ring theory and algebraic geometry
PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey
Zuhair Nashed University of Delaware Newark, Delaware
EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology
Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University
S. Kobayashi University of California, Berkeley
David L. Russell Virginia Polytechnic Institute and State University
Marvin Marcus University of California, Santa Barbara
Walter Schempp Universitat Siegen
W. S. Massey Yale University
Mark Teply University of Wisconsin, Milwaukee
LECTURE NOTES IN PURE AND APPLIED MATHEMATICS
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.
N. Jacobson, Exceptional Lie Algebras L.-A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis /. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch et a/., Differentiab/e Manifolds and Quadratic Forms I. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras 6. R. McDonald et a/., Ring Theory Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus et a/., Calkin Algebras and Algebras of Operators on Banach Spaces £. O. Roxin et a/., Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thornier, Topology and Its Applications J. M. Lopezand K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis L Cesari et a/., Nonlinear Functional Analysis and Differential Equations J. J. Schaffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconcelos, The Rings of Dimension Two R. E. Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology S. K. Jain, Ring Theory B. R. McDonald and R. A. Morris, Ring Theory II R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems H.-C. Wang, Homogeneous Branch Algebras E. O. Roxin et a/., Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altman, Contractors and Contractor Directions Theory and Applications J. S. Go/an, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis R. Gordon, Representation Theory of Algebras M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. I. Arruda et a/., Mathematical Logic F. Van Oystaeyen, Ring Theory F. Van Oystaeyen and A. Verschoren, Reflectors and Localization M. Satyanarayana, Positively Ordered Semigroups D. L Russell, Mathematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory 111 A. Geramita and J. Seberry, Orthogonal Designs J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces P.-T. Liu andJ. G. Sutinen, Control Theory in Mathematical Economics C. Byrnes, Partial Differential Equations and Geometry G. Klambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory 8. Kadem, Binary Time Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L. Sternberg et a/., Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III J. S. Go/an, Structure Sheaves Over a Noncommutative Ring T. V. Narayana et a/., Combinatorics, Representation Theory and Statistical Methods in Groups T. A. Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory
60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102.
J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces O. A. Nielson, Direct Integral Theory J. E. Smith et al., Ordered Groups J. Cronin, Mathematics of Ceil Electrophysiology J. W. Brewer, Power Series Over Commutative Rings P. K. Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L. Herdman et al., Integral and Functional Differential Equations R. Draper, Commutative Algebra W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras R. L Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems J. Van Gee!, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A. Fiacco, Mathematical Programming with Data Perturbations i P. Schultz et al., Algebraic Structures and Applications L Bican et al., Rings, Modules, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry P. Fletcherand W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen et al., Volterra and Functional Differential Equations N. L. Johnson et al., Finite Geometries G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra A. V. Fiacco, Mathematical Programming with Data Perturbations II J.-B. Hiriart-Urruty et al., Optimization A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. Istratescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems J. W. Longley, Least Squares Computations Using Orthogonalization Methods L. P. de Alcantara, Mathematical Logic and Formal Systems C. E. Au/l, Rings of Continuous Functions R. Chuaqui, Analysis, Geometry, and Probability L. Fuchs and L Salce, Modules Over Valuation Domains P. Fischerand W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential Equations
103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120.
C. A. Baker and L. M. Batten, Finite Geometries J. W. Brewer et al., Linear Systems Over Commutative Rings C. McCrory and T. Shifrin, Geometry and Topology D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science B.-L. Lin and S. Simons, Nonlinear and Convex Analysis S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods M. C. Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory D. V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geometry and Topology P. Nelson etal., Transport Theory, Invariant Imbedding, and Integral Equations P. Clement et al., Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications C. M. Dafermos et al., Differential Equations E. O. Roxin, Modern Optimal Control J. C. Diaz, Mathematics for Large Scale Computing
121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183.
P. S. Milojevicj Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations R. M. Shortt, General Topology and Applications R. Wong, Asymptotic and Computational Analysis D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics W. D. Wallis et al., Combinatorial Designs and Applications S. Elaydi, Differential Equations G. Chen et al., Distributed Parameter Control Systems W. N. Everitt, Inequalities H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations O. Anno et a/., Mathematical Population Dynamics S. Coen, Geometry and Complex Variables J. A. Goldstein et a/., Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andima et a/., General Topology and Applications P Clement et a/., Semigroup Theory and Evolution Equations K. Jarosz, Function Spaces J. M. Bayod et a/., p-adic Functional Analysis G. A. Anastassiou, Approximation Theory R. S. Rees, Graphs, Matrices, and Designs G. Abrams et a/., Methods in Module Theory G. L Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M. C. Josh/and A. V. Balakrishnan, Mathematical Theory of Control G. Komatsu and Y. Sakane, Complex Geometry /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. Mabuchi and S. Mukai, Einstein Metrics and Yang-Mills Connections L Fuchs and R. Gobel, Abelian Groups A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum G. Dore et a/., Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedtetal., Functional Analysis K. G. Fischeretal., Computational Algebra K. D. Elworthyetal., Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, et a/., Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions P. Clement and G. turner, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research W. O. Brayetal., Fourier Analysis J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pave!, Optimal Control of Differential Equations M. Ikawa, Spectral and Scattering Theory X. Liu and D. Siegel, Comparison Methods and Stability Theory J.-P. Zolesio, Boundary Control and Variation M. Krizeketal, Finite Element Methods G. Da Prato and L. Tubaro, Control of Partial Differential Equations E. Ballico, Projective Geometry with Applications M. Costabeletal., Boundary Value Problems and Integral Equations in Nonsmooth Domains G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations S. Huggett, Twistor Theory H. Cook et a/., Continua D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings K. Jarosz, Function Spaces V. Ancona et al., Complex Analysis and Geometry E. Casas, Control of Partial Differential Equations and Applications N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability Z. Deng et al., Differential Equations and Control Theory P. Marcellini et al. Partial Differential Equations and Applications A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type M. Maruyama, Moduli of Vector Bundles A. Ursini and P. Agliano, Logic and Algebra X. H. Cao etal., Rings, Groups, and Algebras D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models
184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221.
J. E. /Andersen et al., Geometry and Physics P.-J. Cahen et al., Commutative Ring Theory J. A. Goldstein et a/., Stochastic Processes and Functional Analysis A. Sonb/, Complexity, Logic, and Recursion Theory G. Da Prato and J.-P. Zolesio, Partial Differential Equation Methods in Control and Shape Analysis D. D. Anderson, Factorization in Integral Domains N. L Johnson, Mostly Finite Geometries D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm-Liouville Problems W. H. Schikhofet al., p-adic Functional Analysis S. Sertoz, Algebraic Geometry G. Caristi and E. Mitldieri, Reaction Diffusion Systems A. V. Fiacco, Mathematical Programming with Data Perturbations M. Krizek et a/., Finite Element Methods: Superconvergence, Post-Processing, and A Posteriori Estimates S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups V. Drensky et a/., Methods in Ring Theory W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions P. E. Newstead, Algebraic Geometry D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology Z. Cnen et at., Advances in Computational Mathematics X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs C. Y. Yildirim and S. A. Stepanov, Number Theory and Its Applications D. E. Dobbs et a/., Advances in Commutative Ring Theory F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry J. Kakol et at., p-adic Functional Analysis M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of Algebras R. Costa et a/., Nonassociative Algebra and Its Applications T.-X. He, Wavelet Analysis and Multiresolution Methods H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference J. Kajiwara et a/., Finite or Infinite Dimensional Complex Analysis G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences J. Cagnoletal., Shape Optimization and Optimal Design J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra G. Chen et al., Control of Nonlinear Distributed Parameter Systems F. AH Mehmeti et al., Partial Differential Equations on Multistructures D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra A. Gran/a et al., Ring Theory and Algebraic Geometry
Additional Volumes in Preparation
ring theory and algebraic geometry proceedings of the fifth international conference CSAGBA V3 in Leon, Spain
edited by Angel Granja University of Leon Lean, Spain
Jose Angel Hermida University of Leon Leon, Spain
Alain Verschoren University of Antwerp, RUCA Antwerp, Belgium
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Preface
The Fifth International Conference on Algebra and Algebraic Geometry (SAGA V) was held at the University of Leon (Spain) and contributors and participants originated from more than 15 European and non-European countries. As in earlier versions of these meetings in Antwerp, Santiago de Compostela, Puerto de la Cruz and Brussels, talks concentrated on algebraic and geometric subjects, with particular emphasis on intrinsic links between these domains. The main speakers were P. Ara (Barcelona), A. Bak (Bielefeld), A. Campillo (Valladolid), T. Lenagan (Edinburgh), M.P. Malliavin (Paris VI), F. Van Oystaeyen (Antwerp), A. del Rio (Murcia), M. Spivakovsky (Toronto) and V. W. Vasconcelos (Rutgers). The Scientific Committee consisted of J. L. Bueso, S. Caenepeel, J.A. Hermida, T. Sanchez-Giralda, A. Verschorcn and E. Villanueva. The Organizing Committee consisted of A. de Francisco, A. Granja and J. Susperregui. The organizers would like to thank the University of Leon, the Junta de Castilla Leon and Caja Espana for financial and logistic support. The meeting was also partially sponsored by the grants DGICYT PB95-0603-C02 and LE 36/98. Our particular thanks go to the "locals", M. Carriegos, J. Gomez, M. Lopez and C. Sanchez whose youthful enthusiasm helped to solve all practical problems, large and small during the meeting, to Stef Caenepel, whose capacities as an entertainer played a fundamental role during the meeting (as always) and, last but not least, to the secretaries of Department of Mathematics, Ana del Rio and Ana Robles. We hope the participants enjoyed this meeting as much as we did! Angel Granja Jose Angel Hermida Alain Verschoren
Contents
Preface Contributors Conference Participants
1.
Frobenius and Maschke Type Theorems for Doi-Hopf Modules and Entwined Modules Revisited: A Unified Approach T. Brzezinski, S. Caenepeel, G. Militaru and S. Zhu
1
2.
Computing the Gelfand-Kirillov Dimension II J. L. Bueso, J. Gomez-Torrecillas, and F. J. Lobillo
33
3.
Some Problems About Nilpotent Lie Algebras J. M. Cabezas, L. M. Camacho, J, R. Gomez, A. Jimenez-Merchdn, E. Pastor, J. Reyes, and I. Rodriguez
59
4.
