well known and easy to prove that this turns CL U .F / into a finite complex with augmentation, ˇ called the Cech complex of F with respect to U. Proposition 5.4.1. The augmentation ˛U W F ! CL U .F / is locally a homotopy equivalence on every Ui X . In particular CL U .F / is a finite right resolution of F . ˇ Proof. The restriction of CL U .F / to Ui is simply the Cech complex for the open covering fUi \Uj j j D 1; : : : ; mg of Ui . This covering possesses Ui as one of the open subsets, so it and the covering by Ui alone are equivalent, i.e., they are mutual refinements of ˇ each other. Now by [21, Théorème 5.7.1] the Cech complexes associated to equivalent ˇ open coverings are homotopy equivalent. Since the Cech complex associated to Ui alone is F and the refinement homomorphism to CL U .F / is simply the augmentation, the proposition follows. Clearly the above construction is functorial in F . Thus by applying it to a complex of -sheaves F we obtain a double complex .CL Up .F q // which is bounded in the direction of p. Let CL U .F / denote the associated total complex. The augmentation homomorphism F ! CL U .F / is then locally a homotopy equivalence and hence a quasi-isomorphism. Since the Ui are affine and X is separated, all the UI are affine. Thus all the functors jI and jI are exact. Moreover, if F is bounded above or below, ˇ then so is CL U .F /. It follows that for every symbol 2 fb; C; ; ¿g the Cech complex defines an exact functor CL U W C .QCoh .X; A// ! C .QCoh .X; A// and the augmentation homomorphism a natural quasi-isomorphism id ! CL U . Clearly CL U preserves homotopies, so it descends to a functor of the associated homotopy categories. Finally, it also preserves quasi-isomorphisms; hence both the functor and the natural transformation descend to the derived categories, and likewise to the derived categories of quasi-crystals QCrys.X; A/: ˇ Theorem 5.4.2. For any 2 fb; C; ; ¿g the Cech complex induces exact functors CL U W D .QCoh .X; A// ! D .QCoh .X; A//; CL U W D .QCrys.X; A// ! D .QCrys.X; A//; and in each case the natural transformation ˛U W id ! CL U is an isomorphism. The above construction helps to provide a kind of very flat resolution. For this we first need a functorial construction of free resolutions of modules. Let T be a unitary ring, which may be non-commutative. For any left T -module M we let FT .M / denote the free left T -module with basis f.m/ j m 2 M g. Clearly the T -linear map P P FT .M / ! M; xm .m/ 7! xm m
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5 Derived categories
is surjective. Thus if KT .M / denotes its kernel, we obtain a short exact sequence of T -modules 0 ! KT .M / ! FT .M / ! M ! 0: For all p 0 we define KTp .M / by KT0 .M / ´ M and KTpC1 .M / ´ KT .KTp .M //. Then by setting FTp .M / ´ FT .KTp .M // and splicing together the corresponding short exact sequences we obtain a free left resolution FT .M / ! M . This construction is functorial in T and M in the following sense. Consider a homomorphism of unitary rings ' W T ! T 0 . Then any T 0 -module can be considered a T -module via '. Let M be a left T -module and M 0 a left T 0 -module, and consider a homomorphism of T -modules W M ! M 0 . This data induces a T -linear homomorphism P P FT .M / ! FT 0 .M 0 /; xm .m/ 7! '.xm /. .m// which is compatible with the surjections FT .M / M and FT 0 .M 0 / M 0 , and hence a T -linear homomorphism KT .M / ! KT 0 .M 0 /. By iteration it induces a T -linear homomorphism of the resolutions F .'/ W FT .M / ! FT 0 .M 0 /: Clearly this construction preserves composition and the identity; hence it defines a functor from the category of pairs .T; M / to the category of complexes of abelian groups. For T fixed this functor takes values in the category of T -modules and T linear homomorphisms, but one should note that it is not additive in M , because it preserves neither direct sums of modules nor linear combinations of homomorphisms.
Proposition 5.4.3. (a) For every complex F of very flat quasi-crystals the complex CL U .F / consists of very flat quasi-crystals. (b) For every complex of quasi-crystals F that is bounded above, there exists a quasi-isomorphism G ! CL U .F /, where G is a complex of very flat quasi-crystals that is bounded above. Proof. (a) is a direct consequence of the invariance of the property ‘very flat’ under inverse and direct images via affine open embeddings, see Propositions 4.2.8 (b) and 4.4.8. For (b) we will apply the above free resolution to the constituents of CL U .F /. For any non-empty subset I f1; : : : ; mg we write UI D Spec RI and consider the ring RI ˝ A with the action .u ˝ a/ ´ uq ˝ a. Then by global sections the category QCoh .UI ; A/ is equivalent to the category of left modules over .RI ˝ A/Œ . We let FI W QCoh .UI ; A/ ! C QCoh .UI ; A/ denote the sheaf theoretic equivalent of the free resolution functor above. For any -sheaf F on X we set 8 M ˆ jI FI jI F if p 0, < FUp; .F / ´ jI jDpC1 ˆ :0 if p < 0.
ˇ 5.4 Cech resolution
87
Next consider I D fi0 ; : : : ; ipC1 g with p 0 and 1 i0 < < ipC1 m, and set I 0 D I n fik g for some 0 k p C 1. The functoriality of the resolution functor for the ring homomorphism .RI ˝ A/Œ ! .RI 0 ˝ A/Œ then induces a natural transformation FI0 jI;I 0 ! jI;I 0 FI . By adjunction this determines a natural transformation of functors jI 0 FI0 jI0 ! jI FI jI W QCoh .X; A/ ! C QCoh .X; A/ : Multiplying this by .1/k and summing up over all I and I 0 we obtain a homomorphism of complexes FUp; .F / ! FUpC1; .F /. By functoriality this defines a double complex FU; .F / which is bounded above in both directions. Moreover the augmentation FI ! id induces an augmentation FU; .F / ! CL U .F / which by construction is a quasi-isomorphism. All this is functorial in F and results in complexes in QCoh .X; A/, though it is not additive as a functor of F . In any case we can apply it to a complex F 2 C .QCoh .X; A//, yielding a triple complex which is bounded above in all three direc tions. Let FU .F / denote the associated total complex, which is again bounded above. By functoriality the augmentation induces a quasi-isomorphism FU .F / ! CL U .F /. We are interested in the resulting quasi-isomorphism in C .QCrys.X; A//. Note that we can perform the above construction for any complex in C .QCrys.X; A//, because any such complex is isomorphic to the image of a complex in C .QCoh .X; A// by Proposition 2.3.5 (d). Finally, any free left module over .RI ˝ A/Œ is in particular free as a module over RI ˝ A. Thus its corresponding quasi-crystal on UI is very flat. Since the property ‘very flat’ is invariant under jI by Proposition 4.4.8 and obviously invariant under direct sums, the construction implies that FU .F / consists of very flat quasi-crystals. Thus everything is proved. Theorem 5.4.4. The derived category D .QCrysvflat .X; A// exists and the inclusion induces an equivalence of categories D .QCrysvflat .X; A// ! D .QCrys.X; A//:
Proof. By Proposition 5.4.3 the functor CL U on K .QCrys.X; A// satisfies the assumptions of Theorem 2.8.1 with respect to the additive subcategory QCrysvflat .X; A/ QCrys.X; A/. Thus the theorem follows from Proposition 2.8.3.
Chapter 6
Derived functors
In this chapter we give a systematic account of the derived functors associated to the functors of A-(quasi)-crystals introduced in Chapter 4. For any morphism f W Y ! X we construct derived functors Lf and Rf . We also construct the derived tensor product ˝L and prove the usual relations, including base change and the projection formula. Furthermore, we show how the individual cohomology groups of these functors are calculated in terms of usual quasi-coherent cohomology. Although we are mainly interested in the category D .Crys.X; A// for 2 fb; g, the construction of derived functors requires a larger ambient category. For this we have a choice between D .QCrys.X; A// and D .IndCrys.X; A//, each with their own advantages. In Chapter 5 we have seen that the latter category is equivalent to a strictly full triangulated subcategory of the former. In this chapter we will show that the derived functors with respect to both categories coincide on the smaller category, with an exception for Rf that is dealt with separately in Section 6.5. ˇ The advantage of D .QCrys.X; A// is that Rf can be calculated via Cech coho mology. One advantage of D .IndCrys.X; A// is that Lf is a trivial derived functor, because f is exact on IndCrys.X; A/. Another advantage of D .IndCrys.X; A// is the existence of the functor jŠ for any open embedding j W U ,! X . For compactifiable morphisms f this allows us to define the derived direct image with compact support RfŠ with all the usual properties. We also prove that Lf D f , ˝L , and RfŠ preserve the subcategories D .Crys.: : : //. It seems that reasonable functors of the type RHom and Rf Š do not exist in our setting. One reason for this is the lack of a duality theory for A-crystals, which in turn is due to the fact that the action of on a -sheaf in general has no inverse. Another reason is that already the functor Rf is not so well-behaved. But at least we have three functors out of the usual six, which will suffice for the application to L-functions.
6.1 Inverse image By Theorem 4.1.12 (c) the functor f W IndCrys.X; A/ ! IndCrys.Y; A/ is exact. Consider the associated functor of homotopy categories K .IndCrys.X; A// ! K .IndCrys.Y; A// for any 2 fb; C; ; ¿g. By exactness of f this functor preserves quasi-isomorphisms, so it induces an exact functor f W D .IndCrys.X; A// ! D .IndCrys.Y; A//:
(6.1.1)
6.1 Inverse image
89
For trivial reasons this is a left and right derived functor at the same time. Since f preserves coherence, this functor restricts to an exact functor f W D .Crys.X; A// ! D .Crys.Y; A//:
(6.1.2)
More generally by Proposition 4.1.3 we have a right exact functor f W QCrys.X; A/ ! QCrys.Y; A/: When f is flat, this functor is exact, so it possesses a tautological derived functor, denoted again by f . In general we have: Theorem 6.1.3. This functor possesses a left derived functor Lf W D .QCrys.X; A// ! D .QCrys.Y; A//: Proof. We apply Theorem 2.8.1 to the full additive subcategory A0 ´ QCrysvflat .X; A/ ˇ of A ´ QCrys.X; A/ and the Cech resolution functor CL U associated to any finite open affine covering U of X . By Proposition 5.4.3 this functor satisfies the assumptions 2.8.1 (a) and (b). Next recall from Proposition 5.4.1 that the quasi-isomorphism ˛U W id ! CL U is locally on X a homotopy equivalence. Since exactness is determined locally and f commutes with localization, the induced natural transformation f ! f CL U is again a quasi-isomorphism. Thus the assumption 2.8.1 (c) is satisfied. To verify the remaining assumption 2.8.1 (d) we must show that f maps quasiisomorphisms in K .QCrysvflat .X; A// to quasi-isomorphisms. For this note that a homomorphism of complexes is a quasi-isomorphism if and only if its cone is acyclic. Thus we must prove that f preserves acyclicity. Recall from Proposition 2.3.5 (d) that every complex in K .QCrys.X; A// can be represented by a com plex in K .QCoh .X; A//. So let F be a complex in K .QCoh .X; A// whose constituents define very flat quasi-crystals and whose cohomology is locally nilpotent. Let Lf F be the left derived functor of quasi-coherent sheaves applied to the underlying complex of sheaves, with its semi-linear endomorphism induced by functoriality. As F is bounded above, there are spectral sequences E1ij D Lj f F i H) H iCj Lf F 0 ij E2
D Li f .H j .F // H) H iCj Lf F :
By Proposition 4.2.8 (a) the term E1ij is locally nilpotent whenever j 6D 0. For all indices 0E2ij is locally nilpotent by Proposition 4.1.3 (a). It follows that the complex E10 D f F has locally nilpotent cohomology. The corresponding complex of quasicrystals is therefore acyclic, as desired. Thus all the assumptions in Theorem 2.8.1 are satisfied; hence the desired left derived functor exists.
90
6 Derived functors
Although our derived functors are constructed in a new way as a derived functors of quasi-crystals, their individual cohomology groups turn out to be essentially the same as those of the usual derived functors of quasi-coherent sheaves. To explain this we interpret, as usual, objects of an abelian category as complexes that are concentrated in degree zero. Proposition 6.1.4. For any F 2 QCrys.X; A/ and any integer i , the quasi-crystal H i .Lf F / is isomorphic to Li f F defined in 4.1.2. Proof. The proof of Proposition 5.4.3 shows that F is quasi-isomorphic to a complex of -sheaves G whose underlying sheaves are flat in the usual quasi-coherent sense. This implies that H i .f G / Š Li f F . On the other hand the quasi-crystals associated to G are very flat, so the preceding proof shows that Lf F Š f G in D .QCrys.X; A//. Together this shows that H i .Lf F / Š H i .f G / Š Li f F , as desired. Proposition 6.1.5. The restriction of Lf to D .IndCrys.X; A// coincides with the functor f in (6.1.1) . Proof. Proposition 6.1.4 and Theorem 4.1.12 (b) together imply that Lf F D f F for all F 2 IndCrys.X; A/. Thus the proposition follows from the hypercohomology spectral sequence. g
f
Proposition 6.1.6. For any two morphisms Z ! Y ! X there is a natural isomorphism of derived functors L.fg/ Š Lg Lf . Proof. This follows from the construction of Lg in terms of very flat quasi-crystals and the fact that f preserves the property ‘very flat’ by Proposition 4.2.8 (b).
6.2 Tensor product Theorem 6.2.1. The bi-functor ˝ on quasi-crystals possesses a left bi-derived functor ˝L W D .QCrys.X; A// D .QCrys.X; A// ! D .QCrys.X; A//: Proof. For any complex G 2 K .QCrys.X; A// we will derive the functor ˝ G by applying Theorem 2.8.1 to the subcategory QCrysvflat .X; A/ QCrys.X; A/ ˇ with the Cech resolution functor CL U . The assumptions 2.8.1 (a) and (b) hold by Proposition 5.4.3. As in the proof of Theorem 6.1.3 the assumption 2.8.1 (c) follows from the fact that id ! CL U is locally a homotopy equivalence. To verify the remaining assumption 2.8.1 (d) we must show that ˝ G W K .QCrysvflat .X; A// ! K .QCrys.X; A// maps quasi-isomorphisms to quasi-isomorphisms. Equivalently we must prove that it preserves acyclicity. So let F 2 K .QCrysvflat .X; A// be an acyclic complex. It
6.2 Tensor product
91
suffices to show that F ˝ G is acyclic for any G in QCrys.X; A/. Without loss of generality we may assume that F comes from a complex of -sheaves with locally nilpotent cohomology. As F is bounded above, we then have two spectral sequences with initial terms E1ij D Tor j .F i ; G /; and 0 ij E2
D Tor i .H j .F /; G /;
both abutting to the hyper-Tor on the underlying quasi-coherent sheaves. By the definition 4.2.6 of ‘very flat’ the term E1ij is locally nilpotent whenever j 6D 0. For all indices 0E2ij is locally nilpotent by Proposition 4.1.3 (a). It follows that the complex E10 D F ˝ G has locally nilpotent cohomology; hence its image in K .QCrys.X; A// is acyclic, as desired. Thus all the assumptions in Theorem 2.8.1 are satisfied, and the functor ˝ G possesses a left derived functor. Clearly this derived functor is functorial in G . We claim that it maps quasiisomorphisms in the variable G to quasi-isomorphisms. By the above construction it suffices to show that F ˝ preserves quasi-isomorphisms for any complex F vflat in K .QCrys .X; A//. But this follows at once from the definition of ‘very flat’. Altogether we have now shown that the desired bi-functor exists and is a derived functor in the first argument. By symmetry of the construction it is also a derived functor in the second argument. Again the individual cohomology groups are the old ones: Proposition 6.2.2. For any F , G 2 QCrys.X; A/ and any integer i, the quasi-crystal H i .F ˝L G / is isomorphic to Tor i .F ; G / defined in 4.2.2. Proof. Precisely as in Proposition 6.1.4. Proposition 6.2.3. There are natural functorial isomorphisms for all F , G , H in D .QCrys.X; A//:
(a) F ˝L G Š G ˝L F .
(b) F ˝L .G ˝L H / Š .F ˝L G / ˝L H .
(c) Lf .F ˝L G / Š Lf F ˝L Lf G .
Proof. This follows from Proposition 4.2.5, the construction of the derived functors Lf and ˝L in terms of very flat quasi-crystals, and the fact that f and ˝ preserve the property ‘very flat’ by Proposition 4.2.8. For the comparison with the derived tensor product on IndCrys we need two more technical results: Proposition 6.2.4. Any crystal F on X , whose underlying coherent sheaf on X C is a pullback from the first factor, is flat.
92
6 Derived functors
Proof. By Definition 4.2.6 we must show that Tor i .F ; G / D 0 for all i 1 and G 2 IndCrys.X; A/. As Tor i commutes with filtered direct limits, Tor i .F ; G / lies in IndCrys.X; A/. Thus by Theorem 4.1.7 (b) it suffices to show that ix Tor i .F ; G / D 0 for the embedding ix of any point x 2 X . Since Lix D ix on IndCrys.X; A/ by Proposition 6.1.5, Propositions 6.2.2 and 6.2.3 (c) imply that ix Tor i .F ; G / Š Tor i .ix F ; ix G / in IndCrys.x; A/. Now by assumption the coherent sheaf underlying ix F is a pullback under pr 1 W x C ! x, so it is free. Thus its (quasi-coherent) Tor i vanish for all i 1, as desired. Proposition 6.2.5. Every crystal is the quotient of some flat crystal, and every indcrystal is the quotient of some flat ind-crystal. Proof. For crystals this follows by combining Propositions 4.5.10 and 6.2.4. In general write an ind-crystal F as the union of crystals F i , and every L F i as the quotient of some flat crystal Gi . Then we have an epimorphism G ´ i Gi F , where G is a flat ind-crystal, as desired. Theorem 6.2.6. The bi-functor ˝L from Theorem 6.2.1 induces a functor ˝L W D .IndCrys.X; A// D .IndCrys.X; A// ! D .IndCrys.X; A// which is the left bi-derived functor of ˝ on IndCrys.X; A/. Proof. By Proposition 6.2.5 for every complex F 2 K .IndCrys.X; A// there exists a quasi-isomorphism G ! F with G 2 K .IndCrys.X; A// whose components are flat. Using this it follows from Theorem 2.8.2 that the bi-derived functor of ˝ on IndCrys.X; A/ exists. To prove that it coincides with the restriction of the functor ˝L from 6.2.1 consider any two complexes F , G 2 K .IndCrys.X; A//. To evaluate both derived functors we may replace F by a flat complex. Then the derived functor on IndCrys.X; A/ yields F ˝ G , and we must prove that the natural homomorphism L F ˝ G ! F ˝ G is a quasi-isomorphism. By the hypercohomology spectral sequence it suffices to do this when F D F and G D G both live in degree 0. But then Proposition 6.2.2 and the definition 4.2.6 of flatness imply that F ˝L G ! F ˝ G is an isomorphism, as desired.
Corollary 6.2.7. The bi-functors ˝L from Theorems 6.2.1 and 6.2.6 induce an exact functor ˝L W D .Crys.X; A// D .Crys.X; A// ! D .Crys.X; A//: Proof. Proposition 6.2.2 implies that H i .F ˝L G / is coherent whenever F and G are coherent. The assertion follows from this and the hypercohomology spectral sequence, together with Theorem 5.3.1 (c). Remark 6.2.8. Without further hypotheses one cannot expect that ˝L preserves the subcategories D b .: : : /. Consider, for example, the case X D Spec k and A D kŒ"=."2 /, and take M ´ A=A" with M D id. Then the preceding proposition
6.3 Change of coefficients
93
shows that H i .M ˝L M / Š Tor A i .M; M / Š k for all i 0, with acting as the identity. Thus here the cohomology of M ˝L M in the sense of crystals is not bounded below. We will return to this problem in Section 7.6.
6.3 Change of coefficients Let A ! A0 be as in Section 4.3. By the same methods as in the preceding sections one proves the following results, whose details are left to the reader: Theorem 6.3.1. The functor ˝A A0 on A-quasi-crystals possesses a left derived functor ˝L A0 W D .QCrys.X; A// ! D .QCrys.X; A0 //; A
whose restriction ˝L A0 W D .IndCrys.X; A// ! D .IndCrys.X; A0 // A
is the left derived functor of ˝A A0 on IndCrys.X; A/, and which induces an exact functor ˝L A0 W D .Crys.X; A// ! D .Crys.X; A0 //: A
Proposition 6.3.2. For any F 2 QCrys.X; A/ and any integer i , the quasi-crystal 0 H i .F ˝AL A0 / is isomorphic to Tor A i .F ; A / defined in 4.3.2. Proof. Precisely as in Proposition 6.1.4. Proposition 6.3.3. There are natural isomorphisms of derived functors (a) Lf . (b) .
˝AL A0 / Š Lf
˝AL A0 / ˝AL0 A00 Š
˝AL A0 . ˝AL A00 .
6.4 Direct image I For f affine the functor f on quasi-crystals is exact; hence it induces an obvious exact functor on derived categories, denoted again by f . In general we have: Theorem 6.4.1. For any 2 fb; C; ; ¿g the functor f possesses a right derived functor Rf W D .QCrys.Y; A// ! D .QCrys.X; A//:
94
6 Derived functors
ˇ Proof. Let CL U W K .QCrys.Y; A// ! K .QCrys.Y; A// be the Cech resolution functor associated to any finite open affine covering U of Y . Then the composite functor f CL U W K .QCrys.Y; A// ! K .QCrys.X; A// is in each degree a direct sum of functors of the form .fjI / jI , where jI W UI ,! Y is the embedding of an open affine subscheme. Being composed of exact functors, it preserves quasi-isomorphisms; hence it induces an exact functor between the associated derived categories. That this is the desired right derived functor follows easily from the functorial quasi-isomorphism id ! CL U . (Compare [25, Chapter I, Theorem 5.1].) ˇ Since affine Cech resolutions also serve to calculate the usual derived functor of quasi-coherent sheaves R.f id/ , the individual cohomology groups of our functor Rf can be calculated as follows: Proposition 6.4.2. For any G 2 QCrys.Y; A/ and any integer i, the quasi-crystal H i .Rf G / is isomorphic to Ri f G defined in 4.4.2. f
g
Proposition 6.4.3. For any two morphisms Z ! Y ! X there is a natural isomorphism of derived functors R.fg/ Š Rf Rg . Proof. Left to the reader. Proposition 6.4.4. For 2 fb; g the derived functor Rf of Theorem 6.4.1 is right adjoint to the functor Lf of Theorem 6.1.3. Proof. By Proposition 4.4.5 the functor f on quasi-crystals is right adjoint to the corresponding functor f . The same is therefore true for the associated homotopy categories. To pass to the derived categories recall from Theorem 5.4.4 that the inclusion induces an equivalence of categories ! D .QCrys.X; A//: D .QCrysvflat .X; A// Thus it suffices to test the adjunction on complexes F 2 K .QCrysvflat .X; A// and G 2 K .QCrys.Y; A//. By adjunction on objects we have a functorial isomorphism HomK .::: / f F ; CL U G Š HomK .::: / F ; f CL U G :
By Proposition 2.2.5 (b) this isomorphism induces a functorial isomorphism HomD .::: / f F ; CL U G Š HomD .::: / F ; f CL U G : Thus the quasi-isomorphism id ! CL U and the constructions of Rf and Lf yield the desired isomorphism HomD .::: / Lf F ; G Š HomD .::: / F ; Rf G :
95
6.4 Direct image I
Next we consider a cartesian diagram g0
Y0 f0
/Y
(6.4.5)
f
X0
g
/ X.
Consider the composite natural transformation on D .QCrys.Y; A// Lf 0 Lg Rf D Lg 0 Lf Rf ! Lg 0 ; where the arrow on the right hand side results from the adjunction homomorphism Lf Rf ! id. Its adjoint Lg Rf ! Rf0 Lg 0
(6.4.6)
is called the base change homomorphism. Theorem 6.4.7 (Base Change). The base change homomorphism is an isomorphism. Proof. By Theorem 5.4.4 it suffices to verify the isomorphy on any complex G 2 ˇ D .QCrysvflat .Y; A//. By Theorem 5.4.2 we may also replace G by its Cech complex 0 L CU G . Furthermore, since both Rf and Rf have finite cohomological dimension, we may pass to the individual components of CL U G . Thus it is enough to show the isomorphy on objects of the form j G , where j W V ,! Y is the embedding of an open affine subscheme and G is in QCrysvflat .V; A/. Consider, then, the completed cartesian diagram V 0_
g 00
/V _
g0
/Y
j0
Y0
f0
X0
j
f
g
/ X.
The base change homomorphisms for f , j , and fj form a commutative diagram Lg Rf Rj
base change
/ Rf 0 Lg 0 Rj
/ Rf 0 Rj 0 Lg 00 6.4.3
6.4.3
Lg R.fj /
base change
base change
/ R.f 0 j 0 / Lg 00 .
96
6 Derived functors
It thus suffices to prove the assertion for j or fj in place of f . After replacing Y by V (and, in the second case, X by Y ) we may now assume that Y is affine. Next we may verify the assertion locally on X 0 . After passing to suitable open affines in X and X 0 , and replacing Y and Y 0 by the corresponding restrictions, we may assume that X and X 0 are affine. Note that Y remains affine in this process, because f is separated, and afterwards Y 0 is affine as well. Thus all higher derived functors in question now vanish on very flat quasi-crystals; hence it remains to verify the isomorphy in degree zero. For this let S ˝R R0 DSO 0 o
SO
R0 o
R
be the commutative diagram of affine rings underlying the given diagram of schemes. Let N be the .S ˝ A/Œ -module corresponding to a very flat quasi-crystal on Y . The base change homomorphism then reduces to the well-known isomorphism N ˝R R0 ! N ˝S S 0 ; which finishes the proof. The last main result of this section is the projection formula. For any G 2 D .QCrys.Y; A// and F 2 D .QCrys.X; A// consider the homomorphisms
6.2.3 Lf Rf G ˝L F Š Lf Rf G ˝L Lf F ! G ˝L Lf F ; where the arrow on the right hand side results from the adjunction homomorphism Lf Rf ! id. Via the adjunction between Lf and Rf the composite homomorphism yields a natural transformation (6.4.8) Rf G ˝L F ! Rf G ˝L Lf F : Theorem 6.4.9 (Projection Formula). The homomorphism (6.4.8) is an isomorphism. Proof. Proceeding exactly as in the proof of Theorem 6.4.7, we replace G by its ˇ Cech complex and F by a complex of very flat quasi-crystals, and then we pass to the individual components of both complexes. This reduces us to the case that F D F 2 QCrysvflat .X; A/ and G D j G , where j W V ,! Y is the embedding of an open affine subscheme and G is in QCrysvflat .V; A/. Since the homomorphism (6.4.8) for fj is just the composite of the corresponding homomorphisms for f and j , we can then replace Y by V and thus assume that Y is affine. As the assertion is local on X , we may also replace X by an open affine and Y by its pullback, which remains affine. Now all higher derived functors in question vanish, and it remains to verify the isomorphy in degree zero.
