ring theory and algebraic geometry: proceedings of the fifth international conference

ring theory andalgebraic geometry

Marcel Dekker, Inc. New York • Basel

edited by

Ángel GranjaUniversity of León

León, Spain

José Ángel HermidaUniversity of León

León, Spain

Alain VerschorenUniversity of Antwerp, RUCA

Antwerp, Belgium

TM

ring theory andalgebraic geometry

proceedings of the fifth international conference (SAGA V) in León, Spain

Copyright © 2001 by Marcel Dekker, Inc. All Rights Reserved.

ISBN: 0-8247-0559-9

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PURE AND APPLIED MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS

Earl J. TaftRutgers University

New Brunswick, New Jersey

Zuhair NashedUniversity of Central Florida

Orlando, Florida

EDITORIAL BOARD

M. S. BaouendiUniversity of California,

San Diego

Jane CroninRutgers University

Jack K. HaleGeorgia Institute of Technology

S. KobayashiUniversity of California,

Berkeley

Marvin MarcusUniversity of California,

Santa Barbara

W. S. MasseyYale University

Anil NerodeCornell University

Donald PassmanUniversity of Wisconsin,Madison

Fred S. RobertsRutgers University

David L. RussellVirginia Polytechnic Instituteand State University

Walter SchemppUniversität Siegen

Mark TeplyUniversity of Wisconsin,Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. N. Jacobson, Exceptional Lie Algebras2. L.-Å. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis3. I. Satake, Classification Theory of Semi-Simple Algebraic Groups4. F. Hirzebruch et al., Differentiable Manifolds and Quadratic Forms5. I. Chavel, Riemannian Symmetric Spaces of Rank One6. R. B. Burckel, Characterization of C(X) Among Its Subalgebras7. B. R. McDonald et al., Ring Theory8. Y.-T. Siu, Techniques of Extension on Analytic Objects9. S. R. Caradus et al., Calkin Algebras and Algebras of Operators on Banach Spaces

10. E. O. Roxin et al., Differential Games and Control Theory11. M. Orzech and C. Small, The Brauer Group of Commutative Rings12. S. Thomier, Topology and Its Applications13. J. M. Lopez and K. A. Ross, Sidon Sets14. W. W. Comfort and S. Negrepontis, Continuous Pseudometrics15. K. McKennon and J. M. Robertson, Locally Convex Spaces16. M. Carmeli and S. Malin, Representations of the Rotation and Lorentz Groups17. G. B. Seligman, Rational Methods in Lie Algebras18. D. G. de Figueiredo, Functional Analysis19. L. Cesari et al., Nonlinear Functional Analysis and Differential Equations20. J. J. Schäffer, Geometry of Spheres in Normed Spaces21. K. Yano and M. Kon, Anti-Invariant Submanifolds22. W. V. Vasconcelos, The Rings of Dimension Two23. R. E. Chandler, Hausdorff Compactifications24. S. P. Franklin and B. V. S. Thomas, Topology25. S. K. Jain, Ring Theory26. B. R. McDonald and R. A. Morris, Ring Theory II27. R. B. Mura and A. Rhemtulla, Orderable Groups28. J. R. Graef, Stability of Dynamical Systems29. H.-C. Wang, Homogeneous Branch Algebras30. E. O. Roxin et al., Differential Games and Control Theory II31. R. D. Porter, Introduction to Fibre Bundles32. M. Altman, Contractors and Contractor Directions Theory and Applications33. J. S. Golan, Decomposition and Dimension in Module Categories34. G. Fairweather, Finite Element Galerkin Methods for Differential Equations35. J. D. Sally, Numbers of Generators of Ideals in Local Rings36. S. S. Miller, Complex Analysis37. R. Gordon, Representation Theory of Algebras38. M. Goto and F. D. Grosshans, Semisimple Lie Algebras39. A. I. Arruda et al., Mathematical Logic40. F. Van Oystaeyen, Ring Theory41. F. Van Oystaeyen and A. Verschoren, Reflectors and Localization42. M. Satyanarayana, Positively Ordered Semigroups43. D. L Russell, Mathematics of Finite-Dimensional Control Systems44. P.-T. Liu and E. Roxin, Differential Games and Control Theory III45. A. Geramita and J. Seberry, Orthogonal Designs46. J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach

Spaces47. P.-T. Liu and J. G. Sutinen, Control Theory in Mathematical Economics48. C. Byrnes, Partial Differential Equations and Geometry49. G. Klambauer, Problems and Propositions in Analysis50. J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields51. F. Van Oystaeyen, Ring Theory52. B. Kadem, Binary Time Series53. J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems54. R. L. Sternberg et al., Nonlinear Partial Differential Equations in Engineering and Applied Science55. B. R. McDonald, Ring Theory and Algebra III56. J. S. Golan, Structure Sheaves Over a Noncommutative Ring57. T. V. Narayana et al., Combinatorics, Representation Theory and Statistical Methods in Groups58. T. A. Burton, Modeling and Differential Equations in Biology59. K. H. Kim and F. W. Roush, Introduction to Mathematical Consensus Theory60. J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces

61. O. A. Nielson, Direct Integral Theory62. J. E. Smith et al., Ordered Groups63. J. Cronin, Mathematics of Cell Electrophysiology64. J. W. Brewer, Power Series Over Commutative Rings65. P. K. Kamthan and M. Gupta, Sequence Spaces and Series66. T. G. McLaughlin, Regressive Sets and the Theory of Isols67. T. L. Herdman et al., Integral and Functional Differential Equations68. R. Draper, Commutative Algebra69. W. G. McKay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Repre-

sentations of Simple Lie Algebras70. R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems71. J. Van Geel, Places and Valuations in Noncommutative Ring Theory72. C. Faith, Injective Modules and Injective Quotient Rings73. A. Fiacco, Mathematical Programming with Data Perturbations I74. P. Schultz et al., Algebraic Structures and Applications75. L Bican et al., Rings, Modules, and Preradicals76. D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry77. P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces78. C.-C. Yang, Factorization Theory of Meromorphic Functions79. O. Taussky, Ternary Quadratic Forms and Norms80. S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications81. K. B. Hannsgen et al., Volterra and Functional Differential Equations82. N. L. Johnson et al., Finite Geometries83. G. I. Zapata, Functional Analysis, Holomorphy, and Approximation Theory84. S. Greco and G. Valla, Commutative Algebra85. A. V. Fiacco, Mathematical Programming with Data Perturbations II86. J.-B. Hiriart-Urruty et al., Optimization87. A. Figa Talamanca and M. A. Picardello, Harmonic Analysis on Free Groups88. M. Harada, Factor Categories with Applications to Direct Decomposition of Modules89. V. I.Istra'tescu, Strict Convexity and Complex Strict Convexity90. V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations91. H. L. Manocha and J. B. Srivastava, Algebra and Its Applications92. D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic

Problems93. J. W. Longley, Least Squares Computations Using Orthogonalization Methods94. L. P. de Alcantara, Mathematical Logic and Formal Systems95. C. E. Aull, Rings of Continuous Functions96. R. Chuaqui, Analysis, Geometry, and Probability97. L. Fuchs and L. Salce, Modules Over Valuation Domains98. P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics99. W. B. Powell and C. Tsinakis, Ordered Algebraic Structures

100. G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and TheirApplications

101. R.-E. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications102. J. H. Lightbourne III and S. M. Rankin III, Physical Mathematics and Nonlinear Partial Differential

Equations103. C. A. Baker and L. M. Batten, Finite Geometrics104. J. W. Brewer et al., Linear Systems Over Commutative Rings105. C. McCrory and T. Shifrin, Geometry and Topology106. D. W. Kueke et al., Mathematical Logic and Theoretical Computer Science107. B.-L. Lin and S. Simons, Nonlinear and Convex Analysis108. S. J. Lee, Operator Methods for Optimal Control Problems109. V. Lakshmikantham, Nonlinear Analysis and Applications110. S. F. McCormick, Multigrid Methods111. M. C. Tangora, Computers in Algebra112. D. V. Chudnovsky and G. V. Chudnovsky, Search Theory113. D. V. Chudnovsky and R. D. Jenks, Computer Algebra114. M. C. Tangora, Computers in Geometry and Topology115. P. Nelson et al., Transport Theory, Invariant Imbedding, and Integral Equations116. P. Clément et al., Semigroup Theory and Applications117. J. Vinuesa, Orthogonal Polynomials and Their Applications118. C. M. Dafermos et al., Differential Equations119. E. O. Roxin, Modern Optimal Control120. J. C. Díaz, Mathematics for Large Scale Computing

121. P. S. MilojevicÚ, Nonlinear Functional Analysis122. C. Sadosky, Analysis and Partial Differential Equations

123. R. M. Shortt, General Topology and Applications124. R. Wong, Asymptotic and Computational Analysis125. D. V. Chudnovsky and R. D. Jenks, Computers in Mathematics126. W. D. Wallis et al., Combinatorial Designs and Applications127. S. Elaydi, Differential Equations128. G. Chen et al., Distributed Parameter Control Systems129. W. N. Everitt, Inequalities130. H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differ-

ential Equations131. O. Arino et al., Mathematical Population Dynamics132. S. Coen, Geometry and Complex Variables133. J. A. Goldstein et al., Differential Equations with Applications in Biology, Physics, and

Engineering134. S. J. Andima et al., General Topology and Applications135. P Clément et al., Semigroup Theory and Evolution Equations136. K. Jarosz, Function Spaces137. J. M. Bayod et al., p-adic Functional Analysis138. G. A. Anastassiou, Approximation Theory139. R. S. Rees, Graphs, Matrices, and Designs140. G. Abrams et al., Methods in Module Theory141. G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications

and Computing142. M. C. Joshi and A. V. Balakrishnan, Mathematical Theory of Control143. G. Komatsu and Y. Sakane, Complex Geometry144. I. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations145. T. Mabuchi and S. Mukai, Einstein Metrics and Yang–Mills Connections146. L. Fuchs and R. Göbel, Abelian Groups147. A. D. Pollington and W. Moran, Number Theory with an Emphasis on the Markoff Spectrum148. G. Dore et al., Differential Equations in Banach Spaces149. T. West, Continuum Theory and Dynamical Systems150. K. D. Bierstedt et al., Functional Analysis151. K. G. Fischer et al., Computational Algebra152. K. D. Elworthy et al., Differential Equations, Dynamical Systems, and Control Science153. P.-J. Cahen, et al., Commutative Ring Theory154. S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions155. P. Clément and G. Lumer, Evolution Equations, Control Theory, and Biomathematics156. M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathematical Research157. W. O. Bray et al., Fourier Analysis158. J. Bergen and S. Montgomery, Advances in Hopf Algebras159. A. R. Magid, Rings, Extensions, and Cohomology160. N. H. Pavel, Optimal Control of Differential Equations161. M. Ikawa, Spectral and Scattering Theory162. X. Liu and D. Siegel, Comparison Methods and Stability Theory163. J.-P. Zolésio, Boundary Control and Variation164. M.Kr'íz''ek et al., Finite Element Methods165. G. Da Prato and L. Tubaro, Control of Partial Differential Equations166. E. Ballico, Projective Geometry with Applications167. M. Costabel et al., Boundary Value Problems and Integral Equations in Nonsmooth Domains168. G. Ferreyra, G. R. Goldstein, and F. Neubrander, Evolution Equations169. S. Huggett, Twistor Theory170. H. Cook et al., Continua171. D. F. Anderson and D. E. Dobbs, Zero-Dimensional Commutative Rings172. K. Jarosz, Function Spaces173. V. Ancona et al., Complex Analysis and Geometry174. E. Casas, Control of Partial Differential Equations and Applications175. N. Kalton et al., Interaction Between Functional Analysis, Harmonic Analysis, and Probability176. Z. Deng et al., Differential Equations and Control Theory177. P. Marcellini et al. Partial Differential Equations and Applications178. A. Kartsatos, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type179. M. Maruyama, Moduli of Vector Bundles180. A. Ursini and P. Aglianò, Logic and Algebra181. X. H. Cao et al., Rings, Groups, and Algebras182. D. Arnold and R. M. Rangaswamy, Abelian Groups and Modules183. S. R. Chakravarthy and A. S. Alfa, Matrix-Analytic Methods in Stochastic Models

184. J. E. Andersen et al., Geometry and Physics

185. P.-J. Cahen et al., Commutative Ring Theory186. J. A. Goldstein et al., Stochastic Processes and Functional Analysis187. A. Sorbi, Complexity, Logic, and Recursion Theory188. G. Da Prato and J.-P. Zolésio, Partial Differential Equation Methods in Control and Shape

Analysis189. D. D. Anderson, Factorization in Integral Domains190. N. L. Johnson, Mostly Finite Geometries191. D. Hinton and P. W. Schaefer, Spectral Theory and Computational Methods of Sturm–Liouville

Problems192. W. H. Schikhof et al., p-adic Functional Analysis193. S. Sertöz, Algebraic Geometry194. G. Caristi and E. Mitidieri, Reaction Diffusion Systems195. A. V. Fiacco, Mathematical Programming with Data Perturbations196. M. Kr'íz''ek et al., Finite Element Methods: Superconvergence, Post-Processing, and A

Posteriori Estimates197. S. Caenepeel and A. Verschoren, Rings, Hopf Algebras, and Brauer Groups198. V. Drensky et al., Methods in Ring Theory199. W. B. Jones and A. Sri Ranga, Orthogonal Functions, Moment Theory, and Continued Fractions200. P. E. Newstead, Algebraic Geometry201. D. Dikranjan and L. Salce, Abelian Groups, Module Theory, and Topology202. Z. Chen et al., Advances in Computational Mathematics203. X. Caicedo and C. H. Montenegro, Models, Algebras, and Proofs204. C. Y. Yéldérém and S. A. Stepanov, Number Theory and Its Applications205. D. E. Dobbs et al., Advances in Commutative Ring Theory206. F. Van Oystaeyen, Commutative Algebra and Algebraic Geometry207. J. Kakol et al., p-adic Functional Analysis208. M. Boulagouaz and J.-P. Tignol, Algebra and Number Theory209. S. Caenepeel and F. Van Oystaeyen, Hopf Algebras and Quantum Groups210. F. Van Oystaeyen and M. Saorin, Interactions Between Ring Theory and Representations of

Algebras211. R. Costa et al., Nonassociative Algebra and Its Applications212. T.-X. He, Wavelet Analysis and Multiresolution Methods213. H. Hudzik and L. Skrzypczak, Function Spaces: The Fifth Conference214. J. Kajiwara et al., Finite or Infinite Dimensional Complex Analysis215. G. Lumer and L. Weis, Evolution Equations and Their Applications in Physical and Life Sciences216. J. Cagnol et al., Shape Optimization and Optimal Design217. J. Herzog and G. Restuccia, Geometric and Combinatorial Aspects of Commutative Algebra218. G. Chen et al., Control of Nonlinear Distributed Parameter Systems219. F. Ali Mehmeti et al., Partial Differential Equations on Multistructures220. D. D. Anderson and I. J. Papick, Ideal Theoretic Methods in Commutative Algebra221. Á. Granja et al., Ring Theory and Algebraic Geometry222. A. K. Katsaras et al., p-adic Functional Analysis223. R. Salvi, The Navier-Stokes Equations224. F. U. Coelho and H. A. Merklen, Representations of Algebras225. S. Aizicovici and N. H. Pavel, Differential Equations and Control Theory226. G. Lyubeznik, Local Cohomology and Its Applications227. G. Da Prato and L. Tubaro, Stochastic Partial Differential Equations and Applications228. W. A. Carnielli et al., Paraconsistency229. A. Benkirane and A. Touzani, Partial Differential Equations230. A. Illanes et al., Continuum Theory231. M. Fontana et al., Commutative Ring Theory and Applications232. D. Mond and M. J. Saia, Real and Complex Singularities233. V. Ancona and J. Vaillant, Hyperbolic Differential Operators

Additional Volumes in Preparation

Preface

The Fifth International Conference on Algebra and Algebraic Geometry (SAGA V)was held at the University of Leon (Spain) and contributors and participants origi-nated from more than 15 European and non-European countries. As in earlier ver-sions of these meetings in Antwerp, Santiago de Compostela, Puerto de la Cruz andBrussels, talks concentrated on algebraic and geometric subjects, with particularemphasis 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 Committeeconsisted of J. L. Bueso, S. Caenepeel, J.A. Hermida, T. Sanchez-Giralda, A. Ver-schorcn and E. Villanueva. The Organizing Committee consisted of A. de Francisco,A. Granja and J. Susperregui. The organizers would like to thank the University ofLeon, 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-C02and 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 practicalproblems, large and small during the meeting, to Stef Caenepel, whose capacities asan entertainer played a fundamental role during the meeting (as always) and, lastbut not least, to the secretaries of Department of Mathematics, Ana del Rio andAna Robles. We hope the participants enjoyed this meeting as much as we did!

Angel GranjaJose Angel Hermida

Alain Verschoren


PrefaceContributorsConference Participants

1. Frobenius and Maschke Type Theorems for Doi-Hopf Modules andEntwined Modules Revisited: A Unified ApproachT. Brzezinski, S. Caenepeel, G. Militaru and S. Zhu

2. Computing the Gelfand-Kirillov Dimension IIJ. L. Bueso, J. Gomez-Torrecillas, and F. J. Lobillo

3. Some Problems About Nilpotent Lie AlgebrasJ. M. Cabezas, L. M. Camacho, J, R. Gomez, A. Jimenez-Merchdn,E. Pastor, J. Reyes, and I. Rodriguez

4. On L*-Triples and Jordan //*-PairsA. J. Calderon-Martin and C. Martin-Gonzdlez

5. Toric Mathematics from Semigroup ViewpointA. Campillo and P. Pison

6. Canonical Forms for Linear Dynamical Systems over CommutativeRings: The Local CaseM. Carriegos and T. Sdnchez-Giralda

1. An Introduction to Janet Bases and Grobner BasesF. J. Castro-Jimenez and M. A. Moreno-Frias

8. Invariants of CoalgebrasJ. Cuadra and F. Van Oystaeyen

9. Multiplication ObjectsJ. Escoriza and B. Torrecillas

10. Krull-Schmidt Theorem and Semilocal Endormorphism RingsA. Facchini

11. On Suslin's Stability Theorem for R[xr,..,xm]J. Gago-Vargas

12. Characterization of Rings Using Socle-Fine and Radical-Fine NotionsC. M. Gonzdlez, A. Idelhadj, and A. Yahya

13. About Bernstein AlgebrasS. Gonzdlez and C. Martinez


14. About an Algorithm of T. OakuM. /. Hartillo-Hermoso

15. Minimal Injective Resolutions: Old and NewM. P. Malliavin

16. Special Divisors of Blowup Algebras5. E. Morey and W. V. Vasconcelos

17. Existence of Euler Vector Fields for Curves with Binomial IdealA. Nunez and M. J. Pisabarro

18. An Amitsur Cohomology Exact Sequence for Involutive Brauer Groupsof the Second KindA. Smet and A. Verschoren

19. Computation of the Slopes of a D-Module of Type D'/NJ. M. Ucha-Enriquez

20. Symmetric Closed Categories and Involutive Brauer GroupsA. 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 Universidad de Cadiz, Cadiz, Spain.

L. M. Camacho Universidad de Sevilla, Sevilla, Spain.

A. Campillo Universidad de Valladolid, Valladolid, Spain.

M. Carriegos Universidad de Leon, Leon, Spain.

F. J. Castro-Jimenez Universidad de Sevilla, Sevilla, Spain.

J. Cuadra 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 Universidad de Granada, Granada, Spain.

S. Gonzalez Universidad de Oviedo, Oviedo, Spain.

M. I. Hartillo-Hermoso Universidad de Cadiz, Cadiz, Spain.

A. Idelhadj Universite Abdelmalek Essad, Tetouan, Morocco.

A. Jimenez-Merchan Universidad de Sevilla, Sevilla, Spain.

F. J. Lobillo Universidad de Granada, Granada, Spain.

M. P. Malliavin Universite Paris VI, Paris, France.

C. Martm-Gonzalez Universidad de Malaga, Malaga, Spain.


C. Martinez Universidad de Oviedo, Oviedo, Spain.

G. Militaru University of Bucharest, Bucharest, Romania.

M. A. Moreno-Frfas Universidad de Cadiz, Cadiz, Spain.

S. E. Morey 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 Universidad de Valladolid, Valladolid, Spain.

A. Smet University of Antwerp (RUGA), Antwerp, Belgium.

B. Torrecillas Universidad de Almeria, Almeria, Spain.

J. M. Ucha-Enriquez Universidad de Sevilla, Sevilla, Spain.

F. Van Oystaeyen University of Antwerp (UIA), Antwerp, Belgium.

W. V. Vasconcelos Rutgers University, New Jersey, U.S.A..

A. Verschoren University of Antwerp (RUGA), Antwerp, Belgium.

C. Vidal Universidad de La Corufia, La Coruna, Spain.

A. Yahya Universite Abdelrnalek Essad, Tetouan, Morocco.

S. Zhu Fundan University, Shanghai, China.


Conference Participants

J.N. Alonso Alvarez, Dpto. de Matematicas. Universidad de Vigo. Lagoas-Marco-sende. Vigo. E-36280. Spain.

P. Ara, Departament de Matematiques Edifici C. Universitat Autonoma de Barce-lona 08193 Bellaterra. Barcelona. Spain.E-mail: [email protected]

A. Bak, Department of Mathematics. University of Bielefeld. 33501 Bielefeld.Germany.E-mail: [email protected]

J. C. Benjumea, Dpto. Geometria y Topologia. Facultad de Matematicas. Univer-sidad 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 PuertoReal, Cadiz. Spain.E-mail: [email protected]

L.M. Camacho, Dpto. Matematica Aplicada I. Univ. Sevilla. Avda. Reina Mer-cedes 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.


E-mail: [email protected]

P. Carvalho, Dpto. de Maternatica Pure. Fac. de Ciencias. 4099-002-Porto. Por-tugal.

F. J. Castro Jimenez. Dpto. de Algebra. Universidad de Sevilla. Apdo. 1160-Sevilla. 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 deValladolid. Pdo. De la Magdalena s/n. 47005- Valladolid. Spain.E-mail: [email protected]

F. J. Echarte, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universidadde 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 Topolo-gie et Dynamique. Batiment 425. 91405. Orsay, France.E-mail: [email protected]

D. Ferndndez, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universi-dad 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 Com-postela. 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 Facultadde Matematicas. 41012-Sevilla. Spain.E-mail: [email protected]


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. de Algebra. Universidad de Granada. E-18071Granada. Spain.

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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. Fuentenuevas/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. Lagoas-Marcosende. Vigo. E-36280. Spain.

A. Granja Baron, Dpto. de Matematicas. Universidad de Leon. Leon. Spain.E-mail: [email protected]

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M. I. Hartillo Hermoso, E. U. Empresariales. c/ Por-Vera 54. 11403 Jerez de laFrontera. Cadiz. Spain.


E-mail: [email protected]

J. A. Hermida Alonso, Dpto. de Matematicas. Universidad de Leon. Leon. Spain.E-mail: [email protected]

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A. Jimenez Merchdn, Dpto. Matematica Aplicada I. Facultad de Informatica. Uni-versidad 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 EH93JZ. Scotland.E-mail: [email protected]

C. Lamp, Dpto. de Matematica Pure. Fac. de Ciencias. 4099-002 Oporto. Portu-gal.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. Universi-dad 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 deCadiz. Apartado de Correos. 40. 11510 Puerto Real. Cadiz. Spain.E-mail: [email protected]

/. Musson, Department of Mathematical Sciences. University of Wisconsin-Milwau-kee. 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 laUniversidad s/n. 10071 Caceres. Spain, [email protected]

A. Nunez, Dep. de Algebra, Geometrfa y Topologfa. Universidad de Valladolid.Spain.E-mail: [email protected]


J. Niinez, Dpto. Geometria y Topologfa. Facultad de Matematicas. Universidadde Sevilla. Aptdo 1160. 41080-Sevilla. Spain.

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P. Pison Casares, Dpto. de Algebra. Facultad de Matematica. Apartado 1160.41080-Sevilla. Spain.E-mail: [email protected]

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J. Reyes, Dpto. de Matematica. Escuela Politecnica Superior. Universidad deHuelva. Huelva. Spain.E-mail: [email protected]

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/. Rodriguez Garcta, Dpto Matematicas. Escuela Politecnica Superior. Universidadde Huelva. Huelva. Spain.E-mail: [email protected]

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T. Sdnchez Giralda, Dpto. de Algebra, Geometria y Topologfa. Universidad deValladolid. Spain.E-mail: [email protected]


J. Seto, Dpto. de Matematicas. Universidad de Oviedo. Oviedo. Spain.E-mail: [email protected]

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A. Smet, Department of Mathematics and Computer Science. University of Ant-werp. 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 Topolo-gie et Dynamique. Batiment 425. 91405 Orsay. France.E-mail: [email protected]

M. Sudrez Alvarez, Universidad de Buenos Aires-Universite de Paris 11. Equipe deTopologie 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.

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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. [email protected]

V. W. Vasconcelos, Department of Mathematics. Rutgers University. 110 Fre-linghuysen 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 ofAntwerp. RUGA. Antwerp. Belgium.E-mail: [email protected]

P. Vicente Matilla, Dpto. de Matematicas. Universidad de Leon. Leon. Spain.E-mail: [email protected]


C. Vidal Martin, Universidad de La Coruna. Facultad de Informatica. Campus deElvina 15071. La Coruna. Spain.E-mail: [email protected]

A. Vieites, Universidad de Vigo. E.T.S.Ingenieros Industrials. Lagoas-Marcosen-de. 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.P21.21 Tetouan. Morocco.


Frobenius and Maschke Type Theorems for Doi-Hopf Modules and Entwined Modules Revisited: AUnified 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-70109Bucharest 1, Romania.

S. ZHU,3 Institute of Mathematics. Fudan University. Shanghai. 200433, China

Abstract

We study when induction functors (and their adjoints) between categoriesof Doi-Hopf modules and, more generally, entwined modules are separable,resp. Frobenius. We present a unified approach, leading to new proofs ofold results by the authors, as well as to some new ones. Also our methodsprovide a categorical explanation for the relationship between separability andFrobenius properties.

1 INTRODUCTION

Let H be a Hopf algebra, A an F-comodule algebra, and C an if-module coalge-bra. Doi [17] and Koppinen [21] independently introduced unifying Hopf modules,nowadays usually called Doi-Koppinen-Hopf modules, or Doi-Hopf modules. Theseare at the same time yi-modules, and C-comodules, with a certain compatibility

'EPSRC Advanced Research Fellow2Research supported by the bilateral project "Hopf algebras and co-Galois theory" of the Flem-

ish 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 gov-ernments


2 Brzeziriski et al.

relation. Modules, comodules, graded modules, relative Hopf modules, dimodulesand Yetter-Drinfel'd modules are all special cases of Doi-Hopf modules. Propertiesof Doi-Hopf modules (with applications in all the above special cases) have beenstudied 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 anexact 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] tellsus that we have a Maschke Theorem for the functor F if C is finitely generatedprojective and there exists an A-bimodule C-colinear map A <8> C —-> C* <8> A satis-fying a certain normalizing condition. In [13], we have seen that F is Frobenius ifC is finitely generated and projective and A <g> C and C* <8> A are isomorphic as A-bimodules 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, theright adjoint of F (see [10]). This connection was not well understood at the timewhen [12] and [13] were written. The aim of this paper is to give a satisfactory expla-nation; in fact we will present a unified approach to both problems, and solve themat the same time. We will then apply the same technique for proving new Frobeniustype properties: we will study when the other forgetful functor forgetting the A-action is Frobenius, and when a smash product A#ftB is a Frobenius extension ofA 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 func-tors for which a "functorial" type of Maschke Theorem holds. A key result dueto 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 ge-ometry generalize Doi-Hopf modules. The most interesting examples of entwinedmodules 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-Hopfmodules. Many results for Doi-Hopf modules can be generalized to entwined mod-ules, see e.g. [3], where the results of [12] and [13] are generalized to the entwinedcase.