On L*-Triples and Jordan //*-Pairs A. J. Calderon-Martin and C. Martin-Gonzdlez
87
5.
Toric Mathematics from Semigroup Viewpoint A. Campillo and P. Pison
95
6.
Canonical Forms for Linear Dynamical Systems over Commutative Rings: The Local Case M. Carriegos and T. Sdnchez-Giralda
113
1.
An Introduction to Janet Bases and Grobner Bases F. J. Castro-Jimenez and M. A. Moreno-Frias
133
8.
Invariants of Coalgebras J. Cuadra and F. Van Oystaeyen
147
9.
Multiplication Objects J. Escoriza and B. Torrecillas
173
10.
Krull-Schmidt Theorem and Semilocal Endormorphism Rings A. Facchini
187
11.
On Suslin's Stability Theorem for R[xr,..,xm] J. Gago-Vargas
203
12.
Characterization of Rings Using Socle-Fine and Radical-Fine Notions C. M. Gonzdlez, A. Idelhadj, and A. Yahya
211
13.
About Bernstein Algebras S. Gonzdlez and C. Martinez
223
vi
14.
Contents
About an Algorithm of T. Oaku M. /. Hartillo-Hermoso
241
15. Minimal Injective Resolutions: Old and New M. P. Malliavin
251
16.
257
Special Divisors of Blowup Algebras 5. E. Morey and W. V. Vasconcelos
17. Existence of Euler Vector Fields for Curves with Binomial Ideal A. Nunez and M. J. Pisabarro 18.
An Amitsur Cohomology Exact Sequence for Involutive Brauer Groups of the Second Kind A. Smet and A. Verschoren
289
297
19. Computation of the Slopes of a D-Module of Type D'/N J. M. Ucha-Enriquez
311
20.
325
Symmetric Closed Categories and Involutive Brauer Groups A. Verschoren and C. Vidal
Contributors
T. Brzeziriski
University of Wales Swansea, Swansea, U. K.
J. L. Bueso
Universidad de Granada, Granada, Spain.
J. M. Cabezas
Universidad del Pais Vasco, Vitoria-Gasteiz, Spain.
S. Caenepeel
Free University of Brussels, Brussels, Belgium.
A. J. Calderon-Martm L. M. Camacho A. Campillo
Universidad de Cadiz, Cadiz, Spain.
Universidad de Sevilla, Sevilla, Spain. Universidad de Valladolid, Valladolid, Spain.
M. Carriegos
Universidad de Leon, Leon, Spain.
F. J. Castro-Jimenez J. Cuadra
Universidad de Sevilla, Sevilla, Spain.
Universidad de Almerfa, Almerfa, Spain.
J. Escoriza
Universidad de Almeria, Almerfa, Spain.
A Facchini
Universita di Padova, Padova, Italy.
J. Gago-Vargas
Universidad de Sevilla, Sevilla, Spain.
C. M. Gonzalez
Universidad de Malaga, Malaga, Spain.
J. R. Gomez
Universidad de Sevilla, Sevilla, Spain.
J. Gomez-Torrecillas S. Gonzalez
Universidad de Granada, Granada, Spain.
Universidad de Oviedo, Oviedo, Spain.
M. I. Hartillo-Hermoso
A. Idelhadj
Universite Abdelmalek Essad, Tetouan, Morocco.
A. Jimenez-Merchan F. J. Lobillo
Universidad de Cadiz, Cadiz, Spain.
Universidad de Sevilla, Sevilla, Spain.
Universidad de Granada, Granada, Spain.
M. P. Malliavin
Universite Paris VI, Paris, France.
C. Martm-Gonzalez
Universidad de Malaga, Malaga, Spain.
Contributors
C. Martinez
Universidad de Oviedo, Oviedo, Spain.
G. Militaru
University of Bucharest, Bucharest, Romania.
M. A. Moreno-Frfas S. E. Morey
Universidad de Cadiz, Cadiz, Spain.
Southwest Texas State University, Texas, U.S.A..
A. Nunez
Universidad de Valladolid, Valladolid, Spain.
E. Pastor
Universidad del Pais Vasco, Vitoria-Gasteiz, Spain.
M. J. Pisabarro
Universidad de Leon, Leon, Spain.
P. Pison
Universidad de Sevilla, Sevilla, Spain.
J. Reyes
Universidad de Huelva, Huelva, Spain.
I. Rodriguez
Universidad de Huelva, Huelva, Spain.
T. Sanchez-Giralda
A. Smet
University of Antwerp (RUGA), Antwerp, Belgium.
B. Torrecillas
Universidad de Almeria, Almeria, Spain.
J. M. Ucha-Enriquez
F. Van Oystaeyen W. V. Vasconcelos
A. Verschoren C. Vidal
A. Yahya S. Zhu
Universidad de Valladolid, Valladolid, Spain.
Universidad de Sevilla, Sevilla, Spain.
University of Antwerp (UIA), Antwerp, Belgium. Rutgers University, New Jersey, U.S.A..
University of Antwerp (RUGA), Antwerp, Belgium.
Universidad de La Corufia, La Coruna, Spain.
Universite Abdelrnalek Essad, Tetouan, Morocco. Fundan University, Shanghai, China.
Conference Participants
J.N. Alonso Alvarez, Dpto. de Matematicas. Universidad de Vigo. Lagoas-Marcosende. Vigo. E-36280. Spain. P. Ara, Departament de Matematiques Edifici C. Universitat Autonoma de Barcelona 08193 Bellaterra. Barcelona. Spain. E-mail:
[email protected] A. Bak, Department of Mathematics. University of Bielefeld. Germany. E-mail:
[email protected] 33501 Bielefeld.
J. C. Benjumea, Dpto. Geometria y Topologia. Facultad de Matematicas. Universidad de Sevilla. Aptdo 1160. 41080-Sevilla. Spain. E-mail:
[email protected] J. Bernad Lusilla, Dpto. de Matematicas. Universidad de Oviedo. Oviedo. Spain. E-mail:
[email protected] J. Bueso Montero, Dpto. Algebra. Universidad de Granada. Granada. Spain. E-mail:
[email protected] J.M. Ca&ems, Dpto. de Matematica Aplicada. Universidad del Pais Vasco. E.U.I.T. Industrial y Topografia. C/ Nieves Cano 12, 01006 Vitoria-Gasteiz. Spain. E-mail:
[email protected] S. Caenepeel, Faculty of Applied Sciences. Free University of Brussels. VUB, Pleinlaan 2. B-1050 Brussels. Belgium. E-mail:
[email protected] A.J. Calderon Martin, Dpto. de Matematicas. Universidad de Cadiz. 11510 Puerto Real, Cadiz. Spain. E-mail:
[email protected] L.M. Camacho, Dpto. Matematica Aplicada I. Univ. Sevilla. Avda. Reina Mercedes s/n. 41012-Sevilla. Spain. E-mail:
[email protected] A. Campillo, Dpto. Algebra, Geometria y Topologia. Universidad de Valladolid. C/ Prado de la Magdalena, s/n. 47005-Valladolid. Spain. E-mail:
[email protected] M. Carriegos, Dpto. de Matematicas. Universidad de Leon. 24071-Leon. Spain.
x
Conference Participants
E-mail:
[email protected] P. Carvalho, Dpto. de Maternatica Pure. Fac. de Ciencias. 4099-002-Porto. Portugal.
F. J. Castro Jimenez. Dpto. de Algebra. Universidad de Sevilla. Apdo. 1160Sevilla. Spain. E-mail:
[email protected] M. Company Cabezos, Dpto. de Algebra, Geometria y Topologfa. Fac. de Ciencias. Universidad de Malaga. 29071 Malaga. Spain. J. Cuadra Diaz, Dpto. de Algebra y Analisis Matematico. Universidad de Almeria. 04120-Almeria. Spain. E-mail:
[email protected] F. Delgado de la Mata, Dto. de Algebra, Geometria y Topologfa. Universidad de Valladolid. Pdo. De la Magdalena s/n. 47005- Valladolid. Spain. E-mail:
[email protected] F. J. Echarte, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universidad de Sevilla. Aptdo 1160. 41080 Sevilla. Spain. E-mail:
[email protected] J. Escoriza, Universidad de Almeria, Dpto. de Algebra y Analisis Matematico, Crtra. Sacramento s/n. 04120 Almeria. Spain. E-mail:
[email protected] M. Farinati, Universidad de Buenos Aires-Universite de Paris 11. Equipe de Topologie et Dynamique. Batiment 425. 91405. Orsay, France. E-mail:
[email protected] D. Ferndndez, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universidad de Sevilla. Aptdo 1160. 41080 Sevilla. Spain. E-mail:
[email protected] J. Ferndndez Sucasas, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] J.M. Ferndndez Vilaboa, Dpto. de Alxebra. Universidad de Santiago de Compostela. Santiago de Compostela. E-15771. Spain. P. Florez Valbuena, Dpto. de Matematicas. Universidad de Leon. Leon. Spain.
A. de Francisco Iribarren, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] J. Gago Vargas, Dpto. de Algebra. Universidad de Sevilla. C/ Tarifa s/n Facultad de Matematicas. 41012-Sevilla. Spain. E-mail:
[email protected] Conference Participants
xi
J. /. Garcia Garcia, Dpto. de Algebra. Universidad de Granada. E-18071 Granada. Spain. E-mail:
[email protected] M.A. Garcia Muniz, Dpto. de Matematicas. Universidad de Oviedo. Oviedo. Spain. P. A. Garcia Sdnchez, Dpto. Granada. Spain.
de Algebra. Universidad de Granada.
E-18071
P. Gimenez, Dto. de Algebra, Geometrfa y Topologfa. Fac. de Ciencias. Universidad de Valladolid. Spain. E-mail:
[email protected] M. Gomez Lozano, Dpto. de Algebra, Geometrfa y Topologfa. Fac. de Ciencias. Universidad de Malaga. 29071 Malaga. Spain. J.R. Gomez Martin, Dpto. Matematica Aplicada I. Universidad de Sevilla. Avda. Reina Mercedes s/n. 41012 Sevilla. Spain. E-mail:
[email protected] J. Gomez Perez, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] Jose Gomez Torrecillas, Dpto. de Algebra. Universidad de Granada. Fuentenueva s/n. 18071 Granada. Spain,
[email protected] C.M. Gonzdlez, Dpto. de Algebra Geometrfa y Topologfa. Universidad de Malaga. Apartado 59. 29080-Malaga. Spain. S. Gonzdlez, Dpto. de Matematicas, Universidad de Oviedo. Oviedo. Spain. E-mail: santos@pinon. ecu. uniovi. es
J.R. Gonzdlez Martinez. Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] M. F. Gonzdlez Rodriguez, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] R. Gonzdlez Rodriguez, Dpto. de Matematicas. Universidad de Vigo. LagoasMarcosende. Vigo. E-36280. Spain. A. Granja Baron, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] F. Gudiel, Dpto. de Algebra. Universidad de Sevilla. Sevilla. Spain. E-mail:
[email protected] M. I. Hartillo Hermoso, E. U. Empresariales. c/ Por-Vera 54. 11403 Jerez de la Frontera. Cadiz. Spain.