6.5 Direct image II
97
For this let R ! S be the ring homomorphism corresponding to f , let M be the .R ˝ A/Œ-module corresponding to F , and let N be the .S ˝ A/Œ -module corresponding to G . The homomorphism (6.4.8) then amounts to the well-known isomorphism N ˝R M ! N ˝S .S ˝R M /: This finishes the proof. Proposition 6.4.10. There is a natural isomorphism of derived functors: Rf
˝AL A0 ! Rf .
˝AL A0 /:
Proof. The homomorphism is constructed as in (6.4.8). The proof follows that of Theorem 6.4.9, where G can also be made very flat, so that G is in particular acyclic for ˝A A0 .
6.5 Direct image II When f is proper, the functor f preserves the subcategory IndCrys.: : : /, and we can study the derived functor of the restriction. As in the preceding sections we will show that this derived functor coincides with the restriction of the derived functor Rf on QCrys.: : : /. The main point is a certain spectral sequence, which exists even when f is not proper. When f is not proper, we can still do the following. Recall from Proposition 5.3.3 that the inclusion i W IndCrys.X; A/ ,! QCrys.X; A/ possesses a right adjoint splitting ind W QCrys.X; A/ ! IndCrys.X; A/. This functor is left exact, and in place of f we can look at the composite left exact functor ind f i W IndCrys.Y; A/ ! IndCrys.X; A/:
(6.5.1)
Recall from Proposition 5.3.3 that the derived functor R ind induces a quasi-inverse of the natural functor i W D .IndCrys.X; A// ! Dindcrys .QCrys.X; A// for every 2 fb; C; ; ¿g. Theorem 6.5.2. For any 2 fb; C; ; ¿g the functor (6.5.1) possesses a right derived functor making the following diagram commute: D IndCrys.Y; A/
R.ind f i/
i
D QCrys.Y; A/
/ D IndCrys.X; A/ O R ind
Rf
/ D QCrys.X; A/ .
98
6 Derived functors
Proof. It suffices to show that R ind Rf D ind f on any injective in IndCrys.Y; A/. We first translate this into a statement on -sheaves. By Proposition 2.4.9 (c) any injective in IndCrys.Y; A/ can be represented by an injective in IndCoh .Y; A/. Next the functors ind and R ind are induced from the functors ind and R ind by Proposition 5.3.3. Only the functor Rf was constructed directly on quasi-crystals, without defining it first on -sheaves. But clearly the proof of Theorem 6.4.1 translates verbatim to -sheaves, showing that Rf on quasi-crystals is induced from a derived functor Rf ´ f CL U W D .QCoh .Y; A// ! D .QCoh .X; A//: Therefore it suffices to show that R ind Rf D ind f on any injective object of IndCoh .Y; A/. We claim that this assertion is local on X . First, injectivity in IndCoh .: : : / is ˇ invariant under localization by Proposition 5.2.7. Next the Cech resolution commutes with localization; hence so does Rf . Thirdly the derived functor R ind commutes with localization by Theorem 5.2.10. Thus all ingredients commute with localization, so after shrinking X and Y we may assume that X is affine. Now by Proposition 2.4.7 any injective in IndCoh .Y; A/ can be embedded into ind. / for an injective 2 QCoh .Y; A/. Being injective, it is then a direct summand of ind. /. Thus it suffices to prove that R ind Rf ind. / D ind f ind. /. For this note that Theorem 5.2.10 gives a short exact sequence 0 ! ind. / ! ! S. / ! 0:
(6.5.3)
Lemma 6.5.4. We have R ind Rf S. / D 0 in D .IndCoh .X; A//. ˇ Proof. By construction Rf D f CL U , where CL U denotes the Cech resolution corresponding to some finite affine open covering U of Y . Thus if jI W UI ,! Y are the associated open embeddings, the components of Rf S. / are direct sums of f jI jI S. / for varying I . By Proposition 5.2.3 these summands are isomorphic to .fjI / S.jI /. Thus it suffices to show that R ind.fjI / S.F / D 0 for F ´ jI . But the canonical homomorphism S.F / ! S.S.F // is an isomorphism by the definition 5.2.3 of S. Thus the canonical homomorphism .fjI / S.F / ! S..fjI / S.F // is an isomorphism by Proposition 5.2.6 (b). By Theorem 5.2.10 this implies that R ind.fjI / S.F / D 0, as desired. Lemma 6.5.5. We have Ri ind f D 0 in IndCoh .X; A/ for all i 1. Proof. Choose a finite affine open covering of X , with open immersions ji W Ui ,! X . Then we can embed every jL i into an injective -sheaf i , which by adjunction induces an embedding ,! i ji i . Since is injective, this embedding splits. It suffices therefore to prove the assertion for every ji i in place of . In other words we must show that Ri ind.fji / i D 0 in IndCoh .X; A/ for all i 1. Now since i is injective, it is strongly ind-acyclic by Proposition 5.1.11. Thus by Proposition 5.2.6 (a) so is the -sheaf .fji / i . Proposition 5.1.10 (a) now implies that Ri ind.fji / i D 0 for all i 1, as desired.
6.6 Extension by zero
99
Finally, applying the functor R ind Rf to the exact sequence (6.5.3), Lemma 6.5.4 implies that R ind Rf ind. / Š R ind Rf . Since is injective and Rf is the right derived functor of f , we have Rf D f . Thus Lemma 6.5.5 implies that R ind Rf is concentrated in degree 0. Together it follows that R ind Rf ind. / is concentrated in degree 0; hence it is equal to ind f ind. /, as desired. Theorem 6.5.6. If f is proper, then for any 2 fb; C; ; ¿g the functor Rf from Theorem 6.4.1 induces a functor Rf W D .IndCrys.Y; A// ! D .IndCrys.X; A// which is the right derived functor of f on IndCrys.Y; A/. Proof. As f is proper, Proposition 6.4.2 implies that the derived functors Ri f of quasi-crystals preserve coherence, and hence also ind-coherence, for all i 2 Z. Thus the hypercohomology spectral sequence implies that Rf i lands in the subcategory Dindcrys .QCrys.X; A//. It is therefore equal to i R‹f for some exact functor R‹f W D .IndCrys.Y; A// ! D .IndCrys.X; A//: From Proposition 5.3.3 (c) and Theorem 6.5.2 we deduce that R‹f Š R ind i R‹ f D R ind Rf i D R.ind f i /: But ind f i D f when f is proper, so the theorem follows. Corollary 6.5.7. If f is proper, then for any 2 fb; g, the functors Rf in Theorems 6.4.1 and 6.5.6 induce an exact functor Rf W D .Crys.Y; A// ! D .Crys.X; A//: Proof. As in the preceding proof this follows from the fact that Ri f preserves coherence, together with Theorem 5.3.1 (c).
6.6 Extension by zero The remaining two sections concern only the categories IndCrys.: : : /. Consider an j
i
open embedding U ,! X and let Y ,! X be a complementary closed embedding. By Proposition 4.5.7 the functor jŠ is exact. Thus for any 2 fb; C; ; ¿g it induces an exact functor jŠ W D .IndCrys.U; A// ! D .IndCrys.X; A//;
(6.6.1)
which restricts to an exact functor jŠ W D .Crys.U; A// ! D .Crys.X; A//:
(6.6.2)
100
6 Derived functors
Proposition 6.6.3. For any 2 fb; C; ; ¿g the following assertions hold within D .IndCrys.: : : //: (a) The functor jŠ is left adjoint to j . (b) The adjunction homomorphism id ! j jŠ is an isomorphism. (c) The composite i jŠ is zero. (d) There is a natural distinguished triangle jŠ j ! id ! i i ! jŠ j Œ1: j0
j
(e) For any open embeddings U 0 ,! U ,! X there is a natural isomorphism .jj 0 /Š Š jŠ jŠ0 . (f) For any morphism g W X 0 ! X consider the pullback diagram U0
j0
/ X0
g0
U
g
/ X.
j
The base change homomorphism jŠ0 g 0 ! g jŠ is an isomorphism. Proof. On IndCrys.: : : / the functor jŠ is left adjoint to j by Proposition 4.5.7 (a). The same is therefore true on the associated homotopy category. By Proposition 2.2.5 (a) the same follows on the derived category; this proves assertion (a). The remaining assertions follow directly from Propositions 4.5.7, 4.5.8, and 4.5.9. Note that in (d) we have Ri D i , because i is affine. Proposition 6.6.4. There are natural isomorphisms of derived functors (a) jŠ .
˝L j / Š jŠ
(b) jŠ .
˝
(c) jŠ .
˝AL A0 / Š jŠ
L
/ Š jŠ
˝L
.
˝ jŠ . L
˝AL A0 .
Proof. We work out (a); the other cases are exactly analogous. The homomorphism is defined as the adjoint of the isomorphism G ˝L j F Š j jŠ G ˝L j F Š j .jŠ G ˝L F /
given by Proposition 6.6.3 (b) and 6.2.3 (c). By construction it is an isomorphism over U , so by the definition of jŠ it remains to show that i .jŠ G ˝L F / vanishes. L But by Proposition 6.2.3 (c) this is isomorphic to i jŠ G ˝ i F , where i jŠ G D 0, so it vanishes, as desired.
101
6.7 Direct image with compact support
The following lemma will be used in the next section. Lemma 6.6.5. Consider a commutative diagram
j0
X
j
Y
/ Yx o
i0
fN
f
/ Xx o
? _ @Y @f
i
? _ @X
where the vertical arrows are proper morphisms, where j and j 0 are open embeddings, where i and i 0 are the respective complementary closed embeddings, and where both squares are cartesian. Then there is a natural isomorphism of derived functors jŠ Rf Š RfN jŠ0 . Proof. Since both rectangles are cartesian, we have isomorphisms j RfN jŠ0 D Rf j 0 jŠ0 6.4.7
6.6.3 (b)
6.4.7 i RfN jŠ0 D R.@f / i 0 jŠ0
D
Rf ;
6.6.3 (c)
D
0:
Composing the distinguished triangle 6.6.3 (d) with the functor RfN jŠ0 we obtain jŠ j RfN j 0 ! RfN jŠ0 ! i i RfN jŠ0 ! ; „ ƒ‚ …Š „ ƒ‚ … k k jŠ Rf 0 whence the desired isomorphism.
6.7 Direct image with compact support Now we assume that the morphism f is compactifiable, i.e., that it lies in a commutative diagram j / N Y {Y {{ { (6.7.1) f {{ }{{{ fN X, where j is an open embedding and fN is proper. (Any morphism of finite type has this property, see [40], [41], [33].) For any such diagram and any 2 fb; C; ; ¿g we define, using Corollary 6.5.6 and (6.6.1), an exact functor RfŠ ´ RfN jŠ W D .IndCrys.Y; A// ! D .IndCrys.X; A//:
(6.7.2)
102
6 Derived functors
By 6.5.7 and (6.6.2) for any 2 fb; g it induces an exact functor RfŠ ´ RfN jŠ W D .Crys.Y; A// ! D .Crys.X; A//:
(6.7.3)
As usual, one should be aware that RfŠ is not the derived functor of a functor fŠ . The following results are deduced by standard techniques from those of the preceding sections. Compare [SGA4, Exposé XVII], [19, Chapter I, § 8], or [37, Chapter VI, § 3]. Proposition 6.7.4. The functor RfŠ depends only on f , and not on the chosen compactification, up to a natural isomorphism of functors. Proof. Since any two compactifications are dominated by a third one, it suffices to compare compactifications that dominate each other. So consider a commutative diagram j0 / x0 Y Y p
/ Yx zz zz z f zz fN z |z p X,
Y
j
fN 0
where fN, p, and fN0 D fNp are proper and j and j 0 are open embeddings. We may also replace Yx 0 by yet another compactification, so after passing to the closure of j 0 .Y / we may assume that j 0 .Y / is dense in Yx 0 . Then one easily shows that the upper square is cartesian and that it can be extended to a diagram as in Lemma 6.6.5. By Lemma 6.6.5 there is therefore a natural isomorphism of functors jŠ ! Rp jŠ0 . Its composite N with Rf is the desired isomorphism 6.4.3 RfN jŠ ! RfN Rp jŠ0 D RfN0 jŠ0 : f
g
Theorem 6.7.5. Consider two morphisms Z ! Y ! X such that f and fg are compactifiable. Then g is compactifiable and there is a natural isomorphism of functors R.fg/Š Š RfŠ RgŠ . Proof. By assumption, we have compactifications
j
/ YN z zz f zz z z } z fN X
Y
and
/ O zZ z zz gf zz z z} X.
Z
6.7 Direct image with compact support
103
x be the closure in Yx ZO of the graph of g. This is proper over Yx . Let ZQ Z x Let Z be the inverse image of Y Yx . Then we have a commutative diagram / ZQ j / Z x { { { { gN {{ {{ g { gQ {{x { {{ { { }{ gQ }{ j / YN Y { {{ {{ f { }{{{ fN X,
Z
j0
Q
where the oblique morphisms are proper and the horizontal ones open embeddings. xQ and j is cartesian. Furthermore, by construction the parallelogram formed by jQ, g, N g, xQ jQŠ . By Lemma 6.6.5 we therefore have a natural isomorphism of functors jŠ RgN Š Rg 0 N Its composite with Rf and jŠ is the desired isomorphism def N xQ jQŠ j 0 def xQ jQŠ j 0 6.4.3 RfŠ RgŠ D RfN jŠ RgN jŠ0 Š RfN Rg Š D R.f g/ Š D R.fg/Š :
Theorem 6.7.6 (Proper Base Change). In the cartesian diagram (6.4.5) assume that f is compactifiable. Then f 0 is compactifiable and there is a natural isomorphism of functors g RfŠ Š RfŠ0 g 0 . Proof. Choose a diagram (6.7.1) and define the rest of the following diagram by pullback: g0 /Y p Y 0 Eq CC EEj 0 CCj EE CC E" C! gN / x f0 Yx 0 {Y y { y { y f yy 0 {{ |yy fN }{{ fN / X. X0 g Then fN0 is again proper and we have def 6.4.7 g RfŠ D g RfN jŠ D RfN0 gN jŠ
6.6.3 (f)
D
def RfN0 jŠ0 g 0 D RfŠ0 g 0 ;
as desired. Theorem 6.7.7 (Projection Formula). For compactifiable f W Y ! X there is a natural ˝L f . isomorphism of functors RfŠ ˝L Š RfŠ Proof. In terms of the factorization (6.7.1) we have RfŠ
˝L
RfN jŠ ˝L 6.4.9 D RfN jŠ ˝L fN 6.6.4 (a) D RfN jŠ ˝L j fN def D RfŠ ˝L f ; def
D
104
6 Derived functors
Proposition 6.7.8. For compactifiable f W Y ! X there is a natural isomorphism of functors RfŠ . ˝AL A0 / Š RfŠ ˝AL A0 . Proof. Combine Propositions 6.4.10 and 6.6.4 (c).
Chapter 7
Flatness
The notion of flatness was introduced for quasi-crystals in Definition 4.2.6. In this chapter our main focus is on flatness of crystals. Many of the results obtained here, notably from Sections 7.5 and 7.6, will be needed in Chapter 9 to study of the Lfunctions associated to flat crystals. We begin this chapter with a short Section 7.1, which collects some essentially known facts on flatness of modules over commutative rings. Section 7.2 provides some basic formulas for Tor i of crystals and establishes that flatness of crystals is a pointwise condition. Section 7.3 links the flatness of a crystal F to the flatness of the quasi-coherent sheaf underlying its canonical representative Fy in the category of all -sheaves. This is the main technical tool, from which many of the subsequent results are deduced. In Section 7.4 we show that flatness is preserved by most of our functors from Chapter 4. The preservation by f can be discussed only for the corresponding derived notion of complexes of finite Tor-dimension, which is studied in Section 7.6. In Section 7.5 we analyze to what extent a flat crystal can be represented by a coherent -sheaf whose underlying sheaf is free, or locally free. Finally Section 7.7 discusses finite flat resolutions when A is regular. Throughout, we consider a morphism f W Y ! X and a homomorphism A ! A0 , as in Chapter 4. We let kc denote the residue field at a point c 2 C .
7.1 Flatness of modules As a preparation we first recall, respectively provide several results on modules over a noetherian commutative ring T . Proposition 7.1.1. For any finitely generated T -module N and any filtered direct system of T -modules Mi we have Hom N; lim Mi Š lim Hom.N; Mi /: ! ! i
i
Proof. The statement is clear if N is free of finite type. The general case follows from this using a presentation T ˚m ! T ˚n ! N ! 0, the left exactness of Hom, and the exactness of lim. ! The next statement is a reformulation of [36, Theorem 7.6]:
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7 Flatness
Proposition 7.1.2. A T -module M is flat if and only if every homomorphism N ! M for every finitely generated T -module N factors through homomorphisms N ! T ˚r ! M for some r 0. The following result is well documented for finitely generated modules, for example in [47, Proposition 4.4.11]. For arbitrary modules, we do not know a reference and thus include a proof. Let kp denote the residue field of T at a prime ideal p. Proposition 7.1.3. A T -module M is flat if and only if Tor Ti .M; kp / D 0 for all i 1 and all prime ideals p. Proof. If M is flat, then of course Tor Ti .M; kp / D 0 for all i 1 and all p. Conversely, assume this. To prove that M is flat, it suffices to show that Tor Ti .M; T =a/ D 0 for all ideals a T . Let S be the set of ideals a such that there exists i 1 with Tor Ti .M; T =a/ 6D 0. We must prove that S is empty. Because T is noetherian, S possesses a maximal element a. We claim that a is a prime ideal. Indeed, if not, there exist t, u 2 T n a with t u 2 a. Then we have .a C tT /=a Š t T =.t T \ a/ Š T =b with b ´ fw 2 T j wt 2 ag, and thus a short exact sequence 0 ! T =b ! T =a ! T =.a C t T / ! 0: The ideals a C t T and b a C uT both strictly contain a; hence they do not lie in S. applied to the above short exact sequence The long exact Tor-sequence for M ˝T therefore shows that a itself cannot lie in S , which is a contradiction. Thus a is a prime ideal. So let us rename p ´ a. The residue field kp can be written as a filtered direct limit lim T =p, taken over all elements u of the multiplicative system T n p. We define ! u N by the short exact sequence 0 ! T =p ! kp ! N ! 0; so that N D lim T =.uT C p/. By the maximality of p, the ideals uT C p do not lie ! u in S. In the limit we can therefore deduce that Tor Ti .M; N / D 0 for all i 1. The long exact Tor-sequence for M ˝T now shows that Tor Ti .M; T =p/ Š Tor Ti .M; kp / D 0 for all i 1, contradicting our assumption on p. Thus S must have been empty in the first place, as desired. Let Tp and Mp denote the localizations of T and M at p. Recall that M is said to be flat at p if Mp is a flat Tp -module. Since Tor i commutes with localization, Proposition 7.1.3 immediately implies: Corollary 7.1.4. Let T be a noetherian local ring and M a T -module which is flat at all prime ideals different from the maximal ideal m. Then M is flat over T if and only if Tor Ti .M; T =m/ D 0 for all i 1. We also need a variant of a theorem of Kunz:
7.2 Basic properties
107
Proposition 7.1.5. The morphism W X ! X is flat if and only if X is regular. Proof. The assertion is local on X , so we may assume that X D Spec R for some local noetherian ring R. Set Rq ´ fx q j x 2 Rg. Then Kunz’s theorem[35, Theorem 107] asserts that R is regular if and only if R is reduced and Rq ,! R is flat. Now if R is regular, it is reduced; hence induces an isomorphism R ! Rq . By the above, the inclusion Rq ,! R is flat, and so is flat. Conversely assume that is flat. By Kunz’s theorem, we only need to show that R is reduced. Let a R be the kernel of W R ! R. Because is flat, the induced homomorphism a ˝R; R ! R ˝R; R Š R must be injective. But it is given by a ˝ u 7! aq u, which is zero by definition of a. Thus a˝R; R is zero. Because is flat and a bijection on the topological space Spec R, it is faithfully flat. This shows that a D 0. Since all nilpotent elements are annihilated by a power of , this implies that R is reduced, as desired. The above result can be combined generically with the following result, which can be found for instance in [36, Theorem 24.4]. Proposition 7.1.6. Any reduced noetherian scheme contains a regular dense open subscheme.
7.2 Basic properties Let us summarize some equivalent characterizations of flatness of (ind-) crystals: Proposition 7.2.1. The following properties of F 2 IndCrys.X; A/ are equivalent: (a) F is flat. (b) Tor i .F ; G / D 0 for all i 1 and all G 2 IndCrys.X; A/. (c) Tor i .F ; G / D 0 for all i 1 and all G 2 Crys.X; A/. (d) The functor F ˝
W IndCrys.X; A/ ! IndCrys.X; A/ is exact.
Proof. The equivalence (a),(b) is just Definition 4.2.6. Next (b),(c) follows from the fact that Tor i commutes with filtered direct limits. Finally (b),(d) results from Proposition 6.2.2 and Theorem 6.2.6. We write Crysflat .X; A/ Crys.X; A/ for the full subcategory of flat crystals. Proposition 7.2.2. (a) Any direct summand of a flat crystal is flat. (b) Any ind-crystal, whose underlying quasi-coherent sheaf is flat, is flat. (c) Any crystal whose underlying coherent sheaf is a pullback from the first factor, is flat.
108
7 Flatness
Proof. Parts (a) and (b) are clear from the definition of flatness, part (c) is a restatement of Proposition 6.2.4. Proposition 7.2.3. For any F , G 2 Crys.X; A/ and any i 0 there are natural isomorphisms: (a) f Tor i F ; G Š Tor i f F ; f G . A 0 0 (b) f Tor A i F ; A Š Tor i f F ; A . Proof. By Theorem 4.1.12 (c) and Proposition 6.1.5 the functor f on ind-crystals is exact and coincides with the restriction of the derived functor Lf . Together with Proposition 6.2.3 (c) this implies that f H i F ˝L G Š H i Lf .F ˝L G / D H i f F ˝L f G : Using Proposition 6.2.2 this proves (a). The same argument using Propositions 6.3.2 and 6.3.3 (a) proves (b). 0 Proposition 7.2.4. If F is flat, then Tor A i .F ; A / D 0 as a crystal for all i 1.
Proof. The assertion is local on X , so we may assume that X D Spec R. Let M be the .R ˝ A/Œ-module corresponding to F . We must prove that the induced on 0 .M; R ˝ A0 / Š Tor A Tor R˝A i .M; A / i
is locally nilpotent. For this we write A0 as a filtered direct limit of finitely generated A-modules Nj , obtaining isomorphisms R˝A 0 A Tor A .M; R ˝ Nj /: i .M; A / Š lim Tor i .M; Nj / Š lim Tor i ! ! j
j
These isomorphisms are -equivariant, if the action on R ˝Nj is defined as ˝id. But then R ˝ Nj defines a coherent -sheaf on X; hence the last Tor i is locally nilpotent by the flatness of F . The proposition follows. To show that flatness of crystals is a pointwise property, we need the following lemma. Lemma 7.2.5. For any x 2 X and any crystal Gx on x, there exists a crystal G on X with ix G Š Gx . Proof. Let i W Y ,! X denote the embedding of the scheme-theoretic closure of x. Then x is the generic point of Y ; hence the coherent sheaf on x C underlying Gx can be extended to a coherent sheaf G 0 on U C for some open neighborhood U Y of x. After shrinking U , we may also extend Gx to a homomorphism G 0 W . id/ G 0 ! G 0 . Thus G 0 is a crystal on U with generic stalk Gx . With j W U ,! Y denoting the open embedding, the crystal G ´ i jŠ G 0 then has the desired property.
7.3 Flatness of the canonical representative
109
Proposition 7.2.6. A crystal F on X is flat if and only if ix F is flat for every x 2 X . Proof. For any x 2 X , any G 2 Crys.X; A/, and any i 1, we have ix Tor i F ; G Š Tor i ix F ; ix G by Proposition 7.2.3 (a). If F is flat, the left hand side vanishes. In view of Lemma 7.2.5 this implies that ix F is flat, as desired. Conversely, if ix F is flat, the right hand side vanishes. If this holds for all x 2 X , Theorem 4.1.7 (b) implies that Tor i F ; G vanishes; hence F is flat.
7.3 Flatness of the canonical representative Throughout this section we fix a coherent -sheaf F on X . In (3.3.12) we defined the -sheaf Fy ´ lim . n id/ F 2 IndCoh .X; A/; ! n
which is a canonical representative of the crystal associated to F . We let F W F ! Fy denote the natural nil-isomorphism and Fy the quasi-coherent sheaf on X C underlying Fy . (This is a slight abuse of notation, because Fy depends not only on F , but also on F .) In the present section we study how the flatness properties of F as a crystal are reflected in flatness properties of Fy as a quasi-coherent sheaf. Let ˆF denote the set of points z 2 X C at which F possesses a factorization ˚r y Fz ! OXC ;z ! Fz
(7.3.1)
for some r 0. Lemma 7.3.2. (a) The set ˆF is open. (b) If Fy is flat at z 2 X C , then z 2 ˆF . Proof. To prove (a) consider a point z 2 X C with a factorization (7.3.1). Since the modules on the left and in the middle are finitely generated, both homomorphisms extend to a neighborhood of z (using Proposition 7.1.1). Moreover, since the composite coincides with F at z, it does so on some other, smaller neighborhood. By definition this whole neighborhood is then contained in ˆF , proving (a). Assertion (b) is a direct consequence of Proposition 7.1.2. Lemma 7.3.3. If X is regular, then: (a) . id/.ˆF / D ˆF . (b) If z 2 ˆF , then Fy is flat at z.
110
7 Flatness
Proof. To prove (a) take any point z 2 ˆF . Applying . id/ to the factorization (7.3.1) yields a factorization of F at z 0 ´ . id/.z/: ˚r Fz 0 ! . id/ F z 0 ! . id/ OXC ! . id/ Fy z 0 Š Fyz 0 : z0 Since X is regular, Proposition 7.1.5 shows that id is flat; hence the module in the middle is flat over OXC ;z 0 . As Fz 0 is finitely generated, Proposition 7.1.2 shows ˚r 0 that the composite of the two homomorphisms on the left factors through OXC ;z 0 for some r 0 0. By the definition of ˆF this implies that z 0 2 ˆF , proving the inclusion . id/.ˆF / ˆF . To show equality recall that ˆF is open by Lemma 7.3.2 (a). Its closed complement then satisfies the opposite inclusion; which in turn is an equality by Lemma 4.6.1. Therefore the inclusion for ˆF is an equality, proving (a). To prove (b) let again z 2 ˆF . Then (a) implies that zn ´ . n id/1 .z/ 2 ˆF for all n 0. Applying . n id/ to the factorization (7.3.1) at zn yields a factorization of F at z ˚rn Fz ! . n id/ F z ! . n id/ OXC ! . n id/ Fy z Š Fyz z
for some rn 0. By Proposition 7.1.5 the module in the middle is flat over OXC;z . Consider any homomorphism Nz ! Fyz with Nz any finitely generated OXC;z -module. By Proposition 7.1.1 this homomorphism factors through a homomorphism Nz ! ˚rn /z .. n id/ F /z for some n. The composite homomorphism Nz ! .. n id/ OXC then factors through a free OXC;z -module by Proposition 7.1.2. It follows that the original homomorphism Nz ! Fyz factors through a free module. Thus Proposition 7.1.2 implies that Fyz is flat, as desired. The combination of Lemmas 7.3.2 and 7.3.3 yields: Proposition 7.3.4. If X is regular, the set of points z 2 X C at which Fy is flat is open and invariant under id. As a first application we derive the following useful characterization of flatness: Theorem 7.3.5. For any coherent -sheaf F on X the following assertions are equivalent: (a) F is flat as a crystal.