3) In [8], we look at separable and Frobenius algebras from the point of viewof nonlinear equations; also here we have a connection between the two notions:both separable and Frobenius algebras can be described using normalized solutionsof the so-called FS-equation. But the normalizing condition is different in the twocases.Let F : C —» T> be a covariant functor having a right adjoint G. From Rafael'sTheorem, it follows that the separability of F and C is determined by the naturaltransformations 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 Vand W can also be used to decide when G is a left adjoint of F. This is what wewill do in Section 4; we will find new characterizations for (F, G) to be a Frobeniuspair, and we will recover the results in [13] and [3]. In Section 5, we will applythe same technique to decide when the other forgetful functor is Frobenius, andin Section 6, we will study when the smash product of two algebras A and B is aFrobenius extension of A and B. This results in necessary and sufficient conditionsfor the Drinfel'd double of a finite dimensional Hopf algebra H (which is a specialcase 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 func-tors. We will explain our approach in the most classical situation: we consider aring 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 scalarsfunctor 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 presentthe results in such a way that they can be extended to more general situations inthe subsequent Sections.Let us remark at this point that the relationship between Frobenius extensions andseparable extensions is an old problem in the literature. A classical result, dueto Eilenberg and Nakayama, tells us that, over a field k, any separable algebra isFrobenius. Several generalizations of this property exist; conversely, one can givenecessary 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 elegantand more general than that of the Doi-Hopf modules; several left-right conventionsare possible and there exists a dictionary between them. In [12] and [13], we haveworked 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-Heynemannotation for comultiplications and coactions. For the comultiplication A on a coal-gebra 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 FUNC-TORS

Let F : C — - > T> be a covariant functor. Recall [23] that F is called a separablefunctor if the natural transformation

induced by F splits. From [25] and [26], we recall the following characterisation inthe case F has an adjoint.


Brzeziriski et al.

PROPOSITION 2.1 Let G : V -> C be a right adjoint ofF. Let 77 : lcand e : FG — > lp 6e i/ie unit and counit of the adjunction. Then1) 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 inverseinC .A pair of adjoint functors (F, G) is called a Frobenius pair if G is not only a rightadjoint, but also a left adjoint of F. The following result can be found in anybook on category theory: G is a left adjoint of F if and only if there exist naturaltransformations v 6 V = ]^(GF, lc) 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, orwhether (.F, G) is a Frobenius pair, one has to investigate the natural transforma-tions V — ̂ j|(G.F, lc) and W = ^^(l-p,FG). It often happens that the naturaltransformations in V and W are determined by single maps. In this Section weillustrate this in a classical situation and recover well-known results. In the comingSections more general situations are considered.Let i : R — » 5 be a ring homomorphism, and let F = • ®/j S : MR — » Ms bethe induction functor. The restriction of scalars functor G : MS — + MR is a rightadjoint of F. The unit and counit of the adjunction are

VM e MR, T]M r)M(m)=m®l,

s — ns.Let us describe V and W. Given v : GF — > IMR in V, it is not hard to prove thatP = VR : S — > _R is left and right .R-linear. Conversely, given an .R-bimodule mapv : S — > .R, a natural transformation z/ € V can be constructed by

VM € MR, s) = rnV(s).

Thus we have

Now let C : IMS

(3)

be in W. Then e = X) e1 <g> e2 = Cs(l) e 5 <S>/j 5 satisfies

^se1 <g>e2 = ̂ e1 <g> e2s, (4)

for all s e 5. Conversely if e satisfies (4), then we can recover C

Gv : 5,

In the sequel, we omit the summation symbol, and write e = e1 ® e2, where it isunderstood implicitely that we have a summation. So we have

W ^ Wl = {e = e1 ® e2 € S <8>R S se1 s, for all s e S}.

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 aFrobenius pair, then S is finitely generated and protective as a (right) R-module.

PROOF.- For all s G S, we have s = se1z/(e2) = e1z7(e2s), hence {e1,F(e2*)} is adual 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 algebraover a commutative ring R, S is projective as an R-module and G is separable, thenS is finitely generated as an R-module.

Using other descriptions of V and W, we find other criteria for F and G to beseparable 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:

( r f s ) ( t ) = r f ( t s ) , ( 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 notationintroduced above. Then

V = JM(GF, lc) = V2= Horn RiS(S, Horn R(S, R)).

PROOF.- Define aj : Vi —» V2 as follows: for v G Vi, let ot\(v} = <fi : S —>Hom/j(5, R) be given by

f ( s ) ( t ) = v(st).

Given </> € 1/2, puta-1(^)=^(l).

We invite the reader to verify that a\ and a^1 are well-defined and that they areinverses of each other. D

Brzeziriski et al.

PROPOSITION 2.6 Let i : R — > S be a ringhomomorphism and assume that Sis finitely generated and projective as a right R-module. Then, with the notationintroduced 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

THEOREM 2.7 Let i : R — > S be a ringhomomorphism. We use the notationintroduced above.1) F : MR — > MS *s separable if and only if there exists <f> 6 V<2 such that

2) Assume that S is projective as a right R-module. Then G is separable if and onlyif S is finitely generated as a right R-module and there exists 4> e Wi such that

3) (F, G) is a Frobenius pair if and only if S is finitely generated and projective asa 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, usingProposition 2.4 (for 2)) and Corollary 2.3 (for 3)). We prove one implication of3). Assume that (F,G) is a Frobenius pair. From Corollary 2.3, we know that Sis finitely generated and projective. Let v G V\ and e e Wi be as in part 3) ofTheorem 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 Sweedler-Heyneman notation:

tfj(c®a) = a,/, ®c*.

If the map ijj occurs more than once in the same expression, we also use \I> or \&' assummation 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 aresatisfied for all a € A and c € C,

c**, (7)

£c(c*)a^ = ec(c)a, (8)

a,/,® Ac (c*) = 0^(8)0^®^, (9)

1^®C* = 1®C . (10)

A /c-module M together with a right A-action and a right C-coaction satisfying thecompatibility 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 comesfrom Doi-Koppinen-Hopf structures. A (right-right) Doi-Koppinen-Hopf structureconsists of a triple (H,A,C), where H is a /c-bialgebra, A a right H-comodulealgebra, and C a right //-module coalgebra. Consider the map ijj : C ® A -^> A®Cgiven by

a) = a[0] ® ca[j].

Then (A, C1, tj}) is an entwining structure, and the compatibility relation (11) takesthe 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 Doi-Koppinen-Hopf module or a Doi-Hopf module. Doi-Koppinen-Hopf modules wereintroduced independently by Doi in [17] and Koppinen in [21]. Properties of thesemodules 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 Galoisextensions, see [6] for details. Entwining structures were introduced in [7]. Manyproperties 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-Hopfmodules, their properties can be formulated more elegantly in the language of en-twined modules.The functor F : C = A^VOS ~~> MA forgetting the C-coaction has a right adjointG — • <g> C. The structure on G(M) = M <S> C is given by the formulae

pr(m®c) — m®^!) ®C( 2 ) , (13)(m<g>c)a = ma^igic*. (14)

For later use, we list the unit and counit natural transformations describing theadjunction,

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-actionis given by a(b ® c) = ab <S> c. This makes A ® C into an object of A-M^)^, thecategory of entwined modules with an additional left A-action that is right ^-linearand right C-colinear.The other forgetful functor G' : M(i^)c

A -> Mc has a left adjoint F' = • ® A. Thestructure on F'(N) — N ® A is now given by

pr(n®a) — np] <8> a^, ® nnp (15)ia)6 = n®ab. (16)

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 <8> A — > A, ^LM (m ® a) = ma,TIN'- N —> N ® A, ?7w(n) = n <g> 1.

In particular G'(C) = C0A 6 M(4>)^- The map T/> : C®A — > /ligjC1 is a morphismin A/!(V')^- C® A is also a left C-comodule, the left C-coaction being induced bythe comultiplication on C. This coaction is right A-linear and right C-colinear, andthus G ® A is an object of c ' M(tp)^ the category of entwined modules togetherwith 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 functorforgetting the coaction, and G = • <g> C its adjoint. In [13] necessary and sufficientconditions for (F, G) to be a Frobenius pair are given (in the Doi-Hopf case; theresults were generalized to the entwining case in [3]), under the additonal assump-tion that C is projective as a fc-module. In this Section we give an alternativecharacterization that also holds if C is not necessarily projective, and we find a newproof 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 thefc-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 <8> A ® C- *A ~A

IA

>C-

10 Brzezirtski et al.

Write A = (I A ® ?c} ° v and A = (EC <8> I A) ° K- Then it follows that

0(c <g> d) = A(c ® 1 <8> d) = A(l <S> c <g> d).

We have seen before that A <g> 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 C-colinear. Take c,d £ C, and put

Writing down the condition that 17 is left C-colinear, and then applying EC to thesecond factor, we find that

C(i) ® ^(c(2) ® <0 = ̂ Ci <g) at = v(c ® d <S> 1). (21)z

Since T> is also right C-colinear,

I7(c <g> 1 <g> d (1)) ® <i(2) = ̂ ci(1) ® a^ ® cf(2)i

and, applying EC to the second factor, we find

d ( 2 ) = ^ ( c i ® a i ) , (22)

and (18) follows from (21) and (22). This proves that there is a well-defined mapa : V -> 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 rightA-linear and right (7-colinear, and that v is a natural transformation. The rightyl-linearity follows from (17), and the right C-colinearity from (18). Given anymorphism / : 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 leftto 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 <g) c; G VFj if and only if

cf . (23)

PROPOSITION 4.2 Let (A, C, •0) be a right-right entwining structure. Then thereis an isomorphism of k-modules (3 : W — » W\ given by

0(0 = Ci(l). (24)

The inverse of (3 is /3~1(^; ai ® c;) = C; wii/i (jv : -^ — > N <£> C given by

(25)

PROOF.- We leave the details to the reader; the proof relies on the fact that £4 isleft and right A-linear. D

In [10], Propositions 4.1 and 4.2 are used to determine when the functor F andits adjoint G are separable.

THEOREM 4.3 Let F : M(ip)% -> MA be the forgetful functor, and G = • 0 Cits adjoint.F is separable if and only if there exists 9 6 V\ such that

9 o AC = EC-

G is separable if and only if there exists z = ̂ ; a; <g> 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 equivalentto the existence of 6 € Vj and z G Wi, but now satisfying different normalizingconditions.

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\ andz = y^; ai ® ci G W\ such that the following normalizing condition holds, for all

ec(d)l = Y^ aid(ci <g> d) (26)

PROOF.- Suppose that (F, G) is a Frobenius pair. Then there exist v € V andC € W such that (1-2) hold. Let 9 = a(v) € Vj, and z = ]T^ a; ® c; - /?(() 6 Wi.Then (1) can be rewritten as

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

for all n € JV e Al/i and d & C. Taking N = A and n = I , one obtains

1 <g> d = Y

(29)

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 flatas a fc-module, and C is projective as a fc-module, then C is finitely generated. Thenext proposition shows that, in fact, one does not need the assumption that C isprojective.

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 <g> Q be as in Theorem 4.4. Then for all d G C,

(27)

(7)

(9)

(18)

Write C((i) ® C;(2) = Y^j^i cij ® cij an<^ f°r a^ ^ 3 consider the map

aij : A®C-*A, ay (a® d)

Then for all a e /I and d e C,

so {<J/j, 1 ® C; • | Z = 1, • • • , n, j = I , • • • , mi} is a finite dual basis for A ® C as a leftA-module. D

In some situations, one can conclude that C is finitely generated and projectiveas a fc-module.

COROLLARY 4.6 Let (A,C,tl>) be a right-right entwining structure, and assumethat (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 generatedprojective 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 generatedby 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 projectiveafter a faithfully flat commutative base extension, then it is itself finitely generatedand 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 thatC 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* <g> A can be made into an object ofas follows: for all a,b,b' £ A, c* € C* ,

b(c*®a)b' =

pr(c*®a) = d J * c ' ( 8 i a ^ ® d . (31)

This can be checked directly. An explanation for this at first sight artificial structureis given in Section 6. We now give alternative descriptions for V and W. Recallfrom [10] that there are many possibilities to describe V. As we have seen, a

natural transformation v & V is completely determined by 9. Nevertheless, themaps 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 isthe map we need. At some place it is convenient to use C* <g> A as the image space,at some other we prefer Horn (C,A). Note that (/> is given by

~4>(a <g> c)(d) = A(d ® a ® c) = \(a^ <g> d^ <g> c) = a^6(d^ & c),

or(j)(a®c) = 2^d*®ai,e(<% <g>c) . (32)

It turns out that <f> is a morphism in AM.CA. More specifically, one has

PROPOSITION 4.7 Let (A,C,ip) be a right-right entwining structure. If C is afinitely generated and protective k-module, then

The isomorphism is a\ : V\ —> V-2, with cti(0) = is left A-linear. It is also right A-linear because

^(aOc)6 = ^d*(S)a^e(df ® c)bi

(17) = 3dt®<ty6*/*0(df*®c*')

(7) =

Notice that the dual basis for C satisfies the following equality (the proof is left tothe reader):

Using this equality one computes

(31) = V d* * d*Z — / J

(34) =

(7) =

(9) -

(18) =

This proves that ^ is^right C"-colinear. Conversely, given <f> E.V2, first one needs toshow that 9 = aJ~1((/>) € Vi. It is now more convenient to work with Horn. (C, A)rather than C* <g> A. For / e Horn (C, A), 6,6' e /I, (30) can be rewritten as

(bfb')(c) = btf(c*)b'. (35)

Take any c, d 6 C, a G /I and compute

(35) =is right A-linear) =

^ is left A-linear) = a^4>(l ® d*) (c)

(35) =

This proves that ^ satisfies (17). Before proving (18), we look at the right C-coactionpr on/ = 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)

16 Brzezinski et al.

This can be used to show that 9 satisfies (18). Explicitly,

6(c(2)®d}^®c^l} = V(c(i) <8> 0(c(2) <g> d))

(36) = 0(1 ® d)[0](c) ®^(1

(4> is right C-colinear) = <j>(\ ® d^)(c) iS> d(2)

It remains to be shown that a\ and a^1 are inverses of each other. First take06 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) — <f> is left and right ^.-linear and right C-colinear. For all c* 6 C* and a, 6 6 A,

l(7) = 53 a/ a^, 6* <g) {c* , c.,- (2) ) c**

c* (gi a)6,

proving that <£ is right A-linear. The proof of left A-linearity goes as follows:

*, cl(2j)cf(l}

(7) =

=

(9) =

(23) =

=

Next one needs to show that <£> is right C-colinear. Using (36), one finds

Ei (c*,

i

(9) = ^aza^,® (c*,C( (2 ))(cf) (1 ) ® (cf)(2)

Conversely, let </> e W2 and put z = 0(e ® 1) = ^^ a; ® c;. Using (30), we see thata(£<g>l) = (e<g)l)a, for all a € A, hence az = a00(e (E> l)a = za, and z 6 Wi.Take z = '£llai®ci€Wl. Then

Finally, take </> £ W2, and write j3l l($) = <p(e ® 1) = ^ a; ® c;. C* ® A and

A ® C* are right C-comodules and left C'*-modules. Since (j> is right A-linear, rightC-colinear and left C*-linear,

i

and it follows that )) , as required. D

Suppose that C is finitely generated and projective as a /c-module. From Propo-sition 4.7, it follows that F is separable if and only if there exists a map (f> € V^ =Horn ̂ A(A <8> C, C* <g> A) such that (/>(! <g> C(2))(c(i)) = e(c)l, for all c e C. In theDoi-Hopf case, this implies the Maschke Theorem in [12]. Now we apply the sameprocedure to determine when (F, G) is a Frobenius pair.

THEOREM 4.9 Consider an entwining structure (A, C, ijj), and assume that C isfinitely generated projective as a k-module. Let F : A/t('0)5 ~* MA be the functorforgetting the C-coaction, and G — • ® C be its right adjoint. Then the followingstatements 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 < j 6 : A ® C —> C* <S> A,

given by

l

i

are inverses of each other.3) C* <g> A and A Cg> C are isomorphic as objects in AA /t(V')5-

PROOF.- 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*®A-Similarly, for all c e C,

(18)

(26)

Since </> and <j> are left A-linear, 0 o <f> =2) =» 3). Obvious, since c/> and (j> are in3) =» 1). Let (f> : C* ® A — > A <S> C be the connecting isomorphism, and putz = <t>(e ® 1) = E; ai ® ci e Wi, 6 = a^-1) e Vi. Applying (32) and (37), onefinds

e ig i l = <^)-1 ((£(£ 01)) =

Evaluating this equality at c € C, one obtains (27). For all c £ C,

1 0 c = <j)(4>~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) isa Frobenius pair. D

REMARK 4.10 Recently M. Takeuchi observed that entwined modules can beviewed 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 quiteclear that the procedure applied in Section 2 to extension and restriction of scalarscan be adapted to functors associated to corings, leading to a generalization of theresults 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 theadjunction 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

G'F' 6 C.A/1 (•</>) ̂ . The left structures are given by

a(m 06)= 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 leftC-colinear.

Now write V = ^(G'-F', IA^C). W7' = MM(lc,^'G')- Following the philoso-phy of the previous Sections, we give more explicit descriptions of V and W . Wedo not give detailed proofs, however, since the arguments are dual to the ones inthe 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.

PROOF.- Details are left to the reader. Given $ € V, for N & M.c, the naturalmap v'N : N <g> A —> IV is

n

For any /c-linear map e : C —> A <8> A and c 6 C, we use the notation e(c) =e1(c) ®e2(c) (summation understood). Let W[ be the fc-submodule of Horn (C, A <8>A) consisting of maps e satisfying

e1(c ( 1 ))(g>e2(c ( 1 ))(8)c (2 ) = e1

PROPOSITION 5.3 The map 8 : W -> W{ given by

"( I ) ' (41)

(42)

is an isomorphism. Given e £ W[, C,1 = /3~^(e) is recovered from e as follows: forM

PROOF.- We show that 8 is well-defined, leaving other details to the reader. Con-sider a commutative diagram

A-

C-

The map A = CC^A ° (7c Co,- ® a!i • Then

•~A <8> C J^^..^ ® C ® A

3ft and right C-colinear. Write A(c)

C(i) <8> A(c(2)) = ̂ ci(1) ig) ci(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 *

and hence proves (41). To prove (42) note that A = CA®C ° (nA <S> IG) is leftright A-linear, hence

e1(c)®e2(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[ suchthat 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 forall c<=C,

e1(c)e2(c)=e(c)l. (44)

3) (F',G') is a Frobenius pair if and only if there exist •& G V{ and e & W( suchthat

e(c)l = 79(C ( 1 )®e1(C ( 2 )))e2(c ( 2 )) (45)

(46)

PROOF.- We only prove 3). If (F' , G') is a Frobenius pair, then there exist i/ G Vand C' e W such that (1) and (2) hold. Take $ e V{ and e € W{ corresponding toi/ and ^' and write down (1) applied to n ® 1 with n G N e Mc ,

(47)

Taking N = C, n = c, and applying £c to the first factor, one obtains (45).Conversely, if -& e V{ and e e W^i satisfy (45), then (47) is satisfied for all AT e Mc ,and (1) follows since v'N ® I A and C/v®/l are right A-linear.Now write down (2) applied to m e M e A1(V>)2i

(48)

Take M = C ® A, m = c (g> 1, and apply £c to the first factor. This gives (46).Conversely, if $ € V{ and e e VFj satisfy (46), then application of (46) to the secondand third factors in TOpi ® mli} ® 1> an<^ then EC to the second factor shows that(48) holds for all M e M(t/>)%. Finally note that (48) is equivalent to (2). D

Inspired by the results in the previous Section, we ask the following question:assuming (F',G!) is a Frobenius pair, when is A finitely generated projective as afc-module. We give a partial answer in the next Proposition. We assume that ip isbijective (cf. [2, Section 6]). In the Doi-Hopf case, this is true if the underlying Hopfalgebra H has a twisted antipode. The inverse of ijj is then given by the formula

ip~l(a <g> c) = c5(a(!)) (g> a(0).

PROPOSITION 5.5 Let (A,C,ip) be a right-right entwining structure. With no-tation as above, assume that (F', G') is a Frobenius pair. If there exists c & C suchthat e(c) = I , and if tjj is invertible, with inverse </? = ifj~l : A <8> C —> C (x) A, thenA is finitely generated and projective as a k-module.

PROOF.- Observe first that (A, C, <p) is a left-left entwining structure. This meansthat (7-10) hold, but with A and C replaced by A°p and Ccop. In particular,

£(c<p)av> = e(c)a, (49)

This can be seen as follows: rewrite (8) and (9) as commutative diagrams, reversethe arrows, and replace i/j by <p. Then we have (49) and (50) in diagram form. Nowfix c e C such that E(C) = 1. Then for all a € A,

a = e(c)a = e(cv)av

(42) = 0(c* <

Write (7®e)A(c) = Y.T=\ci®bi®ai £ C®A®A. For i = l , - - - , m , define a* e A*by

(a*, a) = •&(cf <8> a^bij.

Then {ctj, a* i = I , • • • , m} is a finite dual basis of A as a k-module. D

From now on we assume that A is finitely generated and projective with finitedual basis {a,, a* | i = 1, • • • , m}. The proof of the next Lemma is straightforward,and therefore left to the reader.

LEMMA 5.6 Let (A, C, i]j) be a right-right entwining structure, and assume thatA is finitely generated and projective as a k-module. Then A* (g> C € cM.($)C

A. Thestructure is given by the formulae

pr(a*®c] = a* <S>c (1 ) <8>C( 2 ) , (52)I/ * ,~~ \ _ / * \ y^ /o * /o\ T^ /CCQ°\p \Q> 09 Cj ^ \d , &iih )C( -\\ 09 a^ Qv C / r j \ . i oo i

We now give alternative descriptions of V and W.

PROPOSITION 5.7 Let (A,C,il>) be a right-right entwining structure, and as-sume that A is finitely generated and projective as a k-module. Then there is anisomorphism

Pi '• W[ —> W!} = Horn kA(A* ®C,C® A),

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

\^/ — \ 5 ̂ "ibib

(42) = (a*,e1(c ( 2 ))*}cf1 )®e2(c ( 2 ))fe

b) /?i(e) = fi is right C-colinear: for all a* 6 A* and c e C1, we have

(9) , ( ^ ^ , ( l ) ( 3 ) $

(41) = (a*,e1(c (2))^)cj'1 ' )®e2(c (2))®c (3 )

) <8>C(2) .c) /?i(e) = 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*,e1(c))a l(g)e2(c) = el(c) ® e2(c),

) ((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 afinitely generated projective k -module, then the map

<*i • V{ -» V2' - Horn g£(C7 ® A, A* ® C1),

defined by a\ ($} = SI, with

fi(c ® a) = (i9, C(!) (g) a^ai)a* (g) c*2)

is on isomorphism. The inverse of ai is given by aj~1(fi) = i? with

PROOF.- We first show that ai is well-defined. Take •& e V{, and let ai(i?)a) fi is right A-linear since for all a, b G A and c € C,

(2)

b) 0 is right C*-colinear since for all a G A and c G C,

/3 r(Q(c®a)) =

(9) =

=

c) f2 is left C'-colinear since for all a € A and c € C1,

(9) = *

\ ^ f\ f „ \ li)' ̂ „ *) a -j !

(7)

(40)

Conversely, given fi, we have to show that a^^fi) — $ satisfies (40). Take anyc ig> a e C ® .̂ and write Q(c ® a) = ̂ 6* ®dL £ A* ®C. Since O is right and leftC-colinear, we have

Therefore we can compute

a),

Thus (40) follows. Finally, we show that c*i and aj"1 are inverses of each other.

c }, 1

We know that ai(a^1(O)) is right A-linear. Hence suffices it to show that

for all c € C. From (51), we compute

((a* <S> c)b, IA 0 £c) = (a*, 6}e(c) = (a* g) c, 6 ® ec).

Now write fi(c ® 1) = X^r- a* ® cr anc^ compute

26 Brzezinski et al.

THEOREM 5.9 Let (A,C,t{j) be a right-right entwining structure, and assumethat A is finitely generated and protective as a k-module. With notation as above,we have the following properties:1) F' is separable if and only if there exists SI 6 V^ such that for all c £ C,

A®£c) =£c(c).

2) G' is separable if and only if there exists 51 £ W^ such that for all c 6 C,

]T ai(ec ® lA)tt(a*i <E> c) = ec(c}l.i

3) The following assertions are equivalent:a) (F',G') is a Frobenius pair.b) There exist e £ W[, •& € V{ such that Q, = /?i(e) and O = cti(i)) are inverses ofeach other.c) A* <g> C and C ® A are isomorphic objects in

PROOF.- We only prove a) => 6) in 3). First we show that £7 is a left inverse of fi.Since fi o Q is right ^4-linear, it suffices to show that

(40) = ^(c (2)®e1(c(3))^)cj'1)®e2(c (3 ))

(45) = c ® l .

To show that f2 is a right inverse of O we use that fl o Jl is right (7-colinear andconclude that it suffices to show that for all c e C and a* G A* ,

(I A' ® ec)(n(n(a* ® c))) = £C(c)a*.

Both sides of the equation are in A*, so the proof is completed if we show that bothsides are equal when evaluated at an arbitrary a & A. Observe that

) ) ( i ) ® e2(c(2))*al)a* ® (cf1})f2)

hence

(I A- <8>e c)(n(n(a*®c)))(a)

(42) = (a

(7) - {âê^VKcJf ®e2(c*2)))

(9) - (a*,a^1((C*)(2)V)^((^)f1 ')®e2((^)(2)))

(46) = (a*)a^)e(C*) = <a*,o)e(C),

as required. D

6 THE SMASH PRODUCT

Let (B, A, R) be a factorization structure (sometimes also called a smash or twistedtensor product structure, cf. [27] [22, pp. 299-300] [16]). This means that A and Bare /c-algebras and that R : A®B—+B(><)Aisa fc-linear map such that for alla,c<E A, b, de B,

= bRr®arcR, (54)R(a®bd) = bRdr®aRr} (55)R(a®lB) - IB® a, (56)R(lA®b) = b®lA. (57)

We use the notation R(a ® b) = bR <g> aR. B$RA is the fc-module B <g> A with newmultiplication

= bdR#aRc.

is an associative algebra with unit IB#Â if and only if (54-57) hold. Inthis Section we want to examine when B#RA/A and B$RA/B are separable orFrobenius. This will be a direct application of the results in the second part ofSection 2.In Section 2, take R = A, S = B#RA. For v € Vi = Horn #^(5, R), defineK : B -> A by

Then 17 can be recovered form K, since z/(6#a) = «(6)a. Furthermore

a«(6) = al/(6#l) = v(bR#aR) = K,(bR)aR

and we find that

V ^ Vl =* V3 = {K : B-+A o/e(6) = K,(bR)aR}. (58)

Now we simplify the description of W = Wi C (B#RA) ®A (B#RA). Note thatthere is a /c-module isomorphism

defined by

Let W3 = j(Wi) C B ® B ® A. Take e = bl <8> 62 ® a2 & B ® B ® A (summationimplicitely understood). Then e e W$ if and only if (4) holds, for all s = 6#1 ands = l#a with 6 e B and a e A, if and only if

bb1 ®b2 ®a2 = b1 <g> 626B ® a^, (59)(61) R ® (62)r (8) aRra2 = b1 ® 62 <g> a2a, (60)

for all a € yl, fe € S. This implies isomorphisms

W7 ^ M/i ^ VK3 = {e = b1 <g> 62 ® a2 e B ® B & A | (59) and (60) hold}. (61)

Using these descriptions of V and W, we find immediately that Theorem 2.2 takesthe following form.

THEOREM 6.1 Let (B, A, R) be a factorization structure over a commutative ringk.1) B#RA/A is separable (i.e. the restriction of scalars functor G : A/ls#RA -^ MAis separable) if and only if there exists e = b1 ® 62 <S> a2 G Ws such that

6162<g)a2 = 1B® 1A e B ® A. (62)

l?j B^ftA/A is split (i.e. the induction functor F : MA ~* MB#RA *s separable)if and only if there exists K 6 V-j suc/i i/ia^

K(!B) = IA- (63)

5j B^nA/A is Frobenius (i.e. (F,G) is Frobenius pair) if and only if there existK e Vs, e e W3 such that

(b2)R ® K^^jja2 = 61 ® «(62)a2 = 1B ® 1A. (64)

Theorem 2.7 can be reformulated in the same style. Notice that

Homfl(5',^) = RomA(B#RA,A') ^Eom(B,A).

Rom(B,A) has the following (A,B#rtA)-l>imodule structure (cf. (6)):

( c f ( b # a ) ) ( d ) = c f ( d b ) 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 Propo-sition 2.6

W ^ W2 <* W4 = KomA>B#RA(Rom(B,A),B#RA). (66)Theorem 2.7 now takes the following form:

THEOREM 6.2 Let (B,A,R) be a factorization structure over a commutativering k, and assume that B is finitely generated and projective as a k-module. Let{bi, b* | i — 1, • • • , 771} be a finite dual basis for B.1) B^RA/A is separable if and only if there exists an (A, B#RA}-bimodule map(j>: Horn (B, A) ^ B* <g> A -> B#RA such that

2) B^RA/A is split if and only if there exists an (A, B^RA)-bimodule mapB#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 <8> 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 twoways to proceed: as above, but applying the left-handed version of Theorem 2.7(left and right separable (resp. Frobenius) extension coincide). Another possibilityis to use "op" -arguments. If R : A®B — > B®A makes (B, A, R) into a factorizationstructure, 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 thereis an algebra isomorphism

(A0v#ABop)op * B#RA.