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Conference Participants
E-mail:
[email protected] J. A. Hermida Alonso, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] A. Idelhadj, Departement de Mathematiques Faculte des sciences de Tetouan. B.P 21.21 Tetouan. Morocco.
A. Jimenez Merchdn, Dpto. Matematica Aplicada I. Facultad de Informatica. Universidad de Sevilla. 41012-Sevilla. Spain. E-mail:
[email protected] T. Lenagan, Department of Mathematics and Statistics. University of Edinburg. James Clerk Maxwell Building King's Buildings. Mayfield Road. Edinburgh EH9 3JZ. Scotland. E-mail:
[email protected] C. Lamp, Dpto. de Matematica Pure. Fac. de Ciencias. 4099-002 Oporto. Portugal. E-mail:
[email protected] M. Lopez Cabeceira, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] M. C. Lopez Diaz, Dpto. de Matematicas. Universidad de Oviedo. Oviedo. Spain. M.P. Malliavin, Universite Paris VI. Place Jussieu en l'Ile-75004. Paris. France. E-mail:
[email protected] M.C. Mdrquez, Dpto. Geometrfa y Topologfa. Facultad de Matematicas. Universidad de Sevilla. Aptdo 1160. 41080-Sevilla. Spain. E-mail:
[email protected] C. Martmez Lopez, Dpto. de Matematicas. Universidad de Oviedo Oviedo. Spain. E-mail:
[email protected] M. A. Moreno Frias, Dpto. de Matematicas. Facultad de Ciencias. Universidad de Cadiz. Apartado de Correos. 40. 11510 Puerto Real. Cadiz. Spain. E-mail:
[email protected] /. Musson, Department of Mathematical Sciences. University of Wisconsin-Milwaukee. P.O. Box 413. Milwaukee WI 53201-0413. U.S.A. E-mail:
[email protected] R.M. Navarro, Dpto. de Matematicas. Univ. de Extremadura. Avda. de la Universidad s/n. 10071 Caceres. Spain,
[email protected] A. Nunez, Dep. de Algebra, Geometrfa y Topologfa. Universidad de Valladolid. Spain.
E-mail:
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J. Niinez, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universidad de Sevilla. Aptdo 1160. 41080-Sevilla. Spain. /. Ojeda, Dpto. de Algebra. Universidad de Sevilla. Spain. F. Van Oystaeyen, Department of Mathematics and Computer Science. University of Antwerp. UIA. 2610 Antwerp. Belgium. E-mail:
[email protected] M.A. Olalla Acosta, Dpto. de Algebra. Universidad de Sevilla. Sevilla. Spain. R. Piedra, Dpto. de Algebra. Universidad de Sevilla. Spain. E-mail:
[email protected] M. J. Pisabarro Manteca, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] P. Pison Casares, Dpto. de Algebra. Facultad de Matematica. Apartado 1160. 41080-Sevilla. Spain. E-mail:
[email protected] F. Ramirez, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universidad de Sevilla. Aptdo 1160. 41080-Sevilla. Spain. E-mail:
[email protected] J. Reyes, Dpto. de Matematica. Escuela Politecnica Superior. Universidad de Huelva. Huelva. Spain. E-mail:
[email protected] A. del Rio Mateos, Dpto. de Matematicas. Universidad de Murcia. 30100 Murcia. Spain. E-mail:
[email protected] /. Rodriguez Garcta, Dpto Matematicas. Escuela Politecnica Superior. Universidad de Huelva. Huelva. Spain. E-mail:
[email protected] C. Rodriguez Sdnchez, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] J. C. Resales, Dpto. de Algebra. Universidad de Granada. E-18071 Granada. Spain. E-mail:
[email protected] A. Saez, Dpto. de Algebra, Geometria y Topologfa. Universidad de Valladolid. Spain.
T. Sdnchez Giralda, Dpto. de Algebra, Geometria y Topologfa. Universidad de Valladolid. Spain. E-mail:
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Conference Participants
J. Seto, Dpto. de Matematicas. Universidad de Oviedo. Oviedo. Spain. E-mail:
[email protected] /. P. Shestakov, Sobolev Institute of Mathematics. Novosibirsk. Russia.
M. Siles Molina, Dpto. de Algebra, Geometrfa y Topologia. Facultad de Ciencias. Universidad de Malaga. 29071 Malaga. Spain. E-mail:
[email protected] A. Smet, Department of Mathematics and Computer Science. University of Antwerp. RUGA. Antwerp. Belgium. E-mail:
[email protected] M. Spivakovsky, Universidad de Toronto. Toronto. Canada. E-mail:
[email protected] A. Solotar, Universidad de Buenos Aires-Universite de Paris 11. Equipe de Topologie et Dynamique. Batiment 425. 91405 Orsay. France. E-mail:
[email protected] M. Sudrez Alvarez, Universidad de Buenos Aires-Universite de Paris 11. Equipe de Topologie et Dynamique. Batiment 425. 91405 Orsay. France. J. Susperregui Lesaca, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] J.M. Tornero Sdnchez, Dpto. de Algebra. Universidad de Sevilla. Sevilla. Spain. B. Torrecillas, Dpto. de Algebra y Analisis Matematico. Universidad de Almeria. Crtra. Sacramento s/n. 04120 Almeria. Spain. E-mail:
[email protected] M. T. Trobajo de las Matas, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] J. M. Ucha Enriquez, Dpto. de Algebra. Universidad de Sevilla. Sevilla. Spain.
[email protected] V. W. Vasconcelos, Department of Mathematics. Rutgers University. 110 Frelinghuysen RD Piscataway. N..I. 08854-8019. U.S.A.
E-mail:
[email protected] S. Vega Casielles, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
[email protected] A. Verschoren, Department of Mathematics and Computer Science. University of Antwerp. RUGA. Antwerp. Belgium. E-mail:
[email protected] P. Vicente Matilla, Dpto. de Matematicas. Universidad de Leon. Leon. Spain. E-mail:
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C. Vidal Martin, Universidad de La Coruna. Facultad de Informatica. Campus de Elvina 15071. La Coruna. Spain. E-mail:
[email protected] A. Vieites, Universidad de Vigo. E.T.S.Ingenieros Industrials. Lagoas-Marcosende. 36200 Vigo. Pontevedra. Spain. A. Vigneron-Tenorio, Dpto. de Matematicas. E.U.E. Empresariales. Por-Vera 54. 11403 Jerez de la Frontera. Cadiz. Spain. E-mail:
[email protected] E. Villanueva Novoa, Dpto. de Algebra. Universidad de Santiago de Compostela. Santiago de Compostela. Spain. E-mail:
[email protected] A. Yahya, Departement de mathematiques. Faculte des sciences de Tetouan. B.P 21.21 Tetouan. Morocco.
Frobenius and Maschke Type Theorems for DoiHopf Modules and Entwined Modules Revisited: A Unified Approach T. BRZEZINSKI, 1 Department of Mathematics, University of Wales Swansea. Swansea SA2 8PP, UK.
S. CAENEPEEL, Faculty of Applied Sciences. Free University of Brussels, VUB. B-1050 Brussels, Belgium.
G. MILITARU,2 Faculty of Mathematics. University of Bucharest. RO-70109 Bucharest 1, Romania. S. ZHU, 3 Institute of Mathematics. Fudan University. Shanghai. 200433, China Abstract
We study when induction functors (and their adjoints) between categories of Doi-Hopf modules and, more generally, entwined modules are separable, resp. Frobenius. We present a unified approach, leading to new proofs of old results by the authors, as well as to some new ones. Also our methods provide a categorical explanation for the relationship between separability and Frobenius properties.
1
INTRODUCTION
Let H be a Hopf algebra, A an F-comodule algebra, and C an if-module coalgebra. Doi [17] and Koppinen [21] independently introduced unifying Hopf modules, nowadays usually called Doi-Koppinen-Hopf modules, or Doi-Hopf modules. These are at the same time yi-modules, and C-comodules, with a certain compatibility 'EPSRC Advanced Research Fellow Research supported by the bilateral project "Hopf algebras and co-Galois theory" of the Flemish and Romanian governments ^Research supported by the bilateral project "New computational, geometric and algebraic methods applied to quantum groups and differential operators" of the Flemish and Chinese governments 2
1
2
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relation. Modules, comodules, graded modules, relative Hopf modules, dimodules and Yetter-Drinfel'd modules are all special cases of Doi-Hopf modules. Properties of Doi-Hopf modules (with applications in all the above special cases) have been studied extensively in the literature. In [12], a Maschke type theorem is given, telling when the functor F forgetting the C-coaction reflects the splitness of an exact sequence, while in [13], it is studied when this functor is a Frobenius functor, this means that its right adjoint • Cg> C is at the same time a left adjoint. The two problems look very different at first sight, but the results obtained in [12] and [13] indicate a relationship between them. The main result of [12] tells us that we have a Maschke Theorem for the functor F if C is finitely generated projective and there exists an A-bimodule C-colinear map A C —-> C* A satisfying a certain normalizing condition. In [13], we have seen that F is Frobenius if C is finitely generated and projective and A C and C* A are isomorphic as Abimodules and C-comodules. This isomorphism can be described using a so-called //-integral, this is an element in A ® C satisying a certain centralizing condition. The same //-integrals appear also when one studies Maschke Theorems for G, the right adjoint of F (see [10]). This connection was not well understood at the time when [12] and [13] were written. The aim of this paper is to give a satisfactory explanation; in fact we will present a unified approach to both problems, and solve them at the same time. We will then apply the same technique for proving new Frobenius type properties: we will study when the other forgetful functor forgetting the Aaction is Frobenius, and when a smash product A#ftB is a Frobenius extension of A and B. Let us first give a brief overview of new results obtained after [12] and [13].