(b) For any c 2 C and any i 1 the crystal Tor A i F ; kc is zero. (c) Fy is flat over A. If X D Spec K for a field K, the above assertions are also equivalent to: (d) Fy is flat.
7.3 Flatness of the canonical representative
111
Proof. The implication (a))(b) follows from Proposition 7.2.4. Next we prove the equivalence (b),(c). Since F ! Fy is a nil-isomorphism, the crystal Tor A F ; k c i A y y is represented by the -sheaf Tor i F ; kc . But is an isomorphism on F , so by y functoriality it is also an isomorphism on Tor A i F ; kc . Thus this object vanishes as a crystal if and only if it vanishes as a -sheaf. The equivalence (b),(c) now follows from Proposition 7.1.3. Next we assume that X D Spec K for a field K. Then Proposition 7.3.4 and Lemma 4.6.1 imply that the locus of non-flat points for Fy is of the form X D for some closed subscheme D C . To prove the implication (c))(d) we must show that D D ¿. So suppose otherwise, and let c be a generic point of D. Then any generic point z of X c is a generic point of X D. Thus Fyz is an OXC;z -module which is non-flat at the maximal ideal of this local ring, but flat at all other prime ideals. But (c) implies that O K˝A y y 0 D Tor A .Fz ; K ˝ kc / D Tor i X C;z .Fyz ; kz /; i .Fz ; kc / D Tor i
where kz denotes the residue field at z. We deduce from Corollary 7.1.4 that Fyz is flat, after all, which yields a contradiction. Therefore D D ¿, proving the implication (c))(d). Since the implication (d))(a) is a direct consequence of Proposition 7.2.2 (b), this proves everything in the case X D Spec K. It remains to show the implication (b))(a) for general X . But Theorem 4.1.7 and Propositions 7.2.3 (b) and 7.2.6 reduce this to the case X D Spec K, where it has already been proved. Because over a field every module is flat, Theorem 7.3.5 implies: Corollary 7.3.6. If A is a field, then every A-crystal is flat. The next result extends Theorem 7.3.5 to a neighborhood of a generic point: Theorem 7.3.7. Let 2 X be a generic point where X is reduced and the crystal associated to i F is flat. Then there exists an open neighborhood j W U ,! X of
b
such that the quasi-coherent sheaf underlying j F is flat. Proof. After shrinking X we may, by Proposition 7.1.6, assume that it is regular. Proposition 7.3.4 then shows that the locus Z X C , where Fy is non-flat, is closed and invariant under id. By Lemma 4.6.1 it is therefore a finite union of subsets of the form Yi Di with Yi X and Di C closed. After shrinking X further we have for all i either Yi D X or Yi D ¿; and hence Z D X D for some closed subset D C . Passing to the generic point of X , we find that D C is the locus where the sheaf underlying i F is non-flat. Thus Theorem 7.3.5 (a))(d) implies that D D ¿, and the theorem follows.
b
We record the following consequence of the above theorem, whose proof is straightforward using noetherian induction and Proposition 7.2.6.
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7 Flatness
Corollary 7.3.8. For any flat crystal F on X , there exists a finite partition of X into locally closed reduced subschemes Xj , with immersions ij W Xj ,! X , such that for each j the quasi-coherent sheaf underlying ij F is flat.
b
7.4 Functoriality and constructibility We now describe the behavior of flatness under most of our usual functors. The preservation under f can be discussed only in a derived setting, cf. Propositions 7.6.7 and 7.6.9. Proposition 7.4.1. Let j W U ,! X be an open immersion. If F and G are flat crystals, then so are (a) f F , (b) F ˝ G , (c) F ˝A A0 , (d) jŠ F . Proof. Part (a) follows directly from Theorem 7.3.5 (a),(b) and Lemma 7.2.3 (b). For (b) take a third crystal H and consider the isomorphism .F ˝L G / ˝L H Š F ˝L .G ˝L H / in the derived category D .Crys.X; A// from Proposition 6.2.3 (b). Because F and G are flat, this reduces to an isomorphism .F ˝ G / ˝L H Š F ˝ .G ˝ H /: It follows that Tor i .F ˝ G ; H / D 0 in Crys.X; A/ for all i 1; hence F ˝ G is flat, proving (b). For (c) we argue similarly: Take any point c 0 2 C 0 , say with residue field kc 0 . Since F is a flat crystal, the isomorphism F ˝L A0 ˝L kc 0 Š F ˝L kc 0 A
A0
A
from Proposition 6.3.3 (b) reduces to an isomorphism F ˝ A0 ˝L kc 0 Š F ˝ kc 0 : A
0
A0
A
0 It follows that Tor A i .F ˝A A ; kc 0 / D 0 for all i 1. By the criterion of Theorem 7.3.5 (b) this shows that F ˝A A0 is flat, proving (c). Finally, part (d) is an immediate consequence of Proposition 7.2.6 and the characterization 4.5.5 (a) of jŠ F .
7.5 Representability
113
We also note the following result on constructibility. Theorem 7.4.2. For any F 2 Crys.X; A/ the set fx 2 X j ix F is flatg is constructible. Proof. By Theorem 4.4.7 we may assume that X is reduced. By noetherian induction it suffices to show that every generic point of X possesses an open neighborhood U X such that the crystal ix F is either flat for all x 2 U , or non-flat for all x 2 U . Assume first that i F is flat. Then by Theorem 7.3.7 there is an open neighborhood
b
j W U ,! X of such that the quasi-coherent sheaf underlying j F is flat. By Proposition 7.2.2 (b) this implies that the ind-crystal j F is flat; hence j F is flat, and Proposition 7.2.6 shows that ix F is flat for all x 2 U , as desired. On the other hand assume that i F is non-flat. Then by Theorem 7.3.5 there exist c 2 C and i 1 such that Tor A i .i F ; kc / is a non-zero crystal. Now by Proposition 7.2.3 (b) we have an isomorphism of crystals
b
A Tor A i .ix F ; kc / Š ix Tor i .F ; kc /
for all x 2 X. Moreover, the set of x for which the right hand side is non-zero is constructible by Theorem 4.6.2. Since it is non-zero for x D , it is so for all x in a neighborhood U of . Theorem 7.3.5 then implies that ix F is non-flat for all x 2 U , as desired.
7.5 Representability In this section we discuss the question of whether a given flat A-crystal F can be represented by a coherent -sheaf whose underlying sheaf is locally free, or free. The difference is marginal by the following observation: Lemma 7.5.1. Suppose that X is affine. Then every coherent -sheaf on X , whose underlying sheaf is locally free, is nil-isomorphic to a coherent -sheaf whose underlying sheaf is free. Proof. If X is affine, then so is X C . Thus if F is locally free, it is a direct summand of a free coherent sheaf on X C . The lemma follows by setting ´ 0 on the other direct summand. We will see by several examples in Section 9.7 that the answer to the question is no for general A, even in the simplest case X D Spec k. To remedy this, we introduce the notion of iterates of a crystal and show that every sufficiently high iterate of a flat crystal F generically possesses a locally free representative. We also derive generically affirmative answers for F itself whenever dim X C dim C 1. Another amenable case is that of finite A, which will be addressed in Section 10.6. Consider an integer i 1.
114
7 Flatness
Definition 7.5.2. The category formed by all pairs .F ; i /, consisting of a coherent sheaf F on X C and an OXC -linear homomorphism i W . i id/ F ! F ; with the obvious notion of homomorphisms, is denoted Coh.i/ .X; A/. Its objects are called coherent i -sheaves over A on X . This defines an A-linear abelian category in the same way as Coh .X; A/ D Coh.1/ .X; A/. Following Section 3.3, the notion of nilpotence and nil-isomorphism are defined in an obvious way. The localization of Coh.i/ .X; A/ by all nil-isomorphisms is denoted Crys.i/ .X; A/. Definition 7.5.3. The i th iterate of a -sheaf F D .F ; F / is defined as F .i/ ´ .F ; Fi /. Clearly this defines an exact A-linear functor Coh .X; A/ ! Coh.i/ .X; A/. This functor preserves nilpotence and nil-isomorphisms; hence it induces an exact A-linear functor Crys.X; A/ ! Crys.i/ .X; A/. For technical reasons we will also need the following notion. Definition 7.5.4. A free 0 -representative of F 2 Crys.X; A/ consists of a free coherent sheaf G on X C , an integer i0 1, homomorphisms Gi W . i id/ G ! G for all i i0 , and a homomorphism ' W F ! G , such that Gi B . i id/ Gj D GiCj for all i, j i0 , and ' induces a nil-isomorphism F .i/ ! G .i/ ´ .G ; Gi / for all i i0 . Theorem 7.5.5. Suppose that X is affine and regular. Then for every coherent -sheaf F on X the following assertions are equivalent: (a) The quasi-coherent sheaf underlying Fy is flat. (b) Some iterate F .i/ is nil-isomorphic to a coherent i -sheaf G .i/ whose underlying coherent sheaf is free. (c) F possesses a free 0 -representative. Proof. The implication (c))(b) is obvious. For the implication (b))(a) assume that F .i/ is nil-isomorphic to a coherent i -sheaf G whose underlying sheaf G is free. Then as in Proposition 3.3.13 (a) we find that Fy .i/ Š lim . ni id/ G . Since X is regular, ! n the underlying sheaves . ni id/ G are flat by Proposition 7.1.5; hence so is their direct limit Fy .i/ . As Fy .i/ and Fy have the same underlying sheaf, (a) follows. It remains to prove the implication (a))(c). So assume that Fy is flat. Since X C ' is affine, Proposition 7.1.2 implies that F possesses a factorization F ! G ! Fy , where G is a free coherent sheaf. Next, Proposition 7.1.1 and the definition of Fy imply that factors through a homomorphism n W G ! . n id/ F for some
7.5 Representability
115 '
n 0. Moreover, again by Proposition 7.1.1 the composite homomorphism F ! n
G ! . n id/ F coincides with the adjoint of Fn after enlarging n. Thus we have constructed a commutative diagram F
/ . n id/ F /8* y F NN F p O NNN p p p NNN p n ppp ' NNN NN& ppppp G.
By adjunction it induces a commutative diagram n F
/ t9 F t t t . n id/ ' tt˛tn t t tt . n id/ G ,
. n id/ F
(7.5.6)
where ˛n denotes the adjoint of n . For every integer i n define Gi as the composite homomorphism in the commutative diagram . i id/ F . i id/ '
. i id/ G
n . i n id/ F
/ . in id/ F j5 j jj j j j j i n jjj . id/ ˛n jjjj i G
i n F
/F '
/ G.
Then as in Proposition 3.3.9 we deduce that ' defines a nil-isomorphism of i sheaves F .i/ D .F ; Fi / ! .G ; Gi /. Furthermore, a direct calculation shows that Gi B . i id/ Gj D GiCj whenever i , j n. Thus .G ; .Gi /in ; '/ constitutes a free 0 -representative of F , proving (b). Corollary 7.5.7. If X is reduced and F is a flat crystal on X , then the assertions in Theorem 7.5.5 are true for the restriction of F to some open dense subscheme of X . Proof. Direct consequence of Theorems 7.3.7 and 7.5.5. Next we turn to the case that dim X C dim C 1, where the geometry is still relatively simple. Suppose first that X is a point, i.e., the spectrum of a field. The examples in Section 9.7 show that, in order to improve on Theorem 7.5.5, we must assume something about A. Proposition 7.5.8. Suppose that X D Spec K for a field K and that C is regular of dimension 1. Then every flat A-crystal on X can be represented by a coherent -sheaf whose underlying sheaf is free.
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7 Flatness
Proof. Consider any F 2 Coh .X; A/ whose associated crystal is flat. Then the sheaf Fy underlying its canonical representative is flat by Theorem 7.3.5. Let G denote the image of the nil-isomorphism F W F ! Fy . Then F ! G is a nil-isomorphism by construction. We claim that the underlying sheaf G is locally free; this implies the proposition by Lemma 7.5.1. To prove the claim recall that A is the localization of a finitely generated k-algebra, and k is a finite field. Since C D Spec A is regular, it is therefore smooth over k. It follows that X C is smooth of dimension 1 over X D Spec K. This implies that every flat quasi-coherent sheaf on X C is torsion free, and that every torsion free coherent sheaf on X C is locally free. Now Fy is flat and hence torsion free. Therefore its coherent subsheaf G is torsion free, and thus locally free, as desired. Finally we suppose that dim C D 0, i.e., that A is artinian. This includes the cases where A is finite or a field, but is more general. Theorem 7.5.9. Suppose that X is reduced of dimension 1 and that A is artinian. Then for every flat A-crystal F on X there exists an open dense embedding j W U ,! X such that j F can be represented by a coherent -sheaf whose underlying sheaf is free. Proof. Since F is coherent, its length at each of the finitely many generic points of X C is finite. Thus if F W . id/ F ! F is not surjective at these points, replacing F by the image of F decreases the sum of these lengths. After iterating this, we may assume that F is generically surjective. Since X is reduced, the inverse image under id preserves generic lengths; hence F is then generically an isomorphism. We claim that then F is locally free at all generic points of X C . To prove this note that all generic points of X C lie above generic points of X. Thus for this purpose we may localize at any generic point of X ; hence we may assume that X D Spec K for a field K. Then Theorem 7.3.5 shows that Fy is flat, and so the proof of Theorem 7.5.5 yields a commutative diagram (7.5.6) with G locally free of finite rank on X C . By construction F is an isomorphism; hence so is Fn . Thus the commutative diagram shows that F is a direct summand of a locally free sheaf. It is therefore itself locally free, proving the claim. NowT let T X C be the closed subset where F is not locally free, and set T1 ´ n . id/n .T /. By the claim T and hence T1 is nowhere dense. On the other hand Lemma 4.6.1 implies that T1 is a finite union of Yi Di with Yi X and Di C closed. Since A is artinian, we have dim C D 0, and so Di C is open. Thus the Yi X must be nowhere dense. After removing their union from X we may therefore assume that T1 is empty. The arguments so far hold for dim X arbitrary, but the rest requires the assumption dim X 1. Then T , being nowhere dense, is finite. We will shrink F until T is empty. If it is non-empty, the assumption T1 D ¿ implies that T contains a point t with . id/.t / 62 T . This means that F is locally free at . id/.t /, and so . id/ F is locally free at t . Now recall that F is generically an isomorphism, that t is a nowhere dense closed point of X C , and that dim.X C / 1. These
7.6 Complexes of finite Tor-dimension
117
facts together imply that F is an isomorphism in a punctured neighborhood of t . We can therefore construct a coherent subsheaf G F which coincides with F outside t and with the image of F at t . Then G is locally free outside T and at t . Moreover, by construction F factors through G , so the inclusion defines a nil-isomorphism of -sheaves G ,! F . Thus we may replace F by G , which shrinks T to T n ft g. After iterating this procedure we get T D ¿, so that F is locally free. Finally, we can shrink X further, if necessary, and then apply Lemma 7.5.1, finishing the proof.
7.6 Complexes of finite Tor-dimension The substitute for flatness in the context of derived categories is that of complexes of finite Tor-dimension. We introduce and clarify this notion in the present section. The first result is inspired by Hartshorne [25, Chapter II, Proposition 4.2]:
Proposition 7.6.1. For any complex F 2 C b .Crys.X; A// the following statements are equivalent. (a) There is a quasi-isomorphism G ! F with G 2 C b .Crysflat .X; A//.
(b) The functor F ˝L W D .Crys.X; A// ! D .Crys.X; A// is way out right in the sense of [25, Chapter I, § 7]. In other words, for any n1 there exists an n2 such that for all H 2 D .Crys.X; A// satisfying H i .H / D 0 for all i < n2 , the cohomology groups H i F ˝L H vanish for all i < n1 .
(c) There is a constant n such that for all H 2 Crys.X; A/ and all i < n we have H i .F ˝L H / D 0. Proof. To prove (a))(b) suppose that G i D 0 for i < n, and set n2 ´ n1 n. Consider the spectral sequence E1ij D H j G i ˝L H H) H iCj G ˝L H Š H iCj F ˝L H : Since G i is flat, we have E1ij Š G i ˝ H j .H / for all i and j . Thus E1ij D 0 whenever i < n or j < n1 n; hence the abutment vanishes whenever i C j < n1 , as desired. Next observe that if (b) holds for a given pair of integers .n1 ; n2 /, by dimension shift it also holds for the pair .n1 n2 ; 0/. Thus the implication (b))(c) follows at once with n ´ n1 n2 . It remains to prove (c))(a). Proposition 6.2.5 implies that for every complex F 2 C .Crys.X; A// there exists a quasi-isomorphism G ! F with G 2 i flat C .Crys .X; A//. Choose an integer m n such that F D 0 for all i m. Let d i W G i ! G iC1 denote the i-th differential of G , and consider the complexes d mC1 d mC2 s>m G ´ 0 ! im.d m / ! G mC1 ! G mC2 ! : : : ; d mC1 d mC2 t>m G ´ 0 ! 0 ! G mC1 ! G mC2 ! : : : :
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As F i D 0 for all i m, the composite homomorphism s>m G ,! G ! F is a quasi-isomorphism. Since all G i are flat, to prove (a) it suffices to show that im.d m / is flat. For this we consider the short exact sequence of complexes
0 ! t>m G ! s>m G ! im.d m /Œm ! 0: For any H 2 Crys.X; A/ and any integer i we obtain an exact sequence H i s>m G ˝L H o
H F ˝L H i
/ H i im.d m /Œm ˝L H 6.2.2 Tor mi .im.d m /; H
/ H iC1 t>m G ˝L H H iC1 t>m G ˝ H .
By assumption (c) the term on the left vanishes whenever i < n. The flatness of the G i implies the equality on the right hand side; hence the term on the right vanishes whenever i C 1 m. Since m n, it follows that the term in the middle vanishes whenever i C 1 m. Therefore Torj im.d m /; H D 0 for all j 1, and so im.d m / is flat, as desired.
Definition 7.6.2. A complex F 2 D b .Crys.X; A// is called of finite Tor-dimension if any of the conditions in Proposition 7.6.1 is satisfied. The full subcategory of D b .Crys.X; A// of complexes of finite Tor-dimension is denoted D b .Crys.X; A//ftd . The condition 7.6.1 (c) shows at once that for any distinguished triangle in D b .Crys.X; A//, if two objects are of finite Tor-dimension, then so is the third. Thus D b .Crys.X; A//ftd is a triangulated subcategory of D b .Crys.X; A//. Theorem 7.6.3. The inclusion Crysflat .X; A/ ,! Crys.X; A/ induces an equivalence of categories D b .Crysflat .X; A// ! D b .Crys.X; A//ftd : Proof. Proposition 7.6.1 (a) shows that condition 2.2.8 (b0 ) is satisfied, which in turn implies the result. In the remainder of this section we show that the property ‘of finite Tor-dimension’ is preserved by various functors. Proposition 7.6.4. The functor f induces an exact functor f W D b .Crys.X; A//ftd ! D b .Crys.Y; A//ftd : Proof. Direct consequence of 7.6.1 (a) and Proposition 7.4.1 (a). Proposition 7.6.5. The derived tensor product induces an exact bi-functor ˝L W D b .Crys.X; A//ftd D b .Crys.X; A//ftd ! D b .Crys.X; A//ftd :
7.6 Complexes of finite Tor-dimension
119
Proof. Consider F , G in D b .Crys.X; A//ftd . By the characterization 7.6.1 (a) we may assume without loss of generality that both complexes are in C b .Crysflat .X; A//. Then the derived tensor product F ˝L G is represented by the single complex associated to the double complex F ˝ G , all of whose objects are flat by Pro L position 7.4.1 (b). Therefore F ˝ G lies in D b .Crysflat .X; A//, and hence in D b .Crys.X; A//ftd , as desired.
Proposition 7.6.6. The functor
˝AL A0 induces an exact functor
˝L A0 W D b .Crys.X; A//ftd ! D b .Crys.X; A0 //ftd : A
Proof. Analogous to the preceding proof, using Proposition 7.4.1 (c) instead of 7.4.1 (b). Proposition 7.6.7. If f W Y ! X is proper, then the functor f induces an exact functor Rf W D b .Crys.Y; A//ftd ! D b .Crys.X; A//ftd : Proof. We use the characterization from Proposition 7.6.1 (c). Consider any complex F 2 D b .Crys.Y; A//ftd and H 2 Crys.X; A//. The projection formula, Theo rem 6.4.9, gives an isomorphism Rf F ˝L H Š Rf F ˝L f H . Thus we have a spectral sequence E2ij D Ri f H j .F ˝L f H / H) H iCj Rf F ˝L H : As Rf is a right derived functor, we have E2ij D 0 whenever i < 0. By 7.6.1 (c) there exists a constant n, independent of H , such that we have H j .F ˝L f H / D 0 for all j < n. Thus E2ij D 0 whenever j < n, and hence the abutment vanishes whenever i C j < n. Thus Rf F ˝L H has the desired property 7.6.1 (c). Proposition 7.6.8. For any open embedding j W U ! X the functor jŠ induces an exact functor jŠ W D b .Crys.U; A//ftd ! D b .Crys.X; A//ftd : Proof. Direct consequence of 7.6.1 (a) and Proposition 7.4.1 (d). Proposition 7.6.9. If f is compactifiable, then the functor RfŠ induces an exact functor RfŠ W D b .Crys.Y; A//ftd ! D b .Crys.X; A//ftd : Proof. Direct consequence of Propositions 7.6.7 and 7.6.8.
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7 Flatness
7.7 Regular coefficient rings It is known that every finitely generated module over a regular noetherian ring has finite Tor-dimension (for example, see [36, Theorem 19.2]). In this section we prove a similar result for A-crystals. We include this result for completeness; it is not needed in the remainder of this book. Proposition 7.7.1. Assume that A is regular and that X is regular and affine. Then any F 2 Coh .X; A/ possesses a finite resolution by coherent -sheaves whose underlying sheaves are locally free. Proof. Recall that A is the localization of a finitely generated k-algebra A0 . Since A is regular, after enlarging A0 we may assume that A0 is regular. Then Spec A0 is smooth over Spec k; hence X Spec A0 is smooth over X . It follows that X Spec A0 and hence X C is regular. As X C is affine and regular, the coherent sheaf underlying F possesses a finite projective resolution by coherent sheaves G i . Moreover, as in Lemma 4.1.9 we can equip the G i inductively with a . id/-linear operation making G a resolution of F by coherent -sheaves. Theorem 7.7.2. If A is regular, we have D b .Crys.X; A//ftd D D b .Crys.X; A//: Proof. It suffices to prove that every F 2 Crys.X; A/ has finite Tor-dimension. We will show this by noetherian induction on X . First consider the closed embedding k W X red ,! X. By Theorem 4.4.7 we have F Š k k F . Thus if k F has finite Tor-dimension, then so does F by Proposition 7.6.7. It is therefore enough to prove the assertion with X replaced by X red . Then X is generically regular by Proposition 7.1.6. Let j W U ,! X be an open embedding with U non-empty, regular and affine. Let i W Y ,! X be a complementary closed embedding and consider the short exact sequence 0 ! jŠ j F ! F ! i i F ! 0: By the induction hypothesis i F has finite Tor-dimension; hence so does i i F by Proposition 7.6.7. On the other hand, Proposition 7.7.1 yields a finite resolution of j F by flat crystals. Thus j F has finite Tor-dimension, and so does jŠ j F by Proposition 7.6.8. The same now follows for F , as desired.
Chapter 8
Naive L-functions
In this chapter and the next we work only over schemes X of finite type over k. In this chapter we also restrict ourselves to coherent -sheaves F whose underlying sheaves are locally free. We define their associated ‘naive L-functions’ and establish a kind of Lefschetz trace formula, i.e., the invariance of this L-function under derived direct image with compact support, in a special case. In particular, we obtain the rationality of the naive L-function in this case. For the trace formula we follow a method of Anderson [2], who gave a simple proof of the rationality of L-functions in the case A D k. Taguchi and Wan [45] noticed that his proof carries over with minor modifications to the case that A is any field containing k. We extend this result to arbitrary k-algebras A and give it a cohomological interpretation. The term ‘naive’ refers to the fact that this L-function is in general not invariant under nil-isomorphisms and therefore cannot be associated to crystals. The natural concept of L-functions of crystals will be developed and studied in Chapter 9. Only in that framework a full cohomological interpretation can be given. The naive L-function is defined in Section 8.1. For the rest of the chapter we assume that X is affine and smooth. In Section 8.2 we relate -sheaves F with their Cartier duals D.F / which possess several conceptual advantages. One effect is that the ˝idlinear endomorphism is translated into a 1 ˝ id-linear map which, as explained by Proposition 8.3.8 (b), enjoys a strong contractivity property. This property is the basis of Anderson’s crucial concept of nucleus and to his trace formula, Theorem 8.3.10, which expresses the L-function of F in terms of the characteristic polynomial on a nucleus for its dual D.F /. In Section 8.3 we develop all the definitions and tools necessary to generalize Anderson’s result from coefficients in k to coefficients in a k-algebra A. However, we give only those proofs which differ significantly from those in [2]. Another effect of dualization is that nuclei for D.F / correspond via Serre duality to the highest cohomology group with compact support of F in the sense of crystals. This is one of the deeper reasons why Anderson’s elementary approach to the trace formula was possible. In Section 8.4 we work out this duality and deduce a cohomological trace formula in Theorem 8.4.2. Finally, in Section 8.5 we calculate some explicit examples over a base of dimension 1. By X we denote a scheme of finite type over k. The set of closed points of X is denoted jXj. The residue field at x 2 jX j is denoted by kx and its degree over k by dx .