Using the left-right symmetry again, we find that B#RA/ B is Frobenius if and onlyif (A0P#^B°P)°P/B is Frobenius if and only if (Aop#RBop)/Bop is Frobenius, andwe can apply Theorems 6.1 and 6.2. We invite the reader to write down explicitresults.Our final aim is to link the results in this Section to the ones in Section 4, atleast in the case of finitely generated, projective B. Let (A,C,if)) be a right-rightentwining 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 correspondencebetween 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* ,c f )cJ (g ia V ) , tp(c® a) = ((c*)fl,c)Ci <g> aR.

Moreover, there are isomorphisms of categories

and

In particular, B^^A can be made into an object of ^.A/f ('!/') 5, and this explains thestructure on C* ® A used in Section 4. Combining Theorems 4.9 and 6.2, we findthat the forgetful functor M^}1^ — » MA and its adjoint form a Frobenius pair ifand only if C* ® A and A <8> C are isomorphic as (A, (C*)op#flJ4)-bimodules, if andonly 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-Baxterequation, Trans. Amer. Math. Soc. 349 (1997), 3823-3836.

[2] T. Brzezinski, On modules associated to coalgebra-Galois extensions, J. Alge-bra 215 (1999), 290-317.

[3] T. Brzezinski, Frobenius properties and Maschke-type theorems for entwinedmodules, 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 the-orem, and Frobenius and Galois-type properties, preprint math.RA/0002105.

[6] T. Brzeziriski and P. M. Hajac, Coalgebra extensions and algebra coextensionsof 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 andseparable algebras, K-theory, to appear.

[9] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Smash biproducts ofalgebras and coalgebras, Algebras and Representation Theory 3 (2000), 19-42.

[10] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for thecategory of Doi-Hopf modules, Applications, Adv. Math. 145 (1999), 239-290.

[11] S. Caenepeel, B. Ion, G. Militaru, and Shenglin Zhu, Separable functors for thecategory of Doi-Hopf modules II, in "Hopf algebras and quantum groups" , S.Caenepeel and F. Van Oystaeyen (Eds.), Lect. Notes Puere Appl. Math. 209Marcel Dekker, New York, 2000.

[12] S. Caenepeel, G. Militaru, and S. Zhu, A Maschke type theorem for Doi-Hopfmodules, J. Algebra 187 (1997), 388-412.

[13] S. Caenepeel, G. Militaru, and S. Zhu, Doi-Hopf modules, Yetter-Drinfel'dmodules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997),4311-4342.


[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 unifiedHopf 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.

[19] L. Kadison, Separability and the twisted Frobenius bimodules, Algebras andRepresentation Theory 2 (1999), 397-414.

[20] L. Kadison, "New examples of Frobenius extensions", University Led. Series14, Amer. Math. Soc., Providence, 1999.

[21] M. Koppinen, Variations on the smash product with applications to group-graded rings, J. Pure Appl. Algebra 104 (1995), 61-80.

[22] S. Majid. Foundation of Quantum Group Theory. Cambridge University Press1995.

[23] C. Nastasescu, M. van den Bergh, and F. van Oystaeyen, Separable functorsapplied to graded rings, J. Algebra 123 (1989), 397-413.

[24] R. Pierce, Associative algebras, Grad. Text in Math. 88, Springer Verlag,Berlin, 1982.

[25] M. D. Rafael, Separable functors revisited, Comm. in Algebra 18 (1990), 1445-1459.

[26] A. del Rio, Categorical methods in graded ring theory, Publ. Math. 72 (1990),489-531.

[27] D. Tambara. The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ.Tokyo Sect. IA, Math. 37 (1990), 425-456.


Computing the Gelfand-Kirillov Dimension II

J. L. BUESO, J. GOMEZ-TORRECILLAS and F.J. LOBILLO, Departamento deAlgebra, 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 tocompute the Gelfand-Kirillov dimension for finitely generated modules over solvablepolynomial algebras and, from the theoretical point of view, we characterize thesealgebras within the class of all filtered algebras.

When looking for a notion of dimension for modules over a non-commutativealgebra A which allows its effective computation, it is quite natural to look at theexisting algorithms in Commutative Algebra. It seems reasonable to try first thecomputation of the dimension for the algebra A itself. In the case of commutativefinitely generated algebras over a field k, the problem is equivalent to the computa-tion of the degree of the Hilbert polynomial of k [ x j , . . . , xn]/I, where / is an idealof the commutative multivariable polynomial algebra k [ x j , . . . ,xn], and this canbe 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 cyclick xi,... ,xn -modules to finitely generated ones (see [15, Section 4]). From thispoint of view, the notion of dimension for modules which extends properly to non-commutative algebras is the Gelfand-Kirillov dimension. The techniques from thecommutative case can be easily adapted to compute the Gelfand-Kirillov dimensionin the case of quantum affine spaces, namely, finitely generated k-algebras definedby 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 k-algebras). When dealing with more general non-commutative k-algebras, a fruitfulidea is to consider R to be a finitely generated k-algebra, with finite-dimensionalgenerating vector subspace V, in such a way that the graded algebra gr(R) associ-ated to the standard filtration is a multivariable commutative polynomial algebra(see [6]) or a quantum affine space (see [12]).


34 Bueso et al.

However, a very simple example like the Jordan plane defined by two genera-tors x,y, subject to the relation yx — xy + x2 shows that the former techniques,based on standard filtrations with (semi)commutative associated algebras, had tobe improved. In the Spring of 1996, the first author presented in the SAGA4 heldin Antwerp-Brussels an algorithm to compute the Gelfand-Kirillov dimension forcyclic modules over solvable polynomials algebras ([18]) with respect to the de-gree lexicographical order (see [4, 5]), which is applicable to some quantum groupswhose generators are subject to quadratic relations, like quantum matrices, quan-tum symplectic space or quantum euclidean space). A similar approach appeared inthe 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 oversolvable polynomial algebras (called there PBW algebras) with respect to a weightedgraded lexicographical order in [23, 7], which allows to handle algebras with non-quadratic relations (the simplest example is given by the commutation relationsyx = xy + x3). Here we show that this algorithm can be used to compute theGelfand-Kirillov dimension of any finitely generated module over any solvable poly-nomial algebra, without restrictions on the given term order.

During our search for that algorithm we have discovered some interesting resultson filtered algebras which, in particular, locate the solvable polynomial algebras (or,equivalently, the PBW algebras) in the theory of noetherian rings: they are preciselythose algebras having a filtration (in most cases, non standard) whose associatedgraded algebras are graded quantum spaces. We also show that these algebras canalways be re-filtered by finite-dimensional vector subspaces keeping a quantum spaceas associated graded algebra. This allows to use the well-known graded-filteredtechniques to obtain that any solvable polynomial algebra R is an Auslander-regularnoetherian 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 SUB-SETS

Let N denote the additive monoid of all positive integers (including the neutralelement 0). Let n be a strictly positive integer. We consider Nra as additive monoidwith the sum defined componentwise. Let 61,... ,en be the canonical basis of thisfree 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]), admissibleorders 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 ~<iex • • • ~<iex en, which isknown to be an admissible order. Given a = (a i , . . . , an), w = (101,... , wn) £ N",

Computing the Gelfand-Kirillov Dimension 35

define \a w = aiWi + - • • anwn, i.e., a\w is the dot product (a, w). The w-weighteddegree lexicographical order ^w is defined by

and a ~<iex /3.

The degree lexicographical order ^degiex is obtained as ^w with w = ( 1 , . . . , 1).For any pair of positive integers n, m, let the monoid Nn act on the set Nn>(m) =

Nn x {!,...,m} by a + (/3,i) = (a + /3,i). It is clear that any (a,i) e Nn '(m) maybe written as (a, z) = a + (0, z) for any 1 < z < m and ct € N™. In the case m = 1,we shall identify N"'(1) with N".

DEFINITION 2.3 A non-empty subset E of N"'(m) is said to be stable if E =E + N™. The stable subsets of Nn are called monoideals of Nn.

The following lemma suggests that the study of some aspects of stable subsetsof N™1*-"1-1 can be reduced to monoideals. This is the case of the notion of dimensionthat we will consider later.

LEMMA 2.4 Let E be a stable subset o/N™' ( m ) . For every i = 1,... ,m, the setEi = {a e N"; (a,z) 6 E} is a monoideal o/N™. Moreover, E may be written as adisjoint union

where Ei = (0,i) + Ei.

PROOF. Straightforward. D

The following result was pointed out in [15].

PROPOSITION 2.5 Let E be a stable subset o/N">(m). There is a set (a^ii),. . . , (as,is) of elements of E such that

s

^)+Nn) (1)

Every admissible order X on Nn induces a total order on Nn>^m\ which will alsobe denoted by ;< This order is defined by putting

(2)

for any (a,i) and ((3,j) in N™' ( m ) . Dickson's Lemma extends to Nre'(m), thus X isa well-ordering on N™1'"1-1. Moreover, it enjoys the following properties

(a) (a,i) ^ 7 + (a,i) for every (a,i) € N"'W and every 7 € N";

36 Bueso et al.

(b) for all (a,i), ((3,j) with (a,i) ^ (/3, j ) , and every 7 6 Nn, it follows 7 + (c*,z) ^7+C/3J)

For any function / : N — > 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 befound in [24], [27, Chapter 8] or [20].

DEFINITION 2.6 For any stable subset E C N">(m) and any weight vector w withstrictly positive integer components, we define the Hilbert function HF^ : N — » Nof E relative to w by putting

HF£(s) = card{(a, i) e Nn>(m> \E \a w ^ s}

for every s € N. In the case that w = (1, . . . , 1) we shall use the notation HF# forthe Hilbert function.

In the proof of the next Lemma we shall use the notation |(o;,z)|w = |a w. ByN™ we denote the subset of N" consisting of those vectors with all their componentsstrictly 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 Nn'(m) isdenned as dim(£) = d(HF^), for any w e N£.

The study of the Hilbert function of a stable subset can be reduced to monoide-als, as the following result shows.

PROPOSITION 2.9 Let E be a stable subset o/N™'(m) and consider the decom-position

given in Lemma 2.4- Then

HF£ = HF£ + • - • + HF^m (4)

and

dim(E) = max{dim(B1), . . . , dim(£;m)} (5)

PROOF. The formula (4) is clear, since (a,i) ^ E if and only if a <£ E{. Theequality (5) follows now from [27, Lemma 8.1.7]. D

It follows from Proposition 2.9 that the computation of the dimension of stablesubsets of Nn'(m' reduces to the case of monoideals. Let E be a monoideal of N™.In the case that E is not proper, the dimension can be computed easily as

,. ,„ / 0dim(.E) = < .f „ rtv ' \ n if E = 0

For proper ideals, we shall need the notion of support of a vector 0 ^ ex. 6 Nn,defined as

supp(a) = {z € {!,..., n} at ^ 0}

Consider the set

V(E) = {cr C {1, . . . , n} a n supp(a) ^ 0 Va e E}

which can be computed from a set of generators ai, . . . ,ccs of E since, as can bechecked,

V(E) = {ff C {!,..., n} an supp(afc) ^ 0 V/c = 1, . . . , s}

The following result allows the effective computation of the dimension of any mo-noideal, and hence, of any stable subset. A proof, inspired on the material of [2,Section 9.3], can be found in [5, Section 4].

THEOREM 2.10 Let E be a proper monoideal o/N™.

(1) dim(£) = n - min{card(cr) a 6 V(E)}

(2) If m is the maximum of the entries of all vectors in a finite set of generatorsfor E, then there is an unique polynomial h(x] G Q[z] such that HF£(s) = h(s)for every s ^ mn.

REMARKS 2.11 Equivalent descriptions for the dimension dim(B) can be de-duced from [11, Theorem 3.1]. The fact that HFB(S) coincides with a polynomialfor s big enough was proved in [19, Lemma 16, p. 51].

3 PBW ALGEBRAS, QUANTUM RELATIONS AND FILTRATIONS

Let R be an algebra over a commutative field k, and let xj , . . . ,xn be elements inR. A standard monomial in xi, . . . ,xn is an expression xa = x"1 . . .x^™, wherea = (ai, . . . , an) & N". Assume that an element r € R can be written in the form

r = Y, r<*xa (ra e k) (6)aSN"

The expression (6) is called a standard representation of r. Of course, ra ^ 0 onlyfor a finite subset A/"(r) of Nn. We will often refer as polynomials to the elementsof R having a standard representation.

38 Bueso et al.

DEFINITION 3.1 Let ;< be an admissible order on N™, and consider an elementr of R having a standard representation (6). The exponent exp(r) of (the standardrepresentation of) r is defined as the maximum with respect to ;< of the finite setJV'(r). We note that exp(r) depends on the given order ^ and on the standardrepresentation (6) of r.

The type of relations that are satisfied by many interesting algebras suggeststhe following definition.

DEFINITION 3.2 Let R be a k-algebra generated by elements xi,...,xn. Ifthere are nonzero scalars qji e k and polynomials pji (1 sC i < j ^ n) such that therelations

Q - {xjXi = qjiXiXj +pji, 1 ̂ i < j ^ n}

hold in R, then we will say that the generators x\ , . . . , xn satisfy a set Q of quantumrelations. If in addition,

for every 1 ^ i < j ^ n for some admissible order ^ on Nn, then the quantumrelations Q are said to be ^-bounded.

DEFINITION 3.3 An algebra R over a field k is said to be a Poincare-Birkhoff-Witt algebra if R is generated by finitely many elements x\, . . . , xn such that

(PBW1) The standard monomials xa with a € N™ form a basis of R as a k-vectorspace.

(PBW2) There is an admissible order X on Nn such that the generators xi, . . . , xnsatisfy a set Q of ^-bounded quantum relations.

Notice that R can then be thought of as the algebra generated by x\ , . . . , xn subjectto the relations Q = XjXi = q^XiXj +PJI, 1 ̂ i < j ^ n, such that no further latentrelations appear. We will then say that R — k{xi, . . . ,xn; Q, :<} is a PBW algebra.

Here, some pertinent comments about Definition 3.3 arise. The notion of PBWalgebra is a restatement of the concept of solvable polynomial algebra introducedby Kandri-Rody and Weispfenning in [18]. It follows from [10, Theorem 1.2] thatthe condition (PBW2) is equivalent to

exp(/sO = exp(/) + exp(g) (7)

for every /, g £ R. Finally, if R = k{xi , . . , , xn; Q, ̂ } is a PBW algebra and •<' isa different admissible order on N™, we cannot expect in general that R is a PBWwith respect the new order -<' .

DEFINITION 3.4 ([26, Definition 3.7]). We will say that two elements x, y of thealgebra R are semi- commuting if there is 0 ̂ q & k such that yx = qxy. A gradedk-algebra A is called semi-commutative if R is generated as k-algebra by a finiteset of homogeneous semi-commuting elements.

If A is a semi-commutative graded algebra with homogeneous generators j/i, . . . ,yn, then there are non zero scalars q^ G k such that y^y, = qjiUiyj, 1 ̂ i < j ^ n.Clearly, the standard monomials ya = y"1 .. .y^n with a € N™ span A as a k-vector space. Therefore, every element of A has at least a standard representation.A special interesting case occurs when the standard representation of every elementin A is unique. This leads to the following definition.

DEFINITION 3.5 A semi-commutative graded algebra is said to be a graded quan-tum affine space if the monomials ya are k-linearly independent (and hence, forma k-basis for A).

Of course, every graded quantum affine space is a PBW algebra with respect toany admissible order. Our aim is to show that the PBW algebras are, precisely, thosefiltered algebras which have a quantum affine space as associated graded algebra.The first step will be Proposition 3.6. First, let us recall that an algebra R over k is(positively) filtered, if it is endowed with an ascending chain FR = {FnR \ n ^> 0}of vector subspaces, the filtration of R, satisfying for all n,m ^ 0

f . 1 e FQR,2. FnR C Fn+\R,3. (FnR)(FmR) C Fn+mR,4- R=Un>0FnR-

It obviously follows from the definition that FoR is a subalgebra of R. For anyfiltered algebra R with filtration FR, let us introduce the graded vector space

where F-iR = {0}. If r € FiR \ Fi-iR, then r + Fi^\R has degree i and a(r) =r + Fi-iR is the principal symbol of r.

The multiplication on gr(R) is based, via the distributive laws, on the rule

f cr(rs) i rs £ i+j_i0 otherwise

where r G Fj/? \ Fi-iR and s € K,-J? \ Fj_iR. This well-defined multiplicationmakes gr(jf?) into a k-algebra, called the associated graded algebra.

PROPOSITION 3.6 Let R be a filtered k-algebra with filtration FR such thatgr(.R) is semi-commutative generated by homogeneous elements y\,. .. ,yn withdeg(j/i) = Ui ̂ 0 for I ^ i ^ n. Put u = (MI, . . . , un) e Nn and, for 1 ̂ . i < j ^ n,let 0 ̂ <jjj G k such that j/jj/j = qjiyiyj-

(1) If Xi,... ,xn g J? are swc/i i/iat Xj = j/j + FUi-\R for 1 ^. i ^ n then

F,R =

, therefore, R satisfies (PBW2) with respect to ^<U: that is, x\,..., xn satisfythe ^<u-bounded quantum relations

Q = XjXi — qjiXiXj +pji with exp(pjj) -<u et + €j

40 Bueso et al.

for some polynomials pji G R.

(2) If gf(R) is a graded quantum affine space, then R = k{xi, . . . , xn; Q, ̂ u} is aPBW algebra.

PROOF. 1. We will prove that FSR = Jîai <« â?" by induction on s. Let Zbe the set of indices 1 <J i <J n such that Ui = 0. So, assume s = 0. If Z = 0,then Fo-R — k and there is nothing to prove. If Z ^ 0, then we can assume thatFoR = gr(.R)o and, therefore, j/j = a;, for every i & Z. Since every element in gr(.R)has a standard representation in the monomials ya, we get

F0R=

Now, assume that s > 0 and let r G -FS.R \ Fs_iR. Since r + Fs-iR € gr(/?)s andthe elements yi, . . . , yn are homogeneous generators for gr(/?) we obtain that

|a|u =s |a|u =s

By the induction hypothesis,

f \ T T*'~* £^ P -i /-? — \ ITT*^/ 7 I QtJL> tz X g _ 1 -* ^ — / .IS.**'

And, therefore,

HU ^n

The equality follows from the fact that FR = {FsR}sô is a filtration and Xi 6 -FUi-Rfor z = 1 , . . . , s. In particular, we have proved that xi,..., xn generate R as analgebra over k. For i,j such that 1 ̂ i < j ^ s, there is a non-zero scalar qji suchthat 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) û e^ + e.,-.2. We just need to prove that the monomials xa are linearly independent overk. Assume that the monomials xa are not k-linearly independent. We derive acontradiction as follows. There is a non trivial expression 5ZQewn caXa = 0, thatcan be chosen of minimal exponent (3. Let s = \/3\u, then

0 =|a|u =s |a|u <s

which gives in gr(_R)s the following non trivial relation

E v——^caxa + Fs_lR= 2_, caya

\a\a=s \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 di-mensional then, necessarily, the vector u has all its components strictly positive.

The following result, although not explicitly stated there, was first proved byV. Weispfenning in [33, proof of Theorem 1]. We include a slightly different proofhere.

PROPOSITION 3.8 Let R be a k-algebra finitely generated by x i , . . . , x n andassume that there are non zero qji € k, polynomials pji € R and an admissibleorder ^ on N™ such that the following ^-bounded quantum relations are satisfiedfor 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 aPBW algebra.

PROOF. Let i,j be such that 1 ̂ i < j <C s and write Aji = Af(pji). Then

XjXi — qjiX^Xj — Pji t ^/ K.X

Write

where Cji — € j + €j — Aji. Then C is a subset of Z" such that (the unique extensionto Z™ of) ^ is positive on C. By [29, Corollary 2.2] or [33, Proposition 2.1], thereis some vector w = (wi, . . . , wn) £ N™ with w^ > 0 such that (w, /3} > 0 for every(3 € C. But this means that, for every 1 <C i < j <J n,

a\w = (w, a) < Wi + Wj for every a € Aji

n

After Proposition 3.8, it is possible to decide effectively if a given set of quantumrelations Q is bounded by some admissible order on N" and, in the case of affirmativeanswer, to compute explicitly a bounding order of the form ^.w for some vectorweight w = (wi, . . . , wn) £ N™ . To do this, write

c= C

42 Bueso et al.

where Cji = e, + €j — N(pji) for every 1 ̂ i < j ^ n, consider the linear program-ming problem

minimize /(to) = 101 + • • • + wswith the constraints

PROPOSITION 3.9 The set of quantum relations Q is bounded for some admissi-ble order on N™ if and only if the linear programming problem (9) has a solution. Inthis case, <£> contains some vector w with integer components which gives a boundingordering ^<w.

Some explicit examples of computation of w by means of Proposition 3.9 canbe found in [8].

PROPOSITION 3.10 Let R be a \i-~algebra generated by finitely many elementsXi,...,xn and assume that there are a vector w = (wi,...,wn) € Nn, non-zeroelements qji e k and polynomials pji e R such that the following ^<w-boundedquantum relations are satisfied for 1 ̂ i < j sC n

Q EE XjXi = XiXj + pji with exp(pij) -<w e.^ + €j

Define, for every positive integer s, F™R = ]Ciai <« kcca. Then

(1) The algebra R is filtered with filtration FWR = {F™R}.

(2) The associated graded algebra grw(R) is generated by the principal symbols j/j =Xi + F^.^R forl^is^n.

(3) If hji = Pji + F™i+w._1R then the following ^<w-bounded quantum relations aresatisfied in grw(R) for 1 ̂ i < j ^ n

Qw = yjVi = qjiViVj + hj{ with exp(%) -<w €i + 6j

(4) If R = k{o;i, . . . , xn; Q, ̂ <w} is a PBW algebra, then

is a PBW algebra.

PROOF. 1. By [5, Proposition 1.7 (see also [18, Lemma 1.4]) we have that forevery a, (3 6 N™ there exists qa^ € k such that

xaxf3 = qa3Xa+P + c

Obviously, 1 6 F^R. By (10), the monomials xa generate R as a vector space overk. This gives R = Us>0 F™R. Clearly, if a, j3 € Nn are such that a\w < \(3\w then« ~<w ft- This implies that F™R C F™ R whenever s < t. For any pair of positive

integers s,t, let / 6 F™R and g e F?R. We want to prove that fg e F™+tR.Clearly, it suffices to prove that xax@ € F^_tR for every a.,/3 & N™ such thata\w ^ s and \/3\w ^ t. By (10), it is enough to check that \*Y\W ^ s + t for every

7 ±>«, 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) isgenerated as a k-vector space by xa + F^.1R, with |o: w = n, and we know thatxa + F^R = ya, where y > = X i + F^^R for i = 1, . . . , s. Therefore, yi, . . . , ynare 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. Sincegrw(R) is a graded algebra, it suffices to prove that every linear combination ofhomogeneous monomials of the form Xliaj =s caya = 0 is necessarily trivial. Butthis 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 wecompute a standard representation of hji with respect to the standard monomialsya. 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 thefiltration FWR is standard if and only if w = (1, . . . , 1), i.e., -^w is just the degreelexicographical order. ^4. There is an analog to Proposition 3. 10. (4) for the Rees algebra R associatedto 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 relationsare

where

— /T •T • — ^ - • T . ' r - 4- \ ,->, o^l— X j J — yjz^i^j T^ / ^A"' j

The foregoing propositions can be restated in a more compact form which isinteresting from a theoretical point of view. We give two theorems. The first oneexplains, from the point of view of algebras given by generators and relations, whenit is possible to find a filtration with semi-commutative associated graded algebra.

44 Bueso et al.

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 associ-

ated 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 <; n. This implies that, in Proposition 3.10.(3), all hjiSare zero and, hence, giw(R) is semi-commutative. D

DEFINITION 3.13 An algebra satisfying the conditions of Theorem 3.12 is saidto be a somewhat semi-commutative algebra.

The following Theorem characterizes the solvable polynomial algebras intro-duced in [18] in terms of nitrations. This gives a neat connection between thesolvable polynomial algebras and the filtered structures on noncommutative ringsstudied in [28].

THEOREM 3.14 The following conditions are equivalent for a k-algebra R.(i) R is filtered by a finite-dimensional filtration with gr(fl) a graded affine quan-

tum space.(ii) R is filtered with gr(R) a graded affine quantum space.(iii) R is a PB W algebra.(iv) R is a PB W algebra with respect to some admissible order of the form ^w

with w 6 NJ

PROOF. This follows from propositions 3.6, 3.8 and 3.10 in a way similar to thatof Theorem 3.12. D

4 CONSEQUENCES AND EXAMPLES

Everyone working in the field of non-commutative algebras would greatly appreciatesome way to get properties of a given algebra just checking some properties on thedefining relations. This is the case for the algebras satisfying a set of quantumrelations. You need just to check if the relations are bounded for some admissibleorder on N™ and the concluding theorems in Section 3, in conjunction with the welldeveloped theory of filtered algebras, will take care of everything. We will recallsome of these interesting properties. Let R be a noetherian k-algebra. We say thatthe Gelfand-Kirillov dimension is exact if for every finitely generated left or right-R-module M, GKdim(M) is an integer and

GKdim(M) = sup{GKdim(AO,GKdim(M/AO}

for every submodule N of M. R is left (resp. right) finitely partitive if, given givenany finitely generated left (resp. right) .R-module M, there is an integer n > Q suchthat, 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 propertyif every element in the Jacobson radical is nilpotent. R has the endomorphismproperty 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. Afinitely generated left or right .R-module M satisfies the Auslander condition if forevery integer n and every submodule TV of Ext^(M, R), we have Ext^(Ar, R) = 0for every i < n. A noetherian ring with finite injective dimension is said to beAuslander-Gorenstein if every finitely generated left or right .R-module satisfies theAuslander condition. If in addition R has finite global homological dimension, wesay that R is Auslander regular. Finally, if M is a left or right .R-module, the gradenumber 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 generatedleft 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 rela-tions. 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 finitelygenerated 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 itsGelfand-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 anAuslander-regular and Cohen-Macaulay algebra.

PROOF. By Theorem 3.12, R has a finite-dimensional filtration with semicommu-tative associated graded algebra. The noetherianity of R follows from [25, 1.6.9].The exactness of Gelfand-Kirillov dimension on short exact sequences of finitelygenerated left or right .R-modules follows from [32, 4.4 Theoreme] or [26, 3.8]. Ris left and right finitely partitive by [16, Theorem 2.9]. The bound on the Krulldimension can be obtained from [25, 8.3.18] (see [31] for the original citation). TheNullstellensatz follows from [26, 3.8]. Let R now be a PBW algebra. Then R has afinite-dimensional filtration such that gr(.R) is an affine quantum space. Using [22,

46 Bueso et al.

3.6 Theorem] (see also [13, Theorem 4.2]), it is easy to see that gr(/?) is Auslander-Gorenstein, 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 regularby [25, 7.6.18] (see also [30]). Finally, by [3, 3.8 Theorem] and [25, 8.6.5], R isCohen-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 prop-erties listed in Theorem 4.1. In each case, we include an explicit vector weight wwith strictly positive integer components in such a way that the filtration given inProposition 3.10 yields a graded quantum affine space as associated graded algebra.

• Enveloping algebras of finite-dimensional Lie algebras. Let 0 be afinite-dimensional Lie k-algebra with basis xi,... ,xn. Then the envelopingalgebra U(Q) = k{xi,..., xn; 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 relationsare

+ (A - l)pjiXif}Xja if j > i, /3 > aif j > z, a < j3if j = i, (3 > a

and where the weight vector can be expressed here as a weight matrix:

w= . . (2i+Q)1<:. <

op+i 2P+2 . . . 22p i

• Quantum Weyl algebras. Let Q = (<?i, • • • ,<?«) be a vector in kn withno zero components, and let F = (lij) be a multiplicatively antisymmet-ric n x n matrix over k. The multiparameter Weyl algebra ^4^>r(k) =k{yi,...,yn,xi,...,xn;Q,^,w} is a PBW algebra, where

(i < 3)(i < 3}(i > J)

and io = ( l , l , . . . , l , l , 2 , . . . , n ) .