1) In [10] and [11] the notion of separable junctor (see [23]) is used to reprove (and generalize) the Maschke Theorem of [12]. In fact separable functors are functors for which a "functorial" type of Maschke Theorem holds. A key result due to Rafael [25] and del Rio [26] tells us when a functor having a left (resp. right) adjoint is separable: the unit (resp. the counit) of the adjunction needs a splitting (resp. a cosplitting). 2) Entwined modules introduced in [2] in the context of noncommutative geometry generalize Doi-Hopf modules. The most interesting examples of entwined modules turn out to be special cases of Doi-Hopf modules, but, on the other hand, the formalism for entwined modules is more transparent than the one for Doi-Hopf modules. Many results for Doi-Hopf modules can be generalized to entwined modules, see e.g. [3], where the results of [12] and [13] are generalized to the entwined case. 3) In [8], we look at separable and Frobenius algebras from the point of view of nonlinear equations; also here we have a connection between the two notions: both separable and Frobenius algebras can be described using normalized solutions of the so-called FS-equation. But the normalizing condition is different in the two cases. Let F : C —» T> be a covariant functor having a right adjoint G. From Rafael's Theorem, it follows that the separability of F and C is determined by the natural transformations in V = Nat(GF, lc) and W ~ Nat(lx>, FG). In the case where F
Doi-Hopf Modules and Entwined Modules
3
is the functor forgetting the coaction, V and W are computed in [10]. In fact V and W can also be used to decide when G is a left adjoint of F. This is what we will do in Section 4; we will find new characterizations for (F, G) to be a Frobenius pair, and we will recover the results in [13] and [3]. In Section 5, we will apply the same technique to decide when the other forgetful functor is Frobenius, and in Section 6, we will study when the smash product of two algebras A and B is a Frobenius extension of A and B. This results in necessary and sufficient conditions for the Drinfel'd double of a finite dimensional Hopf algebra H (which is a special case of the smash product (see [14]) to be a Frobenius or separable over H. We begin with a short section about separable functors and Frobenius pair of functors. We will explain our approach in the most classical situation: we consider a ring extension R —> S, and consider the restriction of scalars functor. We derive the (classical) conditions for an extension to be separable (i.e. the restriction of scalars functor is separable), split (i.e. the induction functor is separable), and Frobenius (i.e. restriction of scalars and induction functors form a Frobenius pair). We present the results in such a way that they can be extended to more general situations in the subsequent Sections. Let us remark at this point that the relationship between Frobenius extensions and separable extensions is an old problem in the literature. A classical result, due to Eilenberg and Nakayama, tells us that, over a field k, any separable algebra is Frobenius. Several generalizations of this property exist; conversely, one can give necessary and sufficient conditions for a Frobenius extension to be separable (see [19, Corollary 4.1]). For more results and a history of this problem, we refer to [1], [20] and [18]. We use the formalism of entwined modules, as this turns out to be more elegant and more general than that of the Doi-Hopf modules; several left-right conventions are possible and there exists a dictionary between them. In [12] and [13], we have worked with right-left Doi-Hopf modules; here we will work in the right-right case, mainly because the formulae then look more natural. Throughout this paper, fc is a commutative ring. We use the Sweedler-Heyneman notation for comultiplications and coactions. For the comultiplication A on a coalgebra C, we write A
( c ) = c (l) ® C ( 2 ) .
For a right C-coaction pr and a left C-coaction pl on a /c-module N, we write
pT(ri) =n [ 0 ] We omit the summation symbol ^T\
2
SEPARABLE FUNCTORS AND FROBENIUS PAIRS OF FUNCTORS
Let F : C —-> T> be a covariant functor. Recall [23] that F is called a separable functor if the natural transformation
induced by F splits. From [25] and [26], we recall the following characterisation in the case F has an adjoint.
Brzeziriski et al.
PROPOSITION 2.1 Let G : V -> C be a right adjoint ofF. Let 77 : lc and e : FG —> lp 6e i/ie unit and counit of the adjunction. Then 1) F is separable if and only if 'there exists v € V = Nat(G_F, lc) SMC/Z i/ia£ 1/077 = lc,
the identity natural transformation on C. 2) G is separable if and only if there exists £ 6 W = Nat(lx>, -FG) such that e o £ = ID, the identity natural transformation on C. The separability of F implies a Maschke type Theorem for F: if a morphism / e C is such that F(f) has a one-sided inverse in T>, then / has a one-sided inverse inC. A pair of adjoint functors (F, G) is called a Frobenius pair if G is not only a right adjoint, but also a left adjoint of F. The following result can be found in any book on category theory: G is a left adjoint of F if and only if there exist natural transformations v 6 V = ]^(GF, l c ) and C € W - HM(!D, -FG) such that
(1) (2)
for all M e C, AT e P. In order to decide whether F or G is separable, or whether (.F, G) is a Frobenius pair, one has to investigate the natural transformations V — ^j|(G.F, l c ) and W = ^^(l-p,FG). It often happens that the natural transformations in V and W are determined by single maps. In this Section we illustrate this in a classical situation and recover well-known results. In the coming Sections more general situations are considered. Let i : R —» 5 be a ring homomorphism, and let F = • ®/j S : MR —» Ms be the induction functor. The restriction of scalars functor G : MS —+ MR is a right adjoint of F. The unit and counit of the adjunction are VM e MR,
r)M(m)=m®l,
T]M
s — ns. n
Let us describe V and W. Given v : GF —> IMR i V, it is not hard to prove that P = VR : S —> _R is left and right .R-linear. Conversely, given an .R-bimodule map v : S —> .R, a natural transformation z/ € V can be constructed by s) = rnV(s).
VM € MR,
Thus we have (3)
Now let C : IMS
1
2
be in W. Then e = X) e e = Cs(l) e 5 <S>/j 5 satisfies ^se1 e 2 = ^e1 e2s,
(4)
for all s e 5. Conversely if e satisfies (4), then we can recover C 5,
Gv :
In the sequel, we omit the summation symbol, and write e = e1 ® e 2 , where it is understood implicitely that we have a summation. So we have W ^ Wl = {e = e1 ® e2 € S R S
se1
s, for all s e S}.
Doi-Hopf Modules and Entwined Modules
5
Combining all these data, we obtain the following result (cf. [23] for 1) and 2) and [8] for 3))
THEOREM 2.2 Let i : R —> S be a ringhomomorphism, F the induction functor, and G the restriction of scalars functor. 1) F is separable if and only if there exists a conditional expectation, that is 17 g V\ such that Z'(l) = 1, i.e. S/R is a split extension. 2) G is separable if and only if there exists a separability idempotent, that is e G W\ such that e1e2 = 1, i.e. S/R is a separable extension. 3) (F, G) is a Frobenius pair if and only if there exist 17 6 V\ and e € W\ such that
v(el)e2 = elv(e2) = 1.
(5)
Theorem 2.2 2) explains the terminology for separable functors. Theorem 2.2 3) implies the following
COROLLARY 2.3 We use the same notation as in Theorem 2.2. If (F, G) is a Frobenius pair, then S is finitely generated and protective as a (right) R-module. PROOF.- For all s G S, we have s = se 1 z/(e 2 ) = e 1 z7(e 2 s), hence {e1,F(e2*)} is a dual basis for 5 as a right J?-module. D We have a similar property if G is separable. For the proof we refer to [24].
PROPOSITION 2.4 With the same notation as in Theorem 2.2, if S is an algebra over a commutative ring R, S is projective as an R-module and G is separable, then S is finitely generated as an R-module. Using other descriptions of V and W, we find other criteria for F and G to be separable or for (F, G) to be a Frobenius pair. Let Horn R(S, R) be the set of right -R-module homomorphisms from S to R. Hom/j(5, R) is an (R, 5)-bimodule:
(rfs)(t)=rf(ts),
(6)
for all / G EomR(S,R), r G R and s, t £ S. PROPOSITION 2.5 Let i : R —» S be a ringhomomorphism and use the notation introduced above. Then
V = JM(GF, lc) = V2= Horn RiS(S, Horn R(S, R)). PROOF.- Define a j : Vi —» V2 as follows: for v G Vi, let ot\(v} =
Given € 1/2, put a- 1 (^)=^(l). We invite the reader to verify that a\ and a^1 are well-defined and that they are inverses of each other. D
Brzeziriski et al.
PROPOSITION 2.6 Let i : R —> S be a ringhomomorphism and assume that S is finitely generated and projective as a right R-module. Then, with the notation introduced above,
W =^(lTj,FG) ^W2 = RomRtS(~H.omR(S,R),S). PROOF.- Let {sj, Ui \ i = 1, • • • ,m} be a finite dual basis of 5 as a right fi-module. Then for all s 6 S and / e Horn R(S, R),
s - E SiO-i(s]
and / =
i
Define ft : Wl -> W2 by /^(e) = ^, with
for all / G Horn j? (5,7?). To show that is a left .R-linear and right 5-linear map, take any r € R, s e 5 and compute
Conversely, for (^ 6 W% define
Then for all s e 5
SSj®CTj,
i.e., e G V7i. Finally, /3i and /?j~ 1 are inverses of each other since
D
Doi-Hopf Modules and Entwined Modules
7
THEOREM 2.7 Let i : R —> S be a ringhomomorphism. We use the notation introduced above. 1) F : MR —> MS *s separable if and only if there exists 6 V e Wi such that
3) (F, G) is a Frobenius pair if and only if S is finitely generated and projective as a right R-module, and Horn f i ( S , R) and S are isomorphic as (Ry S)-bimodules, i.e.
S/ R is Frobenius. PROOF.- The result is a translation of Theorem 2.2 in terms of V% and W2, using Proposition 2.4 (for 2)) and Corollary 2.3 (for 3)). We prove one implication of 3). Assume that (F,G) is a Frobenius pair. From Corollary 2.3, we know that S is finitely generated and projective. Let v G V\ and e e Wi be as in part 3) of Theorem 2.2, and take ~$ = ai(u) e V2, 4> = fti(e) e W2. For all / 6 Horn #(£,#) and s € 5, we have
and
(4> o 4>)(s) — (s)(ei)e2 = "P(se1)e2 = I?(e1)e2s = s. D
3
ENTWINED MODULES AND DOI-HOPF MODULES
Let k be a commutative ring, A a /c-algebra, C a (flat) fc-coalgebra, and tj} : C®A —> A ® C a fc-linear map. We use the following notation, inspired by the SweedlerHeyneman notation: tfj(c®a) = a,/, ®c*. If the map ijj occurs more than once in the same expression, we also use \I> or \&' as summation indices, i.e., t/>(c ® a) = a>j( ® c
= a^,i ® c
(A, C, ij}) is called a (right-right) entwining structure if the following conditions are satisfied for all a € A and c € C, c**,
(7)
£c (c*)a^ = e c (c)a,
(8)
a,/,® Ac (c*) = 0^(8)0^®^,
(9)
1^® C * = 1® C .