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8 Naive L-functions
8.1 Basic properties In this section we define local and global naive L-functions and prove some basic properties. As a preparation we briefly recall the theory of the dual characteristic polynomial for endomorphisms of projective modules. For this fix a commutative unitary ring A, a finitely generated projective A-module M , and an A-linear endomorphism ' W M ! M . The first two proofs are left to the reader. Lemma–Definition 8.1.1. Let M 0 be any finitely generated projective A-module such that M ˚ M 0 is free over A. Let ' 0 W M 0 ! M 0 be the zero endomorphism and t a new indeterminate. Then the expression ˇ detA id t .' ˚ ' 0 / ˇ M ˚ M 0 2 1 C tAŒt is independent of the choice of M 0 . It is called the dual characteristic polynomial of .M; '/ and denoted detA .id t ' j M /. Lemma 8.1.2. The assignment .M; '/ 7! detA .id t ' j M / is multiplicative in short exact sequences. L Lemma 8.1.3. Suppose that M D j mod d Mj for some positive integer d and that '.Mj / Mj C1 for all j . Then for all j we have ˇ ˇ detA id t ' ˇ M D detA id t d ' d ˇ Mj : Proof. By a suitable variant of Lemma 8.1.1 we may assume that all Mj are free Amodules of equal rank n. Then the assertion is a universal identity for certain matrices of size d n d n, so by taking indeterminate coefficients it suffices to prove it when A is a polynomial ring over Z. After embedding A into an algebraic closure of its quotient field it also suffices to prove it when A is an algebraically closed field. We can then find bases of the Mj such that each ' W Mj ! Mj C1 is represented by an upper triangular matrix. The determinant being multiplicative, the identity now reduces to the case that dim Mj D 1. It then amounts to the straightforward equality 0
t a1
1
B B Bt a2 B det B B B @ t ad
1 C C C C C D 1 t d a1 : : : ad C C A
1
for all a1 ; : : : ; ad 2 A. Lemma 8.1.4. Let A be an algebra over a field k. Let k 0 be a finite cyclic Galois extension of k of degree d , and a generator of Gal.k 0 =k/. Let M be a finitely
8.1 Basic properties
123
generated projective module over k 0 ˝ A (the tensor product is always taken over k) and ' W M ! M an A-linear endomorphism satisfying '.xm/ D x '.m/ for all x 2 k 0 and m 2 M . Then ' d is k 0 ˝ A-linear and ˇ ˇ detA id t ' ˇ M D detk 0 ˝A id t d ' d ˇ M ; where A is identified with its image in k 0 ˝ A under the map a 7! 1 ˝ a. In particular, both sides lie in 1 C t d AŒt d . Proof. We first note that both sides are invariant under base extension in A. Thus after replacing .A; M; '/ by .A ˝ k 0 ; M ˝ k 0 ; ' ˝ id/ we may assume that there exists a k-linear embedding i W k 0 ,! A. We then have an isomorphism M j k 0 ˝ A ! A; x ˝ a 7! i. x/ a j : (8.1.5) j mod d
L Let M D Mj be the corresponding decomposition of M . Then the assumption on ' implies that '.Mj / Mj C1 for all j , and the detA id t d ' d j Mj are the constituents of detk 0 ˝A id t d ' d j M via the isomorphism (8.1.5). Thus the first assertion follows from Lemma 8.1.3. The second follows from the first and the equality AŒt \ k 0 ˝ AŒt d D AŒt d . Now we return to -sheaves. First let x D Spec kx for a finite field extension kx =k of degree dx . Let F be a coherent -sheaf on x over A whose underlying sheaf is locally free. Then by Lemma 8.1.4 we have ˇ ˇ detA id t ˇ F x D detkx ˝A id t dx dx ˇ F x 2 1 C t dx AŒt dx : Thus the following definition makes sense: Definition 8.1.6. The naive L-function of F over x is ˇ 1 2 1 C t dx AŒŒt dx : Lnaive .x; F ; t / ´ detA id t ˇ F x For later use we record: Proposition 8.1.7. For any F over x as above and the structure morphism sx W x ! Spec k we have Lnaive .x; F ; t / D Lnaive .Spec k; sx F ; t /: Proof. Let M be the .kx ˝ A/Œ -module corresponding to F . Then the AŒ -module underlying M corresponds to the -sheaf sx F . Thus both sides of the equality are by definition equal to detA id t j M /1 . Next let X be a scheme of finite type over k. Let F be a coherent -sheaf on X over A whose underlying sheaf is locally free. Then for any x 2 jX j the L-function of the stalk ix F lies in 1 C t dx AŒŒt dx . As the number of points in jX j of any given degree dx is finite, we can form the product over all x within 1 C tAŒŒt :
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8 Naive L-functions
Definition 8.1.8. The naive L-function of F over X is Y Lnaive .X; F ; t / ´ Lnaive .x; ix F ; t / 2 1 C tAŒŒt : x2jXj
Finally we note the following obvious fact: Proposition 8.1.9. Let W A ! A0 be an algebra homomorphism and denote the induced homomorphism 1 C tAŒŒt ! 1 C tA0 ŒŒt again by . Then for any F over X as above we have Lnaive .X; F ; t / D Lnaive X; F ˝ A0 ; t : A
8.2 Duality Throughout this section, we assume that X is smooth of equidimension n over k. For such X we describe the rudiments of a duality between Frobenius and Cartier linear coherent sheaves. A formulation for general X could be based on the duality theory for (quasi)-coherent sheaves, as described in [25]. We begin by reviewing some of the terminology from Anderson [2]. Definition 8.2.1. A Cartier linear endomorphism of a coherent sheaf V on X C is an OXC -linear homomorphism V W . id/ V ! V . The pair V ´ .V ; V / is then called a Cartier sheaf on X over A. A homomorphism of Cartier sheaves V ! W is a homomorphism of the underlying sheaves ' W V ! W for which the following diagram commutes: V /V . id/ V . id/ '
. id/ W
' W
/ W.
We denote the category of Cartier sheaves on X over A by Cart.X; A/. In the case X D Spec R these notions are expressed on modules as follows. A Cartier linear map on a finitely generated R ˝ A-module V is a homomorphism of additive groups V W V ! V such that V ..x q ˝ a/v/ D .x ˝ a/ V .v/ for all x 2 R, a 2 A, and v 2 V . For simplicity such a pair V ´ .V; V / is called a Cartier module. Example 8.2.2. The standard and motivating example is the Cartier endomorphism of the dualizing sheaf. To explain this, note first that by assumption the sheaf of differentials X ´ X= Spec k of X is locally free of rank n, and so the dualizing sheaf !X ´ ^n X of X is locally free of rank 1. As explained below, it possesses a natural Cartier linear endomorphism ! W !X ! !X , making .!X ; ! /an object of Cart.X; k/.
8.2 Duality
125
More generally, we let !X;A denote the inverse image of !X under the projection X C ! X. The OXC -linear homomorphism . id/ !X;A ! !X;A induced by ! is again denoted by ! , making !X;A ´ .!X;A ; ! / an object of Cart.X; A/. Remark 8.2.3. The homomorphism ! can be constructed in two different ways. For the first let sX denote the structure morphism X ! Spec k. Following the general formalism of local and global duality (see [25], especially Ch. III Theorem 8.7), we have !X D sXŠ OSpec k Œn, and the equation sX D sX yields a natural isomorphism !X Š Š !X . The functor Š is right adjoint to the functor , by which the isomorphism corresponds to the homomorphism ! W !X ! !X , which is also called a trace map. For an alternative construction let Fabs denote the absolute Frobenius x 7! x p on Spec k as well as on X . We then have a commutative diagram whose right hand side is cartesian: Fabs
/ X .p/ XE EE EE EE EE " Spec k F
V
Fabs
/& X / Spec k.
The Cartier operator (see [11] or [28, §7]) induces a natural surjective homomorphism C W F !X ! !X .p/ . If q D p e , the e th iterate of Fabs is the morphism from Section 3.1 and the e th iterate of C is the desired homomorphism ! W !X D Fe !X ! !X : It is well known that the two definitions agree. For lack of a suitable reference we briefly indicate one possible proof. Since the assertion is local on X , we may assume that there exists an étale morphism u W X ! Pkn . Then u !Pkn D !X , and the compatibility of the two constructions with u reduces us to the case X D Pkn . Consider the two homomorphisms tzr, Ce W !X Š !X that are adjoint to the above two candidates for ! . By construction tzr is an isomorphism, so it suffices to show that 1 tzr Ce W !X ! !X is the identity. Since Hom.!X ; !X / D k, it suffices to show that the endomorphisms H n .tr/, H n .C e / of H n .Pkn ; !X / Š k agree. By the invariance of trace maps under composition we have H n .tr/ D id. The other equality H n .C e / D id follows easily from the explicit description of the Cartier operator (see [28, §7]) and the standard calculation of H n .Pkn ; !X / (see [26, Chapter III §5]). Thus the two definitions agree. The dualizing sheaf allows us to relate Cartier sheaves and -sheaves on X whose underlying sheaves are locally free, as follows. For any F D .F ; F / 2 Coh .X; A/ let QF W F ! . id/ F denote the homomorphism adjoint to F . Then on D.F / ´ Hom.F ; !X;A / we obtain a Cartier linear endomorphism D.F / , defined as the com-
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8 Naive L-functions
posite . id/ Hom F ; !X;A
/ Hom F ; !X;A O
o Q ! B. /
. id/ Hom F ; . id/Š !X;A
(8.2.4)
. /BQ F
Hom . id/ F ; !X;A ,
where Q ! W !X;A ! . id/Š !X;A is the isomorphism adjoint to ! , and the isomorphism at the bottom is the local analogue of the adjunction isomorphism between . id/Š and . id/ (compare [25, Chapter III, Theorem 6.7]). This construction yields a Cartier sheaf D.F /, which is clearly functorial in F . Let Cohlocfree .X; A/ Coh .X; A/ and Cartlocfree .X; A/ Cart.X; A/ denote the full subcategories formed by those objects whose underlying coherent sheaves are locally free. Proposition 8.2.5. The functor F 7! D.F / induces an anti-equivalence of categories Cohlocfree .X; A/ ! Cartlocfree .X; A/. Proof. The construction F 7! D.F / clearly induces an anti-equivalence from the category of locally free coherent sheaves on X C to itself. Thus for any fixed locally free coherent sheaf F on X C , we must prove that the ˝ id-linear endomorphisms of F correspond bijectively to the Cartier linear endomorphisms of D.F /. But this follows at once from the construction (8.2.4). Remark 8.2.6. It follows from Proposition 8.2.5 that the naive L-function of any F 2 Cohlocfree .X; A/ depends only on D.F /. In fact it can be associated directly to D.F / in a way similar to that in Section 8.1, which is implicit in [2, p. 300, Proposition 12]. We refrain from working out the details here, because we will content ourselves with quoting the main result of [2].
8.3 Anderson’s trace formula The aim of this section is to generalize Anderson’s trace formula [2, Theorem 1] slightly from coefficients in k to coefficients in a k-algebra A. We develop all the definitions and tools necessary to cover this more general situation. However, proofs are given only when they differ significantly from those in [2]. Let V0 be a k-vector space, typically of countably infinite dimension. Set V ´ V0 ˝ A and consider an A-linear operator V W V ! V . Definition 8.3.1. A k-subspace W0 V0 is called a nucleus for V if it is finitedimensional and there exists an exhaustive increasing filtration of V0 by finite dimensional k-vector spaces W0 W1 W2 such that V .WiC1 ˝ A/ Wi ˝ A for all i 1. The following three results are easy consequences of the definition:
8.3 Anderson’s trace formula
127
Proposition 8.3.2. If W0 is a nucleus for V , then for any j 0 the exterior power Vj V V V V W0 jk V0 is a nucleus for j V W Aj V ! Aj V . k Proposition–Definition 8.3.3 ([2, p. 294]). If V possesses a nucleus W0 , the expressions Tr.V / ´ TrA .V jW0 ˝ A/ 2 A;
.1 t V / ´ detA .1 t V jW0 ˝ A/ ˇV X V ˇ D .1/j Tr j V ˇ jk W0 ˝ A t j 2 AŒt j
are independent of W0 . They are called the trace and the dual characteristic polynomial of V , respectively. Proposition 8.3.4 ([2, Proposition 5]). Let V00 V0 be a k-subspace such that V maps V 0 ´ V00 ˝ A to itself. Set V000 ´ V0 =V00 and V 00 ´ V000 ˝ A. If V has a nucleus, then so do the induced operators V 0 on V 0 and V 00 on V 00 , and Tr.V / D Tr.V 0 / C Tr.V 00 /;
.1 t V / D .1 t V 0 / .1 t V 00 /: Remark 8.3.5. The k-structure V0 of V plays a vital role in the above results, in that the trace and the dual characteristic polynomial are taken only with respect to A-submodules coming from subspaces of V0 . Without such an additional structure on V one cannot define nuclei in general, such that the trace and the dual characteristic polynomial are well defined. Next let R be a finitely generated k-algebra. (It takes the rôle of A in [2], while the ring A here is simply Fq there.) In the following we assume that V0 is the k-vector space underlying a finitely generated R-module. To give examples of operators possessing a nucleus, and to deduce the existence of suitable nuclei, Anderson introduces two notions, which are extended to arbitrary A as follows: Definition 8.3.6. Let r1 ; : : : ; rs be generators of R as a k-algebra, and let v1 ; : : : ; v t be generators of V0 as an R-module. For every integer n let V0;n V0 denote the subspace of all elements admitting a representation t X
fj .r1 ; : : : ; rs / vj
j D1
with polynomials fij 2 kŒX1 ; : : : ; Xs of total degree n. Then the function W V ! Z0 [ f1g; is called a gauge on V .
v 7! inf fn j v 2 V0;n ˝ Ag
128
8 Naive L-functions
Definition 8.3.7. An A-linear operator V W V ! V is called of trace class if for every gauge on V there exist constants 0 K1 < 1 and 0 K2 such that .V .v//
K1 .v/ C K2 for all v 2 V . Proposition 8.3.8 ([2, Propositions 3, 6]). Let V be as above. (a) Any composite of an operator on V of trace class with an R ˝ A-linear operator is of trace class. (b) Suppose that V is of trace class with constants K1 , K2 for some gauge . Then V0;n from 8.3.6 is a nucleus for V for any n 1 C K2 =.1 K1 /. (c) Any Cartier linear operator on V is of trace class with K1 D 1=q. The following technical result is crucial in Anderson’s proof of the trace formula. We give a complete proof, because it is different in our setting. Proposition 8.3.9 ([2, Proposition 8]). Let V W V ! V be an A-linear operator of trace class and '0 W V0 ! V0 an R-linear endomorphism. Set ' ´ '0 ˝ idA . Then both V ' and 'V are of trace class, and Tr.V '/ D Tr.'V /. Proof. By Proposition 8.3.8 (a) the operators V ' and 'V are of trace class. Proposition 8.3.8 (b) implies that they have a common nucleus, say W0 . Then W0 C '0 .W0 / S0 ´ .W0 C is another nucleus for 'V . Abbreviate W00 ´ ker.'0 / \ W0 and W '0 .W0 //=W0 . Then we have the following commutative diagram with exact rows: 0
0
/ W0˝A 0 I / W0˝A 0
I
/ W0 ˝ A I
I
'
V '
I$ / W0 ˝ A
/ .W0 C '0 .W0 // ˝ A P P P P ' V
'
/ .W0 C '0 .W0 // ˝ A
/W S0 ˝ A
/0
P P ' /W S0 ˝ A
/ 0.
Since the dashed arrows are zero, the outer vertical arrows must be zero as well. As the trace is additive in short exact sequences, we deduce that Tr.V '/ D TrA .V 'jW0 ˝ A/ D TrA .'V j.W0 C '.W0 // ˝ A/ D Tr.'V /; as desired. Adapting the remainder of Anderson’s article [2] to our setting is straightforward; so we content ourselves with stating the resulting generalization of his main theorem [2, Theorem 1]. As explained in Remark 8.3.5, the Cartier module V requires a k-structure V0 in order for the dual characteristic polynomial to be well defined. Theorem 8.3.10. Let X D Spec R be affine and smooth of equidimension n over k. Let F 2 Coh .X; A/ be such that F D pr 1 F0 for a locally free coherent sheaf F0 on X.Set V0 ´ Hom.F0 ; !X / and V D V0 ˝ A, so that the Cartier module corresponding to D.F / has the form .V; V /. Then V is of trace class and n1
Lnaive .X; F ; t / D .1 t V /.1/ In particular Lnaive .X; F ; t / is a rational function.
:
8.4 A cohomological trace formula
129
8.4 A cohomological trace formula Let X and F with F D pr 1 F0 be as in Theorem 8.3.10, and sX W X ! Spec k the structure morphism of X . The aim of this section is to express the right hand side of 8.3.10 in terms of the naive L-function of a -sheaf on Spec k which represents the complex RsXŠ F in D b .Crys.X; A//. The main reason why this is possible is the Serre duality on a compactification of X . To construct the desired -sheaf we fix a dense open embedding j W X ,! Xx into a normal projective variety over k. Since X is affine, there is an effective ample Cartier divisor Y on Xx whose support is the complement Xx n X . Its ideal sheaf 0 D OXx .Y / OXx is then locally free of rank one. Next, as X is affine, we can realize F0 as a direct summand of a free coherent sheaf on X. Replacing F0 by the direct sum and setting ´ 0 on the complement, we may, and do, assume without loss of generality that F0 D OX˚r . For any integer m 1 we consider the locally free sheaf F0;m ´ .0m /˚r D ˚r OXx .mY / on Xx extending F0 . Then Fm ´ pr 1 F0;m extends F , and by the construction in Proposition 4.5.1 there is an integer m0 such that for any m m0 the homomorphism F W . id/ F ! F extends to a homomorphism Fm W . id/ Fm ! FmC1 Fm :
(8.4.1)
In particular the -sheaf Fm ´ .Fm ; Fm / represents the crystal jŠ F . Note that the extensions Fm are unique, because Xx is normal and Fm locally free. Thus the restriction of Fm to FmC1 coincides with FmC1 , and we have nil-isomorphisms FmC1 ,! Fm . i For any integers i and m m0 consider the coherent cohomology group G0;m ´ i i x i i x H .X; F0;m /. Set Gm ´ G0;m ˝ A Š H .X C; Fm /, and let Gm i denote its x endomorphism induced by Fm . Let sXx W X ! Spec k denote the structure morphism of Xx . Then Proposition 6.4.2 implies that the -sheaf Gmi on Spec k corresponding to i Gm represents the crystal def
i H i .RsXŠ F / D H i .RsX x jŠ F / Š H .RsX x Fm /:
Our main result extends this description to the derived category and relates it with naive L-functions. Since Gmi is free, the naive L-function of Gmi is defined. Theorem 8.4.2. Let X D Spec R be affine and smooth of equidimension n over k and with structure morphism sX W X ! Spec k. Let F 2 Coh .X; A/ be such that F D pr 1 F0 for F0 D OX˚r . Then with the Gmi defined as above, for all m m0 we have (a) Gmi is nilpotent for all i 6D n, and n
(b) Lnaive .X; F ; t / D Lnaive .Spec k; Gmn ; t /.1/ .
130
8 Naive L-functions
The proof of Theorem 8.4.2 extends over the rest of this section. Since it is rather technical, we first sketch its main features in the special case that Xx is smooth and i A D k. Let D.Gm / denote the Cartier module corresponding to the Cartier sheaf i D.Gm / on Spec k. Then Serre duality says that ˚r i D.Gm / D H i .Xx ; Fm /_ Š H ni Xx ; Hom.Fm ; !Xx / Š H ni Xx ; !X x .mY / for all m and i . As Y is ample, this vanishes for all i 6D n and m 0, yielding (a). On the other hand we have ˚r ˚r n 0 D.Gm / Š H 0 Xx ; !X D H 0 .X; D.F //; x .mY / H X; !X n / form an exhaustive increasing filtration of V ´ and so for m ! 1 the D.Gm 0 H .X; D.F // by finite dimensional subspaces. The factorization (8.4.1) implies n n n n / maps D.G that D.Gm mC1 / to D.Gm /, which shows that D.Gm / is nucleus for V ´ H 0 .X; D.F / / for any m m0 . Thus (b) follows from the equalities 8.3.10
n1
Lnaive .X; F ; t / D .1 t V /.1/ n1 8.3.3 n .1/ n / jD.G / D detA 1 t D.Gm m n1 n .1/ n jG D detA 1 t Gm m 8.1.6
n
D Lnaive .Spec k; Gmn ; t /.1/ :
In the general case we deal with the possible non-smoothness of X as follows. Š x Let !Xx ´ sX x OSpec k denote the dualizing complex of X (cf. [25]). Its cohomology i sheaves H .!Xx / are coherent and vanish in degrees i 62 Œn; 0, because Xx has equidimension n. (If Xx is Cohen–Macaulay, they vanish for all i 6D n, but we can avoid that special assumption.) In any case we have j !Xx D sXŠ OSpec k D !X Œn, and so in particular j H i .!Xx / D 0 for all i 6D n. Lemma 8.4.3. For all i and all m 0 there is a natural isomorphism i i .G0;m /_ Š Hom.F0;m ; H i .!Xx // DW W0;m :
Proof. Serre duality induces an isomorphism i .G0;m /_ D H i .Xx ; F0;m /_ Š Exti F0;m ; !Xx : The right hand is described further by the hyper-Ext spectral sequence E2ij D Exti F0;m ; H j .!Xx / H) ExtiCj F0;m ; !Xx : ˚r Since Y is ample, the equality F0;m D OX x .mY / implies that
_ ˝ H j .!Xx / Š H i Xx ; H j .!˚r /.mY / E2ij Š H i Xx ; F0;m x X
8.4 A cohomological trace formula
131
vanishes for all i 6D 0 and m 0. The spectral sequence then degenerates to E20j D Hom F0;m ; H j .!Xx / Š Extj F0;m ; !Xx for all j , which for j D i yields the desired isomorphism. Next, as Gmi lives on Spec k, its associated Cartier sheaf is simply the dual D.Gmi / D i i / D .G0;m /_ ˝ A .Gmi /_ together with the endomorphism dual to H i .Fm /. Let D.Gm with D.Gm i / denote the corresponding Cartier module. Another such endomorphism arises from the equation sXx D sXx . As in Remark 8.2.3, it yields a natural isomorphism !Xx Š Š !Xx . For the inverse image sheaf !X;A ´ pr 1 !Xx we obtain the isomorphism !X;A Š . id/Š !X;A on Xx C , and so x x x ! !X;A . The same definition as by adjunction a homomorphism ! W . id/ !X;A x x in (8.2.4) yields a Cartier linear endomorphism of RHom Fm ; !X;A , denoted D.Fm / , x 0 x i and hence, after applying H X C; H . / , a Cartier linear endomorphism Wmi of i i H 0 Xx C; H i .RHom.Fm ; !X;A x // D Hom.Fm ; H .!X;A x // D W0;m ˝ A: i i /_ ˝ A Š W0;m ˝ A deduced from 8.4.3 Lemma 8.4.4. The isomorphism .G0;m identifies the endomorphisms D.Gm i / and W i . m
Proof. The construction of D.Fm / amounts to the commutativity of the diagram . id/ Fm ˝L . id/ RHom Fm ; !X;A x O
eval
/ . id/ !x
X;A
QFm ˝id
Fm ˝L . id/ RHom Fm ; !X;A x
!
id˝D.Fm /
Fm ˝L RHom Fm ; !X;A x
eval
/ !x . X;A
Taking (hyper)-cohomology we obtain from this the commutative diagram H i .Xx C; Fm / Ext i Fm ; !X;A x / O
[
H i .Fm /id
H i .Xx C; Fm / Ext i Fm ; !X;A x / idHi .D.Fm / /
H i .Xx C; Fm / Ext i Fm ; !X;A x /
[
/ H0 .Xx C; !x / KX;A KK KKtrace KK Š KKK KK % H0 .! / s9 A s ss Š sss s ss ss trace / H0 .Xx C; ! x /. X;A
132
8 Naive L-functions
Here the triangle on the right hand side commutes by the invariance of the trace map under composition. Thus the vertical arrow H0 .! / is the identity. By Serre duality it i _ i i x C; Fm /_ follows that the endomorphism D.Gm i / D H .Fm / of D.Gm / Š H .X corresponds to the endomorphism Hi .D.Fm / / of Exti Fm ; !X;A x / . Finally, under the isomorphism i Exti Fm ; !X;A Š Hom Fm ; H i .!X;A x / Š W0;m ˝ A x obtained from the proof of Lemma 8.4.3 the Hi .D.Fm / / automatically corresponds to Wmi . The lemma follows. Next the equality j H n .!Xx / D !X induces a restriction homomorphism n Resm W W0;m D Hom.F0;m ; H n .!Xx // ! Hom.F0 ; !X / DW V0 :
The construction of Wmi implies that n ˝ A ! V0 ˝ A DW V Resm ˝ idA W W0;m
is compatible with the endomorphisms Wmn and V on the two sides. Lemma 8.4.5. For all i and all m 0 the endomorphism Wmi is zero for i 6D n, and zero on ker.Resm ˝ id/ for i D n. Proof. Since 0 D 0q , the defining property (8.4.1) of Fm0 implies that for any ` 0 the homomorphism Fm0 C` factors through the inclusion Fm0 Cq` ,! Fm0 C` . factors through the natural map Therefore W i m0 C`
i W0;m ˝A 0 C` o
Hom.Fm0 C` ; H
/ Wi 0;m
0 Cq`
˝A
o i
.!X;A x //
/ Hom.Fm0 Cq` ; H i .!x //. X;A
In suitable local coordinates this map is simply multiplication by 0.q1/` . For i ¤ n we have j H i .!Xx / D 0; hence the coherent sheaf H i .!Xx / is supported on Y . It is therefore annihilated by 0 .q1/`0 for some `0 . Thus for all m m0 C `0 the endomorphism Wmi vanishes, proving the first assertion. For i D n we apply the same argument to the kernel of the natural homomorphism H n .!Xx / ! j !X . We find that Wmn annihilates all elements with support on Y for all m m0 C `0 , proving the second assertion. Proof Theorem 8.4.2 (a). Lemmas 8.4.3, 8.4.4, and 8.4.5 imply that D.Gmi / D 0 for all i 6D n and m 0. Thus the corresponding Gmi are nilpotent. Since for i 6D n fixed all Gmi represent the same crystal, they are nilpotent for all m m0 .
8.4 A cohomological trace formula
133
n Lemma 8.4.6. For any m m0 the subspace Resm .W0;m / V0 is a nucleus for V .
Proof. Clearly the n D Hom.F0;m ; H n .!Xx // Š H 0 Xx ; H n .!˚r W0;m x /.mY / X are finite dimensional for all m, and the union of their images under Resm is V0 . On n the other hand, for all m m0 the factorization (8.4.1) implies that WmC1 factors n through W0;m ˝ A, which shows that n n V Resm .W0;mC1 / ˝ A Resm .W0;m / ˝ A: n / satisfy the conditions of Definition 8.3.1. Thus all Resm .W0;m
Proof Theorem 8.4.2 (b). For any m 0 we have a commutative diagram n / Resm .W0;m /˝A n n n
n W0;m ˝A
/ V0 ˝ A
V n wn n n n / Resm .W0;m W0;m ˝A /˝A W n
m
V
/ V0 ˝ A,
where the dashed arrow is induced by Lemma 8.4.5. Using Lemma 8.4.6 and combining everything, including the fact that the characteristic polynomial of an endomorphism is equal to that of its dual, we deduce that n1
8.3.10
Lnaive .X; F ; t / D .1 t V /.1/ ˇ .1/n1 8.3.3 n D detA 1 t V ˇ Resm .W0;m /˝A 8.4.5
n1
8.4.4
n1
D detA .1 t Wmn /.1/ .1/ n / D detA .1 t Gm
n
8.1.6
D Lnaive .Spec k; Gmn ; t /.1/ :
This proves Theorem 8.4.2 (b) for all m 0. For m m0 the factorization (8.4.1) yields a commutative diagram n GmC1
n Gm
G n
n / GmC1 MMM MMM MMM MM& G n m / Gn . m mC1
n / D detA .1 t G n By linear algebra this implies that detA .1 t Gm /, proving mC1 Theorem 8.4.2 (b) for all m m0 by descending induction on m.