• Quantum symplectic spaces. For every nonzero element q in k, the algebraOq(s)f>k2n) = k{j/i, . . . ,yn,xi,... ,xn;Q,^.w} is a PBW algebra, where

j X i = q-xixj,lyi - fyiXi =

i-l2 - i) E (?i=\• Iterated skew differential operators algebras. The foregoing quantized

algebras and the enveloping algebras in the solvable case are instances of thefollowing more general construction. Quantum euclidean spaces are obtainedin this way. Let

be an iterated Ore extension of k. Assume that <TJ(XJ) = qijXi, for everyi < j ' 5: Pi where the qij's are nonzero scalars in k. The set B — {xa; a € W1}is a k-basis of R. We will construct a weight vector w = (wi, . . . ,wn). Setw\ = 1 and define w\z as the degree in x\ of the polynomial f>i(x\). Put w^ =max{l, 1012} + 1. Suppose we have defined wi, . . . , ifj-i for j > 2. For k =1, . . . ,j — l, set Wfcj = max{aiifi + - • •+a,-_iuij__i; a 6 J\f(6j(xk))} and chooseu>j = maxjljU;^ — Wi; 1 < i , f c < j — 1} + 1. Then /? = k{xj, ... ,xn;Q, ̂ w}is a PBW algebra, where

Q = iXj + 8j(xi] I ^ i < j ^ n}

• Polynomial Ore algebras. For 1 ̂ i ^ s, 1 ̂ j sj r, let a,ji,bji G k witha_jj ^ 0, and let Cjj(cc) e k[x i , . . . , z s ] be commutative polynomials. Thealgebra O = k{xi, . . . , xr, tti, . . . ,us; Q, ̂ w} is a PBW algebra, where thequantum relations are

The vector weight is

for 1for 1

j + bjiUj + Cji(x) for 1

to = (!,...,!, wi,...,wr)

i < j ^ 5i < j ^ ri ^ s, I ^ j

where the weights w\, . . . ,wr are constructed as follows. For each 1 ^ i ^s> 1 ̂ J ^ r> let wji be the total degree of the polynomial Cji(x). Then chooseWj = max{u>ji 1 <C i ^ s}.

The algebra Vq(sl3(C)). Let q be a complex number such that q8 ^ 1.Consider the complex algebra given by ten generators /i2, /is, /23, &i, ^2, ' i j

48 Bueso et al.

/2, 612, 613, 623 subjected to the following relations.

613612 = 9~26l2ei3 /13/12 = 9~2/12/13

623612 = 926l2623 - 9e13 /23/12 = ?2/12/23 ~ 9/13

623613 = 9~26l3623 /23/13 = 9~2/13/23

£2 _ /2 612/01 = 9~

612/12 = /12612 + 6 k = k

612/13 = /13612 + 9/23^ 613*1 = ̂

612/23 = /23612 613fc2 = 9~1/C26l3 &2/13 =

613/12 = /12613 - <?~1^623 e23^1 = 9^1623 &1/23 =

fcf fc| - l\l\ 623fc2 = q~2k2e23 /c2/23 =613/13 = /13613 - —— 5 ————— 5— , 2, 7 ,.2 ~ 2

613/23 = /236i3 + 9^|ei2 ei2,;2 = q-ll2el2 /2/12 = 9"1 /J2

623/12 = /12623 e^/j = qr/ i e i3 /j/^ = q-/13^

623/13 = /13623 ~ 9~ /12^2 C13/2 = g/26l3 ^2/13 = 9/13^2

623/23 — _

By solving the linear programming problem (9) we obtain that the displayedquantum relations are ^TO-bounded, where

w = (3, 5,3,1,1,1,1,1,1,1,)

Bergman's Diamond Lemma shows, after some large computations, that thestandard monomials in /i2, /is, /23, ki, fc2, li, /2,612,613,623 form a C-basis ofVq(sls(C)), and, thus, it is a PBW algebra. Notice that the quantized en-veloping C-algebra of s[s(C) is a factor algebra of V^(sIs(C)).

5 GROBNER BASES FOR MODULES

Let R = k{Qi,..., Qn\ :<,} be a PBW algebra and consider the free left _R-moduleRm with basis ei,...,em. Every finitely generated left .R-module is isomorphicto a factor module of the form Rm/K for some m and some submodule K of Rm

which has to be finitely generated (R is noetherian). The computational treatmentof finitely generated left /?-modules has its basic tool in the notion of Grobner basisof submodules of free modules of finite rank. For this purpose, we will introducesome notions and the corresponding notation. Our point of view is influenced bythe theory developed for the commutative case in [21] and [15]. Every element/ e Rm can then uniquely be expressed as Y^iLi fiei> where /; € R, for 1 < i <m. For every (a,i) g pjn.(m)) consider the element x^a'^ = xaei of Rm, wherexa = x*1 • • -x%». Since B = {xa-} a g N"} is a k-basis of R, it follows that

Bm — {x(a'^; (a,i) £ N"'(m)} is a basis of Rm as a k-vector space. Therefore,every element / £ Rm has a unique standard representation

We define the Newton diagram of / to be

and the exponent of / to be

exp(/) = maxA/"(/).

Here, the maximum is computed with respect to the order on N"'(m) denned in (2).If (a,i) is the exponent of /, then

We will refer to c^a^ as the leading coefficient of / and we will write lc(/) = C(QJJ).The exponent of the submodule K of Rm is denned as the stable subset of Nra '^m^

Exp(tf) = {exp(/) feK}

DEFINITION 5.1 A subset 9l, . . . ,gt of K is said to be a Grobner basis of K if

t+N") (13)

By Proposition 2.5, every submodule K of Rm posseses a Grobner basis, whichis expected to be a set of generators for K. This can be deduced from the DivisionAlgorithm, which is also the fundamental tool to compute effectively Grobner bases.By [10, Theorem 1.2], for every a, j3 £ PP, there is an unique nonzero scalar qa,p € kand a polynomial pa,/3 G R such that

xax13 = qa,0xa+f3 + pa>0 (exp(pQj/3) -< a + (3)

Given / £ Rm with exp(/) = (a,k) e N™'( m ) j we define the scalar exponent of /as sexp(/) = a € N™. We will write (a,i) <n '(m) (/3,j) if and only if i = j and(/3, i) = 7 + (a, z) for some 7 £ N™. As the reader could check by using Dickson'sLemma, the Division Algorithm given in Algorithm 1 is mathematically sound.

The element r obtained in the Division Algorithm is said to be a remainder of77* __

the division of / by the set F = {/lt . . . , fs}. We will denote by / any remainder.Of course, this notation is somewhat ambiguous, as the remainder is not in generaluniquely determined. We have the following consequence of the Division Algorithm.

PROPOSITION 5.2 A finite subset G = {<?i, . . . ,<7 t} of K consisting of nonzerof~t __

elements is a Grobner basis of K if and only if f — 0 for every f £ K . Inparticular, every Grobner basis of K is a set of generators.

50 Bueso et al.

Algorithm 1 Division Algorithm for free modules_______ _____ __INPUT: / , / ! , . . . , /s € Rm with /, ^ 0 (1 < i < s)OUTPUT: hi,... ,hs,r such that / = hlf1 + ••• + hsfs + r

and r = 0 or Af(r) D UI=i(exP(/i)) + Nn) = 0 andmax(exp(/i!) +exp(/1), . . . ,exp(/is) + exp(/J,exp(r)) = exp(/).

INITIALIZATION: hi := 0 , . . . , hs := 0, r := 0, 5 := fWHILE g ^ 0 DO

IF there exists i such that exp(/J <™'(m) exp(g) THENchoose i minimal such that exp(/i) <n'(m) exp(g)O-i = lC(ff)lc(/i)'?Sexp(g)-sexp(/ i),sexp(/ i)/H := hi + aixsexP^)-^p(f,)g :=g - aiX^xp(g)-sexp(fi)fi

ELSEr := r + lm(g)g := g - lm(gr)

The effective computation of the Gelfand-Kirillov dimension of Rm/K we areinterested in needs as an important ingredient the computation of a Grobner basisfor K. We present a version of Buchberger's algorithm that fits our PBW algebras(see Algorithm 2). This algorithm is a special case of the one we proposed in [10]with the purpose of the effective computation of "Ext functors" for modules overthe left PBW rings introduced there.

DEFINITION 5.3 Let f,g be vectors in the free left .R-module Rm, and writeexp(/) = (a,j), exp(g) = ((3,k). Let 7 = (71, ...,7n) be defined by 7* =

*i,/3j} for 1 <C i sC n. The S-vector of / and g is defined as

SP(/ ) f f) = [ g-iaaic(ff)a.-y-a /_g-i lc(/)^-/3g Htkk.

Algorithm 2 Left Grobner Basis Algorithm for ModulesINPUT: F = { / ! , . . . , f,} C Rm with /^ 0 (1 < i < s)OUTPUT: G = {glt... ,gt}, a Grobner basis for Rfi + --- + Rfs.INITIALIZATION: G := F, B := { { f , g } ; f ^ g e G}

WHILE B ^ 0 DOChoose any {/,<?} € BB:=B\{{f,g}}h' := SP(/,g)h:=°SP(f,g)IF h ̂ 0 THEN

B:=Bu{{p,h}; p € G}G:=GU{/ i}

6 HOMOGENEOUS GROBNER BASES

Let R be a filtered k-algebra with filtration FR. A filtration on a left /?-moduleM is a family FM = {FnM n ^ 0} of vector subspaces of M satisfying

1. FnM C Fn+iM for all n > 0,2. (FnR)(FmM) C Fn+mM for all n,m > 0,3- M = \Jn>QFnM.

The graded vector space

is endowed with a structure of graded left gr(/?)-module in the following way. If?7i G FjM \Fj-iM, the principal symbol of m is given by <r(m) = m + Fj-iM. Theaction of gr(.R) on gr(M) is based, via the distributive laws, on the rule

fcr(rm) itrm<£ FJ+i_iMa(r)a(m) — <

I 0 otherwise

where r G ̂ .R \ F^fi and m G FjM \ Fj^M.Now, let # = k{:ri,...,:rn;Q, i^} be a PBW algebra (by Proposition 3.8

any PBW algebra is of this form, after replacing, if needed, the old order ;< bya new weighted order ^w). By Proposition 3.10, grw(R) = k{xi , . . . , xn; Qw, ̂ w}is a PBW algebra. Let Rm be a finitely generated free left /^-module with ba-sis BI, ..., em. This module has a canonical filtration FwRm given by F™Rm =(F™R)Rm for every positive integer s. It is immediate that F™Rm = (F™R)ei ®• • • ® (F™R)em. An immediate consequence is that grw(Rm) is a free graded leftgrw(_R)-module with basis BI, ... em.

Every B-submodule K of Rm inherits a filtration FWK given by F?K = K nF™Rm for every positive integer s. Also, the factor module M = Rm/K is filteredby Ff(M) = (F™Rm + K)/K. With these nitrations, the short exact sequence ofleft /^-modules

0 -+ K -* Rm -> Rm/K -* 0

is strict in the sense of [27], and, hence, we obtain a short exact sequence of gradedleft gr™(.R)-modules

0 -^ g?w(K) -* grw(Rm) -+ grw(Rm/K) -> 0

which implies, in particular, that grw(Rm/K) ^ giw(Rm)/grw(K). Therefore, apresentation of grw(Rm/K) can be constructed from a presentation of Rm/K. Wewill show that this construction can be done effectively.

LEMMA 6.1 For any non-zero vector f G Rm we have exp(/) = exp(<j(/)).

PROOF. First, we will prove that exp(/) = exp(<j(/)) for every / e R. So, leta = exp(/) and write |ct|u, = s. Therefore, / = caxa + 5^/3-< a

C/3X^• Since R isa PBW-algebra with respect to ^w, it follows that

52 Bueso et al.

*(f)=CaXa + Vfi -<w fiI/3|» =s

so exp(<r(/)) = exp(/). Now, let / — ( f i , . . . , fm) <E Rm, and write s = a\w, where(a,i) = exp(/). For each 1 <C j <J TO, define

f = /j if |' 0 i f | ex P ( / J )U< S

and'/ = ( s / i , . . . , s /™) . Clearly,

and, thus,

exp(/) = (CM) = (exp(/ i),i) - (exp^/O),*) = exp(<r(s/)) = exp(a(/)).

D

PROPOSITION 6.2 For every submodule K of the free left R-module Rm , wehave

PROOF. This follows in a straightforward way from Lemma 6.1. D

THEOREM 6.3 1. If {glt.. .,gt} is a Grobner basis for K then

is a Grobner basis for grw(K).2. If {hi, . . . , hs} is a Grobner basis for grw(K) and g1, . . . , gt 6 K are such thata(di) = hi for every 1 ̂ i ^ t, then {gl, . . . , gt} is a Grobner basis for K .

PROOF. 1. Indeed, if {g1, . . . ,gt} is a Grobner basis of K, then

As gr™(.fQ) = Exp(.K') by Proposition 6.2, and exp(gi) — e\p(a(gi)) by Lemma6.1, clearly

so {a(gl\... ,a(gt)} is a Grobner basis of grw(K), indeed.2. Similar to part 1. D

7 THE GELFAND-KIRILLOV DIMENSION

Let us recall the definition of the Gelfand-Kirillov dimension of finitely generated k-algebras (details can be found in [27], [20], [24]). If V is a finite-dimensional vectorsubspace of R and n is a natural number, then Vn denotes the vector subspace ofR generated by all n-fold products v\ • • -vn, where Vi & V. It is understood thatV° = k. Define

y(n) _ V^ yi

i=0

Now, assume that R is finitely generated as k-algebra by V. The Gelfand-Kirillov dimension of R (GKdim(R), for short) measures the asymptotic rate of poly-nomial growth of the 'dimension function' f ( n ) = dim^ V^n\ In fact, GKdim(.R)is the infimum of the real numbers r such that f ( n ) < nr for n 3> 0. It is knownthat this value is independent of the choice of the generating subspace V. The GKdimension of a finitely generated left .R-module M is given by the growth of thefunction /(n) = dimk(V^U), where U is any finite-dimensional vector subspaceof M that generates M as left .R-module. This real number coincides with thedimension of the function f ( n ) as defined in (3).

Our next goal is to show that the Gelfand-Kirillov of Rm/K can be com-puted from a given set of generators { f i , - - - , f s } of the submodule K. Thefirst step is to replace the given admissible order X by some order ^<w for somevector w = ( u > i , . . . ,wn) with strictly positive integer components. This is pos-sible due to Proposition 3.10 and the effective computation of the weight vec-tor with strictly positive components is give by Proposition 3.9. Thus, assumethat R = k{xi,... ,xn; Q, ̂ w} is a PBW algebra, which will be filtered by theiD-filtration FWR. Any finitely generated left R-module can be presented asM = Rm/K, where Rm is a free left .R-module with basis e i , . . . ,e n , and K isa submodule of Rm. We endow M with the to-filtration FWM. The Hilbert func-tion relative to this filtration is defined as

HF^(s) = dimk(F?M) (14)

LEMMA 7.1 With the previous notation we have

GKdim(M) =

PROOF. Let HFgrw (M) denote the Hilbert function of the graded left giw(R)module grw(M), namely,

3=0

By [32, Proposition 3.2, Lemme 2.2],

GKdim(M) = GKdim(gru '(M)) =

Finally, for each positive integer s,

FW(M]'d i r n k ( g r - ( M ) J ) -

j=0 j=0 j=0

54 Bueso et al.

which implies that HFgrw (M) = HF^. n

LEMMA 7.2 For every positive integer s, the set

K s, a,

is a frasis /or i/ie k -vector space F™(M). In particular,

PROOF. Let G — {g1, . . . ,gt} be a Grobner basis for K and consider f + K with|exp(/)|u, < s. By the Division Algorithm, we know that / = qig^ + • • • + qtQt + fwhere exp(r) ^<w exp(/) and A/"(r) n Exp(K ) = 0, i.e.,

It follows from (cc,z) ^w exp(r) for all (a,«) £ A/"(r), that KQ:,?)!^, < s, so

r =

Moreover, if (a,i) <£ Exp(/f), then a;â'^ ^ X. We thus have proved that the set ofall xâ'^ + K with the property that |(a,z)|u, < k for (a,z) ^ Exp(K) is a systemof generators for FS

U)(M) = {/ + K] |exp(/)|TO < s}. Let us now prove their linearindependence. If

then r = E(Q,i)ÊxP(K) c(a,i)X(a^ £ ^> whereM(r)r \Exp(K) = 0. By Algorithm1 and Proposition 5.2, we obtain that r = Gr = Q, so C(a,i) =0. D

THEOREM 7.3 Let M be a finitely generated left R-module with a presentationM = F/K. Then GKdim(M) = dim(Exp(A")).

PROOF. This is a consequence of Lemmas 7.1 and 7.2. D

Let us now describe the algorithm to compute the Gelfand-Kirillov dimensionof a given finitely generated left .R-module M = Rm/K over a PBW algebraR = k{xi,... ,xn;Q,^<}. Given a set of generators {/1;...,/s} for K, proceedas follows.

1. Compute a vector weight w = (wi,..., wn) € N+ such that

R = .

is a PBW algebra.

2. Compute, by using the algorithms described in Section 5, a Grobner basis Gfor K.

3. Compute, from G, the stable subset Exp(K) of N"'(m).

4. The Gelfand-Kirillov dimension of M is the dimension of 'Exp(K), which iscomputed by using Theorem 2.10.(1).

REMARK 7.4 When w = (!,...,!), it follows that the ^-filtration of M =Rm/K is just the standard filtration of M. On the other hand, we have that theHilbert function associated to this filtration is, precisely, WFExp(K)- By using theresults of Section 2, we get that HFEXp(/f)(s) 'ls a polynomial of known degree, fors big enough. By Theorem 2.10.(2) and Proposition 2.9, this polynomial can beexplicitly computed by interpolation. Details can be found in [9]. This appliesin particular to Weyl algebras and enveloping algebras of finite dimensional Liealgebras, which are PBW algebras with respect to -<degiex-

References

[1] M. Artin, W. Schelter, and J. Tate, Quantum deformations of GLn, Commun.Pur. Appl. Math. 44 (1991), 879-895.

[2] T. Becker and V. Weispfenning, Grobner bases. A computational approach tocommutative algebra, Springer-Verlag, 1993.

[3] J.-E. Bjork, The Auslander condition on noetherian rings, Sem. d'Algebre P.Dubreil et M.-P. Malliavin 1987-1988 (M.-P. Malliavin, ed.), Lecture Notes inMathematics, no. 1404, Springer-Verlag, 1989, pp. 137-173.

[4] J. L. Bueso, F. J. Castro, J. Gomez-Torrecillas, and F. J. Lobillo, Com-puting the Gelfand-Kirillov dimension., SAC Newsletter 1 (1996), 39-52,http://www.ugr.es/~torrecil/Sac.pdf/

[5] ____, An introduction to effective calculus in quantum groups, Rings, Hopfalgebras and Brauer groups. (S. Caenepeel and A. Verschoren, eds.), MarcelDekker, 1998, pp. 55-83.

[6] J. L. Bueso, F. J. Castro, and P. Jara, The effective computation of the Gelfand-Kirillov dimension, P. Edinburgh Math. Soc. (1997), 111-117.

[7] J. L. Bueso, J. Gomez-Torrecillas, and F. J. Lobillo, Effective computationof the Gelfand-Kirillov-dimension for modules over Poincare-Birkhoff-Wittalgebras with non quadratic relations., Proceedings of EACA-98, UniversidadComplutense and Universidad de Alcala, 1998.

[8] ____, fiCudndo un Algebra finitamente generada es de tipo Poincare-Birkhoff-Witt?, Proceedings of EACA-99, Universidad de La Laguna, Tenerife,1999, pp. 195-204.

56 Bueso et al.

[9] J. L. Bueso, J. Gomez-Torrecillas, and F.J. Lobillo, Dimension de Gelfand-Kirillov y Polinomio de Hilbert-Samuel de Grupos Cudnticos: Metodos Al-goritmicos, http://www.ugr.es/~torrecil/academia.pdf, Research Awardof the Academia de Ciencias Matematicas, Fisico-Qufmicas y Naturales deGranada, 1999.

[10] J.L. Bueso, J. Gomez-Torrecillas, and F.J. Lobillo, Homological computationsin PBW modules., preprint, 1998.

[11] G. Carra Ferro, Some properties of the lattice points and their applications toDifferential Algebra, Comm. Algebra 15 (1987), 2625-2632.

[12] F. Chyzak, Fonctions holonomes en calcul formel, Ph.D. thesis, Ecole Poly-thecnique, 1998.

[13] E. K. Ekstrom, The Auslander condition on graded and filtered noetherianrings, Seminaire Dubreil-Malliavin 1987-1988, Lecture Notes in Mathematics,vol. 1404, Springer, 1989, pp. 220-245.

[14] K. L. Fields, On the global dimension of skew polynomial rings, J. Algebra 13(1969), 1-4.

[15] A. Galligo, Algorithmes de calcul de base standards, Prepublication de 1'Uni-versite de Nice, 1983.

[16] J. Gomez-Torrecillas and T.H. Lenagan, Poincare series of multi-filtered alge-bras and partitivity, J. London Math. Soc., to appear.

[17] Li Huishi, Hilbert Polynomial of Modules over Homogeneous Solvable Polyno-mial Algebras, Comm. Algebra 27 (1999), 2375-2392.

[18] A. Kandri-Rody and V. Weispfenning, Non-commutative Grobner bases in al-gebras of solvable type, J. Symb. Comput. 9/1 (1990), 1-26.

[19] E.R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, NewYork, 1973.

[20] G.R. Krause and T.H. Lenagan, Growth of algebras and Gelfand-Kirillov di-mension, Research Notes in Mathematics, vol. 116, Pitman Pub. Inc., London,1985.

[21] M. Lejeune-Jalabert, Effectivite des calculs polynomiaux, 1984-85, Cours deD.E.A., Univ. Grenoble.

[22] T. Levasseur, Some properties of noncommutative regular graded rings, Glas-gow Math. J. (1992), 277-300.

[23] F. J. Lobillo, Metodos algebraicos y efectivos en grupos cudnticos., Ph.D. thesis,Universidad de Granada, 1998.

[24] M. Lorenz, Gelfand-Kirillov dimension and Poincare series, Cuadernos de Al-gebra, vol. 7, Universidad de Granada, 1988.



[25] J. McConnell and J. C. Robson, Noncommutative noetherian rings, Wiley In-terscience, New York, 1988.

[26] J. C. McConnell, Quantum groups, filtered rings and Gelfand-Kirillov di-mension, Noncommutative Ring Theory, Lecture Notes in Math., vol. 1448,Springer, 1991, pp. 139-149.

[27] J.C. McConnell and J.C. Robson, Noncommutative Noetherian rings, J. Wileyand Sons, Chichester-New York, 1987.

[28] T. Mora, Seven variations on standard bases, Preprint, 1988.

[29] T. Mora and L. Robbiano, The Grobner fan of an ideal, J. Symbolic Comput.6 (1988), no. 2-3, 183-208.

[30] A. Roy, A note on filtered rings, Arch. Math. 16 (1965), 421-427.

[31] S. P. Smith, Krull dimension of the enveloping algebra o/sl(2,C), J. Algebra71 (1981), 89-94.

[32] P. Tauvel, Sur la dimension de Gelfand-Kirillov, Comm. Algebra 10 (1982),939-963.

[33] V. Weispfenning, Constructing Universal Grobner Bases, Proceedings ofAAECC 5, Springer LNCS, 356, 1987, pp. 408-417.


Some Problems About Nilpotent Lie Algebras

J.M. CABEZAS and E. PASTOR, Dpto. de Matematica Aplicada, Universidaddel Pais Vasco, Vitoria-Gasteiz. Spain.E-mail:[email protected] and [email protected]

L. M. CAMACHO, J. R. GOMEZ and A. JIMENEZ-MERCHAN, Dpto. MatematicaAplicada I, Universidad de Sevilla, Sevilla. Spain.E-mail:[email protected], [email protected] and [email protected]

J. REYES and I. RODRIGUEZ, Dpto. de Matematicas, Universidad de Huelva,Huelva. Spain.E-m&il:[email protected] and [email protected]

Abstract

In this paper we survey the classification of some important families ofnilpotent Lie algebras. We also consider some geometric problems on suchfamilies and several related computational topics. Most of the results havebeen obtained by at least one of the authors of this paper. So, we give anoverview of the work developed on Lie theory over the last years by somemembers of the research group directed by J.R. Gomez.

1 INTRODUCTION

The importance of Lie algebras in different domains of mathematics and physicshas become increasingly evident in recent years. For instance, in applied mathe-matics Lie theory remains a powerful tool for studying differential equations, specialfunctions and perturbation theory.

The study of the constructive aspects of Lie theory has become practical recentlyby the use of computers. Computers provide a valuable new research tool in Lietheory, which makes it possible to test (and even to formulate) various theoreticalconjectures about Lie algebras. One of the first constructive problems to appear is


60 Cabezas et al.

related with the classification of such algebras. It is in this general framework, thatour paper can be considered.

A Lie algebra (Q,/J.) is a vector space 0 over a field IK, with a bilinear mappingM : 0 x 0 —> 0 denoted (X, Y) —> f-i(X, Y) — [X, Y] and called bracket product,verifying

1. [X,X} = 0,2. [X, [Y, Z}\ + [Y, [Z, X}] + (Z, (X, Y]] = 0

for all the elements X, Y, Z of 0. The second condition is called Jacobi Identity.The dimension of the Lie algebra is the dimension of the vector space 0. Most

of the Lie algebras which appear in this paper will be considered over the complexfield C and will have finite dimension. By taking a basis (Xo,Xi,... ,Xn_i) in 0,the algebra is completely determined by its structure constants, that is, by the setof complex constants {C^}, defined by a(Xi,Xj) = X)fc=i ^tjXk- Then, we canidentify the algebra 0 and its law fj,. Thus, the set Cn of laws of Lie algebras is anaffine algebraic set defined by the polynomials expressions

£<(i)(2)

One could think of a program of classifying all Lie algebras by considering theabove equations to be solved for unknown structure constants. This turns out to bea very complicated problem because of the non-linearity of (2). In fact, the generalclassification is an open problem.

Levi's classical and famous theorem decomposes the classification of Lie algebrasinto the classification of semisimple and solvable Lie algebras. The classification ofsemisimple Lie algebra is well-know from the works of Killing and Cartan in 1914.The classification of the solvable Lie algebras can be reduced in a certain senseto obtain the classification of the nilpotent Lie algebras, which can actually beconsidered the unsolved problem.

The descending central sequence of a Lie algebra 0 is defined by C°0 = 0,<^0 = [Q,Ct~lQ}. The first ideal CIQ = [0,0] is the derived algebra of g. IfC f c0 = {0},for some k, the Lie algebra is said to be nilpotent. The smallest integer k such thatCfcg = {0} is called the nilindex of 0. The nilpotent Lie algebra g is filiform ifdimCzg = n — i — 1 for 1 < z < n — 1. Therefore, an n-dimensional filiform Liealgebra 0 has nilindex n— 1. Hence, any filiform Lie algebra has a maximal nilindex,and filiform Lie algebras are considered to be the "less" nilpotent Lie algebras. Thenilpotent Lie algebras with nilindex n — 2 will be called quasi-filiform.

The first non-trivial classifications of some classes of low-dimensional nilpotentLie algebras are due to Umlauf [57]. Some of the lists of nilpotent Lie algebrasobtained by Umlauf contain errors and they are incomplete. Then, several authorshave worked providing new lists (see [23], [24], [49], [52], [53], [54], [55], in thebibliography), but the complexity of the involved computations leads frequently toerrors. We remark that the first exact classification of the nilpotent complex Liealgebras in dimension n < 6 is probably due to Morosov [48]. It is also obtainedin a different way by Vergne [59], who showed the important role of filiform Liealgebras in the study of variety of nilpotent Lie algebras laws.

Nilpotent Lie Algebras 61

Recently a new invariant has been introduced by Goze to study nilpotent Liealgebras. Let g be a nilpotent Lie algebra. Let gz(X) be the ordered sequence ofJordan block's dimensions of the nilpotent operator a,d(X), where X £ g. In the setof these sequences, we consider the lexicographical order. Then the characteristicsequence

gz(g) = max {gz(X}}

is an invariant of 0. An n-dimensional Lie algebra Q is filiform if and only if theGoze invariant gz(g) is (n — 1, 1).

Using this invariant, Ancochea and Goze gave the classification of nilpotent Liealgebras in dimension 7 [3] and the classification of filiform Lie algebras in dimension8 [2]. For a presentation of this problem see [44]. The list [2] contains some errorsand it has been corrected by the authors [4] and by Selley [56].

The geometric approach to study nilpotent Lie algebras is one of the main meth-ods which has been developed over the last years. The set A/"n of nilpotent algebralaws is an affine algebraic variety; two Lie algebras are isomorphic if and only ifthey belong to the same orbit of the natural action of the general linear group. Inthis approach the notion of filiform Lie algebra appears in a natural way; the subsetJ-n of filiform laws is an open set in J\fn-

Among the geometric aspects of the variety J\fn we point out the descriptionof their orbits, their irreducible components, etc. It is known that J\fn is reduciblewhen n > 7 [5] , [46] . The closure of an irreducible component of Tn is an irreduciblecomponent of j\fn. Thus, we can obtain information about J\fn, by studying thevariety Tn. For example, the variety }-n has at least 3 irreducible components ifn > 12 [46], and consequently the variety J\fn as well. In fact, for n sufficientlylarge, the number of components of J\fn is at least of order n [5] . A survey of theproblem is exposed in [44] , where we can find the description of the components ofFn, n < 10.