(10)
8
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A /c-module M together with a right A-action and a right C-coaction satisfying the compatibility relation pr(ma) = miQia^, ® rn,,,
(11)
is called an entwined module. The category of entwined modules and A-linear C-
colinear maps is denoted by C = Ai(-0)5- An important class of examples comes from Doi-Koppinen-Hopf structures. A (right-right) Doi-Koppinen-Hopf structure consists of a triple (H,A,C), where H is a /c-bialgebra, A a right H-comodule algebra, and C a right //-module coalgebra. Consider the map ijj : C ® A -^> A®C given by a) = a[0] ® ca[j]. Then (A, C1, tj}) is an entwining structure, and the compatibility relation (11) takes the form pr(ma) = mjojajo] ®m[i]a[ij. (12) A /e-module with an yl-action and a C-coaction satisfying (12) is called a DoiKoppinen-Hopf module or a Doi-Hopf module. Doi-Koppinen-Hopf modules were introduced independently by Doi in [17] and Koppinen in [21]. Properties of these modules were studied extensively during the last decade, see e.g. [10], [11], [12], [13], [14], [15]. Another class of entwining structures is related to coalgebra Galois extensions, see [6] for details. Entwining structures were introduced in [7]. Many
properties of Doi-Hopf modules can be generalized to entwined modules (see e.g. [3], [4]). Although the most studied examples of entwined modules (graded modules, Yetter-Drinfel'd modules, dimodules, Hopf modules) are special cases of Doi-Hopf modules, their properties can be formulated more elegantly in the language of entwined modules. The functor F : C = A^VOS ~~> MA forgetting the C-coaction has a right adjoint G — • C. The structure on G(M) = M <S> C is given by the formulae pr(m®c) (mc)a
— m ® ^ ! ) ®C( 2 ),
(13)
= ma^igic*.
(14)
For later use, we list the unit and counit natural transformations describing the adjunction, p : Ic^GF and e : FG -» !MA,
pM • M->M®C, eN = IN ® sc : N ® C -^ N. In particular, A® C £ M(IJJ)A. A <S> C is also a left A- module, the left A-action is given by a(b ® c) = ab <S> c. This makes A ® C into an object of A-M^)^, the category of entwined modules with an additional left A-action that is right ^-linear and right C-colinear.
The other forgetful functor G' : M(i^)cA -> Mc has a left adjoint F' = • ® A. The structure on F'(N) — N ® A is now given by pr(n®a) ia)6
— np] a^, ® n np
(15)
=
(16)
n®ab.
Doi-Hopf Modules and Entwined Modules
The unit and counit of the adjunction are
H : F'G' -> \c and r? : lMc -» G'F',
HM '• M A —> A, TIN'- N —> N ® A,
^LM (m ® a) = ma, ?7w(n) = n 1.
In particular G'(C) = C0A 6 M(4>)^- The map T/> : C®A —> /ligjC1 is a morphism in A/!(V')^- C® A is also a left C-comodule, the left C-coaction being induced by the comultiplication on C. This coaction is right A-linear and right C-colinear, and thus G ® A is an object of c' M(tp)^ the category of entwined modules together with a right A-linear right C-colinear left C-coaction.
4
THE FUNCTOR FORGETTING THE COACTION
Let (A, C, -0) be a right-right entwining structure, F : M(I/?)A —> MA the functor forgetting the coaction, and G = • C its adjoint. In [13] necessary and sufficient conditions for (F, G) to be a Frobenius pair are given (in the Doi-Hopf case; the results were generalized to the entwining case in [3]), under the additonal assumption that C is projective as a fc-module. In this Section we give an alternative characterization that also holds if C is not necessarily projective, and we find a new proof of the results in [13] and [3]. The method of proof is the same as in Section 2, i.e., based on explicit descriptions of V and W. These descriptions can be found in [10], [11] and [4] in various degrees of generality. To keep this paper self-contained, we give a sketch of proof. We first investigate V = Na,i(GF, lc). Let V\ be the fc-module consisting of all fc-linear maps 6 : C <S> C —> A such that
'®d^),
(17)
d)^®cf,Y
(18)
PROPOSITION 4.1 The map a: V -> Vl given by a(v) = 9, with
9(c ® d) = (IA ® £c)(yA®c(^-A ® c ® d)),
(19)
is an isomorphism of k-modules. The inverse a~l(d) = v is defined as follows: I'M '• M ® C —> M is given by Vf^(m ® c) = m[o]$(77T,[i] ® c).
(20)
PROOF.- Consider v_ = VA®C and v — VC®A- Due to the naturality of v and (7) there is a commutative diagram
C A ® C-
A*
~A
IA >C-
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Write A = (I A ® ?c} ° v and A = (EC I A) ° K- Then it follows that
0(c d) = A(c ® 1 d) = A(l <S> c d). We have seen before that A C* e AM(ip)^. It is easy to prove that GF(A ®C) = A <S> (7 ® C1 e ,4.M(^)5 - the left A-action is induced by the multiplication in A and v_ is a morphism in ^.A/^VOS- Thus v_ and A are left and right A-linear, and *
*
proving (17). To prove (18), we first observe that C®A, GF(C®A) = c ' M(i})}% the left C*-coaction is induced by comultiplication in C in the first factor. Also I? is a rnorphism in CM(I4>)(£, and we conclude that ~D is left and right Ccolinear. Take c,d £ C, and put
Writing down the condition that 17 is left C-colinear, and then applying EC to the second factor, we find that C(i) ® ^(c(2) ® is also right C-colinear, I7(c 1 d ( 1 )) ® Vi. To show that the map a"1 defined by (20) is well-defined, take 0 e Vi, M e C, and let Z/M be given by (20). It needs to be shown that VM G C, i.e., i/^ is right A-linear and right (7-colinear, and that v is a natural transformation. The right yl-linearity follows from (17), and the right C-colinearity from (18). Given any morphism / : M —> N in C, one easily checks that for all m e M and c € C ) ® c) = /(m [0 ])0(m[i] ® c) = /(m [0 ]0(m[ 1 ] ® c)) = j(vM(m ® c)), i.e., z/ is natural. The verification that a and a~l are inverses of each other is left to the reader. D Now we give a description of W — Nat(l/^ A , FG). Let VFi = {z e A ® C \ az = za, for all a 6 A},
i.e., 2 = ^; ai N C given by (25)
PROOF.- We leave the details to the reader; the proof relies on the fact that £4 is left and right A-linear. D In [10], Propositions 4.1 and 4.2 are used to determine when the functor F and its adjoint G are separable.
THEOREM 4.3 Let F : M(ip)% -> MA be the forgetful functor, and G = • 0 C its adjoint. F is separable if and only if there exists 9 6 V\ such that
9 o AC = ECG is separable if and only if there exists z = ^; a; GI € W\ such that
PROOF.- This follows immediately from Propositions 2.1, 4.1 and 4.2
D
Next we show that the fact that (F, G) is a Frobenius pair is also equivalent to the existence of 6 € Vj and z G Wi, but now satisfying different normalizing conditions.
THEOREM 4.4 Let F : M(^}CA -> A^ be the forgetful functor, and G = • ® C its adjoint. Then (F, G) is a Frobenius pair if and only if there exist 9 G V\ and z = y^; ai ® ci G W\ such that the following normalizing condition holds, for all
ec(d)l
= Y^ aid(ci d)
(26)
PROOF.- Suppose that (F, G) is a Frobenius pair. Then there exist v € V and C € W such that (1-2) hold. Let 9 = a(v) € Vj, and z = ]T^ a; ® c; - /?(() 6 Wi. Then (1) can be rewritten as
12
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for all m € M <E M(^)^. Taking M = C ® A, m = d®l, one obtains (27). For all n £ N G MA and € C1, one has
(18) and (2) can be written as (29) for all n € JV e Al/i and d & C. Taking N = A and n = I , one obtains 1 d = Y
Applying EC to the second factor, one finds (26). Conversely, suppose that 9 G V\ and z G W\ satisfy (26) and (27). (27) implies (28), and (26) implies (29). Let i/ = a~l(0], C = P ~ l ( z ) - Then (1-2) hold, and (F, G) is a Frobenius pair. D In [13] it is shown that if (H, A, C) is a Doi-Hopf structure, A is faithfully flat as a fc-module, and C is projective as a fc-module, then C is finitely generated. The next proposition shows that, in fact, one does not need the assumption that C is projective.
PROPOSITION 4.5 Let (A,C,tjj) be a right-right entwining structure. If (F,G) is a Frobenius pair, then A®C is finitely generated and projective as a left A-module. PROOF.- Let 9 and z = Y', aL Q be as in Theorem 4.4. Then for all d G C,
(27)
(7) (9)
(18)
Doi-Hopf Modules and Entwined Modules
Write C((i) ® C;(2) = Y^j^i cij ® cij
13
an
) be a right-right entwining structure, and assume that (F, G) is a Frobenius pair. 1) If A is faithfully flat as a k-module, then C is finitely generated as a k-module. 2) If A is commutative and faithfully flat as a k-module, then C is finitely generated projective as a k-module. 3) If k is a field, then C is finite dimensional as a k-vector space. 4) If A = k, then C is finitely generated projective as a k-module. PROOF.- 1) With notation as in Proposition 4.5, let M be the fc-module generated by the c^.. Then for all d £ C,
Since A is faithfully flat, it follows that d € M , hence M = C is finitely generated. 2) From descent theory: if a fc-module becomes finitely generated and projective after a faithfully flat commutative base extension, then it is itself finitely generated and projective. 3) Follows immediately from 1): since k is a field, A is faithfully flat as a fc-module, and C is projective as a fc-module. 4) Follows immediately from 2). D
Now we want to recover [13, Theorem 2.4] and [3, Proposition 3.5]. Assume that C is finitely generated and projective as a fc-module, and let {d^d* \ i = 1, • • • ,m} be a finite dual basis for C. Then C* A can be made into an object of as follows: for all a,b,b' £ A, c* € C* ,
b(c*®a)b'
=
pr(c*®a)
=
dJ*c'(8ia^®d.