134
8 Naive L-functions
Remark 8.4.7. Theorem 8.4.2 is probably true more generally whenever X is affine and Cohen–Macaulay of equidimension n over k, because then j !Xx is again concentrated in degree n (see [25, Chapter III, §1, §7]). The proof in this case should remain basically the same; however, Anderson established Theorem 8.3.10 only in the smooth case. Remark 8.4.8. A formula as in Theorem 8.4.2 can probably also be deduced from the Woodshole fixed point formula for coherent sheaves, as follows. First, by devissage using finite morphisms and fibrations by curves one reduces the problem to the case X D A1 . Next, as in [SGA4 12 , Rapport, Fonction L mod `n et mod p], the formula can then be reduced to a trace formula for the exterior alternating powers of F over the symmetric powers of A1 . Again by induction on dimension, it suffices to prove this trace formula for X D A1 . When A is a field, the desired trace formula should follow from the Woodshole fixed point formula for coherent sheaves [SGA5, Exposé III, Corollaire 6.12]. For general A one has two options. On the one hand, it is likely that the Woodshole formula can be generalized to coefficients in A using [SGA5, Exposé III, Théorème 6.10], much as we have generalized Anderson’s trace formula in Theorem 8.3.10. On the other hand, one can first deduce it for integral domains A by passing to the quotient field, using Proposition 8.1.9. For arbitrary A one can lift F to a -sheaf over kŒT1 ; : : : ; Tn for a suitable homomorphism kŒT1 ; : : : ; Tn ! A. Then again Proposition 8.1.9 reduces the formula to a case already treated. Although the alternative method just sketched has its benefits, following Anderson’s approach seemed more natural in our setting, besides being more elementary. Also, the devissage can be performed better for the crystalline L-functions developed in the following chapter.
8.5 An extended example In this section we work out the content of Theorem 8.4.2 in some explicit cases of dimension n D 1. Throughout we let X ´ A1k D Spec kŒz be the affine line over k. ˚r and consider a homomorphism F W . id/ F ! F . Let B We take F ´ OXC be the .r r/-matrix with coefficients in kŒz ˝ A that represents F . Local terms. Recall that any closed point x 2 jX j corresponds to a maximal ideal of kŒz. We denote the surjection to its residue field by kŒz kx ;
f 7! f .x/:
For example, the image z.x/ of z is a generator of kx over k. Likewise, we write kŒz ˝ A kx ˝ A;
f 7! f .x/
135
8.5 An extended example
for the natural homomorphism and extend this notation componentwise to vector and i matrix valued polynomials. We also abbreviate f .x q / ´ . i ˝ id/.f .x//. Then by Definition 8.1.6 we have 2 dx 1 1 / : Lnaive .x; ix F ; t / D det kx ˝A 1 t dx B.x/B.x q /B.x q / : : : B.x q In the special case r D 1 this simplifies to 1 Lnaive .x; ix F ; t / D 1 t dx Normkx ˝A=k˝A .B.x// :
(8.5.1)
The global term. We use the notation of Section 8.4. Let m 1 be an integer such that all coefficients of B are polynomials of degree < .q 1/m in z. Consider the natural open embedding j W X ,! Xx ´ Pk1 , and let 0 Š OXx .1/ be the ideal sheaf of the point 1. Then F0;m ´ .0m /˚r extends F0 ´ OX˚r and Fm ´ pr 1 F0;m extends F . Since 0n has local generator z m at 1, and .z m / D z q m , the condition on m is precisely enough for F to extend to a homomorphism Fm satisfying (8.4.1). 1 ´ H 1 .Xx ; F0;m / we consider To calculate the coherent cohomology group G0;m the short exact sequence of quasi-coherent sheaves 0 ! 0m ! j OX ! .j OX /=0m ! 0: x m / D H 1 .Xx ; j OX / D 0, the associated long exact cohomology Since H 0 .X; 0 sequence yields a short exact sequence / H 0 .X; OX /
0
/ .j OX /= m 0 1
/ H 1 .Xx ; m / 0
/0
kŒz; z 1 =kŒz 1 z m
kŒz and thus an isomorphism
! H 1 .Xx ; 0m /: V0;m ´ kŒz; z 1 = kŒz ˚ kŒz 1 z m ˚r 1 1 To determine the endomorphism Gm 1 of Gm ´ G0;m ˝ A Š V0;m ˝ A induced by Fm consider the commutative diagram with exact rows
0
/ . id/ Fm
/ . id/ .j id/ F
/ . id/ .j id/ F =Fm
/0
0
/ Fq m
/ .j id/ F
/ .j id/ F =Fq m
/0
B
0
/ Fm
B
/ .j id/ F
B
/ .j id/ F =Fm
/ 0.
136
8 Naive L-functions
The boundary homomorphisms in cohomology yield a commutative diagram ˚r ˝A V0;m ˝id
/ H 1 .Xx C; Fm /
1 Gm
˝id
˚r V0;q m B
˝A
˚r V0;m ˝A
/ H 1 .Xx C; Fq m / B
/ H 1 .Xx C; Fm /
1 Gm
1 . Gm
˚r Thus the endomorphism of V0;m ˝A corresponding to Gm 1 is obtained simply by raising all terms in V0;m to the power q, multiplying the resulting vector by the matrix B, and then reducing the product modulo kŒz ˚ kŒz 1 z m . Let us make this even more explicit in the case m D 2. Here V0;m D k ‘z 1 ’ ˚r 1 has dimension 1, and so Gm Š V0;m ˝ A Š A˚r ‘z 1 ’. On the other hand we have P2q3 B D iD0 Bi z i for .r r/-matrices Bi with coefficients in A. Thus Gm 1 corresponds to the endomorphism of A˚r ‘z 1 ’ that is given by
w ‘z 1 ’ 7! B w ‘z q ’ Bq1 w ‘z 1 ’: In other words, Gm 1 can be represented by the matrix Bq1 , and so the right hand side in Theorem 8.4.2 (b) is equal to (8.5.2) Lnaive .Spec k; Gmn ; t /1 D detA id tBq1 : 2 A special case. Now we assume moreover P thati A D kŒ"=." / and r D 1 and B D 1 C b.z/" for some polynomial b.z/ D i bi z 2 kŒz.
Local terms. The fact that "2 D 0 implies that Normkx ˝A=k˝A 1 C b.x/" D 1 C Tr kx =k .b.x//": Here the trace term has the following interpretation in terms of Frobenius. Consider the curve Y A2k defined by the equation y q y D b.z/. It is an étale Galois covering of X D A1k with Galois group k D Fq , where ˛ 2 k acts by .y; z/ 7! .y C ˛; z/. For any point x 2 jX j choose an element yx of an overfield of kx that satisfies yxq yx D b.x/. Then the Frobenius at x maps yx to yxq
dx
, and so the image of Frobenius ˛x 2 k in the
Galois group of Y over X is characterized by the equation yxq
dx
D yx C ˛x . Therefore
dx
˛x D yxq yx D .yxq yx / C .yxq yx /q C C .yxq yx /q D b.x/ C b.x q / C C b.x q
dx 1
dx 1
/ D Tr kx =k .b.x//:
Thus in this case (8.5.1) simplifies to 1 Lnaive .x; ix F ; t / D 1 t dx .1 C ˛x "/ :
(8.5.3)
8.5 An extended example
137
The global term. Suppose first that deg b.z/ 2q 3. Then we can take m D 2, and since Bq1 D bq1 " in this case, the formula (8.5.2) yields Lnaive .Spec k; Gmn ; t /1 D 1 t bq1 ": Thus Theorem 8.4.2 (b) takes the explicit form Y 1 Lnaive .X; F ; t / D 1 t dx .1 C ˛x "/ D 1 t bq1 ":
(8.5.4)
(8.5.5)
x2jXj
Deligne’s example. Finally, we consider the case B D 1 C z 3 " with q D 2, which corresponds to Deligne’s example [SGA4 12 , p. 127f]. In this case the equation y 2 y D z 3 shows that Y is an affine chart of a supersingular elliptic curve in characteristic 2. On the other hand the inequality deg.B/ < .q 1/m leaves m D 4 as the smallest possible choice. For this value V0;m has dimension 3 and Gm 1 corresponds to the map ‘z 1 ’ 7! .1 C z 3 "/ ‘z 2 ’ ‘z 2 ’; ‘z 2 ’ 7! .1 C z 3 "/ ‘z 4 ’ ‘z 1 ’ "; ‘z 3 ’ 7! .1 C z 3 "/ ‘z 6 ’ ‘z 3 ’ ": It is therefore represented by the matrix over A 0 1 0 " 0 @1 0 0 A 0 0 " whose dual characteristic polynomial is .1 t "/.1 t 2 "/ D 1 C t " C t 2 ": Thus in this case Theorem 8.4.2 (b) takes the form Y 1 Lnaive .X; F ; t / D D 1 C t " C t 2 ": 1 t dx .1 C ˛x "/ x2jXj
(8.5.6)
Chapter 9
Crystalline L-functions
As in the preceding chapter, we work only over schemes X of finite type over k. The aim of this chapter is to attach natural L-functions to flat A-crystals on X and to complexes thereof, and to study their properties. Among the main properties are the rationality, the invariance under RfŠ , and the invariance under change of coefficients. Of course, these L-functions must be defined by recourse to the underlying sheaves, but they must be invariant under nil-isomorphisms. It is quite non-trivial to define them at all, because flat A-crystals cannot in general be represented by coherent -sheaves whose underlying sheaves are locally free, and even when they can, there are obstructions to uniqueness. We succeed in giving a reasonably direct definition only when A is artinian. For certain other ‘good coefficient rings’, in particular for normal integral domains, we show that the L-function defined over their quotient ring actually has coefficients in A. When A is reduced, these L-functions satisfy all the usual cohomological formulas precisely (except duality). When A possesses non-zero nilpotent elements, however, these formulas hold only up to ‘unipotent’ factors. In some sense these factors correspond to nilpotent -sheaves and can therefore not be detected by our theory. Unfortunately, this defect is an unavoidable consequence of the very definition of crystals. We begin this chapter with two sections on polynomials in 1 C tAŒt . Section 9.1 on characteristic polynomials is mostly of technical nature, to be used in connection with L-functions for general A in Sections 9.7 and 9.8. Throughout Sections 9.2 to 9.6 we assume that A is artinian (for example finite or a field). Then any finitely generated A-module M with an endomorphism possesses a unique (primary) decomposition Mss ˚ Mnil , where is nilpotent on Mnil and induces an automorphism on Mss . The corresponding decomposition of the dual characteristic polynomial of is discussed in Section 9.2 on the level of polynomials only. In Section 9.3 we define the local L-factor at x of a flat crystal F in terms of the dual characteristic polynomial of on Mss , where M is the stalk of F at x. In Sections 9.4 and 9.5 the definition is extended to global L-functions of flat crystals and of complexes of finite Tor-dimension. We also compare these L-functions to their naive counterparts from Section 8.1, where possible. In Section 9.6 we investigate their functorial behavior. In particular we prove their invariance under RfŠ , up to unipotent factors, and thus the rationality of these Lfunctions. Here we apply the trace formula for naive L-functions from Chapter 8. The proof is rather technical, because we must deal with several obstructions to representing a flat crystal by a -sheaf whose underlying sheaf is locally free. In Section 9.7 we extend the definition of L-functions to good coefficient rings, and in Section 9.8 we prove their invariance under change of coefficients, again up to unipotent factors. We keep the notation kx and dx for x 2 jX j from the preceding chapter.
9.1 Characteristic polynomials
139
9.1 Characteristic polynomials In this section A denotes an arbitrary unitary commutative ring. We are interested in polynomials and power series over A in one variable t and with constant coefficient 1. The set 1 C tAŒŒt of such power series is a group under multiplication. The subgroup formed by all quotients of polynomials with constant coefficient 1 is called the group of rational functions with constant coefficient 1. Such functions typically arise as the dual characteristic polynomial detA .1 ‰t/ of an endomorphism ‰ of a finitely generated projective A-module. Let nA denote the nilradical of A, i.e., the ideal of all its nilpotent elements. Definition 9.1.1. A polynomial P .t/ 2 1 C tAŒt is called unipotent if its coefficients of t i for all i 1 lie in nA . Proposition 9.1.2. For any nilpotent endomorphism ‰ of a finitely generated projective A-module, the polynomial P .t/ ´ detA .1 ‰t/ is unipotent. Proof. Consider a prime ideal p A with residue field kp , and let Pp denote the image of P in 1 C t kp Œt . Then Pp is the dual characteristic polynomial of a nilpotent endomorphism of a finite dimensional kp -vector space. Thus Pp D 1, in other words, P 1 mod p. Varying p we deduce that P 1 mod nA , so that P is unipotent, as desired. Proposition 9.1.3. The set of unipotent polynomials 1 C t nA Œt is a subgroup of 1 C tAŒŒt. In particular, the inverse of any unipotent polynomial is again a unipotent polynomial. Proof. It suffices to prove the Psecond assertion. So consider a unipotent polynomial P . Then Q.t/ ´ 1 P .t/ D rD1 a t with a 2 nA . Choose an integer n 1 such n that a D 0 for all . Then Qj D 0 for all j nr. Thus the geometric series P 1 P D .1 Q/1 D 1 C j1D1 Qj is really finite; hence it defines a polynomial, as desired. This allows us to define the following equivalence relation on 1 C tAŒŒt : Definition 9.1.4. Two power series P .t/, Q.t / 2 1 C tAŒŒt are equivalent, and we write P Q, if their quotient P =Q is a unipotent polynomial. Note that 1 is the only unipotent polynomial if A is reduced. So in this case we have P Q if and only if P D Q. The remainder of this section deals with dual characteristic polynomials. The basic observation is that the characteristic polynomial of the i -th power of an endomorphism depends only on i and the characteristic polynomial of the original endomorphism. For Pr this let P be any polynomial with constant coefficient 1. Write P .t / D D0 a t
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9 Crystalline L-functions
for some r 0, and let
ˆP;r
0 0 B B B1 B B B ´ B0 B B B B @
ar
0
C C C C C C C C C a2 C C A a1
0
0
0
1
1
be its dual companion matrix of size r r. It satisfies detA .1 ˆP;r t / D P .t /. i t / 2 1 C tAŒt depends Proposition 9.1.5. For i 1 the polynomial detA .1 ˆP;r only on P , but not on r.
Proof. The equality
0
i ˆP;rC1
0 D@
0
0
i ˆP;r
1 A
i i implies that detA .1 ˆP;rC1 t / D detA .1 ˆP;r t /, and the general case follows by induction.
By the above result the following definition makes sense: Definition 9.1.6. The i -th iterate of a polynomial P .t / 2 1 C tAŒt is defined as i P .i/ .t/ ´ detA .1 ˆP;r t / for any r 0. Next for any ring homomorphism W A ! A0 , we denote the induced homomorphism AŒt ! A0 Œt again by . The definition of P .i/ implies at once: Proposition 9.1.7. .P .i/ / D .P /.i/ . Proposition 9.1.8. If P .t/ D detA .1 ‰t/ for an endomorphism ‰ of a finitely generated projective A-module M , then P .i/ .t / D detA .1 ‰ i t /. Proof. By Definition 8.1.1 of the dual characteristic polynomial, we may assume that M is free of finite rank r. Then ‰ can be represented by an r r-matrix over A. This matrix can be obtained by specialization from a matrix with indeterminate coefficients in the polynomial ring Az ´ ZŒx j1 ; r, so by Proposition 9.1.7 it suffices z After passing to the algebraic closure of the quotient to prove the assertion over A. z field of A, it suffices to prove the equality over an algebraically closed field K. Now by assumption ‰ and ˆP;r are matrices over K with the same dual characteristic polynomial. As K is an algebraically closed field, this means precisely that i their semisimple parts are conjugate. Thus the semisimple parts of ‰ i and ˆP;r are conjugate, and so they have the same dual characteristic polynomial, as desired.
9.1 Characteristic polynomials
141
Proposition 9.1.9. .PQ/.i/ D P .i/ Q.i/ . Proof. Suppose that P has degree r and Q has degree s, and let ‰ be the endomorphism of ArCs represented by the block matrix 0 @
ˆP;r
0
0
ˆQ;s
1 A:
Then the definition of P .i/ and Q.i/ shows that detA .1 ‰ i t / D P .i/ .t / Q.i/ .t /. In particular we have detA .1 ‰t/ D P .t/ Q.t /, which by Proposition 9.1.8 implies that detA .1 ‰ i t / D .PQ/.i/ .t /. Next we discuss to what extent a polynomial can be recovered from its higher iterates. For this we must express the iterates in terms of symmetric polynomials. Fix r and let x1 ; : : : ; xr be indeterminates over Z. For i 1 we define symmetric polynomials s.i/ 2 ZŒx1 ; : : : ; xr by r Y
.1 .x /i t / D
D1
r X
s.i/ t :
D0
Clearly s0.i/ D 1, and s ´ s.1/ is the usual -th elementary symmetric polynomial in x1 ; : : : ; xr . In particular any of the polynomials s.i/ can be written in a unique way as a polynomial S.i/ .s1 ; : : : ; sr / in the variables s1 ; : : : ; sr . P Lemma 9.1.10. For any P .t/ D rD0 a t 2 1 C tAŒt we have P .i/ .t / D
r X
S.i/ .a1 ; : : : ; ar /t :
D0
Proof. By Proposition 9.1.7 it suffices to show this when the coefficients of P .t / are independent indeterminates over Z. Since the elementary symmetric polynomials s 2 ZŒx1 ; : : : ; xr are algebraically independent, it suffices to prove it in the case a D s . In this case the equality follows by applying Proposition 9.1.8 to the diagonal matrix with eigenvalues x1 ; : : : ; xr . Lemma 9.1.11. For any j 1, any symmetric polynomial f in the subring Z xi j 1 r; i j ZŒx1 ; : : : ; xr can be written as a polynomial in Z s.i/ j 1 r; i j :
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9 Crystalline L-functions
Proof. It suffices to prove this when f is the symmetrization of a single monomial x1i1 : : : xrir , i.e., the sum of all its distinct conjugates under the symmetric group Sr . We will achieve this by induction on the number of indices with i 6D 0. Without loss of generality we may suppose that the sequence of exponents is monotone decreasing. If all i D 0, the assertion is obvious. Otherwise let 0 s < t r be such that i1 is > isC1 D D i t > i tC1 D D 0. Then the assertion holds already for the symmetrization g of x1i1 : : : xsis . On the other hand we have i 0 ´ i t j , 0 .i 0 / .i 0 / i0 : : : x ti is s ts . Consider the product g s ts , which and the symmetrization of xsC1 i1 is is the sum of all products of one of the distinct Sr -conjugates of x1 : : : xs with one 0 i0 : : : x ti . The sum of all those terms where no of the distinct Sr -conjugates of xsC1 variable occurs in both factors is precisely the polynomial f . All the remaining terms are monomials in at most t 1 of the variables x . By the induction hypothesis, the sum of these terms is already a polynomial in the s.i/ for all 1 r and i j . As .i 0 / is also such a polynomial, the same follows for f , as desired. g s ts Pr Lemma 9.1.12. Let P .t/ D 2 1 C tAŒt , and for any i 1 write D0 a t P .i/ r .i/ P .t/ D D0 a t . Then (a) ar.i/ D .1/r.iC1/ ari , and (b) for any 1 r and j 1, the element a arj can be expressed as a universal .i/ polynomial in the a for all 1 r and i j . Proof. By Lemma 9.1.10 it suffices to prove the analogous assertions (a0 ) sr.i/ D .1/r.iC1/ sri , and (b0 ) s srj 2 Z s.i/ j 1 r; i j . Part (a0 ) follows directly from the definition of sr.i/ . As s srj is the symmetrization of j x1j C1 : : : xj C1 xC1 : : : xrj , part (b0 ) follows from Lemma 9.1.11. Theorem 9.1.13. For any polynomials P .t/, Q.t / 2 1 C tAŒt whose highest non-zero coefficients are units, the following statements are equivalent: (a) P .i/ D Q.i/ for all i 0. (b) P D Q. Proof. The implication (b))(a) is obvious. For the converse we set r ´ deg P and use the notation of Lemma 9.1.12. First 9.1.12 (a) implies that deg P .i/ D r for all i 1. Thus r is determined by the P .i/ for all i 0. Next 9.1.12 (a) shows that all ar.i/ are units and that ar D .1/r ar.iC1/ =ar.i/ . Thus ar can be reconstructed from the P .i/ for all i 0. Thirdly, by 9.1.12 (b) the same also follows for all other coefficients a of P . Altogether we find that P is completely determined by the P .i/ for all i 0. Applying the same arguments to Q, we deduce that (a) implies (b), as desired.
9.2 A primary decomposition for rational functions
143
The same argument also shows the following variant: Proposition 9.1.14. Let A A0 be an inclusion of fields, and let P 2 1 C tA0 Œt be such that P .i/ 2 1 C tAŒt for all i 0. Then P 2 1 C tAŒt . Finally, we have the following characterization of unipotent polynomials: Proposition 9.1.15. A polynomial P .t/ 2 1CtAŒt is unipotent if and only if P .i/ D 1 for all i 0. P Proof. Suppose first that P .t/ D rD0 a t is unipotent. Then by Lemma 9.1.10 we must show that S.i/ .a1 ; : : : ; ar / D 0 for all 1 r and all i 0. For this choose an n D 0 for all 1 r. Then any monomial of degree nr integer n 1 such that a in a1 ; : : : ; ar vanishes. Now by construction S.i/ .s1 ; : : : ; sr / is an isobaric polynomial of weight i , where each s has weight . Thus it is composed of monomials whose usual degree is i =r. For i nr 2 = it follows that S.i/ .a1 ; : : : ; ar / D 0, as desired. Conversely assume that P .i/ D 1 for i 0. Consider a prime ideal p A with residue field kp , and let Pp denote the image of P in 1 C t kp Œt . Then Proposition 9.1.7 shows that Pp.i/ D 1 for all i 0, which by Theorem 9.1.13 implies that Pp D 1. Thus P 1 mod p. Varying p we deduce that P 1 mod nA , so that P is unipotent, as desired.
9.2 A primary decomposition for rational functions From here on, until the end of Section 9.6, we make the following assumption: Assumption 9.2.1. A is artinian. Thus A is a finite direct sum of local artinian rings A1 ˚ ˚ An . For any power series P .t/ 2 1 C tAŒŒt we let Pj denote the component in 1 C tAj ŒŒt . Recall that a polynomial in 1 C tAŒt is called unipotent if all its non-constant coefficients lie in nA . Definition 9.2.2. A polynomial P .t/ 2 1 C tAŒt is called semisimple if the highest non-zero coefficient of each Pj is a unit in Aj . Semisimple polynomials typically arise as follows: Proposition 9.2.3. For any automorphism ‰ of a finitely generated projective A-module, the polynomial detA .1 ‰t/ is semisimple. Proof. Without loss of generality A is local, in which case detA .1 ‰t/ is the dual characteristic polynomial of an invertible matrix. The following decomposition is motivated by the primary decomposition of endomorphisms:
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9 Crystalline L-functions
Proposition 9.2.4. For any polynomial P .t/ 2 1CtAŒt there exist a unique semisimple polynomial Pss and a unique unipotent polynomial Punip in 1 C tAŒt , such that P D Pss Punip . Proof. Without loss of generality A is local, in which case the assertion is a special case of Hensel’s Lemma. Note that in the case nA D 0 every polynomial is semisimple, and so P D Pss and Punip D 1. Next we generalize the decomposition to rational functions. Definition 9.2.5. A rational function with constant coefficient 1 is called semisimple if it is a quotient of semisimple polynomials in 1 C tAŒt . Proposition 9.2.6. A polynomial is semisimple if and only if it is semisimple as a rational function. Proof. Only the ‘if’ part requires a proof. So consider a polynomial P which is semisimple as a rational function. Then P D Q=R for semisimple polynomials Q and R. This means that Q D P R, and so Qj D Pj Rj , where the highest non-zero coefficients of Qj and Rj are units in Aj . The same then follows for Pj , so that P is semisimple, as desired. Proposition 9.2.7. For any rational function P with constant coefficient 1 there exist a unique semisimple rational function Pss with constant coefficient 1 and a unique unipotent polynomial Punip 2 1 C tAŒt , such that P D Pss Punip . Proof. Recall from Proposition 9.1.3 that any quotient of unipotent polynomials is again a unipotent polynomial. Thus the existence follows from Proposition 9.2.4 by taking quotients. For the uniqueness suppose that Ps Pu D Qs Qu for semisimple rational functions Ps , Qs and unipotent polynomials Pu , Qu . Then R ´ Ps =Qs D Qu =Pu is at the same time a semisimple rational function and a unipotent polynomial. Being a polynomial, it is then a semisimple polynomial by Proposition 9.2.6. Being both semisimple and unipotent, it must then be 1. Thus Ps D Qs and Qu D Pu , as desired. The factors Pss and Punip are called the semisimple part, respectively the unipotent part of P . Proposition 9.2.8. We have .PQ/ss D Pss Qss and .PQ/unip D Punip Qunip . Proof. Clearly a product of semisimple polynomials is semisimple, and a product of unipotent polynomials is unipotent. Thus a product of semisimple rational functions is semisimple. Writing P Q D Pss Punip Qss Qunip D .Pss Qss / .Punip Qunip /, we can therefore deduce the desired equalities from the uniqueness part of Proposition 9.2.7. Corollary 9.2.9. For any two rational functions P , Q with constant coefficient 1 we have P Q if and only if Pss D Qss .