In [44], one also finds the description of some components of the variety Afn,n > 12, which are obtained by using some Lie algebras belonging to JT^. Thesefiliform Lie algebras can be graded in a certain way as we will see in Section 6. Theimportance of these graded filiform Lie algebras was shown by Khakimdjanov inthe study of Mn [46].

In the cohomological study of the variety of laws of nilpotent Lie algebras estab-lished by Vergne [59] , the author also considers a class of graded filiform Lie algebrasthat plays a fundamental role. In fact, using such graded filiform Lie algebras, Gozeand Khakimdjanov give in [45] the geometric description of the characteristicallynilpotent filiform Lie algebras. Thus, it is clear that knowing a family of gradedalgebras of a certain class of nilpotent Lie algebras we get valuable information tosolve problems on such a class.

We now describe how this paper is structured, and the problems studied in eachsection.

In Section 2 we present some geometric aspects about J-n. We have also sum-marized the classification of the n-dimensional filiform Lie algebras with n < 11.Finally, as an approach to a general filiform family, we consider the /c-abelian filiformLie algebras, whose classification is given in any dimension.

Section 3 and Section 4 are devoted to the study of nilpotent Lie algebras withsmall nilindex, that is opposite to the filiform ones. Concretely, in the third section

62 Cabezas et al.

we study the p-filiform Lie algebras, defined by the Goze invariant, in order togeneralize the filiform case to nilpotent Lie algebras with given nilindex. We showhow they can be useful to know the variety J\fn when p is close to the dimension, andthere is very little information. In Section 4 we study the metabelian Lie algebras(nilindex 2) and some Lie algebras with nilindex 3 in order to look at relevantalgebras with small nilindex where the classification is not possible.

Section 5 and Section 6 return again to the study of Lie algebras of large nilindex.In fact, we consider quasi-filiform and 3-filiform Lie algebras. However, in thesesections we deal with some graded families of those nilpotent Lie algebras in anydimension n. The obtained classifications are a valuable utility to study the varietyMn • Section 5 provides an extension of the graded algebras used by Vergne to thep-filiform Lie algebras, p = 2,3. In Section 6 we consider the gradation with a largernumber of subspaces that the natural gradation provides, and we study the filiformand quasi-filiform case. The obtained algebras are better than the naturally gradedones to solve some cohomological problems.

Finally, in the last section we show how symbolic calculus can be used to dealwith some similar problems. In particular, how to make adequate use of some ofthe features that the sofware package Mathematica provides.

The authors thank Professor Y. Khakimdjanov for his constructive suggestions.

2 FILIFORM LIE ALGEBRAS

The first non-trivial classifications of some classes of low-dimensional nilpotent Liealgebras are due to K. Umlauf. In his thesis [57] (Leipzig, 1891) he presentedthe list of nilpotent Lie algebras of dimension m < 6. He also gave the list ofnilpotent Lie algebras of dimension m < 9 admitting a basis (Xo,Xi,... ,Xm-i)with [Xo, Xi] = Xi+i for i = 1,..., m - 2. Now, the nilpotent Lie algebras withthis property are called filiform Lie algebras. Umlauf's list of filiform Lie algebrasis exact only for dimensions m < 7; in dimensions 8 and 9 his list contains errorsand it is incomplete.

In the last years, it has been evident to study again nilpotent Lie algebras due tothe importance of them in classification problems and their applications in differentdomains of mathematics and physics. Unfortunately, many of these papers arebased on direct computations (by hand) and the complexity of those computationsleads frequently to errors.

M. Vergne obtained an exact classification of the nilpotent complex Lie algebrasin dimension m < 6, and showed the important role of filiform Lie algebras, termi-nology introduced by herself, in the study of the variety of nilpotent Lie algebraslaws [59]. Indeed, the subset Fn of filiform laws is a Zariski open set in the varietyof nilpotent Lie algebras Afn- In spite of the developed study of filiform Lie alge-bras over the last years the classification problem is still open. In this section wesummarize the results obtained in the last years.

2.1 Obtaining laws of families of filiform Lie algebras

Obtaining a complete list of Lie algebras is a problem whose complexity increaseswith the dimension of the algebra. Thus, an approach to the study of the variety ofnilpotent Lie algebras is the description of the algebraic sets of laws of filiform Lie

algebras and their irreducible components. In [1] it is shown that the set of laws of 8-dimensional nilpotent Lie algebras is the union of 8 irreducible components. Thesehave been determined explicitly using some algorithms of symbolic computationdeveloped ad hoc by Valeiras [58]. A modified implementation developed by himselfin the language Mathematica can be found in [44].

Obtaining an initial handled set of polynomials, which determine the varietysubject to study is an necessary first step in any subsequent study of such a variety.In [35] can be found how to associate a descent vector to each X G CJ0 which letsus write the brackets by using a recursive relation. Then we get an algorithm toobtain families of filiform laws by fixing an adapted basis.

THEOREM 2.1 There exists a polynomially bounded algorithm which generatesthe set of laws of filiform Lie algebras, for each dimension n.

By using the Jacobi identities we can obtain the conditions which the parame-ters have to verify to determine the family of filiform laws. Some of the obtainedequations in the generated family could allow another more simple form and hencesome of the parameters could be eliminated. In this way, and with an implemen-tation of the algorithm stated in the above theorem, the 11-dimensional family offiliform Lie algebra laws was presented in [22]. Finally, in [35] one may find thestudy for dimension 12.

THEOREM 2.2 The sets of the filiform Lie algebra laws over C11 and over C12

can be parametrized (up to isomorphism) respectively by the points of the affinealgebraic sets J^u C C16, of Krull dimension 12, and F\-2 C C21 of Krull dimension13.

Finally, in [31], the polynomial equations which define the variety jFn let usdetermine and describe its two irreducible components.

2.2 Low-dimensional filiform Lie algebras

Ancochea and Goze gave the classification of nilpotent Lie algebras in dimension7 [3] and the classification of filiform Lie algebras in dimension 8 [2],[4], by using thecharacteristic sequence introduced by Goze to study nilpotent Lie algebras. Usingthis invariant, several authors (see [26], [6]) gave the classification of filiform Liealgebras in dimension 9 and 10. The method of Ancochea and Goze, very welladapted to dimension 8, has been transformed in very complex computations indimensions 9 and 10. The lists [6], [26] are again incomplete and many parameterscan be eliminated. According to [46] filiform Lie algebras share some commonproperties starting in dimension 12. This suggests to study separately filiform Liealgebras for dimensions > 12 and to do a detailed study when the dimension is< 11.

Using an approach of Vergne [59], the techniques developed by Khakimdjanov[46], and introducing the notion of elementary changes of bases we have finally givena complete classification up to isomorphism of all complex filiform Lie algebras ofdimension m with m < 11 [33], In this way, for instance, the 130 families of thelist [6] may be reduced to less than 65. Moreover, some families of Lie algebra lawsobtained in [33] do not appear among those 130 families. Hence, the list [6] is notvery useful due to the errors that such a list contains.

64 Cabezas et al.

The simplest (n + l)-dimensional filiform Lie algebra is Ln+i, defined by

where (Xo, Xi, . . . , Xn) is a basis of Ln+i (the undefined brackets being zero). Wenote that any (n + l)-dimensional filiform Lie algebra is isomorphic to a lineardeformation of Ln+i (see [46]).

Let A be the set of pairs of integers (k, r) such that I < k < n—I,2k+l < r <n- if n is odd, one adds to A, also the pair (s^,n). To any element ( k , r ) G A, wecan associate the 2-cocycle for the Chevalley cohomology of Ln with coefficients inthe adjoint module denoted by $k,r and defined by

if 1 < i < k < j < n, and ^k,r(Xi,Xj) = 0 otherwise. Let JFm be the varietyof filiform Lie algebras laws in an rn-dimensional vectorial space. Any (n + 1)-dimensional filiform Lie algebra law fj, G J~n+i is isomorphic to /J,Q + fy, where JJ,Q isthe law of Ln and $ is a 2-cocycle defined by

(fc ,r)€A

and verifying the relation fy o $ = 0 with

* o #(x, y,z) = 3> (*(x, y), 2:) + * (#(?/, z), x) + # (*(z, x), y) .

A basis (^Q; • • • , Xn) of an (n + l)-dimensional filiform Lie algebra with law fj, iscalled adapted, ii^X^Xj) = n0(Xi,Xj) + ̂ > ( X i , X j ) , 0 < i,j < n. Denoting by [a\the integral part of a from now on, all brackets of an (n + l)-dimensional filiformLie algebra in an adapted basis (XQ, Xi, . . . , Xn) are determined by the brackets

+ a fc i lX2fc+3 + • • • + ak>2k-2Xn, l<k<[(n- 2)/2j .

If n is odd we have the supplementary bracket [^(n-i)/2; -X"(n+i)/2] = a(n-i)/2,n -Xn-

The non-uniqueness of an adapted basis for a filiform Lie algebra is clear, so weintroduce adapted and elementary changes of bases. A change of basis / G GL(V) iscalled adapted to the law JJL G J-n+i(V) if the image of an adapted basis is an adaptedbasis. The most important observation here is that to study the closed subgroupof the group GL(V) composed of all adapted changes, denoted by GLaa(V), it isenough to study certain elementary changes of bases.

THEOREM 2.3 Let f be an element ofGL^V). Then f is given by

f ( X o ) = aoXo + aiXi + • • • + anXn

i)l 2<i<n:

where agbi ^ 0. In fact, f is a product of the elementary changes of basesf ( X 0 ) - aX0

i / (a ,6)= { f ( X l ) = bXi a, 6 G C * .

f ( X 0 ) = X0f ( X l ) = Xi

f ( X 0 ) = X0,

f ( X i ) =

Although the factorization of / obtained in the previous theorem is not unique,it may be chosen in a certain order. When a parameterized family of filiform Liealgebras is given, in order to eliminate the unnecessary parameters, the key is tofind a sequence of elementary changes of bases.

In [33] it is shown how the elementary changes of basis act. We can remark herethat if a filiform Lie algebra law is given by its matrix of coefficients {a,k,m}, thenthe application of the change of basis i/(a, 6) gives a new law {bk,m}, with bk,m =bak,m/am~2k. So, we can transform the two nonzero coefficients ak,m,o,r,tt with?7i — Ik ^ t — 2r, to 1 (if such a pair does not exist, we can transform only one of thecoefficients in the law). The changes of bases r(a, k) and a(b, k) are basically usedin parameters elimination. In each application, some coefficients are unaffected,and, under certain restrictions, we can eliminate one of the parameters. Therefore,parameter eliminations must be ordered in such a way that eliminated parametersremain zero in the following changes of bases. In this way the classification for thefiliform Lie algebra with dimension less than or equal to 11 is obtained in [33].

THEOREM 2.4 Every n- dimensional complex filiform Lie algebra law, with n <11, is isomorphic to a law p?n of the List of Laws given in [33].

This approach develops the method of Ancochea and Goze [2] and allows tosimplify the computations, but when dimension increases we cannot obtain reli-able results without the help of computers. The symbolic programming languageMathematica is useful as an assistant to get the classification. This can be seenin [32].

Finally, as an important application, the classification obtained in [33] has beenused to study symplectic structures on the filiform Lie algebras [34].

2.3 /c-abelian filiform Lie algebras

In addition to know some families of filiform Lie algebra laws in concrete dimensions,we also know the classification of filiform Lie algebras g with the condition CIQabelian, obtained by Bratzlavsky [7]. The notion of k-abelian nilpotent Lie algebrais introduced in [27] (if CkQ is abelian) to consider the most general situation.This is a synthesis of two properties in the theory of nilpotent Lie algebras: thefiliformity (the nilpotent Lie algebras with this property are "less" nilpotent) and thecommutativity (the nilpotent Lie algebras with this property are "most" nilpotent).

The classification of 2-abelian filiform Lie algebras is obtained in [27] by usingthe properties of the elementary change of bases in the 2-abelian case.

THEOREM 2.5 Every (n + 1)- dimensional (n > 6) 2-abelian filiform Lie alge-bra law is isomorphic to one of the laws n i i S ^ ( f l ) , H2,a,t(^}> M3,s,t(^)> /-M, 3,3+2(^)1

Mis.t, Mi4,s(^), Mi5,s(fi), fj,ie,s, Mi?, which are defined in [27].

66 Cabezas et al.

A Lie algebra is called characteristically nilpotent if all of its derivations arenilpotent. The definition and first example of such a Lie algebra has been given byDixmier and Lister [25]. Khakimdjanov has shown that the family of characteris-tically nilpotent Lie algebras, which had been considered as narrow, is in fact verylarge [46]. As an application of the obtained classification, and by using the criterionof characteristically nilpotence given by Goze and Khakimdjanov [47],[45], we canconclude that all Lie algebra laws described in Theorem 2.5, except the 2-abelianalgebras of laws fig,Sla+2(^o), Mis.t, Mie.s, Mi?, are characteristically nilpotent.

3 THE FAMILY OF p-FILIFORM LIE ALGEBRAS

The study of the variety of nilpotent Lie algebras through the filiform ones inthe previous section is centered in those algebras of maximal nilindex among thenilpotent Lie algebras having the same dimension. When we want to study thatvariety by considering nilpotent Lie algebras with small nilindex the difficultiesincrease more and more as we can see in Section 4.

To study some problems about nilpotent Lie algebras with small nilindex wehave chosen in each concrete nilindex a family which allows us to use it in a similarway like the filiform family is used to obtain results on the complete variety.

Namely, an n-dimensional nilpotent Lie algebra g is said to be p-filiform ifits Goze invariant is gz(g) = (n — p, 1, /?., 1). Thus, the family of p-filiform Liealgebras is a large family of Lie algebras, including the filiform ones as a particularclass. Indeed, the filiform and quasi-filiform Lie algebras are the p-filiform ones withp — 1,2 respectively.

In this section we are interested in p-filiform families with p close to the dimen-sion of the algebras of such families, i.e., when the family of algebras considered hassmall nilindex and where the complete variety of nilpotent Lie algebras is not verywell-known.

3.1 p-filiform Lie algebras with p > n — 3

When we consider the family of p-filiform Lie algebras with dimension equal to n,it is obvious that 1 < p < n — 1. If p = n — 1 the situation is trivial because thefamily is just reduced to the abelian algebra in the appropriate dimension.

If 0 is a p-filiform Lie algebra, we can always choose an adapted basis of thealgebra g. That is, a basis (Xo, Xi,..., Xn-p, Y j , . . . , Yp-i) such that XQ e g— [g, g]and verifying

[X0,Xi] = Xi+1 l<i<n-p-l{X0,Xn_p} = 0[X^Yj] = 0 l<j<P-l.

We now describe how the family of p-filiform Lie algebras can be determined forthe n-dimensional nilpotent Lie algebras when p = n — 2 and p = n — 3. The resultsobtained in [13] show how it becomes more difficult to obtain these classications asp decreases.

We have adapted techniques already applied to specific dimensions; that is,we have considered appropriate arguments about nilpotentcy (generally, adjoint-nilpotency), characteristic sequences and those conditions which are derived fromassumption that the algebras be p-filiform. Then, with an adapted selection of

changes of bases, we have obtained the explicit classification for the p-filiform fam-ilies with p > n — 3.

Indeed, firstly we have considered some Jacobi identities in which XQ appearsto get a preliminary expression of the brackets which define the family of algebrasconsidered on an adapted basis. Those brackets can be dramatically simplified byconsidering some conditions which the structure constants have to verify in orderto guarantee that Yj £ Im ad(Xo), I < j < p — I , for the vectors in the adaptedbasis considered, and to preserve the appropriate nilindex of the algebras.

Secondly, as the Goze invariant of the algebras considered is (n — p, 1, 1, . . . , 1),there cannot exist a nonzero minor of order n — p in ad(Z), for all Z <£ [g,g].Thus, some appropriate choices for Z (for example: XQ + AYj , Xi + AYj , AXo +Xi, . . . , A € K) with a process of finite induction, lead to some more restrictions.

Finally, some adjoint-nilpotency considerations on any vector X of g (for in-stance, the characteristic polynomial P(A) of the adjoint matrix corresponding tosuch a vector has to be A"), some relations obtained from the remaining Jacobiidentities, and some appropriate changes of basis (from simple change of scale toother, very complicated, ones) we obtain the goal of the classification by eliminatingunnecessary parameters.

In [13] we can see that the (n — 2)-filiform Lie algebras are direct sums of abelianalgebras and Heisenberg algebras.

THEOREM 3.1 In dimension n > 3, there are exactly [(n— 1)/2J nilpotent, real orcomplex, pairwise non-isomorphic (n — 2) -filiform Lie algebras, denoted as gL 1 ^and whose laws in the basis (XQ, X\, . . . , Xn-p, Y\, . . . , ̂ p-i) are given by

4 [X0,*i] = X2

One also finds in [13] the classification of the family of (n — 3)-filiform Lie alge-bras, which are obtained in a similar way. We can see how the different dimensionsof the center and of the derived algebras show that they are pairwise non-isomorphicalgebras.

THEOREM 3.2 In dimension n > 5, there are exactly n — 1 nilpotent, real orcomplex, pairwise non-isomorphic (n — 3) -filiform Lie algebras, denoted as g/3 1 ^,and whose laws in the basis (Xo, Xi, . . . , Xn_p, Yj, . . . , Yp_i) are given by

(X0,Xi} = Xl+1 l<i<2,[*1>^~4] = X3 , w i t h l < S < n

Y?k-i,Y2k = X3 l<k<s-l

— V— ln-4-

68 Cabezas et al.

As an application we have studied some cohomological properties of the p-filiformLie algebras obtained above in which the determination of the algebra of derivationsof each p-filiform Lie algebra is essential. They can be found in [20], [12] where itis also described how symbolic calculus can be used to solve similar problems.

3.2 (n — 4)-filiform Lie algebras

We now show the increasing difficulties involved in the problem when we considerp = n — 4. Obtaining the classification of (n — 4)-filiform algebras by using themethod considered above was not successful. Thus, it has been necessary to developnew techniques. Actually, we have used central extensions of certain algebras withdimension smaller than the dimension of those algebras that we want to study.

Let g be a p-filiform Lie algebra of dimension n. If we obtain its quotient by aone-dimensional central ideal, the resulting quotient algebra has dimension n — 1and its Goze invariant is either (n — p — 1,1,..., 1) or (n — p, 1 , . . . , 1). In thissecond case we can obtain again the quotient algebra by a one-dimensional centralideal, obtaining a quotient algebra of dimension n — 2 and Goze invariant either(n — p — 1 ,1 , . . . , 1) or (n — p, 1 , . . . , 1). If we follow the process (that finishesbecause dim(Z(g)) is finite), it is obvious that we always get a quotient algebraof Goze invariant (n — p — 1,1,...,!) of dimension n — k. Conversely, we invertthe process and from the algebras of characteristic sequence (n — p — 1,1,...,!)and dimension n — 1 we construct the central extensions that have Goze invariantequal to (n — p, 1,..., I ) . These p-filiform Lie algebras will be called algebras of firstgeneration. If we apply the same process to the same algebras, but in dimensionn — 1, we obtain p-filiform algebras that will be called of second generation. Theprocess will continue until an algebra appears in some generation.

In [11] we can find that all nilpotent Lie algebras of dimension n and Gozeinvariant (4,1, (?.~~.4,1) are obtained as central extensions of a first generation ofonly three algebras of dimension n — l and Goze invariant (3,1, ^r.4,1). Specifically,the "first" of each one of the two existing families with dimension of the derivedalgebra 2 and the only one with dimension of the derived algebra 3. Any (n — 4)-filiform nilpotent Lie algebra of dimension n, that is a central extension of secondgeneration, is a central extension of the first generation as well.

THEOREM 3.3 In dimension n > 8, there are exactly 6n — 29 pairwise non-isomorphic complex (n — 4) -filiform nilpotent Lie algebras. They are distributed intwelve families, denoted by 0/4 ^ j \ , 1 < i < 12, whose laws can be found in [11].

We can see in [11] how these algebras are pairwise non-isomorphic by consideringsome invariants as the dimension of several appropriate subspaces: ZQ, C1^, C20,f)i = Cen(ClQ), etc. Finally, it is shown in [11] that the p-filiform Lie algebrascan be obtained as extensions by derivations of filiform Lie algebras of dimensionn — p + I for p > n — 4. The case p = n — 3 appears in [10].

3.3 p-filiform Lie algebras with n — 6 < p < n — 5

In a way similar to the cases studied above, we can obtain a general simplifiedexpression for the families of p-filiform Lie algebras when n — 6 < p < n — 5, but theclassification of such families is by now unsolved. However, we have studied some

interesting subfamilies in each case in order to get an approach to the completefamily.

The generic family of laws of the (n — 5)-filiform Lie algebras is obtained in[8], and the generic family of laws of the (n — 6)-filiform Lie algebras can be foundin [18].

The laws of the family of (n — 5)-filiform Lie algebras allows us to get theclassification when we consider some restrictions on the derived subalgebras of thefamily. Indeed, we can see in [8] or [16] the algebras g obtained when dim[g,g] = 6,that is when such derived subalgebras have dimension as large as possible.

THEOREM 3.4 If Q is an (n — 5)-filiform Lie algebra of dimension n > 9 withdim(C1g) = 6, then it is isomorphic to one of the algebras 0/5 ^ ^, I < i < 5,which are defined in [8].

We have also considered the families of (n — 5)-filiform and (n — 6)-filiform Liealgebras when n = 8. The classification for these families completes the p-filiformfamily in that dimension, which shows more information about the variety A/S. Allof the 3-filiform Lie algebras can be reduced to only three [17]. The classificationof 2-filiform Lie algebras (quasi-filiform Lie algebras) is obtained in [19], where itis shown how symbolic calculus was used to get the goal. Concretely, by usingalgorithms implemented in the Mathematica programming language [60].

4 LIE ALGEBRAS WITH SMALL NILINDEX

We consider here some type of nilpotent Lie algebras with small nilindex. Onlythe abelian algebra is obtained in each dimension having nilindex equal to 1. How-ever, when the nilindex is greater than 1 the difficulties to determine all pairwisenon-isomorphic algebras increase more and more. Indeed, the classification of thenilpotent Lie algebras with nilindex 2 is a very hard problem. The classificationof those (2p + l)-dimensional algebras which have Goze invariant ( 2 , . . . , 2,1) isequivalent to the classification of the bilinear forms on Cp with values in Cp, whichinclude all Lie algebras of dimension p. Thus, it is impossible to obtain a completeclassification of such nilpotent Lie algebras [44], Therefore, to know examples offamilies of such Lie algebras with small nilindex is a valuable utility to study thevariety of nilpotent Lie algebras J\fm.

A nilpotent Lie algebra of nilindex 2 is called a metabelian Lie algebra. Now,we are interested in certain types of metabelian Lie algebras and certain types ofalgebras with nilindex 3 determined by their Goze invariants.

Goze's invariant allows to separate a family of nilpotent Lie algebras with a fixednilindex into some families which we can study more easily. In this way, we canseparate the family of the n-dimensional metabelian Lie algebras by consideringthose having Goze invariant equal to (2,2, .PA,2,1 n7?P\l), where p goes from 1to [(n — 1)/2J. The simplest case to be considered is the family determined by( 2 , 1 , . . . , 1). These are the (n — 2)-filiform Lie algebras and they have been studiedin [13]. The classification can be found in Section 3, where we have surveyed thep-filiform Lie algebras.

In the same way, the nilpotent Lie algebras with nilindex 3 which have Gozeinvariant equal to (3 ,1 , . . . , 1) are the (n — 3)-filiform Lie algebras, and these arealso classified for arbitrary dimension in [13].

70 Cabezas et al.

We present in this section some general results about the metabelian Lie alge-bras, including the classification for a general family of such algebras under certainassumptions. We also study some concrete families of nilindex 2 and 3 in order todemonstrate how and where the difficulty increases in these classification problems.

4.1 Metabelian Lie algebras

To study the metabelian Lie algebras we separate them into families with Gozeinvariant (2 ,2 , .PA,2, 1 n7??\ I). Then, if Q is an n-dimensional Lie algebra withnilindex 2 there is p 6 TL such that 1 < p < [(n - l)/2j and gz(g) = (2,2,.p).,2,lnr.2p\l).

When p = 1, that is gz(g) = (2, 1, . . . , 1), the algebra Q is one of the (n — 2)-filiform Lie algebras determined in [13]. In the general case, the difficulty to obtainthe classification lies in the fact that there are too many parameters with very fewrestrictions.

Let Q be a metabelian Lie algebra. A basis

is called adapted if[Xo,X2i-i] = Xx, l<i<p.

If we consider an adapted basis of a metabelian Lie algebra 0, by using standardnilpotency and adjoint-nilpotency arguments, we get the law of the algebra givenby

n-2p-l

k=l

k=l

Let s denote the rank of the matrix (of), where af is the appropriate coefficienta2t-i 2j-i fr°m the vector Yk in the bracket [ X z i - i ^ X ^ j - i ] above. Then, the derivedalgebra C Jg has no more than s different elements X^i, 1 < i 2p + 1. Then

p<dim(C1Q}<p+P

We can use the inequality in Theorem 4.1 to get some general results about themetabelian Lie algebras. To study families of metabelian Lie algebras g is more

difficult as dim(C10) decreases. In fact, when we consider dim(C10) to have thelargest possible value, the classification of such a family of Lie algebras can beobtained in any dimension. Indeed, let g^ p denote the algebras defined in the basis(Xo,Xi, X2, • • • , X-2P, YI, . . . , Yn_2p-i)) by

where n > 2p + 1 > 5. For every p, the algebra g^ has Goze invariant equal togz(0) = (2, 2, .p)., 2,1 n7??\ 1) and its derived algebra verifies dim(C10jl

l)p) = p+ (%).Then, we can see in [43] how any n-dimensional metabelian Lie algebra g with Gozeinvariant gz(g) = (2, 2, p)., 2,1 n7??\ 1) and dim (Cxg), having the largest possiblevalue, is a trivial extension of an appropriate Lie algebra g^ .

THEOREM 4.2 Let g be an n-dimensional metabelian Lie algebra with Goze in-variant gz(g) = (2, P)., 2,1 n72?\ l},n>2p+l + (P) and dim(CJ0) =P+(%). Thenthe algebra is g = g^p ® Cn~m, where m = (pf2).

It remains an unsolved problem, to obtain the classification when dim(C1g)decreases. Indeed, if the derived algebra has the smallest possible value we alwaysfind a family of algebras in which every algebra is a linear deformation of one of themodel algebras

For instance, a metabelian Lie algebra g with Goze invariant gz(g) = (2, 2 , 1 , . . . , 1)and dim(C1g) = 2 belongs to the family

[X0,X2i-i]=X2i, 1 1 < z < n — 5,

[Yi, Yj] =0^X2 + 4X4, 1 < i < j < n - 5.

But we cannot separate non-isomorphic algebras from the family so far. However, acohomology study about the algebras g° p was introduced in [41], and it can be foundin [51], in order to illustrate the dimension of the orbits of such algebras. Moreover,we can also see in [51] how the number of metabelian Lie algebras obtained increases,when we consider dim(C1g) ^ p.

THEOREM 4.3 Let g be an n-dimensional metabelian Lie algebra, n > 10, withGoze invariant g = ( 2 , 2 , 2 , 1 , . . . , ! ) and dim(C1g) > 3. Then g is isomorphic toa trivial extension of one of the following algebras defined in [51]: 0^ 3, g^ 3, g^, 3;

0n,3> 3n,3> 8n,3> 8n',3> ^ — T — l~T~ \ > Sra',3> ^ — r — {.^~2~ \ ' 3rt'3' ^ — r — l^2~ \ '

0^,3, 0 < r < l1^^; Qn^, 0 < r < L11^12]; g^f3, 0 < r < L2^11] •

We remark that when we consider a metabelian Lie algebra g with Goze invariant( 2 , 2 , 1 . . . , 1) and dim(C1g) > 2, then g has to be a trivial extension of gjj 2-

72 Cabezas et al.

4.2 Lie algebras with nilindex 3

The study of the Lie algebras with nilindex 3 has been started in [51], in orderto check if the results about metabelian Lie algebra could be generalized to otheralgebras with small nilindex. In this case, Goze's invariant could be one of thefollowing (3, .p)., 3, 2, .q)., 2, 1, n-^T2<?), 1). Then, by considering the adapted basis

(Xo, Xi,X%, . . . , X3p_2, -Xsp-lj Xsp, Ui, t/2, . . . U^q-li Ulq, YI, . . . , Yn_3p_29-l)

defined in [51], we can separate the study of these algebras from the possible values(p,q). For example, the pair (1,0) corresponds to the (n — 3)-filiform Lie algebraswhich have been studied in [13]. Although for these algebras we have no relationto bound dim(C1g) as in Theorem 4.1 for the metabelian ones, we can see how thedimension of the derived algebra plays an important role in those algebras withsmall nilindex. Thus, we can express the family determined by (p, q) = (1, 1) in anadapted basis [51], and hence we can obtain the classification of those families ofalgebras with dim(C10) > 3. For the non-split algebras mentioned in Theorem 4.4below, we can also find some cohomological results in [51].