(31)
This can be checked directly. An explanation for this at first sight artificial structure is given in Section 6. We now give alternative descriptions for V and W. Recall from [10] that there are many possibilities to describe V. As we have seen, a
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Brzeziriski et al.
natural transformation v & V is completely determined by 9. Nevertheless, the maps z7, v_^ X or A (with notation as in Proposition 4.1) are possible alternatives. The map A: C®A®C^A induces 0 : A®C-*C*®A^ Horn (C, A). This is the map we need. At some place it is convenient to use C* A as the image space, at some other we prefer Horn (C,A). Note that (/> is given by ~4>(a c)(d) = A(d ® a ® c) = \(a^ d^ c) = a^6(d^ & c), or
(j)(a®c) = 2^d*®ai,e( V-2, with cti(0) =
(l ® d*) (c)
=
This proves that ^ satisfies (17). Before proving (18), we look at the right C-coaction pr o n / = c*(g>a S Horn (C, A) ^ C*(g)A Write pr(f) = /[0](8>/[i] € Horn (C,A)®C. Using (31), we find, for all c e C,
This means that for all / e Horn (C, A) )).
(36)
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Brzezinski et al.
This can be used to show that 9 satisfies (18). Explicitly,
6(c(2)®d}^®c^l}
(36) (4> is right C-colinear)
=
V(c(i) 0(c (2) d))
=
0(1 ® d)[0](c) ®^(1
=
<j>(\ ® d^)(c) iS> d(2)
It remains to be shown that a\ and a^1 are inverses of each other. First take 0 6 Vi. Then for all c, d <E C,
=
6(d®c).
Finally, for 4> eV2, a e A and c, d 6 C:
® c)
a Now we give an alternative description for W%.
PROPOSITION 4.8 Let C be finitely generated and protective as a k-module. Then
W = Wl ^ W2 =Romk£A(C*®A,A®C). The isomorphism j3\ : W\ —> W2 is given by j3i(z) =
) = 4>(e®\}.
(37)
PROOF.- We have to show that @\(z) — is left and right ^.-linear and right Ccolinear. For all c* 6 C* and a, 6 6 A,
l (7)
=
53 a/ a^, 6* is right A-linear, right C-colinear and left C*-linear,
i
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Brzeziriski et al.
and it follows that
€ V^ = Horn ^ A(A C, C* A) such that (/>(! C( 2 ))(c(i)) = e(c)l, for all c e C. In the Doi-Hopf case, this implies the Maschke Theorem in [12]. Now we apply the same procedure to determine when (F, G) is a Frobenius pair.
THEOREM 4.9 Consider an entwining structure (A, C, ijj), and assume that C is finitely generated projective as a k-module. Let F : A / t('0)5 ~* MA be the functor forgetting the C-coaction, and G — • ® C be its right adjoint. Then the following statements are equivalent: 1) (F, G) is a Frobenius pair. 2) There exist z = "^ ai ® ci £ W\ and 9 € Vj such that the maps
A®C and <j6: A ® C —> C* <S> A, given by
l i
are inverses of each other. 3) C* A and A Cg> C are isomorphic as objects in AA / t(V')5PROOF.- 1) => 2). Let z e Wl and d e Vi be as in Theorem 4.4. Then <j) = (3i(z) and (f> = a i (8) are morphisms in A-Mfy)1^, and
(27)
The fact that and are right A-linear and left C*-linear implies that 4>o4> = IC*®ASimilarly, for all c e C,
(18) (26)
Doi-Hopf Modules and Entwined Modules
19
Since and <j> are left A-linear, 0 o = 2) =» 3). Obvious, since c/> and (j> are in 3) =» 1). Let (f> : C* ® A —> A <S> C be the connecting isomorphism, and put
z = (e ® 1) = E; ai ®ci e Wi, 6 = a^-1) e Vi. Applying (32) and (37), one finds e i g i l = ~l(l c)) = ^az6»(cj 0 c (1) ) 0 c (2 ). i Applying e to the second factor, one finds (26). Theorem 4.4 implies that (F, G) is a Frobenius pair. D REMARK 4.10 Recently M. Takeuchi observed that entwined modules can be viewed as comodules over certain corings. This observation has been exploited in [5] to derive some properties of coring counterparts of functors F and G. It is quite clear that the procedure applied in Section 2 to extension and restriction of scalars can be adapted to functors associated to corings, leading to a generalization of the results in this Section. This will be the subject of a future publication.
5
THE FUNCTOR FORGETTING THE A-ACTION
Again, let (A,C,il>) be a right-right entwining structure. The functor
G' : forgetting the A-action has a left adjoint F' . The unit p, and the counit 77 of the adjunction are given at the end of Section 3.
LEMMA 5.1 Let M e AM(^}% N e c' M(if)}cA.
Then F'G'(M) e A.MWOS
and
C
G'F' 6 .A/1 (•) ^. The left structures are given by
a(m 0 6 ) = am 0 b and pl(n 06) = /or a// a, 6 6 ^4, m £ M, n £ AT. Furthermore HM is left A-linear, and VN is left C-colinear.
Now write V = ^(G'-F', IA^ C ). W7' = MM(lc,^'G')- Following the philosophy of the previous Sections, we give more explicit descriptions of V and W . We do not give detailed proofs, however, since the arguments are dual to the ones in the previous Section. Let
V{ = {$ £ (C ® A)*
^(c (1) ®a^)c* 2) =tf(c ( 2 ) 0a)c ( 1 ) , for all c e C,aeA}. (40)
PROPOSITION 5.2 The map a : V -> V{, a(v'} =eovc is an isomorphism.
Brzeziriski et al.
20
PROOF.- Details are left to the reader. Given $ € V, for N & M.c, the natural map v'N : N A —> IV is
n For any /c-linear map e : C —> A A and c 6 C, we use the notation e(c) = e 1 (c) ®e 2 (c) (summation understood). Let W[ be the fc-submodule of Horn (C, A A) consisting of maps e satisfying e 1 (c ( 1 ) )(g>e 2 (c ( 1 ) )(8)c ( 2 )
e1
=
"(I)'
(41) (42)
PROPOSITION 5.3 The map 8 : W -> W{ given by
is an isomorphism. Given e £ W[, C,1 = /3~^(e) is recovered from e as follows: for
M
PROOF.- We show that 8 is well-defined, leaving other details to the reader. Consider a commutative diagram
A-
C-
•~A C J^^..^ ® C ® A
3ft and right C-colinear. Write A(c) The map A = CC^A ° (7c C o,- ® a!i • Then C(i) A(c( 2 )) = ^ ci(1) ig) c i(2 ) ® a, i
Applying e to the second factor, one finds C ( i ) ® e ( c ( 2 ) ) = A(c).
The right C-colinearity of A implies that a
(gi c *
Doi-Hopf Modules and Entwined Modules
21
and hence proves (41). To prove (42) note that A = CA®C ° (nA <S> IG) is left right A-linear, hence e 1 (c)®e 2 (c)a
=
(IA ® £ ® /A)(Cii®c(( 1 ® c ) a ))
D
PROPOSITION 5.4 Let (A,C,ip) be a right-right entwining structure. 1 ) F ' = • ® A : Jv[c —> .M (VOS is separable if and only if there exists "9 G V[ such that for all c G C, i?(c&l)=e(c). (43)
2) G' : M^)^ —> M° is separable if and only if there exists e G W[ such that for all c a (0 ).
22
Brzeziriski et al.
PROPOSITION 5.5 Let (A,C,ip) be a right-right entwining structure. With notation as above, assume that (F', G') is a Frobenius pair. If there exists c & C such that e(c) = I , and if tjj is invertible, with inverse C (x) A, then A is finitely generated and projective as a k-module. PROOF.- Observe first that (A, C, c (1) C( 2 ),
(52)
/ *
iooi
\ y^ /o * /o\ T^
\d , &iih )C( -\\ 09 a^ Qv C / r j \ .
/CCQ°\
We now give alternative descriptions of V and W.
PROPOSITION 5.7 Let (A,C,il>) be a right-right entwining structure, and assume that A is finitely generated and projective as a k-module. Then there is an isomorphism Pi '• W[ —> W!} = Horn kA(A* ®C,C® A),
Doi-Hopf Modules and Entwined Modules
23
Pie = f l , with
The inverse of f3\ is given by J3l l (fi) = e with i
PROOF.- We first prove that (3i is well-defined. a) 0i(e) = ft is right A-linear: for all a* e A*, c e C and 6 6 A, we have
\^/
(42)
—
\
5 ^"ibib
=
(a*,e 1 (c ( 2 ) )*}cf 1 ) ® e 2 (c ( 2 ) )fe
b) /?i(e) = fi is right C-colinear: for all a* 6 A* and c e C1, we have
(9) (41)
, =
(
^^,
( l )
(3)$
(a*,e 1 (c ( 2 ) )^)cj' 1 ' ) ®e 2 (c ( 2 ) )®c ( 3 ) ) C( 2 ).
c
e
) /?i( ) = fi is left C-colinear: for all a* e A* and c e C, we have
(9)
The proof that /3f ^Jl) = e satisfies (41) and (42) is left to the reader. The maps /?i and /9f 1 are inverses of each other since
(a*,e 1 (c))a l (g)e 2 (c) = el(c) ® e 2 (c),
24
Brzeziriski et al.
) ((a*, (ai
At the last step, we used that for all c 6 C and a £ A, ®a) = c®a. D
PROPOSITION 5.8 Let (A,C,^) be a right-right entwining structure If A is a finitely generated projective k -module, then the map
a2 G Ws such that
6 1 6 2 imodule structure (cf. (6)):
(cf(b#a))(d)=cf(db)a, for all a,c G C and b, d e B. From Proposition 2.5, we deduce that
V ^ V2 - 1/4 = Horn ^^(SfeA, Horn (£,>!)).
(65)
If B is finitely generated and projective as a /c-module, then we find using Proposition 2.6
W ^ W2 B#RA(Rom(B,A),B#RA). Theorem 2.7 now takes the following form:
(66)
Doi-Hopf Modules and Entwined Modules
THEOREM 6.2 Let (B,A,R) be a ring k, and assume that B is finitely {bi, b* | i — 1, • • • , 771} be a finite dual 1) B^RA/A is separable if and only
29
factorization structure over a commutative generated and projective as a k-module. Let basis for B. if there exists an (A, B#RA}-bimodule map
(j>: Horn (B, A) ^ B* A -> B#RA such that
2) B^RA/A is split if and only if there exists an (A, B^RA)-bimodule map B#RA -» Horn (B, A) such that
3) B#RA/A is Frobenius if and only if B* ® A and B#RA are isomorphic as (A,B#RA)-bimodules.