9.3 The local L-factor
145
Proof. By Definition 9.1.4 we have P Q if and only if P =Q is a unipotent polynomial. Using Propositions 9.2.8 and 9.2.7 this is equivalent to Pss =Qss D .P =Q/ss D 1, and thus to Pss D Qss , as desired. The decomposition is also functorial in A: Proposition 9.2.10. Let W A ! A0 be a homomorphism of artinian rings, and let again denote the induced homomorphism 1 C tAŒŒt ! 1 C tA0 ŒŒt . Then .Pss / D .P /ss and .Punip / D .P /unip . Proof. Using Proposition 9.2.7, it suffices to show that preserves semisimplicity and unipotence. For unipotence this is obvious. For semisimplicity we first project everything to a local direct summand of A0 , after which we may assume that A0 is local. Then factors through the projection from A to one of its local direct summands; hence we may also assume that A is local. Then is a local homomorphism of local artinian rings, so it preserves units and hence semisimplicity, as desired. Finally, the results of the preceding section imply: Theorem 9.2.11. For any polynomials P .t/, Q.t / 2 1CtAŒt the following statements are equivalent: (a) P .i/ D Q.i/ for all i 0. (b) Pss D Qss . If nA D 0, they are also equivalent to: (c) P D Q. Proof. It suffices to prove this whenever A is local. By Proposition 9.1.9, assertion (a) .i/ .i/ .i/ is equivalent to Pss.i/ Punip D Qss Qunip for all i 0. Proposition 9.1.15 in turn .i/ .i/ shows that this is equivalent to Pss D Qss for all i 0. By Theorem 9.1.13 this is equivalent to assertion (b), proving (a),(b). If nA D 0, then P D Pss and Q D Qss ; this implies (b),(c).
9.3 The local L-factor In this section we let x D Spec kx for a finite field extension kx =k of degree dx . Our goal is to define an L-function for every flat crystal on x. Naturally, this L-function will be defined in terms of an underlying coherent -sheaf, but we must take care to make it invariant under nil-isomorphisms. As shown by the following basic example, the naive L-function defined in Section 8.1 does not share this property. A more subtle example is given in Example 9.7.2.
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9 Crystalline L-functions
Example 9.3.1. Let x D Spec k and suppose that A possesses a non-zero nilpotent element a. Let F be the -sheaf with underlying module A and multiplication by a as endomorphism . Then F is nilpotent, i.e., nil-isomorphic to the zero -sheaf. But its naive L-function is .1 at /1 ¤ 1, while the naive L-function of the zero -sheaf is 1. Let us emphasize again that until the end of Section 9.6 we assume that A is artinian. Under this hypothesis we will now represent any flat crystal on x by a canonical sheaf whose underlying sheaf is locally free. The L-function of a flat crystal on x is then defined to be the naive L-function of this representative. The definition of local L-factors for general A is postponed until Section 9.7. Recall that a -sheaf F is nilpotent if and only if there exists a positive integer n such that Fn D 0. Definition 9.3.2. A -sheaf F is called semisimple if F W . id/ F ! F is an isomorphism. Proposition 9.3.3. Consider F , G 2 Coh .x; A/. (a) There exists a unique direct sum decomposition F D F ss ˚ F nil such that F ss is semisimple and F nil is nilpotent. The summands are called the semisimple part and the nilpotent part of F , respectively. (b) The decomposition in (a) is functorial in F . (c) Any nil-isomorphism F ! G induces an isomorphism F ss ! Gss . (d) The construction induces a functor Crys.x; A/ ! Coh .x; A/ W F 7! F ss : Proof. Note first that Fdx is an endomorphism F ! F , because dx is the identity on kx . Moreover F has finite length, because kx ˝ A is artinian. Thus for n 0 the subsheaves Fss ´ im Fndx and Fnil ´ ker Fndx are independent of n. Clearly F maps them to themselves, so they define -subsheaves F ss and F nil . By construction Fss is surjective and Fnil nilpotent. Since F and hence Fss has finite length, Fss is then also injective. Thus F ss is semisimple and F nil nilpotent. Furthermore, the construction yields a split short exact sequence 0
/ Fnil
/F [
Fss .
ndx
F
/F oo7 ss o o ooo Š
/0
This shows that F D F ss ˚ F nil , proving the existence part of (a). The uniqueness follows from the fact that any semisimple -subsheaf of F is contained in F ss and any nilpotent -subsheaf of F is contained in F nil .
9.3 The local L-factor
147
Part (b) follows directly from the above construction of Fss and Fnil . Also (b) implies that the kernel and cokernel of any homomorphism F ss ! Gss have trivial nilpotent part. This implies (c) and hence (d). Proposition 9.3.4. Consider F 2 Crys.x; A/. (a) F ss is the unique semisimple -sheaf representing F . (b) The functor Crys.x; A/ ! Coh .x; A/ W F 7! F ss is exact. (c) F is flat if and only if the sheaf Fss underlying F ss is locally free. Proof. (a) follows from Proposition 9.3.3, especially from 9.3.3 (c). For the other assertions recall the functor F 7! Fy ´ lim . n id/ F from (3.3.12). Over x !n ! F ss , because the Fss are isomorphisms. Since F ss Š we clearly have F ss Fy , we obtain a functorial isomorphism F ss Š Fy . Thus (b) and (c) follow from Proposition 3.4.8 and Theorem 7.3.5 (a),(d), respectively.
b
b
Definition 9.3.5. The (crystalline) L-function of a flat crystal F on x is Lcrys .x; F ; t / ´ Lnaive .x; F ss ; t / 2 1 C t dx AŒŒt dx : Proposition 9.3.6. Let sx W x ! Spec k denote the structure morphism of x. Then for any F 2 Crys.x; A/ we have Lcrys .x; F ; t / D Lcrys .Spec k; sx F ; t /: Proof. The construction of Fss in Proposition 9.3.3 implies that .sx F /ss D sx .F ss /. Thus the assertion follows from Proposition 8.1.7. Proposition 9.3.7. For any F 2 Coh .x; A/ whose underlying sheaf is locally free, we have Lcrys .x; F ; t / D Lnaive .x; F ; t /ss : Proof. If F D Fss ˚ Fnil is locally free, so are its direct summands. Thus Lnaive .x; F ; t / D Lnaive .x; F ss ; t / Lnaive .x; F nil ; t /: It suffices to show that the first factor is semisimple and the second unipotent. For this we may reduce ourselves to the case that A is local. Since Fss is an isomorphism, its determinant is a unit, and so its dual characteristic polynomial Lnaive .x; F ss ; t /1 is semisimple. To show that Lnaive .x; F nil ; t /1 is unipotent, after dividing by nA we may assume that A is a field. Then all eigenvalues of Fnil are zero, so its dual characteristic polynomial is 1, as desired.
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9 Crystalline L-functions
In Sections 7.3 and 7.5 we have studied the problem of representing a crystal over a global base by a -sheaf whose underlying sheaf is locally free. In spite of several obstructions, we have found a sufficient substitute in the notion of free 0 representative from Definition 7.5.4. For the study of global L-functions we will need to relate these with local L-factors, generalizing Proposition 9.3.7 in a uniform way. For this, the following lemmas will be useful. The first is an adaptation of Proposition 9.3.3. Lemma 9.3.8. Assume that .G ; .Gi /ii0 ; '/ is a free 0 -representative of F 2 Crys.x; A/. Then there exists a unique decomposition G D Gss ˚ Gnil invariant under all Gi , such that (a) Gi induces an isomorphism on Gss for all i i0 , and (b) Gi vanishes on Gnil for all i i0 length.G /. Moreover, ' induces an isomorphism Fss ! Gss . Proof. The decomposition is constructed as in the proof of Proposition 9.3.3 by setting Gss ´ im Gndx and Gnil ´ ker Gndx for any n 0. Then the desired properties follow as in 9.3.3, except that the Gi are only nilpotent on Gnil . To obtain the precise estimate (b) we write i D i1 C ni0 with i0 i1 < 2i0 and consider the decreasing sequence Gnil Gi1 .Gnil / Gi1 Ci0 .Gnil / Gi1 C2i0 .Gnil / : Since both Gi1 and Gi0 are nilpotent on Gnil , this sequence is strictly decreasing until it reaches zero. Thus it reaches zero in at most length.Gnil / steps. But if i D i1 Cni0 i0 length.G /, then nC1 length.G / length.Gnil /; hence Gi .Gnil / D Gi1 Cni0 .Gnil / D 0, as desired. Lemma 9.3.9. The length over kx ˝ A of any coherent sheaf G on x C is bounded by a constant depending only on k, A, and the number of generators of G . Proof. After replacing A by a direct summand we may assume that A is local artinian. Let m denote its maximal ideal. Since A is a localization of a finitely generated kalgebra, its residue field A=m is a finitely generated field extension of k. Thus the algebraic closure of k in A=m is finite, say of degree d over k. Since kx is a finite separable extension of k, it follows that kx ˝ A=m is a direct sum of at most d fields. If e denotes the length of A over itself, we deduce that kx ˝ A has length ed , independent of x. Thus if G has r generators, its length is red .
9.4 The global L-function Let now F be a flat crystal on a scheme X of finite type over k. Since Lcrys .x; F ; t / 2 1 C t dx AŒŒt dx for any x 2 jX j, and the number of points of any given degree dx is finite, we can form the product over all x within 1 C tAŒŒt :
9.4 The global L-function
149
Definition 9.4.1. The (crystalline) L-function of a flat crystal F on X is Y Lcrys .x; ix F ; t / 2 1 C tAŒŒt : Lcrys .X; F ; t / ´ x2jXj
To study the crystalline L-function we will need to relate it with naive L-functions. This in turn requires locally free representatives and a uniformity statement for the local L-factors. Proposition 9.4.2. Assume that .G ; .Gi /ii0 / is a free 0 -representative of F 2 Crys.X; A/. Let Gx and Gi x denote the restrictions to x 2 jX j. Then for all x 2 jX j with dx i0 we have ˇ 1 ; Lcrys .x; ix F ; t / det kx ˝A id t dx Gdxx ˇ Gx i.e., the quotient is a unipotent polynomial. Moreover, we have equality for almost all x. Proof. For any x 2 jX j with dx i0 let Gx D Gx;ss ˚ Gx;nil be the decomposition furnished by Lemma 9.3.8. Then the definition of the local L-factor shows that Lcrys .x; ix F ; t / D Lnaive .x; .ix F /ss ; t / ˇ 1 8.1.6 D det kx ˝A id t dx idxF ˇ .ix F /ss x ˇ 1 9.3.8 D det k ˝A id t dx dx ˇ Gx;ss : 9.3.5
x
Gx
Thus by the multiplicativity of determinants the quotient of the two sides in the proposition is the unipotent polynomial ˇ detkx ˝A id t dx Gdxx ˇ Gx;nil : It remains to prove that this polynomial is 1 for almost all x. For this note that the number of local generators of G is bounded, because G is coherent and X C is noetherian. Thus by Lemma 9.3.9 the length of Gx is bounded independently of x, say by an integer n. Then Lemma 9.3.8 (b) says that Gdxx vanishes on Gx;nil whenever dx i0 n, in which case the above polynomial is 1. As the number of points x 2 jX j with dx < i0 n is finite, we are done. Corollary 9.4.3. Let F be a coherent -sheaf on X whose underlying sheaf is locally free. Then Lcrys .X; F ; t / Lnaive .X; F ; t /; i.e., their quotient is a unipotent polynomial. Moreover, we have equality if A is reduced. Proof. If F is locally free, Proposition 9.4.2 applies to G D F and Gi D Fi for all i 1, showing that Lcrys .x; ix F ; t / Lnaive .x; ix F ; t / for all x 2 jX j, with equality for almost all x. Thus the first assertion follows by taking the product. The second assertion follows from the first and the fact that 1 is the only unipotent polynomial if A is reduced.
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9 Crystalline L-functions
9.5 The L-function of a complex Next we pass to the derived category.
Definition 9.5.1. The (crystalline) L-function of F 2 C b .Crysflat .X; A// is Y i Lcrys .X; F i ; t /.1/ 2 1 C tAŒŒt : Lcrys .X; F ; t / ´ i2Z
Lemma 9.5.2. If F is acyclic, then Lcrys .X; F ; t / D 1. Q Proof. By definition we have Lcrys .X; F ; t / D x2jX j Lcrys .x; ix F ; t /, and the exactness of ix implies that ix F is again acyclic. Thus it suffices to prove the assertion in the case that X D x D Spec kx , where kx is finite. Next, since F ! F ss is an exact functor by Proposition 9.3.4 (b), the complex of -sheaves F ss is again acyclic. Moreover, as the crystals F i are flat, the underlying coherent sheaves Fssi are locally free by Proposition 9.3.4 (c). Thus Lemma 8.1.2 implies that Y i Lnaive .x; F iss ; t /.1/ D 1; Lcrys .x; F ; t / D i2Z
as desired. Lemma 9.5.3. For any short exact sequence 0 ! F C b .Crysflat .X; A// we have
! G ! H ! 0 in
Lcrys .X; G ; t / D Lcrys .X; F ; t / Lcrys .X; H ; t /:
Proof. By the definition of the L-function it suffices to prove this in any single degree. But for a short exact sequence in Crysflat .X; A/ the assertion is simply a special case of Lemma 9.5.2. Lemma 9.5.4. If F , G 2 C b .Crysflat .X; A// are related by a quasi-isomorphism ' W F ! G , we have Lcrys .X; F ; t / D Lcrys .X; G ; t /.
Proof. The mapping cone yields a short exact sequence 0 ! G ! Cone ' ! F Œ1 ! 0 in C b .Crysflat .X; A//. Since Cone ' is acyclic, Lemmas 9.5.2 and 9.5.3 imply that 1 D Lcrys .X; Cone '; t / D Lcrys .X; G ; t / Lcrys .X; F Œ1; t / D Lcrys .X; G ; t / Lcrys .X; F ; t /1 ;
as desired.
Proposition 9.5.5. (a) The L-function Lcrys .X; F ; t / depends only on the isomorphy class of F in D b .Crysflat .X; A//. (b) For any distinguished triangle F ! G ! H ! F Œ1 in D b .Crysflat .X; A// we have Lcrys .X; G ; t / D Lcrys .X; F ; t / Lcrys .X; H ; t /:
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151
Proof. Any isomorphism F Š G in D b .Crysflat .X; A// can be represented by a diagram F H ! G of quasi-isomorphisms in C b .Crysflat .X; A//. Thus (a) follows from Lemma 9.5.4. For (b) recall that the distinguished triangles in D b .Crysflat .X; A// are precisely those that are isomorphic to triangles arising from cones. Thus by (a) it suffices to consider the triangle associated to a short exact sequence of complexes. In this case the assertion reduces to Lemma 9.5.3.
Remark 9.5.6. Via the equivalence of categories ! D b .Crys.X; A//ftd D b .Crysflat .X; A//
from Theorem 7.6.3 we can now extend the definition of Lcrys .X; F ; t / to all F 2 D b .Crys.X; A//ftd . Proposition 9.5.5 again holds in this larger category.
9.6 Functoriality In this section we determine the behavior of L-functions under our various functors. We begin with three easy results. Proposition 9.6.1. Let i W Y ,! X be a closed immersion with open complement j W U ,! X .
(a) For any F 2 D b .Crys.X; A//ftd we have Lcrys .X; F ; t / D Lcrys .Y; i F ; t / Lcrys .U; j F ; t /:
(b) For any F 2 D b .Crys.Y; A//ftd we have
Lcrys .Y; F ; t / D Lcrys .X; i F ; t /:
(c) For any F 2 D b .Crys.U; A//ftd we have
Lcrys .U; F ; t / D Lcrys .X; jŠ F ; t /: Proof. Direct consequence of the pointwise definition of the L-function and the defining properties of i , j , i , and jŠ .
Proposition 9.6.2. For any finite morphism f W Y ! X and any F contained in D b .Crys.Y; A//ftd we have
Lcrys .Y; F ; t / D Lcrys .X; f F ; t /:
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Proof. It suffices to show that the local L-factor of f F at any point x 2 jX j is equal to the product of the local L-factors of F at all points y 2 f 1 .x/. Thus after taking the inverse image via ix W x ,! X , we may assume that X D x D Spec kx for some finite field kx . Then Y is finite, and after passing to its reduced subscheme using Theorem 4.4.7 we may also assume that Y is reduced. Next f F is a direct sum of contributions from all points on Y . Thus by the multiplicativity of dual characteristic polynomials it suffices to prove the equality when Y D y D Spec ky for some finite field ky . In that case the two structure morphisms sx W x ! Spec k and sy W y ! Spec k satisfy sy D sx f , so the assertion follows by applying Proposition 9.3.6 twice:
Lcrys .y; F ; t / D Lcrys .Spec k; sy F ; t / D Lcrys .x; f F ; t /: Proposition 9.6.3. Suppose that f W Y ! X is finite radicial and surjective. Then for any F 2 D b .Crys.X; A//ftd we have Lcrys .X; F ; t / D Lcrys .Y; f F ; t /:
Proof. By assumption f induces a bijection on points and the residue field extensions are totally inseparable. Since the residue fields at closed points are finite, they are isomorphic. Thus the equality is a direct consequence of the pointwise definition of the L-function. Our main result is this: Theorem 9.6.4. Let sX W X ! Spec k denote the structure morphism of X . Then for any complex F 2 D b .Crys.X; A//ftd we have
Lcrys .X; F ; t / Lcrys .Spec k; RsXŠ F ; t /;
i.e., their quotient is a unipotent polynomial. In particular Lcrys .X; F ; t / is a rational function (though not necessarily semisimple). Before proving it we deduce a relative version. Theorem 9.6.5. For any morphism f W Y ! X of schemes of finite type over k and any F 2 D b .Crys.Y; A//ftd we have
Lcrys .Y; F ; t / Lcrys .X; RfŠ F ; t /; i.e., their quotient is a unipotent polynomial.
Proof. Since RsY Š F D RsXŠ RfŠ F by Theorem 6.7.5, on applying Theorem 9.6.4 twice we find that
Lcrys .Y; F ; t / Lcrys .Spec k; RsY Š F ; t / Lcrys .X; RfŠ F ; t /; as desired. For reduced A the only unipotent polynomial is 1, so in this case we have:
9.6 Functoriality
153
Corollary 9.6.6. If A is reduced, then for any morphism f W Y ! X of schemes of finite type over k and any F 2 D b .Crys.Y; A//ftd we have
Lcrys .Y; F ; t / D Lcrys .X; RfŠ F ; t /: Remark 9.6.7. In general we do not have equality in Theorem 9.6.5. An easy example is provided by Section 8.5. Take A D kŒ"=."2 /, X D Spec kŒz, F D OXC and F D multiplication by 1 C z q1 ". Then F is an isomorphism everywhere; hence .ix F /ss D ix F at all points x 2 jX j, and so Lcrys .X; F ; t / D Lnaive .X; F ; t /
(8.5.5)
D
1 t ":
On the other hand the calculation in Section 8.5 shows that Ri sXŠ F is nilpotent for every i . Thus Lcrys .Spec k; RsXŠ F ; t / D 1 6D 1 t ". Another example is Deligne’s (8.5.6). The rest of this section is devoted to proving Theorem 9.6.4. Note that Theorem 9.6.4 is multiplicative in distinguished triangles, by Proposition 9.5.5 (b) and because the derived functor RsXŠ preserves distinguished triangles. Thus we can perform noetherian induction on X . In particular we can achieve the following reductions. Lemma 9.6.8. It suffices to prove Theorem 9.6.4 in the case that (a) X is affine, smooth, and irreducible,
(b) F is a flat crystal F concentrated in degree 0, and (c) the sheaf underlying the canonical representative Fy from (3.3.12) is flat. Moreover, we may further replace X by any non-empty open subscheme. j
i
Proof. Let U ,! X - Y be any complementary open and closed embeddings with U non-empty. Then the distinguished triangle jŠ j F ! F ! i i F from 6.6.3 (d) reduces the theorem for F to the theorem for jŠ j F and i i F . For the latter the theorem follows from Proposition 9.6.1 (b), the relation RsXŠ i D R.sX i /Š , and noetherian induction on X . For the former we may replace X by U and F by j F , using Proposition 9.6.1 (c) and the relation RsXŠ jŠ D R.sX j /Š . In this way we may replace X by any non-empty open subscheme. On the other hand, after passing to the reduced subscheme of X using Theorem 4.4.7, we may assume that X is reduced. Combining this with the preceding reduction, we can therefore achieve (a). Next, by Theorem 7.6.3 we may assume that F is a bounded complex of flat crystals on X . Thus the reduction (b) follows directly from the multiplicativity in distinguished triangles. Finally, the reduction (c) follows from Corollary 7.3.8. Lemma 9.6.9. Theorem 9.6.4 is true in the case dim X 1.
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9 Crystalline L-functions
Proof. For X finite the theorem is a direct consequence of Proposition 9.6.2. So let X and F be as in Lemma 9.6.8 with dim X D 1. Shrinking X , by Theorem 7.5.9 we may assume that the coherent sheaf underlying F is free. Then by Theorem 8.4.2 there exists G 2 Coh .Spec k; A/ whose underlying sheaf is free, such that RsXŠ F Š G Œ1 in D b .Crys.Spec k; A//, and Lnaive .X; F ; t / D Lnaive .Spec k; G ; t /1 : Applying Corollary 9.4.3 to both sides of this equality shows that Lcrys .X; F ; t / Lcrys .Spec k; G ; t /1 : On the other hand, by Proposition 9.5.5 the isomorphism RsXŠ F Š G Œ1 implies that Lcrys .Spec k; G ; t /1 D Lcrys .Spec k; sXŠ F ; t /; The desired formula follows. Now we tackle the general case, using devissage by a fibration. This is not a formal consequence of the preceding results, because it requires a uniformity statement for the crystalline local L-factors similar to Proposition 9.4.2. Let X and F be as in Lemma 9.6.8 with n ´ dim X 2. We begin by setting up the necessary geometry. First we choose a separable dominant morphism f W X ! Z with Z integral and 0 < dim Z < n (for example, with Z D A1k or An1 ). Since X k is smooth, so is the generic fiber of f . Thus by the open nature of smoothness, after shrinking X we may assume that f is smooth of relative dimension n0 ´ n dim Z. Next, since X is affine, f is quasi-projective, i.e., we may choose a commutative diagram j / x X X {{ { { f {{ }{{{ fN Z, where j is an open dense embedding and fN is projective. Thirdly, after replacing Xx by its normalization we may assume that Xx is normal. Then the generic fiber of fN is normal; hence by the constructibility of normality all fibers over some open dense subset V Z are normal. After replacing everything by its restriction to V we may thus assume that all fibers of fN are normal. Furthermore, since X is affine, there exists an effective ample Cartier divisor Y on Xx whose support is the complement Xx n X . We let 0 D OXx .Y / denote its ideal sheaf. Next, we look at the -sheaf F . By the condition 9.6.8 (c) and by Theorem 7.5.5 we may fix a free 0 -representative .G ; .Gi /ii0 / (compare Definition 7.5.4). We identify G D pr 1 G0 for G0 ´ OX˚r . For any integer m 1 we consider the locally free sheaf G0;m ´ .0m /˚r on Xx extending G0 . Then Gm ´ pr 1 G0;m is a locally free coherent sheaf extending G .
9.6 Functoriality
155
Lemma 9.6.10. There is an integer m0 such that for any m m0 and any i i0 the homomorphism Gi W . id/ G ! G extends to a homomorphism Gi m W . id/ Gm ! GmC1 Gm : Moreover .Gm ; .Gi m /ii0 / is locally on X a free 0 -representative of jŠ F . Proof. For any fixed i the construction of Gi m proceeds exactly like that of Proposition 4.5.1. Thus we can find an m0 such that Gi m exists for all m m0 and each of the finitely many values i0 i < 2i0 . For i 2i0 we then obtain the desired 0 extensions by setting inductively Gi m ´ Gi0m B . i0 id/ Gii , thereby proving the m x first statement of the lemma. As X is normal, these extensions are actually unique. They therefore satisfy the relations Gi m B . i id/ Gjm D GiCj from 7.5.4 for all i , m j i0 , and the second statement follows. j In the rest of the proof we keep m m0 fixed. For any integer j let H0;m ´ j N th R f G0;m on Z denote the j derived direct image in quasi-coherent cohomology. Since fN is projective, this sheaf is coherent, so after shrinking Z and replacj ing everything by its restriction we may, and do, assume that H0;m is free. Set j j i Hm ´ pr 1 H0;m on Z C , and for any i i0 let j denote its semilinear
j Hm
Hm
j Hm
endomorphism . id/ ! induced by Recall that by definition Rj fŠ F D Rj fN jŠ F . Thus Proposition 6.4.2 and Lemma 9.6.10 imply: i
j ; . i Lemma 9.6.11. .Hm
j
Hm
Gi m .
/ii0 / is a free 0 -representative of Rj fŠ F .
Next, we consider fibers. For any point z 2 jZj consider the pullback diagram XO
? Xz
j
/ Xx O ? / Xx z
fN
/Z O i
? z / z.
Then Xz is smooth of equidimension n0 . For any coherent sheaf or i -sheaf we will denote by . /z the restriction to the fiber above z. After shrinking Z and replacing everything by its restriction we may assume that dz i0 for all z 2 jZj. Then we can j speak of the dz -sheaves Gz .dz / D .Gz ; Gdzz / on Xz , and Hm;z
.dz /
j D .Hm;z ; dzj / Hm;z
on z. In each of these cases the underlying scheme is defined over the finite field kz of order q dz ; hence all our results on -sheaves extend immediately to these dz -sheaves. j Moreover, since all H0;m are free, base change for coherent cohomology shows that j Hm;z is the sheaf on z C associated to the cohomology group H j .Xxz ; G0;m;z / ˝ A. In particular, we can therefore apply Theorem 8.4.2 to Xz and the dz -sheaves Gz .dz / j and Hm;z
.dz /
in place of X , F , Gmi .
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9 Crystalline L-functions
Lemma 9.6.12. For any j 6D n0 we have Rj fŠ F D 0 in Crys.Z; A/. j j as inverse image implies that Hm;z Proof. Lemma 9.6.11 and the definition of Hm;z
.dz /
represents the crystal iz Rj fŠ F .dz / . By Theorem 8.4.2 (a) the former is nilpotent for all j 6D n0 . Thus iz Rj fŠ F vanishes in Crys.z; A/. Varying z, Corollary 4.6.3 then shows that Rj fŠ F vanishes in Crys.Z; A/. Having collected the necessary prerequisites, we can now do the calculation. First, for all z 2 jZj, Theorem 8.4.2 (b) implies that Y ˇ 1 def Lnaive .Xz ; Gz .dz / ; t / det kx ˝A id t dx Gdxx ˇ Gx x2jXz j
8.4.2 0
n Lnaive .z; Hm;z
.dz /
; t /.1/
n0
def
ˇ n0 .1/n0 C1 det kz ˝A id t dz dzn0 ˇ Hm;z . Hm;z
(9.6.13) Next we will take the product over all z 2 jZj. For this note that the factors in the upper right corner are precisely those in Proposition 9.4.2. By Lemma 9.6.11, the lower right 0 j corner appears in Proposition 9.4.2 applied to Z, Rn fŠ F , .Hm ; . i j /ii0 / in place Hm
of X, F , .G ; .Gi /ii0 /. In the product we therefore obtain Y 9.4.1 Lcrys .x; ix F ; t / Lcrys .X; F ; t / D x2jXj 9.4.2
Y
ˇ 1 det kx ˝A id t dx Gdxx ˇ Gx
x2jXj (9.6.13)
D
Y
ˇ n0 .1/n0 C1 det kz ˝A id t dz dzn0 ˇ Hm;z Hm;z
z2jZj 9.4.2
Y
0
Lcrys .z; iz Rn fŠ F ; t /.1/
n0
z2jZj 9.6.12
D
Y
Lcrys .z; iz RfŠ F ; t /
z2jZj 9.4.1
D Lcrys .Z; RfŠ F ; t /:
Finally, for the structure morphisms sX W X ! Spec k and sZ W Z ! Spec k we have RsXŠ D RsZŠ RfŠ by Theorem 6.7.5. Since Theorem 9.6.4 is true for Z by the induction hypothesis, we deduce that Lcrys .X; F ; t / Lcrys .Z; RfŠ F ; t / Lcrys .Spec k; RsZŠ RfŠ F ; t / D Lcrys .Spec k; RsXŠ F ; t /: This finishes the proof of Theorem 9.6.4.