THEOREM 4.4 LetQ be an n- dimensional Lie algebra, n>8, with Goze invariant(3, 2, 1, . . . , 1) and dim(C1g) > 3. Then g is isomorphic to a trivial extension of oneof the algebras Ql

n iti, 1 < i < 4n — 24, defined in [51].

To classify the Lie algebras having Goze invariant (3, .PA,3, 1, n7??\ 1) seemsa much harder problem than its "similar" (2, .PA,2, 1,n7??\ 1) in metabelian Liealgebras. Indeed, in the simplest case (p,q) = (2,0) we have to consider for aLie algebra 0 its derived algebra verifying 4 < dim(C1g) < 8. However, when weconsider the largest value dim(C10) = 8 the algebra Q just can be a trivial extensionof one determined Lie algebra [42].

THEOREM 4.5 Let Q be an n- dimensional Lie algebra, n > 11, with Goze invari-ant (3,3, 1 n.~.6}, 1) and dim(C10) = 8. Then Q is isomorphic to a trivial extensionof the algebra Q^ 2 0, defined in the basis (Xo, Xi,Xz, • • • , XQ, YI, . . . , Yn-?) by

[Xo,Xi\=Xi+i, i €{1,2,4,5},[X1)x2]=y1,[X1,X4}=Y2,[XltX5=Y3,

To know if the algebras with Goze invariant (3, .PA, 3, l,n7?P\ 1) and havingderived algebra as largest as possible can be determined as extensions of an appro-priate family g^ p 0 is a very interesting question which should be a generalizationof Theorem 4.2.' '

5 NATURALLY GRADED NILPOTENT LIE ALGEBRAS

In the cohomological study of the variety of laws of nilpotent Lie algebras establishedby Vergne [59] the classification of a class of graded filiform Lie algebras plays a

fundamental role. The achieved classification allows for an easy expression of afiliform Lie algebra. The gradation considered by Vergne is provided in a naturalway from the descending central sequence of any nilpotent Lie algebra. Now, thealgebras obtained in this way are called naturally graded Lie algebras.

Vergne proves that, up to isomorphisms, there is only one naturally gradedfiliform Lie algebra in odd dimensions and two of them when the dimension is even.This fact also allows other authors to deal with different aspects of the theory. Forexample, using such graded filiform Lie algebras, Goze and Khakimdjanov give in[45] the geometric description of the characteristically nilpotent filiform Lie algebras.

Thus, it is clear that knowing the graded algebras of a certain class of nilpotentalgebras we get valuable information about the structure of such a class. This mayfacilitate later the study of several problems that can appear within the whole of theclass. In this section we study the naturally graded Lie algebras for the nilpotentLie algebras with nilindex near their dimension. Thus, we concretely are interestedin p-filiform Lie algebras for whose values of p the methods used in the sectionsabove fail.

5.1 Naturally Graded filiform and Quasi-filiform Lie Algebras

If g is a nilpotent Lie algebra of dimension n and nilindex k, it is naturally filteredby the descending central sequence of g, (Cl&)0<i<k, (C°Q = g, Czg = [g, CI~IQ\).We consider the filtration given by (5i+i), where S;+i = g, if i < 0; Si+i = Czg, if1 k. Associated to g there exists a graded Liealgebra grg = ©iezgi, taking g; = 5j/5i+1. Thus, we have

When grg and g are isomorphic, denoted by grg = g, we will say that thealgebra is naturally graded. So, if g is a naturally graded Lie algebra, such anatural gradation is finite, that is grg = gi © g2 © • • • © g^, with [g^, QJ] C Qi+j, fori + 3 £ k, and the number of subspaces jjj is as large as the nilindex of g.

The filiform case, characterized by its maximum nilindex fc = n — 1, in eachdimension was studied by Vergne [59], who introduced the notation, proving thatthere are only two algebras of this type, Ln and Qn, when the dimension of thealgebra is even, and there is only one, Ln, if n is odd. The algebras Ln and Qn arethose defined on the basis (Xo, Xi, . . . , Xn-i), by

[X0,Xi]=Xi+1,

(The undefined brackets, except for those expressing anti-symmetry, are supposedto vanish).

As we have seen in Section 2, any n-dimensional filiform Lie algebra is a lineardeformation of the algebra Ln [44].

Now, we want to study the naturally graded quasi-filiform Lie algebras, that iswe will consider the n-dimensional Lie algebras with nilindex equal to n — 2. In

74 Cabezas et al.

order to make easy the structure of such a family we have to find a basis in whichthe brackets of the algebras of the family are suitable. We can see in [29] how toobtain an homogeneous adapted basis of the algebra 0, i.e., a basis whose propertiesare stated in the theorem below.

THEOREM 5.1 Let Q be an n- dimensional naturally graded quasi-filiform Lie al-gebra. Then, there exists a basis (Xo, Xi, . . . , Xn_2, Y) of the algebra Q such thatXQ, X\ G gi, Xi <G Qi, 2 < i < n — 2, and Y e Qr with I < r < n — 2, verifying

[ X 0 , X i } = Xi+1, z < l < n - 3 ,

Thus, we can use the laws of the family to reduce the unnecessary parameters.When g is an n-dimensional naturally graded filiform Lie algebra such that

dim(C10) = n — 3, we can consider a natural gradation where gi =< Xo,Xi,Y >.Then, the algebras obtained are reduced to a trivial extension of the naturallygraded filiform Lie algebras. In fact, g = Ln-\ © C, if n is even, and either g =Ln_i © C or g = Qn-i © C, if n is odd. These graded quasi-filiform Lie algebraswill be called split, and their only interest is to emphasize the natural gradationunderlying the filiform subalgebra of the extension.

We will denote by Q(n,r} an n-dimensional quasi-filiform Lie algebra in which wecan choose an adapted basis so that the natural gradation verifies Y & Qr (note thatdim(g r) = 2, r ^ 1). Thus, we can view the pair of integers (n,r), I < r < n — 2, asan invariant for any naturally graded quasi-filiform Lie algebra. Then, the result inthe previous paragraph says that if g = g(n , i) , then the algebra g is a split algebra.Moreover, if g = Q(n,r) with r even we can see in [30] that dim(C1g) = n — 3, sog is also a split algebra. Then, we can describe the family of the n-dimensionalnon-split naturally graded quasi-filiform Lie algebras.

LEMMA 5.2 Let g = Q(n,r) be a non-split naturally graded quasi-filiform Lie alge-bra. Then r is odd and g belongs to the parameterized family defined in an adaptedbasis (X0,Xi,...,Xn-2,Y) by

[Xi,Y]=aXi+r

where a £ C if 3 < r < n — 3, and a = 0 if r = n — 2.

We can see how the quasi-filiform Lie algebras L(n>r) and Q(n,r) defined in the basis

If n > 5, and 3 < r < 2 {(n - 1)/2J - 1, r odd,

If n > 7, n odd, and 3 < r < n — 4, r odd,

[X0,Xi} =I

(n,r)

belongs to the family in the lemma above. So, these algebras are naturally gradedalgebras. They play an almost similar role as the algebras Ln and Qn in the filiformfamily. In each dimension n, for each acceptable (n,r), 3 < r < n — 5 we obtainthat the naturally graded quasi-filiform Lie algebras are just L(n>r) when n is even,and I/(n]r) or <5(n>r) when n is odd. Those cases in which r is close to the dimensionn are special. When r = n — 2 (n odd), then g = £(n ) n_2)- When r = n — 3 (n even)or when r — n — 4 (n odd), we obtain also one more algebra noted T(n)7.), which iscalled terminal and is defined in [30], where the algebras £(7,3), £}g 5 \ , and £?g 5^ aredefined as well.

THEOREM 5.3 Every n-dimensional naturally graded quasi-filiform Lie algebrais isomorphic to one of the following algebras:If n is even to Ln-\ ® C, T(n,n-3) or ^(n,r)i with r odd and 3 < r < n — 3.If n is odd to Ln—\ ® C, Qn-i ® C, L(n,n-2)> r(n,n-4)i ^(n,r) or Q(n,r)> with r oddand 3 < r < n — 4. When n = 7 or n = 9 we add the algebras £(jtz)> £}g $\i £?g 5)-

As an application, we refer to [40], [21] for a characterization of the space ofderivations of the naturally graded quasi-filiform Lie algebras and a description ofhow symbolic calculus can be used to solve similar problems [28].

5.2 Naturally Graded 3-filiform Lie Algebras

The complete n-dimensional family of nilpotent Lie algebras with nilindex n — 3 canbe separated in the subfamilies characterized by the Goze invariants (n — 3,1,1,1),and (n — 3,2,1). The first one is the family of the 3-filiform Lie algebras. We arenow interested in the naturally graded Lie algebras for such a family in order tolook for a family of graded algebras which extends the results in the section above.

In fact, the situation for naturally graded 3-filiform Lie algebras is a generaliza-tion of the filiform and quasi-filiform Lie algebras. There is one locally finite familyof terminal algebras (depending on one parameter) for n odd or n even; further-more, there are one or two locally finite families (depending on two parameters) forn odd or n even, respectively. Precisely, for n > 11, there are O(n2) non-split Liealgebras.

THEOREM 5.4 Let Q be a naturally graded 3-filiform Lie algebra of dimension n,grg = ®igs0i and let (Xo, Xi,... ,Xn^3, YI, Yj) be an adapted basis of Q . Then

gi D < X0,Xi >Qi D <Xi> 2<i<n-3,Bi = {0} if i < 0 or i > n - 2.

More precisely, the decomposition obtained is g = gi © 32 © • • • © fln-s with[0i,0j] C Qi+j, for?+j < n-3, verifying that 2 < dim(gi) < 4 and 1 < dim(flj) < 3,2 < z < n-3.

76 Cabezas et al.

The vectors YI, Y% are special. The triple (n, 7-1,7-2) will represent the case ofnaturally graded 3-filiform Lie algebras of dimension n, where the integers r± andr-2 indicate the positions of the coordinates of (dim(gi),dim(02), • • • , dim(gn_3))different from those of (2, !,...,!). And 0(ra,n,r2) represents an algebra of theappropriate family determined by (71,7-1,7-2), 1 < r\ < r% < n — 3. Some generalfacts about (71,7-1,7-2) can be found in [14], [15].• The cases (71,7-1,7-2), where either r\ or r-2 is even are not admissible. Moreover,when r\ = 7-2, then r\ is odd.• The admissible cases (71,1,7-2) produce split Lie algebras: either a direct sum

or a direct sum gn_2 ©C2, where Qn-i and Qn-2 correspond to a naturally0ra_igraded 2- filiform and 1-filiform Lie algebra of dimension n—l and n — 2, respectively.• The case (4, 1, 1) corresponds to the abelian Lie algebra. Moreover, in the case(5, 1, 1) two algebras are obtained: one split and the Heisenberg algebra.

Of course, the general structure of the family of naturally graded 3-filiform Liealgebras is more complicated than that of the quasi-filiform ones.

LEMMA 5.5 Any naturally graded 3-filiform Lie algebra of dimension n is iso-morphic to one whose law can be expressed, with (Xo, X\, . . . ,Xn^s, Yj, Yjj) anadapted basis, by

1 < i < n-4,

+ (- 1 <i< ̂ ,

[Xi,Y2] = fXi+r^ l < z < n - 3 - r 2 ,[Yi , Y2] = hXn_3 if n + r2 = n - 3,

where 3 < ri < r2 < TI — 3, ri,r2 od<i.

We have also determined two bi-parameter locally finite families of pairwisenon-isomorphic algebras, which generalize the algebras Ln and Qn. These algebrasdenoted by i(n,r-1,r2) and Q(n,r1,r2)) are defined in the basis

byIf n > 8, and 3 < r\ < r<2 < n — 3, n, r-2 odd,

r;l = X,A

If n > 10, n even, and 3 < r\ < r^ < n — 5, r\,r-2 odd,

1 < i < n - 4 ,

The naturally graded 3-filiform Lie algebras of dimension n and type (n,ri,r2)when r-2 is "terminal", i.e., when r% is close to n — 3 (so Y? € Qi, Xn-s G QJ ina gradation of the algebra, where jjj is near QJ) has been studied carefully in [14].The terminal families r(n,r, n — 5) (n even, n > 12, r odd, 3 < r < n — 7) andr(n,r,n — 4) (n odd, n > 9, r odd, 3 < r < n — 6) can also be found in [14].As in the quasi-filiform case, we have to consider some exceptional algebras in lowdimensions (n < 10): one for each dimensions 8 and 9 and two infinite families forn = 10 [9].

In the general case (n > 11), we will always suppose to be working with Liealgebras g with n > 11 and 3 < r\ < r%, ri,r2 both odd. We can see in [14] that, forodd dimension, there is one bi-parameter locally finite family of algebras, i(n,r1,r2)(parameters n, r ^ ) , and one terminal one-parameter locally finite family, T(n;r i]n_4).For even dimension another bi-parameter family, <3(ra,r-i,r2) and °ne terminal one-parameter family, T(n;r.1]TJ_5) appear, but T(,i]T.lin_4) disappears (r-2 = n — 4 is evenif n is even and this is impossible).

In order to obtain the desired classification, it is essential that any algebra gshould belong to the family is an extension for ideals of g' = 0/C l™~4(g), whichis in a natural way a graded 3-filiform Lie algebra of dimension n — 1 and type(n — I , r i , r2) . Thus, the general classification is obtained using induction on thedimension of the algebras of such a family. The following theorem shows that thereare no terminal algebras when r^ < n — 7.

THEOREM 5.6 Any naturally graded 3-filiform Lie algebra Q of type (n,TI,^)with n > 11, 7*1, TI both odd, 3 < TI < TI < n — 7, is isomorphic to an algebra of thefamily L(n, TI,^) if n is odd or to an algebra of the families J^(n, r i i r2) or Q(ra,rj,r2)if n is even.

Thus, the results obtained for the naturally graded 3-filiform family are a gener-alization of the filiform and quasi-filiform families. This allows to conjecture aboutthe structure of the naturally graded p-filiform Lie algebras when p is greater than 3.

6 LENGTH OF NILPOTENT LIE ALGEBRAS

The difficulties to obtain the classification of a class of nilpotent Lie algebras leadto study the algebras which can give useful information about such a class. Inthis way, the graded Lie algebras play an important role. We have considered inSection 5 the naturally graded algebras for some families of p-filiform Lie algebras.In that natural gradation on nilpotent Lie algebras, the subspaces in the gradationand the existence of an appropriate homogeneous basis (necessary to obtain theclassification) are a natural consequence of the central descending sequence of theLie algebras considered. However, it introduces a restriction by fixing the numberof subspaces in the gradation by means of the nilindex.

Several authors, for instance Goze, Khakimdjanov, and some of the authors ofthis paper, have considered graded Lie algebras which possess not just one naturalgradation, but a gradation with a large number of subspaces, because this conditionfacilitates the study of some cohomological properties for such algebras (see [13],[44]). For such a "length" of the gradation, the main interest is in algebras whoselength is as large as possible.

78 Cabezas et al.

In this section we are interested in the study of the filiform and quasi-filiformLie algebras with length larger than their nilindex. Thus we have obtained othergraded Lie algebras different from those in the section above. These graded Liealgebras with a greater number of subspaces than the naturally graded ones can beuseful examples of filiform and quasi-filiform algebras to approach some problemsabout the variety of nilpotent Lie algebras.

6.1 Connected gradations

We suppose in this section that all Lie algebras are defined over the field of com-plex numbers. We also consider Z-graded Lie algebras; that is, admitting a de-composition Q = ®j6g Si> where the subspaces Qi satisfy [0i,gj] C Qi+j for alli,j 6 Z. We will say that a nilpotent Lie algebra Q admits a connected gradation0 = Qni © • • • ® Qn2, when each Qi, with n\ < i < 77-2, is nonzero. The number ofsubspaces l(($Q) = n^ — n\ + 1 will be called the length of the gradation.

DEFINITION 6.1 The length /(g) of a Lie algebra g is defined as

I(Q) = max {/(®g) = n2 - n\ + 1 : g = gni ® • • • © gna is a connected gradation} .

This means that /(g) is the greatest number of subspaces from the connected gra-dation which can be obtained in g. Thus, every Lie algebra g has at least lengthequal to 1, because we can consider the connected trivial gradation g = go- On theother hand, an algebra cannot have length greater than its dimension. Thus, forevery nilpotent Lie algebra g we have 1 < /(g) < dimg.

For example, the filiform Lie algebra Ln admits the gradation Ln = Si ® • • -®Qn,where the subspace Qi =<Xj_i >, with 1 < i < n, is generated by the element Xi-\of the basis (Xo, Xi,..., Xn-i) over which the algebra Ln is defined. Although sucha connected gradation on Ln is not natural (QI ^ Q/Cls), it could be more usefulto be considered because each subspace Qi has dimension equal to 1. Moreover, thisgradation has the greatest possible number of nonzero subspaces, and this showsthat the length of such an algebra is l(Ln) = n,

If we consider a family of nilpotent Lie algebras and we look for the graded alge-bras with a given length, then we cannot suppose an adequate basis to express thealgebras in a easy way (as Ln in an adapted basis). This is a first problem to solvein order to obtain the classification of such a graded family of Lie algebras. Thus,firstly we will obtain an homogeneous adapted basis that allows us to determinethe structure of the family of Lie algebras considered. Secondly, we will separatethat family in subfamilies determined by the gradations which we must consider.Finally, we will obtain the non-isomorphic algebras.

6.2 Filiform Lie Algebra of maximum Length

In addition to Ln, there are other filiform Lie algebras that admit this gradation.The algebras Rn and Wn are defined in a homogeneous basis (Xo,..., Xn-i), by

f {X0,Xi} =Xi+1 1 < i < n - 2 ,n~ =X2+J 2 < j < n - 3 .

[X0,Xi]=Xi+l l < i < n - 2 ,r v- y i _ 6 (i-i)! (j-i)! (j-i)- ——— ———

i < j < n — 2 — i.

As for Ln, we can check that l(Rn) = n and l(Wn) = n. Therefore, Rn and Wn arealso maximum length filiform Lie algebras. The importance of these algebras wasshown in the study of reducibility of the variety of laws of the nilpotent Lie algebrasA/"ra. Khakimdjanov [46] describes in the variety j\fn, with n > 12, two irreduciblecomponents which contain an Rn and Wn respectively. Thus, the determination,up to isomorphism, of the filiform Lie algebras with length equal to their dimensioncan provide us, then, with new information about the variety J\fn.

To obtain the naturally graded algebras, Vergne [59] uses the fact that if g is afiliform Lie algebra of dimension n, there exists an adaptedbasis (Xo, Xi,..., Xn_i),such that

[A-0,AVi] = 0

is verified. It can be observed that the basis considered in the definitions of Ln, Rnand Wn are adapted bases. When g is an n-dimensional filiform Lie algebra with1(0) = n, we can see in [37] how to get an adapted basis by choosing homogeneousvectors from the subspaces of any decomposition 0 = ffig such that l(ffig) = n. Thisallows us to obtain the structure of the graded filiform Lie algebras considered.

THEOREM 6.2 If Q is an n-dimensional filiform Lie algebra of maximum lengthI(Q) = n that admits the decomposition g = gni ffi • • • © Sni+n-i, then there existsan adapted and homogeneous basis of g.

Of course, we cannot obtain different Lie algebras from the gradations consid-ered by just reordering the indices. In fact, we can see in [37] that g admits thedecomposition g = g _ n + 2 f f i - • - f f i g i if and only if g = g_i f f i - • - f f i g n _ 2 , and g admitsthe decomposition g = gi ffi • • • ffi gn if and only if g = g_n ffl • • • ffi g_i . In the firstcase the only algebra of the family is g = Ln, but the algebras Kn and Q'n verify0 = 81 ® • • • ffi 0m where Kn (n > 8) and Q'n (n > 7, n odd) are defined in the basis(X0,Xi,...,Xn-.i) as follows:

F V V l V 1 ^ " ^--' ,« O[Ao, A,] = Ai+i, 1 < z < n - 2,

fj , X j n-2 |_.l = (-1)*"1 ( I ̂ ^ I - Z) X I n-2 I , I <J <\ "~4 I ,

r• X •] = ( — ! Y ('~1)("~3~') a v o < 7 < "~3

with a = 0, if n is even and a = 1, if n odd.

I [Xo,Xi] = Xi+i, l<i<n — 2,

Moreover, the algebras Ln, Rn, Wn, Kn and Q'n, which admit the decomposition0 = 0i © • • • ® Qni determine the filiform Lie algebras of maximum length.

80 Cabezas et al.

THEOREM 6.3 Let g be an n- dimensional filiform Lie algebra of maximum length/(g) = n and n > 12. Then, either Q = Ln, g = Rn, Q - Wn, g = Kn or g = Q'n (nodd).

In addition to these filiform Lie algebras, we must consider the algebras withdimension less than 12 and maximum length, obtained in [36].

6.3 Quasi-filiform Lie algebras of length greater than their nilindex

The number of subspaces for the natural gradation in the quasi- filiform case is n— 2,where n is the dimension of the algebra [29]. So, in addition to the Lie algebras withlength n, the algebras with length equal to n — 1 can also provide useful informationto know better the complete quasi-filiform class.

As in the filiform case, to obtain the n-dimensional naturally graded quasi-filiform Lie algebras we can see in [29] how to get an homogeneous adapted basis(X0,Xi, . . . ,Xn-2,Y), such that

[X0,Xi}=Xi+1, i<l<n-3

In [38], we have shown that the adapted basis can also be chosen homogeneousfor any possible gradation with the maximum length on quasi-filiform Lie algebras.

THEOREM 6.4 Let g be an n-dimensional quasi-filiform Lie algebra of maximumlength l(g) = n that admits a decomposition g = gni © • • • © Qm+n-i- Then thereexists an adapted and homogeneous basis (Xo, Xi, . . . , Xn_2i Y) °f 3 such that:(a) If the decomposition o/g is g = g_ n + 2 ®- • -®go®0i , then g =< Y > © < X\ >© < X2 > © • • • ffi < ^o > or Q =< Xi>®<X2>®---®<Y>®<X0>.(b) If the decomposition of Q is g = Q-n+3 © • • • © 01 © 32, then g =< Xi > © <X-2 > © • • • © < ^0 > © < ^ > •

(c) If the decomposition of Q is g = g0 ©gi © • • • ©0n-i, then g =< Y > © < X0 >© < Xi > © • • • © < Xn_2 > •(d) If the decomposition o/g is g = gi©g2©- • -®Qn, then g =< X0>®<Y>®<Xi > © • • • © < Xn-2 > orQ=<X0>®<X1>®---®< Xn^2 > ® <Y > .

Obviously, every trivial algebraic extension g = g'©C of a filiform Lie algebra g'of maximum length is also a quasi-filiform Lie algebra of maximum length, becausewe can join the subspace C at the end of the gradation of g' to get the appropriategradation for g. Thus, the algebras Ln_i © C, Rn~i © C, Wn_i © C, Kn_i © Cand Q'n_i © C are quasi-filiform Lie algebras of maximum length equal to n [37].All these algebras are examples of split Lie algebras g © C, whose decompositionslie actually in g'. The non-split quasi-filiform Lie algebras of maximum lengthhave been determined in [38], where we can also find the study of some geometricproblems about the graded quasi-filiform Lie algebras obtained.

THEOREM 6.5 (Quasi-filiform Lie algebras of maximum length) Let Q bean n-dimensional non-split quasi-filiform Lie algebra of maximum length l(g) = n,with n > 13 . Then the algebra g is either g = gL 2_n-. (n odd), g = g? ^or Q = s f n i ) ; where the algebras 0 L 2 _ r a ) > 0Li ) ; 0L, i) are defined in the basis(Xo, Xi, . . . Xn-2, Y) as follows:

// n > 5 and n is odd,

Ifn>5,2

0(n'1}

' V V I _ V 9 <r" * <" *n ^j\ i , yv i I — yv j_|_3) ^ _ ^ t ^ ( t — <_).

Now we consider an n-dimensional quasi-filiform Lie algebra g of length l(g) =n — 1. We remark that the natural gradation for this algebra has n — 2 subspaces.Moreover, we do not know if such an algebra is naturally graded. If g does, thenthe algebra is one of the algebras obtained in Section 5 with another appropriategradation different from the natural one. Thus, the most interesting point is tostudy families with few naturally graded Lie algebras. Then we could obtain othergraded algebras to study the chosen family. When the derived algebra of g has theminimal dimension dim(C1g) = n — 3, then g = Ln-\ © C or g = <5n_i © C [29], sowe first consider the family of quasi-filiform Lie algebras verifying such a conditionabout their derived algebras.

In [39] we have studied the possible gradations to be considered, and we cansee that if g is an n-dimensional quasi-filiform Lie algebra of length n — 1 anddim(C1g) = n — 3, then it is always possible to choose an homogeneous adaptedbasis of the algebra (Xo,Xi,... ,Xn-2,Y) for any gradation g = ©g such that/(0g) = n — 1. This allows us to get the structure of the family, and finally todetermine the algebras of the family considered. The theorem below shows howhard it is to obtain a classification.

THEOREM 6.6 Let g be an n-dimensional quasi-filiform Lie algebra with length= n— 1, dim(Cllg) = n — 3 and n > 13. Then either g = Qn-\ ffi C, g = gL -,,

Q / _ J _ J _ J A 1 4 / r ' ^ ^ ^ x O 1Q lT~l QllfTL CLTLQi T) Odd OT T) ~~ TL — 4 / 0 —- Q/ \ ITL — t3 "̂ T) "̂ TL — o /

'ra •. (n — 5 < p < n — 3) for 3 < p < n — 3, which are defined in [39].

The low-dimensional cases (n < 13) and an approach to the study of the n-dimensional quasi-filiform Lie family of algebras g with length l(g) = n— 1, verifyingdim(C1g) ^ n — 3, can be found in [50].

7 SYMBOLIC CALCULUS ON LIE ALGEBRAS

Some of the problems studied in this survey require a lot of computations in orderto reach the goal we are aiming at. A family of Lie algebras is usually determinedby a set of polynomial equations involving the structure constants of the family,and it is obtained from some restrictions which have to be considered about suchalgebras arising from Jacobi's relations, nilindex, Goze's invariant, etc. Then, to

82 Cabezas et al.

determine a family of Lie algebras in a reliable way is a very complicated problem.Moreover, the basic object to study is an n-dimensional Lie algebra, whose law isusually given in an adequate basis. Often, we have to change the considered basisand we have to determine how the family is modified. Obviously, a general changeof basis in arbitrary dimension cannot be done. So we have to "guess" the changeof basis which simplifies the law of the family.

When a family is determined in an easy way, for instance when we know thedifferent algebras up to isomorphism, we are often interested in several problemsto approach the complete variety of nilpotent Lie algebras by the family that isconsidered. Thus, we usually want to know the dimension of the orbits of thoserepresentative algebras in the family, their spaces of derivations, etc. In some casesthe theorems come from the algebras whose laws are very simple, but in many caseswe must make a lot of computations to get examples which allow us to find out theright way to the solution of the considered problem.

Thus, we have developed many algorithms to assist us where a computation "byhand" could be either dangerous or almost impossible. Most of them have beenimplemented in the symbolic programming language that the package Mathematicaprovides [60].

The software package Mathematica has been very useful. It has been used for itsprogramming capacities and as a powerful symbolic calculator. It has been consid-ered always as an assistant to obtaining examples from concrete algebras in whichwe are interested for high dimensions. Moreover, we have used the developed Math-ematica packages to check some conjectures that could be made in order to modifythe packages when a general result was stated. Thus, we have used Mathematicain an interactive way, and we often had to solve some problems different from theinitial ones.

Mathematica has become a well-know symbolic software tool to perform compu-tations in a reliable way. We have shown in several sections of this survey how someof the results obtained have been presented in Computational Algebras Conferencesor Mathematical Symposiums. The programs we developed can be found in theappropriate references or directly through the authors of this paper.

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[2] J. M. Ancochea-Bermudez, M. Goze, Classification des algebres de Lie fili-formes de dimension 8, Arch. Math., 50(1988), 511-525.

[3] J. M. Ancochea-Bermudez, M. Goze, Classification des algebres de Lie nilpo-tentes de dimension 7, Arch. Math., 52:2(1989), 157-185.

[4] J. M. Ancochea-Bermudez, M. Goze, On the varietes of nilpotent He algebrasof dimension 7 and 8, J. of Pure and Appl. Algebra, 77(1992), 131-140.