This is also equivalent to the existence of K € ¥3, e =
b1 ig> b2 a2 £ W% such that the maps
and
!>: B#RA^Kom(B,A),
$(b#a)(d) = K(bdR)aR
are inverses of each other. The same method can be applied to the extension B#RA/B. There are two ways to proceed: as above, but applying the left-handed version of Theorem 2.7 (left and right separable (resp. Frobenius) extension coincide). Another possibility is to use "op" -arguments. If R : A®B —> B®A makes (B, A, R) into a factorization structure, then
R: Bop® A°p -> A°p <S> B°p makes (A°P,B°P,R) into a factorization structure. It is not hard to see that there is an algebra isomorphism
(A0v#ABop)op * B#RA. Using the left-right symmetry again, we find that B#RA/ B is Frobenius if and only if (A 0 P#^B°P)°P/B is Frobenius if and only if (Aop#RBop)/Bop is Frobenius, and we can apply Theorems 6.1 and 6.2. We invite the reader to write down explicit results. Our final aim is to link the results in this Section to the ones in Section 4, at least in the case of finitely generated, projective B. Let (A,C,if)) be a right-right entwining structure, with C finitely generated and projective, and put B = (C*)°p. Let {ci,c* i = 1, . . . , n} be a dual basis for C. There is a bijective correspondence between right-right entwining structures (A, C, tj}} and smash product structures (C*op, A,R). R and ip can be recovered from each other using the formulae
J?(a®c*) =
(c*,cf)cJ(gia V ) ,
tp(c® a) =
((c*)fl,c) Ci aR.
30
Brzeziriski et al.
Moreover, there are isomorphisms of categories
and In particular, B^^A can be made into an object of ^.A/f ('!/') 5, and this explains the structure on C* ® A used in Section 4. Combining Theorems 4.9 and 6.2, we find that the forgetful functor M^}1^ —» MA and its adjoint form a Frobenius pair if and only if C* ® A and A C are isomorphic as (A, (C*) op #fl J 4)-bimodules, if and only if the extension (C*)op#RA/A is Frobenius.
REFERENCES [1] K.I. Beidar, Y. Fong and A. Stolin, On Frobenius algebras and the Yang-Baxter equation, Trans. Amer. Math. Soc. 349 (1997), 3823-3836.
[2] T. Brzezinski, On modules associated to coalgebra-Galois extensions, J. Algebra 215 (1999), 290-317. [3] T. Brzezinski, Frobenius properties and Maschke-type theorems for entwined modules, Proc. Amer. Math. Soc., to appear. [4] T. Brzeziriski, Coalgebra-Galois extensions from the extension point of view, in "Hopf algebras and quantum groups", S. Caenepeel and F. Van Oystaeyen (Eds.), Lee. Notes Pure Appl. Math. 209, Marcel Dekker, New York, 2000.
[5] T. Brzezinski, The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties, preprint math.RA/0002105. [6] T. Brzeziriski and P. M. Hajac, Coalgebra extensions and algebra coextensions of Galois type, Comm. Algebra 27 (1999), 1347-1367.
[7] T. Brzeziriski and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), 467-492. [8] S. Caenepeel, B. Ion and G. Militaru, The structure of Frobenius algebras and separable algebras, K-theory, to appear. [9] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Smash biproducts of algebras and coalgebras, Algebras and Representation Theory 3 (2000), 19-42. [10] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules, Applications, Adv. Math. 145 (1999), 239-290. [11] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for the category of Doi-Hopf modules II, in "Hopf algebras and quantum groups" , S. Caenepeel and F. Van Oystaeyen (Eds.), Lect. Notes Puere Appl. Math. 209 Marcel Dekker, New York, 2000.
[12] S. Caenepeel, G. Militaru, and S. Zhu, A Maschke type theorem for Doi-Hopf modules, J. Algebra 187 (1997), 388-412. [13] S. Caenepeel, G. Militaru, and S. Zhu, Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997), 4311-4342.
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[14] S. Caenepeel, G. Militaru, and S. Zhu, Crossed modules and Doi-Hopf modules, Israel J. Math. 100 (1997), 221-247. [15] S. Caenepeel and §. Raianu, Induction functors for the Doi-Koppinen unified
Hopf modules, in "Abelian groups and Modules", A. Facchini and C. Menini (Eds.), Kluwer Academic Publishers, Dordrecht, 1995, p. 73-94.
[16] A. Cap, H. Schichl and J. Vanzura. On twisted tensor product of algebras. Common. Algebra 23 (1995), 4701-4735.
[17] Y. Doi, Unifying Hopf modules, J. Algebra 153 (1992), 373-385. [18] L. Kadison, The Jones polynomial and certain separable Frobenius extensions, J. Algebra 186 (1996), 461-475.
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Computing the Gelfand-Kirillov Dimension II J. L. BUESO, J. GOMEZ-TORRECILLAS and F.J. LOBILLO, Departamento de Algebra, Universidad de Granada. E18071-Granada. Spain. E-mail: [email protected]
1
INTRODUCTION
This paper has a twofold goal and a mixed nature. We propose an algorithm to compute the Gelfand-Kirillov dimension for finitely generated modules over solvable polynomial algebras and, from the theoretical point of view, we characterize these algebras within the class of all filtered algebras. When looking for a notion of dimension for modules over a non-commutative algebra A which allows its effective computation, it is quite natural to look at the existing algorithms in Commutative Algebra. It seems reasonable to try first the computation of the dimension for the algebra A itself. In the case of commutative finitely generated algebras over a field k, the problem is equivalent to the computation of the degree of the Hilbert polynomial of k [ x j , . . . , xn]/I, where / is an ideal of the commutative multivariable polynomial algebra k [ x j , . . . ,xn], and this can be done effectively by means of the computation of a Grobner basis for / (see [2, Section 9.3] and its references). These ideas can be exported from the case of cyclic k xi,... ,xn -modules to finitely generated ones (see [15, Section 4]). From this point of view, the notion of dimension for modules which extends properly to noncommutative algebras is the Gelfand-Kirillov dimension. The techniques from the commutative case can be easily adapted to compute the Gelfand-Kirillov dimension in the case of quantum affine spaces, namely, finitely generated k-algebras defined by relations of the type XjXi = qjiXiXj, for some nonzero scalars QJJ € k (see e.g., [17], where quantum affine spaces are called homogeneous solvable polynomial kalgebras). When dealing with more general non-commutative k-algebras, a fruitful idea is to consider R to be a finitely generated k-algebra, with finite-dimensional generating vector subspace V, in such a way that the graded algebra gr(R) associated to the standard filtration is a multivariable commutative polynomial algebra (see [6]) or a quantum affine space (see [12]). 33
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Bueso et al.
However, a very simple example like the Jordan plane defined by two generators x,y, subject to the relation yx — xy + x2 shows that the former techniques, based on standard filtrations with (semi)commutative associated algebras, had to be improved. In the Spring of 1996, the first author presented in the SAGA4 held in Antwerp-Brussels an algorithm to compute the Gelfand-Kirillov dimension for cyclic modules over solvable polynomials algebras ([18]) with respect to the degree lexicographical order (see [4, 5]), which is applicable to some quantum groups whose generators are subject to quadratic relations, like quantum matrices, quantum symplectic space or quantum euclidean space). A similar approach appeared in the later paper [17, Section 6]. However, there is a mistake in the basic result there [17, Lemma 6.1] which is carried on the whole section. An easy counterexample to [17, Lemma 6.1] is the aforementioned Jordan plane. The algorithm given in [4, 5] was extended to finitely generated modules over solvable polynomial algebras (called there PBW algebras) with respect to a weighted graded lexicographical order in [23, 7], which allows to handle algebras with nonquadratic relations (the simplest example is given by the commutation relations yx = xy + x3). Here we show that this algorithm can be used to compute the Gelfand-Kirillov dimension of any finitely generated module over any solvable polynomial algebra, without restrictions on the given term order. During our search for that algorithm we have discovered some interesting results on filtered algebras which, in particular, locate the solvable polynomial algebras (or, equivalently, the PBW algebras) in the theory of noetherian rings: they are precisely those algebras having a filtration (in most cases, non standard) whose associated graded algebras are graded quantum spaces. We also show that these algebras can always be re-filtered by finite-dimensional vector subspaces keeping a quantum space as associated graded algebra. This allows to use the well-known graded-filtered techniques to obtain that any solvable polynomial algebra R is an Auslander-regular noetherian algebra with exact and finitely partitive Gelfand-Kirillov dimension. Moreover, R is a Cohen-Macaulay algebra which satisfies the Nullstellensatz.
2
ADMISSIBLE ORDERS IN MONOIDEALS AND STABLE SUBSETS
Let N denote the additive monoid of all positive integers (including the neutral element 0). Let n be a strictly positive integer. We consider Nra as additive monoid with the sum defined componentwise. Let 61,... ,en be the canonical basis of this free abelian monoid.
DEFINITION 2.1 An admissible order < on (N™, +) is a total order such that, (a) 0 = ( 0 , . . . , 0) ^ a for every a e N".
(b) For all a, (3,7 e N™ with a ^ f3 it follows 0 + 7 ^ / 3 + 7. REMARK 2.2 By Dickson's Lemma (see, e.g., [2, Corollary 4.48]), admissible orders on Nn are good orders (i.e., any non-empty subset of N™ has a first element). By ^iex we denote the lexicographical order with € j ~ N, consider its dimension or growth degree denned as (3)
logra
which, of course, need not be finite. Some basic properties of this invariant can be found in [24], [27, Chapter 8] or [20]. DEFINITION 2.6 For any stable subset E C N">( m ) and any weight vector w with strictly positive integer components, we define the Hilbert function HF^ : N —» N of E relative to w by putting HF£(s) = card{(a, i) e N n >( m > \E
\a w ^ s}
for every s € N. In the case that w = (1, . . . , 1) we shall use the notation HF# for the Hilbert function.
In the proof of the next Lemma we shall use the notation |(o;,z)| w = |a w. By N™ we denote the subset of N" consisting of those vectors with all their components strictly positive.