9.7 Arbitrary coefficients
157
9.7 Arbitrary coefficients For the remaining two sections we drop the assumption that A is artinian. We let p1 ; : : : ; pn denote the minimal primes of A and call QA ´ Ap1 ˚ ˚ Apn the quotient ring of A. To any coherent -sheaf whose underlying sheaf is locally free we have associated a naive L-function with coefficients in A. We will now discuss conditions on A which guarantee the existence of a crystalline L-function with coefficients in A for all flat A-crystals. Apart from being well defined, our main requirement will be the functoriality under F 7! F ˝A QA . The problems involved are illustrated by the following examples. Example 9.7.1. Let y be an indeterminate and A ´ kŒy 2 ; y 3 kŒy. Let F be the -sheaf on Spec k over A corresponding to the module kŒy with the endomorphism W m 7! y m. Then the crystal associated to F is flat, but cannot be represented by a coherent -sheaf whose underlying sheaf is locally free. Moreover, the (naive and crystalline) L-function of F ˝A QA does not have coefficients in A. Proof. For i 2 the endomorphism i W m 7! y i m maps kŒy into the submodule A. It follows that the iterate F .i/ is nil-isomorphic to its i -subsheaf corresponding to the submodule A kŒy. Thus by Theorems 7.5.5 and 7.3.5 the crystal F is flat. Assume that F can be represented by a coherent -sheaf G whose underlying sheaf is locally free. Then Lnaive x; G ˝ QA ; t D Lnaive .x; G ; t / D detA .id t G jG /1 A
has coefficients in A. On the other hand, since QA D k.y/ is a field, the naive L-function of a -sheaf over QA is invariant under nil-isomorphisms. Thus Lnaive x; G ˝ QA ; t D Lnaive x; F ˝ QA ; t D .1 ty/1 A
A
with y 62 A, yielding a contradiction. Example 9.7.2. Take A ´ kŒy ˚ kŒy" with "2 D 0, and note that A ! QA D k.y/ ˚ k.y/" is injective. Let F be the -sheaf on Spec k over A corresponding to the module kŒy ˚ kŒy y" where is multiplication by y C y" . Then q is multiplication by .y C y" /q D y q , which maps the module into its submodule A. As in Example 9.7.1 1 this implies that the crystal F is flat. The L-function of F ˝A QA is 1 t .y C y" / , which does not have coefficients in A. Example 9.7.3. Take A ´ kŒy; z=.z 2 ; yz/. Then the kernel of A ! QA D k.y/ is the ideal .z/ and its image is kŒy. For any ˛ 2 k let F˛ be the -sheaf on Spec k over A corresponding to the module A where is multiplication by y C ˛z. Since
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9 Crystalline L-functions
y C ˛z annihilates z, the projection map A kŒy induces a nil-isomorphism, where on kŒy is multiplication by y. It follows that all F˛ are nil-isomorphic to each other. 1 On the other hand the naive L-function Lnaive .Spec k; F ˛ ; t / D 1 t .y C ˛z/ depends on ˛. Thus one cannot associate to F˛ a non-trivial natural L-function with coefficients in A that is invariant under nil-isomorphisms. The preceding examples show that, in order to define a natural L-function with coefficients in A for all flat A-crystals, we must impose some conditions on A. Definition 9.7.4. We call A a good coefficient ring if (a) the natural homomorphism A ! QA is injective, and (b) A is closed under any taking finite ring extensions A ,! A0 ,! QA which induce bijections Spec A0 ! Spec A and isomorphisms on all residue fields. Condition (a) means that A has no embedded associated primes; in the literature it is often referred to as the condition S1 of Serre. It allows us to identify A with its image in QA . Clearly any normal integral domain is a good coefficient ring. Also, if A is artinian, it is equal to QA and hence good. The rings kŒx; y=.y 2 x 2 .1Cx// and kŒx; xy; y q are examples of good integral domains that are not normal. The rings A in Examples 9.7.1, 9.7.2 and 9.7.3 are not good. When A is a good coefficient ring, we will define the L-function of a flat A-crystal by working in QA and showing a posteriori that its coefficients lie in A. For this we use the results of Section 9.1 on iterates of polynomials. We first study the local L-factors over x D Spec kx for a finite extension kx of k. Consider any flat F 2 Crys.x; A/. By Theorems 7.3.5 and 7.5.5 it possesses a free 0 -representative .G ; .Gi /ii0 /. We are interested in the dual characteristic polynomials of Gi . To relate them with crystalline local L-factors consider any algebra homomorphism W A ! A0 with A0 artinian. Let also denote the induced homomorphism AŒt ! A0 Œt . Recall from Proposition 7.4.1 (c) that the crystal F ˝A A0 is again flat; hence its L-function is defined. Lemma 9.7.5. In the above situation consider the polynomial 1 P .t/ ´ Lcrys x; F ˝ A0 ; t 2 1 C tA0 Œt : A
Then for all i 0 we have P .i/ .t / D detA .id t Gi j G / 2 1 C t .A/Œt : Proof. Since ˇ detA .id t Gi j G / D detA0 id t Gi ˝ id ˇ G ˝ A0 ; A
9.7 Arbitrary coefficients
159
after replacing F by F ˝A A0 we may assume that A0 D A and is the identity. Then A is artinian, and the definition of local L-factors says that ˇ P .t/ D detA id t F ˇ Fss : Thus Proposition 9.1.8 and Lemma 9.3.8 imply that P .i/ .t /
detA id t Fi 9.3.8 D detA id t Gi 9.3.8 (b) D detA id t Gi 9.1.8
D
ˇ ˇ Fss ˇ ˇ Gss ˇ ˇG
for all i 0, as desired. Lemma 9.7.6. In the above situation, if A is a good coefficient ring, then 1 P .t/ ´ Lcrys x; F ˝ QA ; t 2 1 C tQA Œt A
has in fact coefficients in A. Proof. Let A0 QA be the A-subalgebra generated by the coefficients of P . We must prove that A0 D A. Note that Lemma 9.7.5 applied to the homomorphism A ! QA shows that P .i/ .t / has coefficients in A for all i 0. r r r We first apply this with i D q r for r 0. Since P .q / .t q / D P .t /q , the r coefficients of P .q / are obtained by raising the coefficients of P to the q r -th power. r Thus all coefficients a of P satisfy aq 2 A. In particular they are integral over A, and since there are only finitely many, this implies that A0 is a finite extension of A. r Moreover, we find that A0q A, which shows that the morphism C 0 ´ Spec A0 ! Spec A D C is bijective. Next, let c 0 2 C 0 be the unique point above a point c 2 C , and let kc kc 0 be the inclusion of their residue fields. Let c 0 W A0 ! kc 0 denote the natural homomorphism. Since P .i/ has coefficients in A, Proposition 9.1.7 implies that c 0 .P /.i/ D c 0 .P .i/ / has coefficients in kc , for all i 0. Thus Proposition 9.1.14 shows that c 0 .P / has coefficients in kc . But by the construction of A0 these coefficients generate kc 0 over kc ; hence kc 0 D kc . Now the condition 9.7.4 (b) implies that A0 D A, as desired. Now we can put everything together.
Theorem 9.7.7. If A is a good coefficient ring, then for any F 2 D b .Crys.X; A//ftd the L-function Lcrys .X; F ˝AL QA ; t / has coefficients in A.
Proof. Note that the L-function is defined, because F ˝AL QA is again of finite Tordimension by Proposition 7.6.6. By the multiplicativity of L-functions Theorem 7.6.3 reduces the assertion to the case of a single flat crystal. It also reduces to the case of a local L-factor treated in Lemma 9.7.6. Using this we can now make the following definition:
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9 Crystalline L-functions
Definition 9.7.8. If A is a good coefficient ring, the (crystalline) L-function of F 2 D b .Crys.X; A//ftd is Lcrys .X; F ; t / ´ Lcrys X; F ˝L QA ; t 2 1 C tAŒŒt : A
By the following lemma, all the functoriality results in the preceding sections remain valid in this greater generality: Lemma 9.7.9. Suppose that A is good, and let P and Q be rational functions over A. Then P Q in 1 C tAŒŒt if and only if P Q in 1 C tQA ŒŒt . Proof. Since A ! QA is injective, an element of A is nilpotent, resp. zero, if and only if its image in QA is nilpotent, resp. zero. It follows that P =Q 2 1 C tAŒŒt is a unipotent polynomial if and only if its image in 1 C tQA ŒŒt is a unipotent polynomial. By the definition of the lemma follows.
9.8 Change of coefficients In this last section we study the behavior of L-functions under change of coefficients. Let W A ! A0 be a homomorphism of algebras that are localizations of finitely generated k-algebras. The induced homomorphism 1 C tAŒŒt ! 1 C tA0 ŒŒt is again denoted by . If is the restriction of a homomorphism QA ! QA0 , the desired formulas reduce at once to the case that both A and A0 are artinian. But the general case is somewhat more involved. We begin with the special case where X D x is a point. Lemma 9.8.1. If both A and A0 are good coefficient rings, then we have Lcrys x; F ˝ A0 ; t Lcrys .x; F ; t / A
for any F 2 Crysflat .x; A/. If moreover A is artinian, then equality holds. Proof. Note that both L-functions are defined, because F ˝A A0 is again flat by Proposition 7.4.1 (c). Set P .t/ ´ Lcrys .x; F ; t /1 2 1 C tAŒt ; P 0 .t / ´ Lcrys .x; F ˝A A0 ; t /1 2 1 C tA0 Œt : Let .G ; .Gi /ii0 / be a free 0 -representative of F . By base change to A0 we obtain a free 0 -representative of F ˝A A0 . Thus, applying Lemma 9.7.5 independently shows that for all i 0 we have P .i/ .t / D detA .id t Gi j G /; P 0.i/ .t / D detA .id t Gi ˝ id j G ˝A A0 /:
9.8 Change of coefficients
161
This immediately implies that P 0.i/ D .P .i/ /. On the other hand by Proposition 9.1.7 we have .P .i/ / D .P /.i/ . Thus for all i 0 we have P 0.i/ D .P /.i/ within 1 C tA0 Œt. Applying Theorem 9.2.11 over QA0 now implies that the semisimple parts of P 0 and .P / over QA0 coincide; in other words that P 0 .P / over QA0 . Using Lemma 9.7.9 we conclude that P 0 .P / over A0 , as desired. If A is artinian, after passing from A0 to QA0 we may assume that A0 is artinian as well. Since base change preserves isomorphisms, the -sheaf F ss ˝A A0 is then again semisimple and hence equal to .F ˝A A0 /ss . The equality follows directly from this and Definition 9.3.5. Alternatively, one can use the fact that both P and P 0 are semisimple polynomials, that therefore .P / is semisimple by Proposition 9.2.10, and that thus P 0 .P / implies P 0 D .P /. Theorem 9.8.2. If both A and A0 are good coefficient rings, then for any F D b .Crys.X; A//ftd we have Lcrys X; F ˝L A0 ; t Lcrys .X; F ; t / ;
2
A
i.e., the quotient is a unipotent polynomial. If moreover A is artinian, then equality holds. Proof. It suffices to prove this for a single crystal F 2 Crysflat .X; A/, and by Proposition 9.6.3 we may assume that X is reduced. Using Proposition 9.6.1 (a) and Corollary 7.3.8 we can reduce ourselves to the case that the canonical representative Fy of F is flat and that X is affine and regular. Fix a free 0 -representative .G ; .Gi /ii0 / of F furnished by Theorem 7.5.5. Then by base change to A0 we obtain a free 0 -representative of F ˝A A0 . Thus applying Proposition 9.4.2 twice implies that for almost all x 2 jX j ˇ 1 9.4.2 Lcrys x; ix .F ˝ A0 /; t D detkx ˝A id t dx Gdxx ˝ id ˇ Gx ˝ A0 A A ˇ 1 dx dx ˇ D detkx ˝A .id t Gx Gx / 9.4.2 crys D L .x; ix F ; t / : For the finitely many remaining x 2 jX j, Lemma 9.8.1 yields Lcrys x; ix .F ˝ A0 /; t Lcrys .x; ix F ; t / ; A
and equality if A is artinian. Taking products the theorem follows. Since in a reduced ring implies D, we deduce: Corollary 9.8.3. If both A and A0 are good coefficient rings, and A0 is reduced, then for any F 2 D b .Crys.X; A//ftd we have Lcrys X; F ˝L A0 ; t D Lcrys .X; F ; t / : A
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9 Crystalline L-functions
Finally, we note that in Theorem 9.8.2 we do not have equality in general. The reason for this is that some non-trivial part of a local L-factor over A may become nilpotent over A0 without becoming trivial. But this factor is eliminated in the Lfunction over A0 , because nilpotent -sheaves yield trivial crystals. This problem is an unavoidable consequence of our very definition of crystals. It is illustrated by the following example. Example 9.8.4. Let A ´ kŒy be the polynomial ring in one variable over k. Let F be the -sheaf on Spec k over A corresponding to the module kŒy with the endomorphism W m 7! y m. Let W A A0 ´ kŒy=.y 2 / be the projection map. Then F ˝A A0 is nilpotent and hence Lcrys .Spec k; F ˝A A0 ; t / D 1, while Lcrys .Spec k; F ; t / D .1 ty/1 has image 6D 1 under .
Chapter 10
Étale cohomology
Throughout this chapter we assume that A is a finite k-algebra. In this case we establish a natural equivalence of categories between IndCrys.X; A/ and the category Ét.X; A/ of étale sheaves of A-modules on X, under which Crys.X; A/ corresponds to the full subcategory Étc .X; A/ of constructible sheaves. We show that our functors on indcrystals correspond to the usual functors on étale sheaves, and the same for their derived functors. We also translate our results on L-functions to the étale setting. The relevant functor "N W IndCrys.X; A/ ! Ét.X; A/ is defined in Section 10.1 by generalizing the Artin–Schreier theory described, for instance, in [SGA4 12 , Fonctions L mod `n et mod p, § 3]. In Section 10.2 we show that "N commutes with the usual functors; this fact is used as a main tool in Section 10.3 to prove the indicated equivalences of categories. In Sections 10.4 and 10.5 we extend everything to the respective derived categories, to flat objects, and to complexes of finite Tor-dimension. In Section 10.6 we assume that X is a scheme of finite type over k and show that the crystalline L-function of a complex in D b .Crys.X; A//ftd coincides with the L-function of its image in D b .Étc .X; A//ftd . Thus all our results from Chapter 9 translate into results for the L-functions of étale sheaves. In particular we obtain a slight generalization of a trace formula first proved by Deligne [SGA4 12 , Fonctions L mod `n et mod p]. The results of this chapter should be compared with recent results by Emerton and Kisin [17], [18]. They construct a Riemann–Hilbert correspondence between the category of (formal) étale Zp -sheaves on a variety in characteristic p and a suitable category of quasi-coherent analogues of unit-F -crystals, and prove a trace formula for such objects. This generalizes our Theorem 10.6.5 from finite coefficients to formal coefficients in a complete noetherian Zp -algebra. In order to distinguish étale sheaves from crystals we will denote objects of Ét.X; A/ by the letters F, G, etc. The tensor product over A is denoted F ˝ G.
10.1 Basic Definitions To every -sheaf F D .F ; F / over A on X we will associate an étale sheaf of A-modules on X as follows. First, for any étale morphism u W U ! X we set FeK t .U / ´ .u pr 1 F /.U /:
(10.1.1)
By étale descent of quasi-coherent sheaves this defines a sheaf on the small étale site of X, and it is a sheaf of OeK t ˝ A-modules, where OeK t denotes the étale structure sheaf
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10 Étale cohomology
OeK t .U / ´ OU .U /. Next the adjoint of F induces a homomorphism .u pr 1 F /.U / ! .u pr 1 . id/ F /.U / D .u pr 1 F /.U / ! . u pr 1 F /.U / D .u pr 1 F /.U /: Varying u this defines a . ˝ id/-linear endomorphism of the étale sheaf FeK t , which for simplicity we again denote by . Now we forget the action of OeK t and define an étale sheaf of A-modules on X by (10.1.2) ".F / ´ ker id W FeK t ! FeK t : Clearly this construction is functorial in F , that is, to any homomorphism ' W F ! G it associates a homomorphism ".'/ W ".F / ! ".G /. Thus it defines an A-linear functor " W QCoh .X; A/ ! Ét.X; A/:
(10.1.3)
Its restriction to IndCoh .X; A/ will be denoted by the same symbol. Since the functors pr 1 , u , and ker used in the construction are left exact and commute with filtered direct limits, we deduce that " has the same properties. Example 10.1.4. Recall from Example 4.2.4 that the -sheaf 1lX;A consists of the structure sheaf OXC with its obvious . Thus .1lX;A /eK t D OeK t ˝ A with .u ˝ a/ ´ uq ˝ a; and therefore ".1lX;A / Š AX , the constant sheaf on X with stalk A. In the same way one deduces ".M ˝A 1lX;A / Š M ˝ AX for any finitely generated A-module M . One easily shows that this construction translates into a natural isomorphism ".F /.U / Š HomQCoh .1l U;A ; u F /:
(10.1.5)
Proposition 10.1.6. The natural monomorphism ".ind.F // ,! ".F / is an isomorphism. Proof. Since 1l U;A is a coherent -sheaf, any homomorphism 1l U;A ! u F factors through ind.u F / u F . But this -subsheaf is equal to u .ind.F // by Proposition 5.2.2. Thus the isomorphism (10.1.5) implies that ".F /.U / D ".ind.F //.U /, as desired. Proposition 10.1.7. (a) If F is locally nilpotent, then ".F / D 0. (b) If ' is a nil-isomorphism, then ".'/ is an isomorphism. (c) The functor " induces a unique A-linear functor "N W QCrys.X; A/ ! Ét.X; A/: Its restriction to IndCrys.X; A/ will be denoted by the same symbol. (d) This functor commutes with direct limits. (e) This functor is left exact. (f) The natural monomorphism of functors "N ind ,! "N is an isomorphism.
10.1 Basic Definitions
165
Proof. We first prove (b), and for this we consider any nil-isomorphism of -sheaves ' W F ) G . In the proof of Proposition 5.3.3 we observed that the functor ind preserves nil-isomorphisms. Thus ind.'/ W ind.F / ! ind.G / is a nil-isomorphism. In view of Proposition 10.1.6 we may therefore replace ' by ind.'/; hence we may assume that F and G are ind-coherent. Then Proposition 3.3.8 says that ' is a filtered direct limit of nil-isomorphisms of coherent -sheaves. Since " commutes with filtered direct limits, we may therefore assume that F and G are coherent. Then by adjunction, Proposition 3.3.9 yields a commutative diagram n
/ . n id/ F 9 tt ˛ tt t ' . n id/ ' tt ttttn / . n id/ G G
F
in QCoh .X; A/ for some n 0. Its image under " is a commutative diagram / " . n id/ F o7 ".˛/oooo o ".'/ ".. n id/ '/ oo o o oo". n / / " . n id/ G . ".G / ".F /
". n /
By the very definition of " the two homomorphisms ". n / here are isomorphisms. Thus ".˛/ is an isomorphism inverse to ".'/, and so ".'/ is an isomorphism, proving assertion (b). Applying (b) to a nil-isomorphism F ) 0 proves (a). Part (c) follows from part (b) and Lemma 2.2.3 (b). Assertions (d) and (e) follow from the corresponding assertions for the functor (10.1.3), and finally, assertion (f) follows from Proposition 10.1.6. In the following we concentrate on the restricted functor "N W IndCrys.X; A/ ! Ét.X; A/: In Section 10.3 we will show that this is an equivalence of categories. As a preparation for that result we prove it here in the simplest special case of geometric stalks. The following fact is well known from [SGA7, Exposé XXII, § 1]. Lemma 10.1.8. Let K be a separably closed field containing k, and M a finite dimensional K-vector space together with a -linear endomorphism W M ! M . Set M ´ fm 2 M j m D mg. Then the natural homomorphism P P K ˝ M ! M; xi ˝ mi 7! xi mi is injective, and its cokernel is annihilated by a power of . In particular, we have dimk M dimK M < 1.
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10 Étale cohomology
Let Mod.A/ denote the category of A-modules and Modft .A/ the full subcategory of A-modules of finite type. Theorem 10.1.9. If X D Spec K for a separably closed field K, then: (a) The following functors are equivalences of categories: IndCrys.X; A/
"N
/ Ét.X; A/
F7!F.X/
/ Mod.A/:
(b) They induce equivalences of categories Crys.X; A/
"N
/ Ét .X; A/ c
F7!F.X/
/ Modft .A/:
Proof. All three categories in (a) are the ind-categories of the respective categories in (b). Thus as both functors commute with filtered direct limits, (a) is an immediate categorical consequence of (b). For (b) note that the functor on the right hand side is an equivalence of categories. Thus it suffices to study the composite functor. This composite functor is given by F 7! "N.F /.X / D .F .X C // ; where . / denotes -invariants. If F is coherent, its corresponding .K ˝ A/Œ module F .X C / is of finite type over K ˝ A and hence over K. Thus in that case Lemma 10.1.8 implies that .F .X C // is of finite type over A. Therefore the composite defines a functor Crys.X; A/ ! Modft .A/: 0 denote the We construct a quasi-inverse as follows. For any A-module M0 let M -sheaf on X corresponding to the K ˝ A-module K ˝ M0 together with the endomorphism D ˝ id. Clearly this construction defines a functor Modft .A/ ! Crys.X; A/: Now on the one hand the equality K D k yields a natural isomorphism 0 .X C / D .K ˝ M0 / D K ˝ M0 Š M0 ; M which shows that the latter functor is a right quasi-inverse of the former. On the other hand consider any F 2 Coh .X; A/ and let M ´ F .X C / be its associated .K ˝ A/Œ-module, whose dimension over K is finite. Then the natural homomorphism K ˝ M ! M from Lemma 10.1.8 is .K ˝ A/Œ -linear; hence it corresponds ! F . Lemma 10.1.8 also shows that this homoto a homomorphism of -sheaves M morphism is a nil-isomorphism. It therefore defines an isomorphism in Crys.X; A/, which is functorial in F , proving that the latter functor is a left quasi-inverse of the former. Altogether this shows that the former functor is an equivalence of categories, as desired.
167
10.2 Functors
10.2 Functors In this section we show that the functor "N W IndCrys ! Ét translates our functors from Chapter 4 into the corresponding functors of étale sheaves. For simplicity, the functors on the two sides are denoted by the same symbols. As usual we fix a morphism f W Y ! X. For technical reasons, we first consider the direct image functor on QCrys and then continue our discussion of functors on IndCrys in the usual order. Proposition 10.2.1. There exist natural isomorphisms of functors f "N Š "Nf Š "N ind f W QCrys.Y; A/ ! Ét.X; A/: Proof. The functor F 7! FeK t from (10.1.1) commutes with f up to a natural isomorphism. More precisely, for G 2 QCrys.Y; A/ and F ´ f G we have FeK t Š f GeK t . Taking -invariants and using Proposition 10.1.6 yields isomorphisms f " Š "f Š " ind f W QCoh .Y; A/ ! Ét.X; A/: By Theorem 2.2.3 (d) they descend to QCrys.Y; A/, as desired. Proposition 10.2.2. There exists a natural isomorphism of functors f "N Š "Nf W IndCrys.X; A/ ! Ét.Y; A/: Proof. The functors f and f on quasi-crystals are adjoint by Proposition 4.4.5. Thus the adjunction homomorphism id ! f f and Proposition 10.2.1 induce a natural transformation "N ! "Nf f Š f "Nf : Its adjoint is a natural transformation f "N ! "Nf W QCrys.X; A/ ! Ét.Y; A/; whose restriction to IndCrys.X; A/ we denote by f . We will show that f is an isomorphism. As everything commutes with filtered direct limits, it suffices to do this on Crys.X; A/. Moreover, we may verify the desired isomorphy on stalks. Thus for any geometric point iyN W yN ! Y we must prove that iyN f is an isomorphism. g
f
As a preparation for the next few reductions consider two morphisms Z ! Y ! X . One easily shows that the following diagram commutes: fg
.fg/ "N ko
g f "N
g f
/ g "Nf
/ "N.fg/ g f
ko
/ "Ng f .
(10.2.3)
By the diagram (10.2.3) with g D iyN it suffices to prove that f iyN and iyN are isomorphisms, i.e., that g is an isomorphism for any morphism of the form g W yN ! X.
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10 Étale cohomology
Next set x ´ g.y/ N 2 X , let kxN kyN be the separable closure of its residue field kx , ixN gN and define xN ´ Spec kxN . Then g is the composite of two morphisms yN ! xN ! X, and by (10.2.3) again it suffices to prove that gN and ixN are isomorphisms. For gN recall from the proof of Theorem 10.1.9 that every object of Crys.x; N A/ is isomorphic to kxN ˝ M0 with D ˝ id for some M0 2 Modft .A/. Under gN this is mapped to kyN ˝ M0 with D ˝ id. By the definition of "N, both are mapped to M0 , and gN corresponds to the identity on M0 . Thus gN is an isomorphism. To show that ixN is an isomorphism, let XxN denote the strict henselization of X in x. N iN
uN
Then ixN is the composite of two morphisms xN ,! XxN ! X , and by (10.2.3) again it suffices to prove that iN and uN are isomorphisms. The construction of "N shows at once that u is an isomorphism for any étale morphism u. Thus, since uN is an inverse limit of étale morphisms, a direct limit argument easily proves that uN is an isomorphism. It remains to prove that iN is an isomorphism. For this write XxN D Spec R and kxN D R=m. Let M be the .R ˝ A/Œ -module corresponding to a coherent -sheaf on XxN . By the definition of "N and iN we must show that the obvious homomorphism M ! .M=mM / is bijective, where . / denotes -invariants. To prove injectivity, suppose that m 2 M maps to 0 in M=mM . Then we have i m 2 mM , T and so for every i 0 we deduce that m D i .m/ 2 i .mM / mq M . Thus m 2 n mn M . But by Krull this intersection is zero, because R is noetherian and M is of finite type over R. Therefore m D 0, as desired.