[5] J. M. Ancochea-Bermudez, M. Goze, You. B. Hakimjanov (Khakimdjanov),Sur la reductibility de la variete des algebres de Lie nilpotentes, C. R. A. S.Paris, 313:1 (1991), 59-62.

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[7] F. Bratzlavsky, Classification des algebres de Lie nilpotentes de dimension n,de classe n — I , dont I'ideal derive est commutatif, Acad. Roy. Belg. Bull. Cl.Sci., 560 (1974), 858-865.

[8] J.M. Cabezas, L.M. Camacho, J.R. Gomez, R.M. Navarro, A class of nilpotentLie algebras, to appear in Communications in Algebra, 2000.

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[10] J.M. Cabezas, J.R. Gomez, Las algebras de Lie (n — 3)-filiformes como exten-siones por derivaciones, Extracta Mathematicae, 13(3) (1998) 383-391.

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[18] L.M. Camacho, J.R. Gomez, R.M. Navarro, Family of laws of (n — 6)-filiformLie algebras, Preprint MA1-02-OOVI, Universidad de Sevilla, 2000.

[19] L.M. Camacho, J.R. Gomez, R.M. Navarro, The use of Mathematica for theclassification of some nilpotent Lie algebras, IMACS-ACA'99, Meeting El Es-corial (Spain), 1999.

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[26] J. R. Gomez, F. J. Echarte, Classification of complex filiform nilpotent Liealgebras of dimension 9, Rend. Sem. Fac. Sc. Univ. Cagliari, 61(1) (1991),21-29.

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[34] J. R. Gomez, A. Jimenez-Merchan, Y. Khakimdjanov, Symplectic Structures onthe Filiform Lie Algebras. To appear in Journal of Pure and Applied Algebra.

[35] J. R. Gomez, A. Jimenez-Merchan, J. Nufiez-Valdez, An algorithm to obtainlaws of families of filiform Lie algebras, Linear Algebra and its Aplications 279(1998), 1-12.



[36] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Tratamiento simbolico de algebrasde Lie filiformes graduadas conexas. Actas del Segundo Encuentro de AlgebraComputational y Aplicaciones, EACA'97, 136-142, 1996.

[37] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Filiform Lie Algebras of MaximumLength. Submitted to Acta Mathematica Hungarica.

[38] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Quasi-Filiform Lie Algebras ofMaximum Length. Submitted to Linear Algebra and Applications.

[39] J. R. Gomez, A. Jimenez-Merchan, J. Reyes. Quasi-Filiform Lie Algebras ofLength dim[g, g] + 2. ILAS'99 Meeting Barcelona, 1999.

[40] J.R. Gomez, R.M. Navarro, Espacios de derivaciones de algebras de Lie conMathematica, Proc. of IV Encuentro de Algebra Computational y Alicaciones,(EACA'98). Sigiienza, 1998.

[41] J.R. Gomez, I. Rodriguez. Geometrical properties of model metabelian Lie al-gebras, Submitted to Algebras, Groups, and Geometries.

[42] J.R. Gomez, I. Rodriguez. A special case of family of Lie algebras with nilindex3. ILAS'99 Meeting Barcelona, 1999.

[43] J.R. Gomez, I. Rodriguez. Metabelian Lie algebras with maximal derived.ILAS'99 Meeting Barcelona, 1999.

[44] M. Goze, You. B. Hakimjanov (Khakimdjanov), Nilpotent Lie algebras, KluwerAcademics Publishers, 1996.

[45] M. Goze, You. B. Hakimjanov (Khakimdjanov), Sur les algebres de Lie nilpo-tentes admettant un tore de derivations, Manuscripta Math. 84 (1994), 115-224.

[46] You. B. Hakimjanov (Khakimdjanov), Variete des lois d'algebres de Lie nilpo-tentes, Geometriae Dedicata, 40(1991), 229-295.

[47] Yu. Hakimjanov (Khakimdjanov), Characteristically nilpotent Lie algebras,Mat. Sbornik, 181:5 (1990), 642-655 (russian). English transl. in Math. USSRSbornik, 70:1(1991).

[48] V. Morozov, Classification of nilpotent Lie algebras of sixth order. Izv. Vysch.U. Zaved. Mat., 4:5(1958), 161-171.

[49] O. A. Nielsen, Unitary representations and coadjoint orbits of low-dimensionalnilpotent Lie groups, Queen's Papers in Pure and Appl. Math., 63, 1983.

[50] J. Reyes. Algebras de Lie casifiliformes graduadas de longitud maximal. PhDThesis, Universidad de Sevilla, 1998.

[51] I. Rodriguez. Algebras de Lie con invariante de Goze dado. PhD Thesis, Uni-versidad de Sevilla, 2000.

[52] M. Romdhani, Classification of real and complex nilpotent Lie algebras of di-mension 7, Linear and Multilinear Algebra, 24, 167-189, 1989.


86 Cabezas et al.

[53] Saffiullina, On the classification of nilpotent Lie algebras of dimension 7,Kazan, 1976 (in Russian).

[54] C. Seeley, Seven-dimensional nilpotent Lie algebras over the complex numbers,PhD. Thesis, Chicago, 1988.

[55] C. Seeley, Degenerations of 6-dimensional nilpotent Lie algebras on C, Com-mun. Algebra, 18:10(1990), 3493-3505.

[56] C. Seeley, Some nilpotent Lie algebras of even dimension, Bull. Austral. Math.Soc. 45(1992), 71-77

[57] K. A. Umlauf, Uber die Zusammensetzung der endlichen continuierlichenTransformationsgruppen insbesondere der Gruppen vom Range null, Thesis,Leipzig, 1891.

[58] G. Valeiras, Sobre las componentes irreducibles de la variedad de leyes dealgebras de Lie nilpotentes complejas de dimension 8, PhD. Thesis, Univ.Sevilla, 1992.

[59] M. Vergne, Cohomologie des algebres de Lie nilpotentes. Application a I'etudede la variete des algebres de Lie nilpotentes, Bull. Soc. Math. France 98(1970),81-116.

[60] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer,Addison-Wesley, 1991.


On L*-Triples and Jordan #*-Pairs

A. J. CALDERON-MARTIN, Departamento de Matematicas. Universidad deCadiz. 11510-Puerto Real. Cadiz, Spain.E-mail: [email protected]

C. MARTIN-GONZALEZ, Departamento de Algebra, Geometria y Topologfa. Uni-versidad de Malaga. Apartado 59, 29080-Malaga, Spain.

Abstract

In [12], Lister introduced the concept of Lie triple system. He classified thefinite-dimensional simple Lie triple system over an algebraically closed field ofcharacteristic zero. Neher also studies Lie triple systems and their relationswith Jordan triple systems in [13]. In order to study infinite-dimensional Lietriple systems, we introduced the notion of //-triple and obtained a classifica-tion of //-triples admitting a two-graded //-algebra envelope in [3]. However,the problem on the existence of L*-algebra envelopes is still open. We prove inthis paper, using Jordan //"-pairs techniques, that every infinite-dimensionaltopologically simple //-triple, verifying a purely algebraic additional property,has a two-graded L*-algebra envelope and then we classify them.

1 PREVIOUS RESULTS ON //-TRIPLES

Let A be a C-algebra and * : A —» A a conjugate-linear map, for which (x*)* = x and(xy)* = y*x* hold for any x, y e A. Then * is called an involution of the algebra A.We recall that an H*- algebra A over C is a nonassociative C-algebra, which is alsoa Hilbert space over C with inner product (• •), endowed with an involution * suchthat (xy\z) = (x\zy*) = (y\x*z) for all x,y, z € A. A two-graded H*-algebra, is an//"-algebra which is a two-graded algebra whose even and odd part are selfadjointclosed orthogonal subspaces. We call the two-graded //'"'-algebra A topologicallysimple if A2 ^ 0 and it has no nontrivial closed two-graded ideals. In the sequel an//-algebra will mean a Lie //*-algebra. The classification of topologically simple//-algebras is given in the separable case by Schue (see [14], [15]) and later in the


88 Calderon-Martin and Martin-Gonzalez

general case in [8], [1] and [13]. If HI and H-2 are Hilbert spaces with scalar products( • | - ) i , z = l ,2 and / : H i — > H-2 a linear map such that

for any x, y € H I , then we will say that / is a k-isogenic map.We can define an H*-triple as a complex triple system (T, < -, •, • >) provided

with:

1. A conjugate-linear map * : T — > T such that (x*)* = x and

<x,y,z >* = <z* ,y* , z* >

for any x, y, z G T.

2. A complex Hilbert space structure whose inner product is denoted by ( • [ • ) andsatisfies

(<x,y,z>\t) = (x <t,z*,y* >) = (y| <z*,t,x* >) = (z <y*,x*,t>)

for any x, y, z,t €. T.From [5] it follows that the triple product < - , - , - > of T is continuous.

We define the annihilator of an //"-triple (T, < •, •, • >) as

Ann(T) = {x e T :< x,T,T >=< T,x,T >=< T,T,x >= 0}.

This set Ann(T) turns out to be a closed ideal of T.The definition of topologically simple H "-triple is similar to that used in the case

of //"-algebras. The structure theorems for //"-triples given in [5] reduce the intereston H *-triples to the topologically simple case. Essentially, these theorems claim thatunder suitable conditions, one can split any //"-triple as the orthogonal direct sumof its annihilator plus the closure of an orthogonal direct sum of topologically simple//"-triples.

We define an L* -triple as a Lie //""-triple. If L is an L*-algebra, then it canbe viewed as an L*-triple by defining the triple product [x , j / ,z j = [[x,y],z] for anyx,y,z <E L, this L*-triple will be denoted by LT . Furthermore, for any two-gradedL*-algebra, L = Lo-LLi, its odd part LI is an L*-subtriple of LT . If T is an //"-tripleisometrically *-isomorphic to the L*-triple LI for some two-graded L*-algebra L,we shall say that L is a two-graded L*-algebra envelope of T iff LQ := [Li, L I ] . It isalso easy to check that if T is an L*-triple with two-graded L*-algebra envelope L,then T is topologically simple if and only if L is topologically simple in the gradedsense. We refer to [3, Theorem 1] for the following classification of topologicallysimple L*-triples that admit a two-graded L*-algebra envelope.

THEOREM 1.1 Let T be a topologically simple L* -triple admitting a two-gradedL* -algebra envelope. Then T is one of the following:

1. The L* -triple associated to an L*- algebra L by defining the triple product[a, 6, c] := [[a, 6],c] and the same involution and inner product of L.

L*-Triples and Jordan tf*-Pairs 89

2. Skw(A,r) with A a topologically simple associative H* -algebra, T an involu-tive *- automorphism of A, the involution and inner products induced by theones in A, and the triple product as in the previous case.

3. Sym(A,a) with A as in the previous case but now a is an involutive *-antiautomorphism, and the triple product, involution and inner product in-duced by the ones in A.

4- Skw(A,T)r\Skw(A,a) with A as before, T an involutive *- antiautomorphism,a an involutive ^-automorphism such that ar = ra , and the involution andinner product induced by the ones in A.

The problem on the existence of L*-algebra envelopes is still open. However, ifT is an L*-subtriple of A~ for a topologically simple ternary //"-algebra A, that is,for any x, y, z & T one has

[x, y, z] =< x, y, z > - < y, x, z > - < z, x, y > + < z, y, x >

where < •, •, • > is the triple product of A, then it can be proved that it has a two-graded L*-algebra envelope. Indeed, from the classification of topologically simpleternary //"-algebras (in the complex case) of [7, Main Theorem, p. 226], one may seethat there is an associative topologically simple two-graded //"-algebra B = Bo-LBi(see [6] for classification theorems) such that A is the ternary //"-algebra associatedto BI (with triple product < x,y,z >= xyz for all x,y,z 6 BI). Let // = Lo-L^ibe the two-graded L*-subalgebra of B" generated by T. It is easy to prove thatLI = T and LQ = [T, T], hence the topological simpleness of T implies that of L inthe graded sense.

2 PREVIOUS RESULTS ON JORDAN //"-PAIRS

Let A = (A+ , A~) be a pair of modules over a commutative unitary ring K, and

< - , - , - >: Aa x A~a x A" — » A",

two trilinear maps such that

(x,y,z) H->< x,y,z>

for a g {+,—}• Then A is called an associative pair if the following identities aresatisfied:

« x,y,z >,u,v >=< x, < u,z,y >,v >—< x,y,< z,u,v »

for x,z,v € A" and y, u 6 A~a .Let A = (A+,A ) be a pair of /iT-modules and Q° : A° — > homK(A~CT,A'7)

two quadratic operators for a € {+,—}• We define the trilinear operators

and the bilinear operators

Da : Aa x A~a — > End(Aa)

90 Calderon-Martm and Martin-Gonzalez

as {x,y,z}° = D"(x,y)z := Q° ' (x + z)y - Qa ' (x)y - Q° ' (z)y , for z, z £ Aa , y € A~a

and cr 6 {+,—}. We shall say that A = (A+,A~) is a Jordan pair if the nextidentities and its linearizations are true:

Q°(Q°(x)y) = Q°(x)Qfor x, z e A° , y <E A~a and cr e {+, -}.

One can find a classification of prime, nondegenerate Jordan pairs with nonzerosocle in [10, Teorema 6] or [9, Theorem 7].

If A is an associative pair, then AJ will denote the symmetrized Jordan pair ofA, that is, the Jordan pair whose underlying /('-module agrees with that of A, andwhose quadratic operators are given by Qa (x)y =< x,y,x >a .

Let A = (A+,A~) be a complex pair and * = (*+ ,*~) a couple of conjugate-linear mappings *°" : Aa — > A~a for which *CT o *~CT = Id and

for xa , za 6 Aa and y a G A <T. Then * = (*+,* ) is called an involution of A.We say that A is an H* -pair if A+ and A~ are also Hilbert spaces over the complexnumbers with inner products ( - ( - j o - : Aa x A" —> C, endowed with an involution* = (*+ ,*~) such that

t-crfor xa,za,ter e ̂ and y~a e A"We also recall that an H*-psdr A is said to be tope/logically simple when

< Aa,A~a,A° >^0

and its only closed ideals are {0} and A. If A is an associative H*-pair, then AJ canbe canonically equipped with a Jordan H*-pair structure. Moreover, if £ : A —» Aop

is an involutive *-antiautomorphism, then both Sym(A, £) and Skw(A, ^) can also bestructured as Jordan //"-pairs in obvious ways. Given an infinite dimensional andtopologically simple Jordan //*-pair, we can always describe it from a Jordan H*-algebra or from an associative //*-pair. Indeed, every topologically simple Jordan//"-pair J is prime, non-degenerate and with non-zero socle (see [2, Proposition1]), then, the classification of Jordan pairs given in [10, Teorema 6] implies thatforgetting the H*-structure, the underlying Jordan pair J is one of the following:

Type (i). J = (V, V) with (V, {-, •}) a quadratic Jordan algebra, being the tripleproducts {x, y, z}a = {x, { y , z}} - {y, {x, z}} + {z, {x, y}} for any x, y, z e V.

Type (ii). J is a subpair of (L(X,Y),L(Y,X))J containing (F(X, Y),F(Y,X))J

with the triple products {x,y,z}a := xyz + zyx , where (X,X') and (Y,Y') aredual pairs over an associative C-division algebra A.

Type (iii). J is a subpair of

(Sym(L(X, Y), |), Sym(L(Y, X), H))

L*-TripIes and Jordan H*-Pairs 91

containing (Sym(F(X,Y),$),Sym(F(Y,X),W where (X,YJ), (X,YJ°r) are adual pair and its opposite over an associative C -division algebra (A,r) with invo-lution T, where the triple products are {x, y, z}" = xyz + zyx and with ft the adjointoperator.

Type (iv). J is a subpair of

(Skw(L(X, 7), fl) , Skw(L(Y, X ) , «))

containing (Skw(F(X,Y),$),Skw(F(Y,X),$)) where ( X , Y , f ) , ( X , Y , f ° P ) are adual pair and its opposite over C, where the triple products are {x, y, z}a = xyz +zyx and with ft the adjoint operator.

If J is a topologically Jordan //*-pair of Type (i), then one can define an in-volution and inner product on V so as to obtain that V is a Jordan //"-algebra.We prove, in [2], that any topologically Jordan //"-pair of Type (ii) is of the formJ = AJ , where A is a topologically simple associative //*-pair. Similar argumentsapply to a topologically simple Jordan //*-pair J of types (iii) or (iv), give us thatJ is of the form J = Sym(A, £) or J = Skw(A, £) respectively, with A as above,and £ an involutive *-antiautomorphism from A to Aop, (see [4] for more details).

The concept of polarized triple system of a pair will also be used with the samemeaning as in [13].

3 MAIN RESULTS

Neher finds in [13] the relation between simple polarized Lie triple systems, simplepolarized Jordan triple systems and simple Jordan pairs. We can obtain similarresults in an H "-context so as to give the next

THEOREM 3.1 (a) Let L = L+ J_ L~ be a topologically simple polarized L* -triple.Then J(L) : — L with the product

is a topologically simple polarized Jordan H* -triple.(b) Let J = J+ _L J~ be a topologically simple polarized Jordan H* -triple. Then

L ( J ) :— J with the product x,y,z := {x,y,z} — {y,x,z} is a topologically simplepolarized L* -triple.

(c) The operations L — > J ( L ) and J — > L(J) described in (a) and (b) areinverses of each other.

PROOF.- (a) and (b).According to [13] we have that J(L) := L (resp. L(J) := J) is a Jordan triple

system (resp. Lie). It is easy to check that the involution and inner product of L(resp. J) make J(L) := L (resp. L ( J ) := J) an //"-triple.

Since any ideal of J(L) is an ideal of L, we conclude that J(L) is a topologicallysimple Jordan //"-triple. Our next claim is that L(J) is topologically simple. Wefirst observe that AnnL(J) = 0. Indeed, from [13], AnnL(J) is a IT-invariant idealof L(J), being II the automorphism of L ( J ) defined as H((x+,x~)) : — (x+, -x~),hence [13] shows that AnnL(J) is an ideal of J and therefore AnnL(J) = 0. Finally,let J be a closed ideal of L(J), as [ I , L ( J ) , L ( J ) } is an ideal of J ([13]), either[7, L( J), L(J)\ = 0 and then / = 0, or [I, L ( J ) , L ( J ) ] = J which implies / = L( J ) .


92 Calderon-Martin and Martin-Gonzalez

The item (c) is immediate to check.

It follows, easily, that a Jordan H*-pair is topologically simple if and only if itsassociated polarized Jordan _ff*-triple is also topologically simple.

The next result is clear by [13].

PROPOSITION 3.2 Let J; = (J+, Jr) (i e {1,2}) be topologically simple JordanH*-pairs, and let L ( J i ) be their associated polarized L*-triples. Then, if J\ and J%are *-isometrically isomorphic, L(J\) and L(J^) are also *-isometrically isomor-phic. If L(Ji) and £(1/2) are *-isometrically isomorphic then J\ is *-isometricallyisomorphic with J% or J%p.

THEOREM 3.3 Let (T, [ - , - , • ] ) be an L*-triple. Then there exists an associativealgebra U such that [x, y, z] = xyz — yxz — zxy + zyx for any x,y,z € T.

PROOF.- The proof consists in the construction of L = LQ ® LI, a two graded Liealgebra envelope of T and then, denning U as the universal enveloping algebra ofL. Define LQ to be the span of ad(Li,Li), where a d ( x , y ) ( z ) = [ x , y , z ] is the leftproduct operator. We have that L = LQ ® LI with the product defined by

[ ( a d ( x , y ) , z ) , ( a d ( u , v ) , w ) } =

= ( a d ( [ u , v , y ] , x ) - ad([u, v , x ] , y ) + ad(z,w), [z,y, w] - [ u , v , z ] )

is a two graded Lie algebra envelope of T, that is, {x, y, z] — [[x, y]z] for any x, y, z €T. (In fact, one can construct an involution *, preserving the grading, and a non-degenerate hermitian form /, in such a way that LI is isometrically *-isomorphicto T considered as an L*-triple with the restriction of * and of /, however, it doesnot seem easy to complete this construction so as to produce on L an i*-algebrastructure). The universal enveloping algebra of L, see [11, Chapter V] is the algebraU we are looking for.

THEOREM 3.4 (MAIN THEOREM) Let (T, [•, - , •]) be an infinite dimensional topo-logically simple L*-triple system. Let U be an associative algebra such that [x, y, z] =xyz —yxz — zxy + zyx for any x,y,z e T. If xyx G T for every x,y e T, then T hasa two-graded L*-algebra envelope and therefore is one of the L*-triples described inTheorem 1.1.

PROOF.- By section 1, we only have to prove that T is an L*-subtriple of A~, beingA a topologically simple ternary H ""-algebra.

It is straightforward to prove that T ® T with the product

[(a, 6), (c, d), (u, v)} := (adu + uda — cbu — ubc, bcv + vcb — dav — vad),

involution (a, 6)* := (6*, a*) and inner product < ( a , b ) \ ( c , d ) >=< a\c > + < b\d >is a polarized L*-triple and that

T —> T®Tx H-> (x,x)

is a fc-isogenic *-monomorphism of L*-triples. Now, the structure theorems forpolarized L*-triples (analogous to the ones in [5]), allow us to assert that T is

L*-Triples and Jordan #*-Pairs 93

*-isomorphically /c-isogenic to an L*-subtriple of a topologically simple polarizedL*-tripleT+ LT~.

Let (J = J+ _L J~, { - , - , • } ) be the polarized Jordan H*-triple associated toT+ _L T~ (see Theorem 3.1). Following the description of topologically simpleJordan #*-pairs given in Section 2 we have to study the following four possibilities:

(a) If J = (V, V), where V is a quadratic Jordan //*-algebra, then we have

4> : T+ ±T~ — » L(V ± I/)

an *-isometric isomorphism of //-triples. One easily shows the existence of

<f>+ : T — » V

and 4>~ : T — > V such that 4>(x,y] = (4>+ (x) , <j>~ (y)) for x,y € T. We canalso check that

for any x + ,y~ ,2 + £ T. Therefore <p+ establishes a *-isomorphism from T ontothe L*-triple (V, [ - , - , • ] ' ) with the triple product defined as [a,b, c}' := [a, <r(6),c],where <j = 4>~ (4>+}~l , and a suitable involution. However, (V, [•, •, •]') turns out tobe a nontopologically simple L*-triple (moreover, Ann(V) ^ 0 since [I, a, b]' = 0for a, 6 £ V), which is impossible. Thus, J has to be one of the other types oftopologically simple polarized Jordan //""-triples, that is:

(b) J = (A+ _L A~)J with (A+ A. A~ , < • , - , - > ) a topologically simple ternaryff*-algebra. Then, for any (x+,x~), (y+, y~), (z+,z~) £ T, we have that

<

that is, T is a subtriple of (^4+ ± A~)~, with A+ X A~ a topologically simpleternary ff* -algebra, as we wanted to prove.

(c) and (d). If J = Sym(A+ _L A~,£) or J = Skw(A+ ± A~,£), with (>1+ 1^4"", < - , - , • > ) a topologically simple ternary _ff*-algebra, we can prove as in case(b) that T is also a subtriple of a topologically simple ternary H* -algebra. Thiscompletes the proof.

Note that, following the proof of the last theorem, we have that every non-quadratic topologically simple L*-triple system that embeds in a polarized i*-triplesystem (particularly topologically simple polarized L*-triple system) has a two-graded L*-algebra envelope.

94 Calderon-Martin and Martm-Gonzalez

REFERENCES

[1] M. Cabrera, A. El Marrakchi, J. Martmez and A. Rodrfguez Palacios, AnAllison-Kantor-Koecher-Tits construction for Lie #*-algebras. Journal of Al-gebra. 164 (1994), no.2, 361-408.

[2] A. J. Calderon and C. Martin, Dual pairs techniques in H* -theories, Journalof Pure and Applied Algebra 133 (1998) 59-63.

[3] A.J. Calderon Martin, C. Martin, Two graded L*-algebras, preprint, Univer-sidad de Cadiz (Spain), 1999.

[4] A.J. Calderon Martin, C. Martin, On associative ff*-structures induced byJordan JY*-pairs, preprint, Universidad de Cadiz (Spain), 1999.

[5] A. Castellon, J. A. Cuenca, Associative #*-triple Systems. In "Workshop onNonassociative Algebraic Models'. Nova Science Publishers (eds. Gonzalez S.and Myung H. C.). New York, 1992. pp. 45-67.

[6] A. Castellon, J. A. Cuenca, and C. Martin, Applications of ternary #*-algebrasto associative /f*-superalgebras, Algebras, Groups and Geometries, 10, (1993),181-190.

[7] A. Castellon, J. A. Cuenca, and C. Martin, Ternary H*-algebras. BolletinoU.M.I. (7) 6-B (1992), 217-228.

[8] J. A. Cuenca, A. Garci'a and C. Martin, Structure theory for L*-algebras.Math-Proc. Camb. Phil. Soc. (1990), 361-365.

[9] A. Fernandez Lopez, E. Garcia Rus and E. Sanchez Campos. Prime Nonde-generate Jordan Triple Systems with Minimal Inner Ideals. Nova Science Pub-lishers Inc. (1992) 143-166.

[10] A. Garcia, "Nuevas aportaciones en Estructuras Alternativas", Tesis Doctor-ales/Microficha num. 157. Secretariado de Publicaciones de la Universidad deMalaga. 1995.

[11] N. Jacobson, Lie algebras. Interscience 1962.

[12] W.G. Lister, A structure theory of Lie triple systems. Trans. Amer. Math. Soc.72 (1952), 217-242.

[13] E. Neher, On the classification of Lie and Jordan triple systems. Comm. inAlgebra. Vol. no. 13 (12), 2615-2667.

[14] J. R. Schue, Hilbert Space methods in the theory of Lie algebras. Trans. Amer.Math. Soc. vol 95 (1960), 69-80.

[15] J. R. Schue, Cartan decompositions for I/*-algebras. Trans. Amer. Math. Soc.vol 98 (1961), 334-349.


Toric Mathematics from Semigroup Viewpoint

A. CAMPILLO, Departamento de Algebra, Geometrfa y Topologfa, Universidadde Valladolid. 47005-Valladolid. Spain.E-mail: campillo @agt. uva. es

P. PISON, Departamento de Algebra, Computacion, Geometrfa y Topologfa, Uni-versidad de Sevilla. Aptdo. 1160. 41080-Sevilla. Spain.E-mail:ppison@cica. es

I INTRODUCTION

Toric geometry is a subject of increasing activity. Toric varieties are objects onwhich one usually can check explicitly properties and compute invariants from alge-braic geometry. This happens for the so-called normal toric varieties, i.e. algebraicvarieties which are constructed from rational fans in a euclidean space. In the last10 years the theory of non normal toric varieties has also been developed providinga very different and new scope as well as interesting and beautiful new applications.

Normal toric geometry mainly uses techniques from convex geometry, as it istechnically founded on the concepts of fan and cone. Fans are sets of polyhedralcones in such a way that each cone provides an affine chart of the toric variety.Namely, those charts have, as coordinate algebra, the algebra of the semigroup oflattice points lying inside the dual cone of the corresponding cone of the fan.

To study non normal toric geometry one needs to be more precise than to con-sider only cones. In fact, what one needs is to consider affine charts where coordinatealgebras are semigroup ones for more general classes of semigroups. Thus, convexgeometry should be used only as a tool by taking into account that nice semigroupsgenerate concrete polyhedral cones.

The purpose of this paper is to show how mathematics in toric geometry canbe understood as the theory of appropriate classes of commutative semigroups withgiven generators. This viewpoint involves the description of various kinds of derivedobjects as abelian groups and lattices, algebras and binomial ideals, cones and fans,


96 Campillo and Pison

affine and projective algebraic varieties, simplicial and cellular complexes, polytopes,and arithmetics.

Our approach consists in showing the mathematical relations among the aboveobjects and clarifying their possibilities for future developments in the area. Forthat purpose, we will survey some recent results and concrete applications.

2 SEMIGROUP AND GENERATORS OF TORIC GEOMETRY

The central object we will consider along the paper should be finitely generatedcancellative commutative semigroups with a specified system of generators.

By a commutative semigroup we understand here a set S endowed with aninternal commutative operation denoted by + having a zero element denoted 0.Semigroup homomorphisms are maps preserving the operation + and the element0. Thus, one has the category of semigroups.

Cancellative for 5 means that S is isomorphic to a subsemigroup of an abeliangroup, or in other words that the semigroup homomorphism S —> G(S), whereG(S) is the abelian group generated by S, is injective. Here G(S) is the abeliangroup of classes of pairs (m,n) G 5 x 5 for the relationship (m,n) ~ (m',n') iffm + n! — m' + n.