LEMMA 2.7 Let E C N"'(m' be a stable subset and let w € N™ . Then d(HF£) = d(HFE). PROOF. Consider w = max{wj, . . . ,u>n}. If |(a,i)| < s, then |(a,z)|a, < ws and, thus, HFE(S) < HF^(ws). Moreover, it follows from
that \(a,i)\u < s implies |(a,i)| < s, whence HF%(s) < HFE(s).
D
By Lemma 2.7, the following definition makes sense.
DEFINITION 2.8 The dimension of a non-empty stable subset E of N n '( m ) is denned as dim(£) = d(HF^), for any w e N£. The study of the Hilbert function of a stable subset can be reduced to monoideals, as the following result shows.
PROPOSITION 2.9 Let E be a stable subset o/N™'( m ) and consider the decomposition
given in Lemma 2.4- Then
HF£ = HF£ + • - • + HF^m
(4)
dim(E) = max{dim(B1), . . . , dim(£;m)}
(5)
and
Computing the Gelfand-Kirillov Dimension
37
PROOF. The formula (4) is clear, since (a,i) ^ E if and only if a 0 and let r G -FS.R \ Fs_iR. Since r + Fs-iR € gr(/?)s and the elements yi, . . . , yn are homogeneous generators for gr(/?) we obtain that
|a|u =s
|a|u =s
By the induction hypothesis, f/
\7
T T*'~* £^ Pg _ -i1 -*/-?^ — I QtJL> tz X —
\ /
ITT*^ .IS.**'
And, therefore,
H U ^n
The equality follows from the fact that FR = {FsR}s^o is a filtration and Xi 6 -FUi-R for z = 1 , . . . , s. In particular, we have proved that xi,..., xn generate R as an algebra over k. For i,j such that 1 ^ i < j ^ s, there is a non-zero scalar qji such that yjj/i = qjiyiyj. Therefore,
x-Xi + Fu.+u.-iR = (x- + Fu.-iR)(xi + Fu.-iR) = which implies that ir jit
J
M j *•
'
j
"i i "'J
^
/
^
In particular, pji has a standard representation such that exp(pjj) ^ u e^ + e.,-. 2. We just need to prove that the monomials xa are linearly independent over k. Assume that the monomials xa are not k-linearly independent. We derive a contradiction as follows. There is a non trivial expression 5ZQewn caXa = 0, that can be chosen of minimal exponent (3. Let s = \/3\u, then 0= |a|u =s
|a|u <s
Computing the Gelfand-Kirillov Dimension
41
which gives in gr(_R) s the following non trivial relation
E
v——^
caxa + Fs_lR=
\a\a=s
2_,
c
aya
\a\a=s
which contradicts the linear independence of the monomials ya.
D
REMARK 3.7 If the filtration FR in Proposition 3.6 is assumed to be finite dimensional then, necessarily, the vector u has all its components strictly positive.
The following result, although not explicitly stated there, was first proved by V. Weispfenning in [33, proof of Theorem 1]. We include a slightly different proof here.
PROPOSITION 3.8 Let R be a k-algebra finitely generated by x i , . . . , x n and assume that there are non zero qji € k, polynomials pji € R and an admissible order ^ on N™ such that the following ^-bounded quantum relations are satisfied for 1 ^ i < j ^ n with exp(pjj) -< €j + €j
Then there is w = (wi, . . . , wn) & N™ such that \exp(pji)\w < Wi+Wj. In particular, if R = k{zi, . . . , xn; Q, ^} is a PBW algebra, then R = k{xi, . . . , xn; Q, ^w} is a PBW algebra. PROOF. Let i,j be such that 1 ^ i < j «, a + /3. But this follows from the definition of ^w. 2. For each s Jj 0, the s-th homogeneous component of the graded algebra gr(jR) is generated as a k-vector space by xa + F^.1R, with |o: w = n, and we know that xa + F^R = ya, where y > = X i + F^^R for i = 1, . . . , s. Therefore, yi , . . . , yn are homogeneous generators for gr(R). 3. We have the following computation
yjyi = (Xj + F^.^RKxi + F^R) = x,x{ + F^+W]_,R = (qjiX.Xj + F^.^.^R) + (Pji + F^.^.^R) = q^y-j + %
(11)
Moreover, since N(hji) C N(pji) we get that exp(hji) ^w exp(pji) -<w e* + €j. 4. We just need to prove that the monomials ya are k-linearly independent. Since grw(R) is a graded algebra, it suffices to prove that every linear combination of homogeneous monomials of the form Xliaj =scaya = 0 is necessarily trivial. But this is a consequence of the linear independence of the monomials xa , since we have
REMARKS 3.11 1. When computing the relations Qw it is understood that we compute a standard representation of hji with respect to the standard monomials ya. This is very easy, in fact, if pj% = £QaQa:a then % = Y.\a\v =wi+Wi a «J/ Q 2. For every positive integer s it can easily be proven that
F?R={feR
|exp(/)U<s}
(12)
3. Let R — k{xi, . . . ,xn; Q, ^w} be a PBW algebra. It is easy to see that the filtration FWR is standard if and only if w = (1, . . . , 1), i.e., -^w is just the degree lexicographical order. ^ 4. There is an analog to Proposition 3. 10. (4) for the Rees algebra R associated to R = k{xi, . . . ,xn; Q, ^<w}, which was proved in [9, Teorema 2.6.3]. Thus, R = k[t,Xi, . . . ,xn;Q,^w] is a PBW algebra, where it is a new central variable, Xj = tWiXi for every 1 ^ i ^ n, w = (l,to), and the new ^-bounded quantum relations are
where — / T •TJ• — ^ - • T . ' r - 4— X j — yjz^i^j T^
\
/
,->, o^l ^A"' j
The foregoing propositions can be restated in a more compact form which is interesting from a theoretical point of view. We give two theorems. The first one explains, from the point of view of algebras given by generators and relations, when it is possible to find a filtration with semi-commutative associated graded algebra.
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THEOREM 3.12 The following conditions are equivalent for a k~algebra R. (i) R is filtered by a finite-dimensional filtration with semi-commutative associated graded algebra. (ii) R is filtered with semi-commutative associated graded algebra. (in) R satisfies a set Q of ^.-bounded quantum relations for some admissible order X. (iv) R satisfies a set Q of ^<w-bounded quantum relations for some vector w € N™.
PROOF,
(i) =^> (ii) is obvious.
(ii) => (iii) is given by Proposition 3.6.(1). (iii) => (iv). This is given by Proposition 3.8 (iv) =>• (i). By Proposition 3.8, w can be assumed to be such that \exp(pji)\w < Wi + Wj for every 1 ^ i < j Q such that, for every chain M = M0 3 MI D • • • D Mm
with GKdim(Mi/Mi + i) = GKdim(M), one has m < n. R has the radical property if every element in the Jacobson radical is nilpotent. R has the endomorphism property over k if for each simple left R-module M, End(M) is algebraic over k. If both properties are satisfied by R, then R satisfies the Nullstellensatz over k. A finitely generated left or right .R-module M satisfies the Auslander condition if for every integer n and every submodule TV of Ext^(M, R), we have Ext^(Ar, R) = 0 for every i < n. A noetherian ring with finite injective dimension is said to be Auslander-Gorenstein if every finitely generated left or right .R-module satisfies the Auslander condition. If in addition R has finite global homological dimension, we say that R is Auslander regular. Finally, if M is a left or right .R-module, the grade number of M is defined as
j ( M ) = inf{i I Ext*fl(M,.R) ^ 0} e N U {+00} The noetherian k-algebra R is Cohen-Macaulay if for all non-zero finitely generated left or right .R-modules M,
j ( M ) + GKdim(M) = GKdim(R) THEOREM 4.1 Let R be a k-algebra satisfying a set of bounded quantum relations. Then the following statements hold. (1) The algebra R is left and right noetherian.
(2) The Gelfand-Kirillov dimension is exact on short exact sequences of finitely generated left or right R-modules.
(3) The algebra R is left and right finitely partitive. (4) The Krull dimension of every finitely generated R-module is bounded by its Gelfand-Kirillov dimension. (5) The algebra R satisfies the Nullstellensatz over k.
// R is any PBW algebra, then, in addition to the foregoing properties, R is an Auslander-regular and Cohen-Macaulay algebra.
PROOF. By Theorem 3.12, R has a finite-dimensional filtration with semicommutative associated graded algebra. The noetherianity of R follows from [25, 1.6.9]. The exactness of Gelfand-Kirillov dimension on short exact sequences of finitely generated left or right .R-modules follows from [32, 4.4 Theoreme] or [26, 3.8]. R is left and right finitely partitive by [16, Theorem 2.9]. The bound on the Krull dimension can be obtained from [25, 8.3.18] (see [31] for the original citation). The Nullstellensatz follows from [26, 3.8]. Let R now be a PBW algebra. Then R has a finite-dimensional filtration such that gr(.R) is an affine quantum space. Using [22,
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3.6 Theorem] (see also [13, Theorem 4.2]), it is easy to see that gr(/?) is AuslanderGorenstein, so by [3, 3.9 Theorem] R is Auslander-Gorenstein. Moreover, by [25, 7.5.3] (see also [14]) gr(/?) has global finite dimension, so R is Auslander regular by [25, 7.6.18] (see also [30]). Finally, by [3, 3.8 Theorem] and [25, 8.6.5], R is Cohen-Macaulay if and only if gr(R) is Cohen-Macaulay, but this follows from [22, 5.10]
D
The following are examples of PBW algebras and, hence, they enjoy the properties listed in Theorem 4.1. In each case, we include an explicit vector weight w with strictly positive integer components in such a way that the filtration given in Proposition 3.10 yields a graded quantum affine space as associated graded algebra. • Enveloping algebras of finite-dimensional Lie algebras. Let 0 be a finite-dimensional Lie k-algebra with basis xi,... ,xn. Then the enveloping algebra U(Q) = k{xi,..., x n ; Q, ^.w} is a PBW algebra, where
i < j ^ n} and w = ( 1 , . . . , 1). • Quantum matrices are a special case of the algebra H(p, A) defined in [1]. H(p, A) = k{ziQ | 1 ^ i, a ^ p; Q, ^w} is a PBW algebra, where the relations are
+ (A - l)pjiXif}Xja
if j > i, /3 > a if j > z, a < j3 if j = i, (3 > a
and where the weight vector can be expressed here as a weight matrix:
w=
.
op+i
2
P+2
(2 i+Q ) 1