To prove surjectivity, choose an R-linear epimorphism N ´ Rn M which is an isomorphism modulo m. Since N is free, it possesses a -linear endomorphism O such that becomes a homomorphism of -modules. Thus to prove the desired surjectivity, it suffices to show that the composite homomorphism N O ! M ! .M=mM / Š .N=mN /O is surjective. For this we must consider the system of polynomial equations id O D 0 on Rn . Its Jacobian is the identity matrix. The desired surjectivity is therefore a consequence of [37, Theorem I.4.2], which we recall in Lemma 10.2.4 for the convenience of the reader. Lemma 10.2.4. Let R be a henselian local ring with maximal ideal m. Consider polynomials p1 ; : : : ; pn 2 RŒT1 ; : : : ; Tn and let pNi denote their reductions modulo m. SupN D 0 for all i , and that det.@pNi =@Tj /i;j .u/ N ¤ 0. pose that uN 2 .R=m/n satisfies pNi .u/ Then there exists u 2 Rn satisfying pi .u/ D 0 for all i , such that uN u mod m. Proposition 10.2.5. There exists a natural isomorphism of functors "N ˝ "N Š "N.
˝
/ W IndCrys.X; A/ IndCrys.X; A/ ! Ét.X; A/:
10.3 Equivalence of categories
169
Proof. The construction (10.1.1) yields an A-bilinear pairing FeK t GeK t ! .F ˝ G /eK t which commutes with . Thus it induces a functorial homomorphism ".F / ˝ ".G / ! ".F ˝ G /: By Theorem 2.2.3 (d) it descends to IndCrys.X; A/. One easily shows that this homomorphism is compatible with inverse image. Thus it suffices to verify the isomorphy on stalks. It also suffices to verify it on Crys.X; A/. But if X D Spec K for a separably closed field K, the proof of Theorem 10.1.9 showed that every object of Crys.X; A/ is isomorphic to K ˝ M0 with D ˝ id for some M0 2 Modft .A/. For such objects the isomorphy is obvious. Proceeding as above, one also obtains the following result: Proposition 10.2.6. For any homomorphism A ! A0 of finite k-algebras, there exists a natural isomorphism of functors "N ˝ A0 Š "N ˝ A0 W IndCrys.X; A/ ! Ét.X; A0 /: A
A
For extension by zero, our axiomatic treatment is helpful in the proof of the following: Proposition 10.2.7. For any open embedding j W U ,! X there exists a natural isomorphism of functors jŠ "N Š "NjŠ W IndCrys.U; A/ ! Ét.X; A/: Proof. Let i W Y ! X denote the closed immersion of a complement of U in X . Then Proposition 10.2.2 and the defining property 4.5.5 of jŠ yield natural isomorphisms j "N.jŠ F / Š "N.j jŠ F / Š "N.F /; i "N.jŠ F / Š "N.i jŠ F / Š "N.0/ Š 0: By the characterization of the étale jŠ this provides us with a functorial isomorphism "N.jŠ F / Š jŠ "N.F /.
10.3 Equivalence of categories The aim of this section is to prove that "N induces equivalences of categories between selected subcategories. We begin with a few lemmas. Lemma 10.3.1. The functor "N W IndCrys.X; A/ ! Ét.X; A/ is exact and faithful. Proof. For ind-crystals as for étale sheaves, the exactness of a sequence and the vanishing of a homomorphism can be tested on stalks. Thus by Proposition 10.2.2 it suffices to consider the case X D Spec K for a separably closed field K. For such X both assertions are part of Theorem 10.1.9.
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10 Étale cohomology
Lemma 10.3.2. The functor "N maps Crys.X; A/ to Étc .X; A/. Proof. Let F be an A-crystal on X . We must show that X possesses a finite stratification by locally closed subsets on which "N.F / becomes locally constant with finite fibers. By noetherian induction and the compatibility of "N with inverse image, it will suffice to show this on some dense open subscheme of X . For this note first that by Theorem 4.4.7 and the compatibility of "N with inverse and direct image, we may assume that X is reduced. After possibly shrinking X we may also assume that it is irreducible and regular. On the other hand, as in the proof of Theorem 7.5.9, after replacing F by a nilisomorphic -subsheaf we may assume that F W . id/ F ! F is generically an isomorphism. Since A is finite, this means that F is an isomorphism over some dense open subscheme of X. After shrinking X we may therefore assume that F is an isomorphism everywhere. Then the induced homomorphism pr 1 F ! pr 1 F is an isomorphism. Furthermore, as A is finite, the sheaf pr 1 F on X is coherent. Since X is reduced, this sheaf is thus generically free. After possibly shrinking X we may thus assume that pr 1 F is locally free. Now [29, Theorem 4.1.1] implies that there exists a finite étale covering u W U ! X such that u "N.F / is a constant sheaf with finite stalks, as desired.
Lemma 10.3.3. Let F 2 Crys.X; A/ and G "N.F / be an étale subsheaf. Then there exists a subcrystal G F such that G D "N.G / "N.F /. Proof. We will construct a -subsheaf G F with G D ".G / ".F /. Then G is automatically coherent and defines the desired subcrystal. Assume first that G is locally constant. Choose a finite étale covering u W U ! X which trivializes G and an A-linear surjection ' W A˚r U u G. Then the composite homomorphism (10.1.5)
'
A˚r ! G.U / ! ".F /.U / Š HomQCoh .1l U;A ; u F / corresponds to a homomorphism of -sheaves following diagram commutative: ".1l ˚r U;A /
". /
/
W 1l ˚r U;A ! u F which makes the
10.2.2
".u F / Š u ".F / S
ko '
A˚r U
/ / u G.
Let G denote the image of the composite homomorphism u
trace
u 1l ˚r ! F : U;A ! u u F
10.3 Equivalence of categories
171
Since " commutes with u by Proposition 10.2.1 and with images by Lemma 10.3.1, we deduce that ".G / is the image of the composite homomorphism u '
trace
u A˚r ! u u ".F / ! ".F /: U But this is simply the image of the composite surjection u '
trace
u A˚r ! ! u u G ! ! G; U proving that ".G / D G, as desired. We now prove the lemma in general using noetherian induction on X . Choose a dense open embedding j W U ,! X such that j G is locally constant, and let i W Y ,! X be a closed complement. By the above special case there exists a -subsheaf G 0 j F such that ".G 0 / D j G j ".F /
10.2.2
D
".j F /:
Define G C by the pullback diagram F
/ j j F
GC
/ j G 0 .
S
S
As " is left exact, it preserves pullbacks; this implies that G ".G C /. Note also that, since F is coherent, so is its -subsheaf G C . Thus, by the induction hypothesis, there exists a -subsheaf G 00 i G C such that ".G 00 / D i G i ".G C /
10.2.2
D
".i G C /:
We define G by the pullback diagram GC
/ i i G C
G
/ i G 00 .
S
S
Then by construction we have j ".G /
10.2.2
D
".j G / D ".j G C / D ".G 0 / D j G:
On the other hand, by the coherence of G C and G 00 , and the exactness of i on crystals, the construction of G implies that i G ! G 00 is a nil-isomorphism. Therefore i ".G /
10.2.2
D
".i G /
10.1.7
D
Altogether we deduce that ".G / D G, as desired.
".G 00 / D i G:
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10 Étale cohomology
Lemma 10.3.4. For any G 2 Étc .X; A/ there exists G 2 Crys.X; A/ with "N.G / Š G. Proof. We use noetherian induction on X . If X is non-empty, choose an étale morphism uW U ! X with U non-empty, such that u G is constant with finite stalks. Let 'W u G ! M ˝ AU be a trivialization for a suitable finitely generated A-module M . Consider its adjoint W G ! M ˝ u AU . The composite homomorphism u
adj
u G ! u .M ˝ u AU / ! M ˝ AU is equal to '; hence is injective over the open subset u.U / X . Let iW Y ,! X be a closed complement. Then the natural homomorphism G ! M ˝ u AU ˚ i i G is injective. Now on the one hand we have 10.1.4 10.2.1 M ˝ u AU Š u "N.M ˝A 1l U;A / Š "N ind.M ˝A u 1l U;A / : On the other hand, by the induction hypothesis, there exists G 00 2 Crys.Y; A/ such that "N.G 00 / Š i G. By Proposition 10.2.1 this implies that i i G Š "N.i G 00 /. Thus altogether we obtain an injection G ,! "N.F / for some ind-crystal F . Since "N commutes with filtered direct limits and G is constructible, there also exists an injection G ,! "N.F 0 / for some subcrystal F 0 F . By Lemma 10.3.3 we then have G Š "N.G / for some subcrystal G F 0 , as desired. Lemma 10.3.5. For any F , G 2 Crys.X; A/ the homomorphism HomCrys .F ; G / ! HomÉt .N".F /; "N.G // is bijective. Proof. By Lemma 10.3.1 it is injective. To prove surjectivity consider any homomorphism ' W "N.F / ! "N.G /. Then Lemma 10.3.3 yields a subcrystal H F ˚ G such that "N.H / D graph.'/. Consider the projections 1 W H ! F and 2 W H ! G . Then "N.1 / is an isomorphism. By the exactness from Lemma 10.3.1 this implies that "N.ker.1 // Š ker.N".1 // D 0 and "N.coker.1 // Š coker.N".1 // D 0, which by the faithfulness from Lemma 10.3.1 implies that ker.1 / D coker.1 / D 0. Thus 1 is an isomorphism, and H D graph.2 11 /. Again by exactness "N commutes with taking graphs; hence graph.'/ D graph "N.2 11 / , and so ' D "N.2 11 /, as desired. Theorem 10.3.6. The functor "N induces equivalences of categories IndCrys.X; A/ ! Ét.X; A/; Crys.X; A/ ! Étc .X; A/: Proof. By Lemma 10.3.2 the lower functor exists with the target Étc .X; A/. It is essentially surjective by Lemma 10.3.4, and fully faithful by Lemma 10.3.5, and hence an equivalence of categories. Since IndCrys and Ét are the ind-categories of Crys and Étc , respectively, and "N commutes with filtered direct limits, the equivalence in the upper row is an immediate categorical consequence of the equivalence in the lower row.
10.5 Flatness
173
10.4 Derived categories and functors Passing to derived categories, the following result is immediate from Theorems 5.3.1 (c) and 10.3.6. Theorem 10.4.1. For 2 fb; g, the functor "N induces the following commutative diagram with indicated equivalences of categories:
D .Crys.X; A//
/ Dcrys .IndCrys.X; A//
/ D .IndCrys.X; A//
"N o
"N o
"N o
D .Étc .X; A//
/ D Ét
c
.Ét.X; A//
/ D .Ét.X; A//.
The compatibilities of functors proved in Section 10.2 all extend to the derived setting: Theorem 10.4.2. For any 2 fb; g there are natural isomorphisms of derived functors on D .IndCrys.: : : //: (a) f "N Š "Nf . (b) "N ˝L "N Š "N. (c) "N ˝AL A0 Š "N.
˝L
/.
˝AL A0 /.
(d) Rf "N Š "N R.ind f /. (e) jŠ "N Š "N jŠ . (f) RfŠ "N Š "N RfŠ for f compactifiable. Proof. Assertions (a)–(e) follow directly from the compatibilities from Section 10.2, because "N is an equivalence of categories and derived functors are unique. Part (f) follows from parts (d) and (e), because on both sides RfŠ is defined as RfN jŠ for a compactification f D fNj .
10.5 Flatness Let Étflat c .X; A/ denote the full subcategory of Ét c .X; A/ formed by all constructible étale sheaves of flat A-modules, i.e., those whose geometric stalks are flat over A. Theorem 10.5.1. The functor "N restricts to an equivalence of categories "N W Crysflat .X; A/ ! Étflat c .X; A/:
174
10 Étale cohomology
Proof. By Proposition 7.2.1 a crystal F 2 Crys.X; A/ is flat if and only if the functor F ˝ is exact on IndCrys.X; A/. By Theorems 10.3.6 and 10.4.2 (b) this is equivalent to the functor "N.F / ˝ being exact on Ét.X; A/. But this means precisely that "N.F / is flat, as desired. By D b .Étc .X; A//ftd we denote the strictly full triangulated subcategory of D .Étc .X; A// of complexes with finite Tor-dimension. The following result is an immediate consequence of Theorems 7.6.3, 10.4.2 (b), and 10.5.1. b
Theorem 10.5.2. All functors in the following commutative square are equivalences of categories: / D b .Crys.X; A// D b .Crysflat .X; A// ftd "N
"N
D b .Étflat c .X; A//
/ D b .Ét .X; A// . c ftd
From this and Propositions 7.6.4, 7.6.5, 7.6.6 and 7.6.9 we obtain: Corollary 10.5.3. The derived functors f , ˝L , ˝AL A0 , and RfŠ on complexes of étale sheaves preserve the subcategories D b .Étc .: : : //ftd .
10.6 L-functions We first review the definition of local L-factors. Let x D Spec kx for a finite extension kx of k of degree dx . Set xN ´ Spec kNx , where kNx denotes an algebraic closure of kx . Then the stalk FxN of any sheaf F 2 Étflat c .x; A/ is a locally free A-module of finite type, carrying a natural action of Gal.kNx =kx /. This Galois group is generated by the dx arithmetic Frobenius Frobx , whose action on kNx is given by u 7! uq . Thus, taking Lemma 8.1.1 into account, the following definition makes sense: Definition 10.6.1. The L-function of F 2 Étflat c .x; A/ is ˇ 1 ˇ FxN 2 1 C t dx AŒŒt dx : L.x; F; t / ´ detA id t dx Frob1 x Next let X be a scheme of finite type over k. The embedding of any closed point x 2 jXj is denoted ix W x ,! X . As the number of points x of any given degree dx is finite, we can form the product over all x within 1 C tAŒŒt : Definition 10.6.2. The L-function of a complex F 2 C b .Étflat c .X; A// is L.X; F ; t / ´
Y Y i2Z x2jXj
i
L.x; ix Fi ; t /.1/ 2 1 C tAŒŒt :
10.6 L-functions
175
A simple computation on stalks shows that the L-function of an acyclic complex is trivial. Therefore L is naturally defined on D b .Étc .X; A//ftd . Alternatively, this follows from the proposition below combined with Theorem 10.5.2.
Proposition 10.6.3. For any F 2 D b .Crys.X; A//ftd we have
L.X; "N.F /; t / D Lcrys .X; F ; t /: Proof. It suffices to prove this for a single flat crystal F over a single point x. We may also assume that F is a semisimple coherent -sheaf; its underlying sheaf is then locally free by Proposition 9.3.4 (c). Let M be the corresponding locally free .kx ˝A/module with its induced . ˝ id/-linear automorphism W M ! M . Then the stalk of the étale sheaf F ´ "N.F / is the set of . ˝ /-invariants FxN D .kNx ˝kx M /˝ ;
and by construction the action of Frobx on it is induced by the action of dx ˝ id on kNx ˝kx M . Since ˝ induces the identity on FxN , the action of Frobx is also induced by id ˝ dx . Next, by Lemma 10.1.8 the natural homomorphism kNx ˝ FxN ! kNx ˝kx M;
˛ ˝ m 7! .˛ ˝ 1/ m
is an isomorphism. Moreover, the above description of Frobx implies that the action dx of id ˝ Frob1 on the right x on the left hand side corresponds to the action of id ˝ hand side. Thus from the definition of local L-factors we conclude that ˇ 1 def ˇ FxN L.x; F; t / D detA id t dx Frob1 x ˇ 1 ˇ N D detkNx ˝A id t dx .id ˝ Frob1 x / kx ˝ FxN ˇ 1 D detkNx ˝A id t dx .id ˝ dx / ˇ kNx ˝kx M ˇ 1 D detkx ˝A id t dx dx ˇ M def
D Lcrys .x; F ; t /;
as desired. The equivalence of categories from Theorem 10.5.2 combined with Proposition 10.6.3 shows that each of the functoriality results for crystalline L-functions from Section 9.6 gives rise to a corresponding result for the L-functions of complexes in D b .Étc .X; A//ftd . We state the consequences that seem most important to us, which follow from Theorems 9.6.4 and 9.6.5: Theorem 10.6.4. For any complex F 2 D b .Étc .X; A//ftd the L-function L.X; F ; t / is a rational function. Theorem 10.6.5. For any morphism f W Y ! X of schemes of finite type over k and any F 2 D b .Étc .Y; A//ftd we have L.Y; F ; t / L.X; RfŠ F ; t /; i.e., their quotient is a unipotent polynomial.
176
10 Étale cohomology
For reduced coefficient rings A, the above results were first proved by Deligne in [SGA4 12 , Fonctions L mod `n et mod p, Théorème 2.2]. In [17, Theorem 1.5], Emerton and Kisin give a proof for arbitrary finite A of some characteristic p m . By an inverse limit procedure, in [17, Corollary 1.8], they give a suitable generalization to any coefficient ring A which is a complete noetherian local Zp -algebra with finite residue field. Remark 10.6.6. For examples where the corresponding equality does not hold see Section 8.5 and Remark 9.6.7. In the situation of (8.5.3) let W 1eK t .X; x/ N k denote the homomorphism defining the action on the Galois covering Y ! X with Galois group k. Then F ´ "N.F / is the locally constant étale sheaf on X of rank 1 with stalk A D kŒ"=."2 / and the representation 1eK t .X; x/ N ! A ;
7! 1 C ./":
Here RsXŠ F vanishes, although L.Y; F ; t / may be 6D 1.
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List of notation
Rings and schemes A, 2, 3, 5, 39 C , 3, 39 c 2 C , 105 dx , 121 Fp , Fq , 2 k, 3 kc , 105 kx , 54, 121 .R ˝ A/Œ, 40 S, 72 T , 71, 105 T Œ, 71 jXj, 121 Morphisms A ! A0 , 60 Fabs W X ! X, 125 f W Y ! X, 6, 52, 151, 152, 167 fN W Yx ! X, 101 h W C 0 ! C , 60 i W Y ,! X, 6, 64, 99, 151 ix W x ,! X , 54 j W U ,! X, 6, 64, 88, 99, 151, 169 x 4, 101, 129 j W X ,! X, pr 1 W X C ! X , 39 W X ! X, 39, 107 Modules and sheaves F, 163 FxN , 174
F , 40 FeK t , 163 F , 3, 5, 40 Fy , 46, 109 F .i/ , 114 F nil , 146
F ss , 9, 146 f 1 m, 73 FT .M /, 85 G, 163 G , 40 Gnil , 148 Gss , 148 0 , 39 0 F , 39 V , 124 V , 126 M , 73 Mnil , 138 Mss , 138 M.'/, 50 M.'/, 50 X , 124 !X , 124 !X;A , 125 1lX;A , 59 S 1 M , 73 , 40 F , 5, 40 Categories A, 16, 22 B, 29 C, 17 x 18 C, C .A/, 27 C .A/, 28 CB .A/, 29 Cart.X; A/, 124 Cartlocfree .X; A/, 126 Coh .X; A/, 5, 40 Coh.i/ .X; A/, 114 Cohlocfree .X; A/, 126 Crys.X; A/, 5, 47
182
List of notation
Crysflat .X; A/, 107 D .A/, 29 DB .A/, 29 b D .Crys.X; A//ftd , 118 Ét.X; A/, 163 Étc .X; A/, 11, 163 Étflat c .X; A/, 173 IndCoh .X; A/, 40 IndCrys.X; A/, 47 K .A/, 28 .A/, 29 KB LNil .X; A/, 43 LNilCoh .X; A/, 43 LNilCrys.X; A/, 47 Mod .T /, 72 Modft .T /, 72 Modind .T /, 72 Nil .X; A/, 43 QCoh .X; A/, 40 QCrys.X; A/, 47 QCrysflat .X; A/, 59 QCrysvflat .X; A/, 59 , 17 1 C, 17 qi , 29 Bqi , 29 nilqi , 81 Functors D, 126 ", 11, 164 "N, 11, 164, 165, 172, 173 f , 6, 52, 89 f , 6, 61 ind, 72 ind, 84 jŠ , 6, 67, 99 , 29 `, 78 `M , 73 Lf , 89 Li f , 52 q, 18
r, 24 RfŠ , 101 Rf , 93, 99 Ri f , 7, 62 R ind, 84 Ri ind, 74, 80 R ind, 84 Rr, 37 Ri r, 37 s, 25 snil , 49 S, 78 t, 26 Tor i . ; /, 58 0 Tor A i . ; A /, 61 ˝ ,6 ˝ , 58, 163 ˝A A0 , 6, 60 ˝L , 90 ˝L A0 , 93 A
Polynomials and L-functions
, 127 , 127 Lcrys , 9, 147, 149, 150, 160 Lnaive , 3, 123, 124 P .i/ , 140 P Q, 10, 139 Pss , 144 Punip , 144 Tr, 127 Various ˛U , 85 ch' , 50 CL Up , 84 CL U , 85 deg.a/, 49 d', 50 Endk .L/, 49 L Œ1, 28 nA , 10, 139
List of notation
', 50 'a , 50 ˆF , 109 p, 3 q, 3
. /0 , 39 !, 14, 48 H), 17, 48 K, 18, 48
183
Index
A-crystal, see crystal adjoint functors, 15 f and f , 62 i and r, 24 jŠ and j , 67, 100 Lf and Rf , 94 q and s, 25 A-motive, 2, 5, 51 Anderson, G., v, 2, 124 Anderson trace formula, 4, 121, 126– 128 A-quasi-crystal, see quasi-crystal arrow dotted, 18, 48 double, 17, 48 solid, 14, 48 Artin–Schreier theory, 11, 163 augmentation, 85
coefficient scheme, 3 cohomology with compact support, see direct image compactifiable morphism, 101 Conrad, B., v constructible set, 69, 113 constructible sheaf, 11, 52, 69, 163 crystal, v, 4, 5, 7, 39, 47, see also quasicrystal canonical representative, 46, 109 flat, 8, 59, 91, 92, 105–120, 147, 173 free 0 -representative, 114 homomorphism, 48, 49 locally free representative, 8, 113 pullback from the first factor, 8, 68, 91, 107 semisimple representative, 9, 147 crystal (over function fields), 7
base change, 68, 95, 100 proper, 7, 103 Blickle, M., 11
Deligne, P., 1, 12, 137 direct image, 61, 93, 97, 167, 173 under a proper morphism, 6, 62, 99, 119, 151 with compact support, 4, 101, 102, 119, 152, 153, 173 distinguished triangle, 26, 28 Drinfeld, V., v, 2 Drinfeld modular form, 12 Drinfeld module, 2, 5, 50 characteristic of, 50 dual characteristic polynomial, 122, 127, 139 Dwork, B., v, 1, 2
Cartier linear, 124 Cartier module, 124, 128 Cartier operator, 125 Cartier sheaf, 124, 125 associated to F , 126 category, 13 abelian, 20 additive, 16 derived, 27, 71, 81 Grothendieck, 23, 29, 41, 48, 72 triangulated, 26 ˇ Cech complex, 85, 95, 98 ˇ Cech resolution, 7, 84–87 change of coefficients, 6, 8, 11, 60, 93, 119, 160–162, 169, 173 coefficient ring, 3
Eichler, M., 12 elliptic sheaf, 2, 51 Emerton, M., 11, 163 enough injectives, 23–25, 37, 71, 74, 80
186
Index
equivalence of categories for abstract categories, 13, 15, 20, 30, 33, 36, 37 Cartier sheaves, 126 étale sheaves, 11, 166, 172–174 flat (quasi)-crystals, 87, 118, 151 (quasi)-crystals, 7, 47, 63, 71, 74, 80, 82 étale sheaf, 11, 163–176 of flat A-modules, 173 extension by zero, 4, 6, 8, 64, 67, 99, 119, 169, 173 finite Tor-dimension, 117, 118, 120 for regular coefficients, 120 flat module, 106 fraction, left or right, 17, 18, 73 F -sheaf, 51 functor, 14–16, 21 derived, 30, 32, 88 gauge, 127 good coefficient ring, 10, 158 good cohomology theory, 1, 3 Goss, D., v, 2, 12 Grothendieck, A., v, 1, 3 ind-acyclic, 75, 76, 80 inverse image, 6, 8, 52, 88, 89, 118, 152, 167, 173 iterate, 5, 42, 113, 114, 140, 157, 158 Katz, N., 11 Kisin, M., 11, 163 Kunz, E., 107 Lafforgue, V., 12 Lefschetz trace formula, 121, 129, 152 L-function, 2, 11, 12 crystalline, 9, 138–162, 175 étale, 174, 175 naive, 3, 9, 121–137, 147, 149 rationality, 2, 4, 10, 121, 128, 138, 152, 175 limit, direct or inverse, 15, 16
localization, 5, 13, 18, 47, 73 localization functor, 18, 25 localizing subcategory, 20, 83 locally small, 22, 24 monic polynomial in T Œ , 72 multiplicative system, 17 compatible with the triangulation, 26 essentially locally small, 18 saturated, 19 nil-isomorphism, 5, 44–46 nil-quasi-isomorphism, 81 nilradical, 10, 139 noetherian object, 22, 36 nucleus, 121, 126, 133 Ore localization, 73 '-sheaf, 5 polynomial iterate, 140, 158 semisimple, 143, 144 unipotent, 139, 143, 144 projection formula, 7, 96, 103 quasi-crystal, 7, 47 associated to a -sheaf, 48 flat, 59 very flat, 59, 60, 64, 86 quasi-isomorphism, 28, 33 quotient ring, 157 r-acyclic, 37 radicial morphism, 62 rational function with constant coefficient 1, 139 semisimple, 144 semisimple part, 144 unipotent part, 144 equivalence, 10, 139 Riemann–Hilbert correspondence, 12, 163 Serre, J.-P., 1
Index
Serre duality, 130 Serre subcategory, 22, 24, 36, 41, 43, 47, 72 Shimura, G., 12 shtuka, 2, 51 stalk of a (quasi)-crystal, 6, 8, 54 stalk of a -sheaf, 54 strongly ind-acyclic, 75, 78 Taguchi, Y., v, 2, 4, 5, 12 tensor product, 6, 8, 58, 60, 90, 92, 118, 168, 173 unit object for, 59 t-motive, see A-motive Tor-object, 52, 91, 107, 108, 110 trace class operator, 128 trace formula Anderson, 4, 121, 126–128 Lefschetz, 121, 129, 152 relative, 5, 10, 152, 153 trace map, 125 trace of V , 127 -sheaf, 40 coherent, 3, 5, 7, 40 homomorphism, 40
187
ind-acyclic, 76, 80 ind-coherent, 40 locally nil-coherent, 42 locally nilpotent, 42 nilpotent, 5, 42, 54, 56, 57, 64 nilpotent part, 146 pullback from the first factor, 8, 68, 91, 107, 128, 129, 154 quasi-coherent, 7 semisimple, 146 semisimple part, 146 strongly ind-acyclic, 78 underlying a (quasi)-crystal, 48 i -sheaf, 114 T Π-module, 71 ind-acyclic, 75 strongly ind-acyclic, 75 Wan, D., v, 2, 4, 5, 12 way out right functor, 117 Weil, A., 1, 2 Weil conjecture, v, 1, 2 Woodshole fixed point formula, 134 -function, 1, 2