Thus, our central object should be the data of a semigroup S as above plus asurjective semigroup homomorphism

TTO : Nh -> 5,

where N is the semigroup of nonnegative integers. Notice that TTQ is just the samedata than the choice of a generator system of the semigroup S, namely the generatorsystem ni,... ,nh, where rij = TTQ(CJ), ej being the /i-uple with j-coordinate 1 andother coordinates 0.

Since toric geometry is a subject providing explicit computations and results, onecan think that toric mathematics essentially consists in the detailed study of mapsTTo of the above type. Along the paper it will be shown how the above statementstands when dealing with affine or projective toric objects.

For studying such a map TTQ one needs to understand the structure and behaviorof its fibers 7r^"1(?7i) for m & S. This is an elementary and difficult problem which,for many purposes, becomes the key problem of toric geometry.

A first remark is that one should consider some kind of finiteness hypothesis,namely requiring that the fibers TT^^TO) be finite for every m. The following resultgives some distinct characterizations of that hypothesis.

PROPOSITION 2.1 (see [4]) Let TTQ : N'1 —> 5 be a surjective semigroup homo-morphism where S is a cancellative commutative semigroup. Then, the followingconditions are equivalent:

1. KQ (TO) is finite for every m.

2. There is no infinite sequence m € S, TOI, . . . , m j , . . . € S — {Q}, such thatm — TOI — • • • — mi 6 S for every i.

3. Sn(-S) = {0}.


Toric Mathematics from Semigroup Viewpoint 97

4- There exists a semigroup hornomorphism A : 5m = 0.

N such that A (TO) = 0 iff

Semigroups satisfying conditions in Proposition 2.1 are called, in the literature,of different forms according to the property one wants to emphasize. Thus, theyare said to be combinatorially finite in view of (1) (see [5]), Nakayama in view of(2) (see [18] and [24]), strongly convex in view of (3) (see [11]), or positive in viewof (4). The terminology which will be used along this paper is that of "positive".

The description of the fibers is related to the study of relations among thechosen generators of S. Since the "kernel" of TTQ does not exist in the category ofsemigroups, to describe the relations one needs a different object, the congruenceF of TTo, to define those relations. The congruence F is the binary relation on N^consisting of those pairs (u, v) g N'1 x N'1 such that u, v belong to the samefiber TTQl(rn) for some m e S. Congruences are binary equivalence relations onsemigroups allowing to give a semigroup structure on the quotient, i.e. with theproperty that (u, v) 6 F and w is in the semigroup (i.e. Nh in our case) then(u + w,v + w) e F. Since 5 is a finitely generated semigroup, by [13, 1.6], onehas that the congruence F is finitely generated, i.e. that F is the least congruencecontaining one finite set of elements in it. In other words, one can say that 5 is afinitely presented semigroup.

In the rest of the paper we will show how to treat and exploit the informa-tion in a semigroup with their generators and relations. This will involve severalfields of mathematics on each of which one will derive concrete perspectives andconsequences. The figure below shows the scheme of the spirit of our discussions.

\ TSemigroups

(Generators and relations)

I Arithmetics!

3 ABELIAN GROUPS AND LATTICES

Consider a map TTQ : N'1 —> 5 as in section 1. Since the assignment to a semigroupof the group generated by the semigroup is functorial, one has an induced exactsequence of abelian groups given by

0 -> L -» G(Nh) = Zh -> G(S) -» 0,

where L is a subgroup of Zh, so L is finitely generated and torsion free. We willrefer to L as the lattice associated to the data TTQ. Notice that L is nothing butthe kernel of surjective induced group homomorphism -K : Zh —> G(S), so that Lis the object keeping the information of the group theoretical relations among thesemigroup generators n\,..., n^.


The relation between the congruence F and the lattice L can be described easily.In fact, if (u, v) e F then there is a unique element w € N'1 such that (u — w, v —w) 6 F and the supports of u — w and v — w are disjoint. Here by support of anelement of Zh we mean the set of indices whose coordinates are nonzero for suchan element. Notice that if < denotes the componentwise product ordering on Zh,then w is nothing but the infimum of u, v for that ordering. Thus, one has a welldefined map

b : F -> N'1 x L,

given by 6(u, v) = (w, u — v).If (w,l) € N'1 x L, set 1 = 1+ - I", where 1+ = sup(l,Q), l~ = sup(-l,0) and

sup(,) stands for the supremum relative to the ordering <. Then the assignmentto the element (w, 1) of the couple (1+ + w, 1~ + w) is a map Nh x L —> F which is,by construction, inverse to 6. So one has the following result.

PROPOSITION 3.1 The map b is a bijection.

It follows from Proposition 3.1 that the information in F is just the same as inL and how one can get one from the other.

Moreover, from free abelian groups and their sublattices one can study thesemigroups we are interested in. In fact, if a lattice L C Zh is given, then from theobvious exact sequence

0-> L-+Zh -^Zh/L->0,one can consider the subsemigroup S of the group Zh/L given by the image of Nh

and generators given by the images of the elements ei,... ,e^. Also notice that thecondition on S to be positive is equivalent to the condition L D Nh = (0).

Notice that, in general, the abelian group G(S) = Zh/L can have torsion, sothe semigroup S can also have torsion in the sense that it can contain elements?7i, n € S, m ^ n and integers a G N such that am = an. If T is the torsionsubgroup of G(5), the image of S in G(S)/T is a new semigroup S of the samekind than 5. Notice that S is positive iff S is so. This follows from the fact thatLnN'1 = (0) iff LnN'1 = (0), L being the lattice for the induced map 7f0 : Nh -> 5.

Finally, we remark that S is not only the image of N'1 by TT, but S is also theimage of other subsets of Zh, in particular of the set N^ + L. This new set is alsoa semigroup which has an obvious structure of N^-module. As semigroup it is notpositive (except in the trivial case L = 0 for which 5 = (0)), however if S is positivethen N^ + L has the property analogous to (2) in Proposition 2.1, i.e. there isno infinite sequence of elements m = TOO > TOJ > . . . > TOJ > . . . in N'1 + L. Inother words, if 5 is positive then Nft + L is generated by its minimal elements forthe ordering <. Note that such minimal elements are nothing but the primitiveelements of the set N'1 + L, i.e. those elements which are not a sum of a nonzeroelement of Nh with another element of N'1 + L.

4 SEMIGROUP IDEALS AND ALGEBRAS

In this section we will fix a commutative field k. Then, one has a functor from thecategory of semigroups to that of /c-algebras taking each semigroup to its semigroup/c-algebra. Notice that, for any semigroup S, the semigroup /c-algebra k[S] consists

of the vector space generated (freely) by the symbols xm> one f°r each m e 5,endowed with a multiplication given on symbols by the rule xm ' Xn = Xm+n> f°r

m,n € S.Now, consider a map TTQ : N'1 —> 5 as in section 1, and apply the above functor

to it. One gets an exact sequence

0 -» 7 -> A = k[Nh ?$R = k[S] -» 0,

where / is the kernel of the fc-algebra homomorphism </?o associated to TTQ, whichis called the ideal of the semigroup relative to the generators n j , . . . ,n^. Noticethat, if Xi,..., Xh are variables corresponding to the coordinates in Nfe, one hasa canonical identification A = k[Xi,... ,Xh]. Moreover, both R and A are gradedover the semigroup S (say, S-graded) by giving the obvious degree m to the symbolXm and degree rij to the variable Xi. In particular, one has a decomposition intohomogeneous components A = ®m65 Am, R = ®mes kxm- Here Am is the vectorspace generated by all the monomials of degree TO, i.e. X" = X™1 • • • X^h with*r^h2^i=i Wi/ij = m.

The homomorphism </? becomes 5-graded of degree 0 and, therefore, the semi-group ideal is 5-homogeneous, i.e. one has I = @m€g Im with Im = I n Am forevery m e 5.

Notice that /T! is generated, as a /c-algebra, by the symbols xni > • • •) x"hi so that/ can be understood as the ideal of polynomial relations of such symbols. The ideal7 is binomial as it is generated by the binomials X" — X" for (u, v) ranging over thecongruence F. Using Proposition 3.1 one sees that it is also generated by X' — X'where 1 ranges over the lattice L. Anyway, notice that to generate / it is enough totake a finite number of binomials X" — Xw, where the couples (u, v) generate thecongruence F.

Now, assume that S is positive. Then nice properties occur. First, by 3) inProposition 2.1, one has that the irrelevant MR = ®TOJO kxm and MA = ©m-*;o A™are ideals of R and A respectively. Second, by 1) one has that each Am is afinitely dimensional vector space. Third, by 2), Nakayama's lemma holds for S-graded modules; in particular one can speak about minimal systems of homogeneousgenerators for / which are nothing but those inducing a basis of the vector spaceI/MA!- It is clear that one can consider minimal sets of binomial generators for /.

In fact, one can consider S'-graded free homogeneous resolutions of R as anA-module. If 5 is positive, Nakayama's lemma shows that one can consider theminimal free resolution (which is unique up to isomorphism) which is one of thetype

n ^ p ^ p ^ p ^ l p ^ l p A fS. H ^. r\U —> fp —> • • • —> 1*2 —> f\ —> -TQ = A —> ri —> U,

where each Fi is a free S-graded finite A-module, the ̂ are graded of degree 0 andp is the projective dimension of R as A-module, i.e. the least integer p such thatFp ^ 0. The Auslander-Buchsbaum theorem shows the relation p + r — h, wherer is the depth of R. The integer r ranges over the values 0 < r < d, where d isthe Krull dimension of R. Notice that the Krull dimension of R coincides with therank of the abelian group G(S). The last statement follows from the computationof dimensions in terms of transcendence degrees. It implies, in particular that thedimension of the /c-algebra k[S] does not depend on the field k. This is not the casefor the integer r which could depend on k.

Commutative algebra provides interesting particular cases. First, when r = d,the ring k[S] is said to be Cohen-Macaulay. This property depends on S and k butnot on the map TTQ. If k[S] is Cohen-Macaulay and, moreover, Fp has rank 1 as anA-module, then k[S] is said to be Gorenstein. This is a case in which the minimalresolution is self-dual, i.e., by applying the functor Hom( — ,A) and considering thenatural grading, the induced exact sequence

0 -> Eom(FQ, A) -> Hom(Fi, A) -> • • •

• • • -* Eom(Fp, A) -» Coker(<4) -> 0,

is S-graded isomorphic to the minimal resolution of R. Again the Gorenstein prop-erty depends on S and k, not on TTQ. Finally, k[S\ is said to be a complete intersectionif / can be generated by h — d homogeneous elements (in fact binomials). Equiv-alently, complete intersection means that the congruence F can be generated byh — d pairs. The complete intersection property only depends on 5 and implies theGorenstein property.

5 CONES AND FANS

With assumptions as above, the next object one can associate to a semigroup 5 isthe cone C(S) generated by S, i.e. the cone generated by the image of S in theQ-vector space VQ := G(S) (8>z Q. Since the base ring extension from Z to Q killsthe torsion, the cone C(S) obviously coincides with that of its image 5 in G(S)/T.

If 5 is not positive, then C(5) is equal to the whole VQ, so it contains trivialinformation. Thus, the interesting case turns out to be the case in which S ispositive. Note that S is positive if and only if C(S) is a strongly convex cone (i.e.,if one has C(S) n —C(S) = 0). This fact justifies the terminology in (3) section 1.

Now, if one takes into account the generators of 5, then one has that the coneC(S) is the rational polyhedral one generated by (i.e. it is the convex hull of) theimages in VQ of the generators n\,..., n^. Thus, convex geometry occurs as a usefultechnique of toric mathematics.

There is a very important case, in which the cone C(S) determines the semigroup5. In fact, a semigroup is said to be normal if it is torsion free and if, moreover,one has 5 = C(S) n G(S). It is well known that 5 is a normal semigroup if andonly if the semigroup fc-algebra A;[5] is an integrally closed domain, i.e. a normalring. Hochster's theorem [14] shows that if S is normal then, in fact, k[S] is Cohen-Macaulay.

A trivial example of normal semigroups are the free semigroups, i.e., those whichare isomorphic to N* for some integer t. In fact, free semigroups are the onlyones such that the fc-algebra k[S] is a regular ring. The terminology "regular" iscoherently used also in convex geometry, being applied to a cone on Q* which isgenerated by a basis of the lattice Z*. Notice that a semigroup is free if and only ifit is normal and if the cone it generates is regular.

Toric geometry appears initially as the study of normal toric varieties. Thus, thedevelopment of normal toric geometry is based on convex geometry and, therefore,one can say that normal toric mathematics is convex geometry mathematics.

Coming back to the general case, the convex cone C(S) provides a new interest-ing invariant for a semigroup 5, namely the number of edges e of C(S). Comparing

with the dimension, one has e > d, and the equality holds whenever the cone C(S)is simplicial. Thus, semigroups for which e = d will be called simplicial along thepaper. Free semigroups are a very special case of simplicial semigroups.

Toric varieties also include non affine ones. Affine toric varieties are nothing butthe affine varieties X with coordinate fc-algebra equal to a semigroup fc-algebra withthe assumptions of section 1. General toric varietes are algebraic varieties whichcan be covered by affine toric varieties with overlappings which are also affine toric.

Normal toric varieties are usually given in terms of convex geometry. The dataconsists in a fan $ of rational polyhedral cones in Q™, i.e, a set {crj^g^, where <& isa finite set, each a a strongly convex polyhedral cone in Qn, the faces of each a in$ are also in $, and the intersection of every couple of two cones in $ is a commonface of both of them. The variety is constructed in the following way. For each ain <&, consider the semigroup 5CT of integer coordinates points which lie inside thedual cone of a, and let Xa be the affine toric variety given by Sa. Then the toricvariety X is the join of the affine varieties Xff, the intersection of any two Xa, XT

of those affine charts being the toric variety Xanr.Thus, for a normal toric variety, the fan <& not only determines the variety but it

represents and exhibits its geometry. In fact, cones in the fan correspond to affinecharts in such a way that intersection of cones correspond to the overlapings of thecorresponding charts.

For non normal toric varieties one can proceed in a similar way, but taking afurther precision on the semigroups. Thus, one needs a fan 4> as above plus, foreach cone a, a subsemigroup S'a of Sa generating the same cone as Sa and in sucha way that the intersection of two charts with respective coordinate algebras k[S'a]and k[S'T] is the affine chart with coordinate algebra /c[S^.nr]. Thus, one sees that,also in the global case, toric mathematics is not only convex geometry but againinvolves finitely generated cancellative semigroups.

The support of a fan <£> is defined to be the union of the supports of the conesin the fan. The fan is said to be complete if its support is Qn. Toric varieties builtfrom complete fans are complete algebraic varieties. The next section is devoted tothe particular case of projective varieties, a subclass of complete toric varieties.

6 AFFINE AND PROJECTIVE TORIC VARIETIES

Toric varieties are algebraic ones, so algebraic geometry is naturally related totoric mathematics. Particularly interesting algebraic varieties are the affine andprojective ones. When some data TTQ : N^ —-> S, is given, the semigroup 5 givesrise to the (abstract) affine toric variety X = Spec(k[S]), whereas the choice ofgenerators provided by TTQ gives rise to an embedding of X into the affine space A.h.The dimension of X is just the rank d of the abelian group G(S). Below, we discussand emphasize how abstract and embedded projective toric varieties can also bedescribed in nice terms.

Let S be a finitely generated cancellative commutative semigroup. Assumethat 5 is endowed with a semigroup map A : 5 —> N such that the semigroup isgenerated by the elements in the set 5j = A -1(l). Then, for any choice of thefield k, the couple (S, A) gives rise to an (abstract) (d — l)-dimensional projectivealgebraic scheme, namely Z = Proj(k[S}), where k[S] is now viewed as an N-graded algebra by relaxing its natural 5-grading via the map A (in other words,


degree i 6 N homogeneous elements are the sum of homogeneous elements of S-degrees in A~ 1(z)) . Along the paper, couples (S, A) as above will be referred to aspolarized semigroups.

For a polarized semigroup (5, A), one has the property that m G 5 is a sumof i > 0 elements of Si if and only if one has A(m) = i. This property has twoimmediate consequences. First, the set Si (and hence any fiber X ~ 1 ( i ) ) is finite, as5*1 is nothing but the set of irreducible elements in S. Second, the semigroup S is,a fortiori, positive. For the last statement, notice that to prove positiveness, whena map A as above already exists, one only needs to check A~ x (0) = 0 which followsfrom the afore mentioned property.

Now, assume that 5 is torsion free. Then, since k[S] is a domain, the projectivealgebraic scheme Z is, in fact, a projective algebraic variety.

PROPOSITION 6.1 Let (S, A) be a polarized semigroup such that S is torsionfree. Then Z = Proj(k[S}) is a projective toric variety.

In fact, since S is torsion free, it can be viewed as a subset of VQ. On the otherhand, the map A extends to a group homomorphism AZ : G(S) —> Z and to anR-linear map AR : VR —» R, where VR = G(S) ®z R» Now, let QI be the convexhull of the set Si in VR, and let S° C Si be the vertex set of Sx. Notice, that S°,Siand QI lie in the affine hyperplane in VR given by A^^l).

Fix m° € S°. Then, the semigroup S(m°) generated by the set of elements oftype TO — TO° with m G Si is a new positive finitely generated semigroup whoseassociated group is A^^O). In particular, it follows that the dimension of the affinetoric variety X(m°) given by S(m°) is d— 1 where d = rankG(S), i.e. the dimensionof the projective variety Z. Moreover, X — Spec(k[S}) being the projecting coneof Z, the construction shows that the affine toric varieties X(m°), when TO° rangesover Sf, form a covering of Z as affine charts, making Z into a projective toricvariety. This shows the proposition.

For projective normal varieties it is possible to describe which Cartier divisorsare ample and very ample ones. By a polarization of a projective variety one meanspicking a very ample Cartier divisor class. It provides an embedding of the varietyin a projective space. When the variety is toric, one sees that the polarizationproduces a polarized semigroup (S, A) in such a way that the variety is isomorphicto the one given by the couple (S, A). See [11] for details.

Thus, it is equivalent to give an embedded projective toric variety and to give apolarized semigroup. Notice that, for a given polarized semigroup (S, A), the set Siis the set of irreducible elements of S, so it is the only generator set contained in Siwhich gives the embedding of the affine toric variety X = Spec(k[S\) which is theprojecting cone of Z. Thus, a polarized semigroup provides a canonical embeddingof the projective toric variety into ph~l where h is the cardinality of Si.

We remark that the fan giving rise to the projective variety Z lies in the dualspace of the hyperplane Aq^O). Namely, the cones of the fan are exactly the dualsof the cones generated by the semigroups S(m°). By construction, it is easy to seethat such a fan is a complete one which corresponds to the algebraic geometric factthat any projective variety is complete.

Finally, as it occurs for affine toric varieties, the main algebraic geometric char-acteristics of projective toric varieties are recognized in terms of the polarized semi-group (S, A). Thus, Z = Proj(k[S}) is said to be arithmetically Cohen-Macaulay


(resp, Gorenstein) if and only if the algebra k[S] is Cohen-Macaulay (resp. Goren-stein). In the same way, Z is projectively normal if and only if k[S] is normal, i.e.,if the semigroup 5 is normal. Finally the variety Z is normal (resp. regular) if andonly if each semigroup S(m°) is normal (resp. free).

Notice that to be projectively normal means that Si = 5,, where Si = A~1(z)and Si = C(S) n X ~ 1 ( i ) , i.e. if every element in 5, is a sum of i elements of Si forall i's. Normalness can be characterized in rather similar terms, using the Ehrhartand Hilbert functions. The Ehrhart (resp. Hilbert) function is the map E (resp.H): N —> N given by E(i) = card(Si) (resp. H(i) = card(Si)), which coincideswith a polynomial map of degree d—1 with coefficients in Q for i big enough. Then,under the most general conditions, the leading terms of the polynomials for E andH are equal, and the variety Z is normal exactly when both polynomials are equal.Obviously, in those terms, projective normalness is characterized by the propertyE = H.

7 POLYTOPES, SIMPLICIAL AND CELLULARCOMPLEXES

Once one has an embedded affine or projective toric variety, one looks at describingand computing, when possible, equations and syzygies for the embedding. Mostresults in this direction are recent and they use combinatorial objects such as sim-plicial and cellular complexes or polytopes. Note, from section 5, that the projectivecase is reduced to the affine one, as for a given polarized semigroup, the equations(and syzygies) of the embedded projective variety it defines are the same as theequations (and syzygies) for its projecting cone affine variety. Such an affine varietyis nothing but the (affine) toric variety given by the semigroup S of the polarizationwith Si as chosen system of generators.

Along this section, we will assume that a map TTQ : Nh —> S, as in section 1, isfixed, and that 5 is a positive semigroup. Denote by A the generator system of Sgiven by TTQ, by II the set of primitive elements of the N^-module M = N'1 + L,and, for every m £ S, by Tm the set of monomials of 5-degree equal to m. Noticethat the set Tm can be identified with the fiber iiQl(m). Recall that the fact thatS is positive implies that M is generated by II and that each TTO is finite. Then,there are several combinatorial objects with vertex set one of A, II or Tm which arenaturally associated to TTQ as described below.

Associated to any fixed element TO in S one has the simplicial complexes ATO,Qm and the polytope Qm defined, respectively as follows. First, Am is the simplicialsubcomplex of parts F of A such that m — np € 5", where np = X^eF n- Second,@m is the simplicial subcomplex of parts G of Tm such that all the monomials of Ghave a non unit greatest common divisor (i.e. those monomials share at least onevariable). Third, fim is the polytope in VR = Zh ®z R- given by the convex hull ofthe set Tm = 7r^1(rn).

Notice that on the set S one can consider an ordering ^ defined by TO' ^ m ifand only if TO — m' € 51, and that, if TO' ;< TO then one has Am/ C Am, 0m/ times amonomial of degree TO — m! is a subcomplex of Qm and, finally, the translation offlm/ by any vector in the fiber i^Ql(m — m!) is a subset of f2m.

Associated to the whole of 5, one has two useful regular cellular subcomplexesof parts of II. Namely, on one side one has the so-called Taylor complex H which

is nothing but the (simplicial) complex of all parts of II, and, on the other hand,the hull complex S which is the subcomplex whose faces are the subsets of II whichcorrespond with some unbounded face of the convex hull of the set of points ofVR. of type ta = (tai,...,tah) for a = (ai,...,a^) € M where t is any big enoughreal number. The mentioned correspondence is the obvious one taking into accountthat any vertex of the above convex hull is necessarily one of type tb with b e II.See [2] for details on the construction and properties of the hull cellular complex.Sometimes, a subcomplex of E, the so-called Scarf complex is considered. It is, infact, a simplicial complex which is defined to be the set of parts H of II satisfyingthe property a# ^ a/// for every H' ^ H where, a# stands for the supremum ofthe elements in H for the ordering < of section 2. The hull and the Scarf complexescoincide when the data TTQ is generic, i.e. when the congruence F can be generated bycouples (u, v) such that the unions of the supports of u and v is the set {1, 2 , . . . , h}.

In the sequel, we will often use reduced homology with values in the field k forsimplicial and cellular complexes. The corresponding z'-th reduced homology vectorspaces will be denoted by Hi.

The description of equations has to do, in practice, with the determination ofsets of binomial generators of the semigroup ideal / (section 3) which are either aminimal set of generators or a Grobner basis. For each monomial ordering (i.e. atotal order on the set of monomials for which the monomial 1 is the minimum andwhich is closed under multiplication by constant monomials) one has a well dennedreduced Grobner basis with respect to such an ordering, which also happens tobe generated by binomials (see [22] for details). Thus, each such reduced Grobnerbasis can be understood either as a subset of the congruence F or of the lattice L(sections 1 and 2). The union of reduced Grobner bases for all the possible monomialorderings is called the universal Grobner basis, and it has the property of being,simultaneously, a Grobner basis for all monomial orderings. Again the universalGrobner basis can be seen as a subset of F or L. A reduced Grobner basis withrespect to a concrete ordering can be computed from any other generator systemby means of the well known Buchberger algorithm. The description of the universalGrobner basis becomes more difficult and it will be stated precisely just below.

To find the universal Grobner basis, consider the subset U of S consisting ofthose elements m G S such that the polytope fim has an edge which is not parallelto some edge of some fim/ for some TO' -< m. Then, for each m 6 U consider thebinomials of type Xu — Xv, where the coordinates of u — v are relatively prime andthe segment [u,v] is an edge of £2m. A result by Sturmfels, Weismantel and Ziegler[21] shows that the set of all binomials one obtains in this way when m ranges overU is exactly the universal Grobner basis of /. Such universal basis is finite as onecan see that it is contained into the so-called Graver basis which is itself finite. TheGraver basis consists of the binomials corresponding to the primitive elements ofthe lattice L, i.e. those elements 1 = 1+ — 1~ in L for which there are no other1' = 1'+ - I'" in L such that 1 ̂ 1' and 1'+ < 1+ and l'~ < 1~.

To find minimal sets of homogeneous generators of / one can proceed as follows.Consider the set C of elements m £ S such that Ho(Qm) ^ 0, i.e. those elements forwhich the complex Qm is not connected. The set C is finite. For each TO 6 C picka monomial Xu in each connected component of Qm and distinguish the monomialXv picked for one concrete component. Then the binomials Xu — Xv, where Xu

ranges over the picked monomials for the other components, are the degree m terms


of a minimal system of homogeneous generators of /. Thus, when TO ranges over theset C, the whole set of obtained binomials Xu — Xv is a minimal set of homogeneousgenerators for the ideal.

A different way to find homogeneous generators for /, which involves the com-plexes Am, is also available for higher order syzygies, and it will be discussed nextin this higher order context. We do not know if such discussion could also be rea-sonably done in terms of the complexes 0m as well as of the complexes Am orwhether Qm could be used to describe the universal Grobner basis.

The description of syzygies consists in obtaining either the minimal 5-gradedresolution (section 3) or concrete resolutions with other special properties, for ex-ample, the property of preserving the symmetries relative to the action of the latticeL.

With notations as in section 3, the z-th order syzygy module is the 5-gradedmodule A^ = ker((fi). Notice that one has A^o = /. For each degree TO G S, thenumber of generators of degree m in any minimal set of generators for Ni is, byNakayama's lemma, the dimension of the /c-vector space Vi(m) = (Ni}m/(MANi)m.A first and key connection between syzygies and toric geometry is a result dueinitially to Hochster, [15], and considered again by several authors in [7], [1], [5],which asserts that one has an explicit and natural vector space identification of type

V-(m) = ffi(Am),

where Hi stands for the reduced simplicial homology with coefficients in the field k.Moreover, computations of direct and inverse images by the isomorphisms givingrise to the above identification are available.

This result illustrates how combinatorics play a natural role also for describingsyzygies, and, therefore, how one has many reasons to include combinatorics amongtoric mathematics. The first direct applications of the above result are given by Bri-ales, Campillo, Marijuan and Pison in [4] to give an effective algorithm to computeminimal systems of binomial generators of the ideal /.

To apply for i > I the above natural isomorphisms, the main difficulties whicharise are first to compute those values of m such that ffj(Am) is nonzero, and,second, to determine the homology. If one is able to avoid these two difficultiesin concrete cases, then from the fact that the isomorphisms are explicit, one canderive successive methodic constructions of minimal sets of generators for the syzygymodules in the minimal resolution of R (see [5] for details).

To approach the first difficulty, we will mention that, recently, Briales, Pison andVigneron [6] ( [19] for the case i = 1) determine appropriate finite subsets Cj of Swith the property that m $. Ci implies //j(Am) ^ 0. As a consequence, they obtainan algorithm for computing the minimal resolution, (see [6] for details), becausethe second difficulty is quite well understood from a computational viewpoint, asconcrete homologies can be calculated by means of linear algebra and integer linearprogramming as pointed out in [18], [19] and [6].

However, integer programming being also a technique to whose developmenttoric geometry also is contributing (as we will show at the end of the paper), itis convenient to try to better understand the explicit structure of the homologiesH_(Am). This is treated by Campillo and Gimenez in [8]. For it, one considers apartition A = £ U C, where £ is a subset of generators whose image in Vq generatesminimally the cone C(S), in the sense that, for each edge of C(S), £ contains exactly


one element whose image generates such an edge. Notice, that one has e = card(£),e being nothing but the invariant of S in section 4. Algebraically, one deduces thatk[S] becomes a finite extension of k[£]. Thus, the minimal graded resolution ofk[S] as an ^4-module can be compared with its minimal resolution as a B-module,where, now, B = fc[Ne] corresponds, as in section 1, to the semigroup generated bythe set £.

This situation puts in evidence two kinds of objects. First, one has the Aperyset relative to £, which is nothing but the set Q of elements q 6 S such thatq — n $_ S for every n E £. In other words, the Apery set is nothing but the set ofexponents whose corresponding symbols generate minimally k[S] as a fc[£]-module,and, therefore, it is a finite set. Second, for each m € S one has the analog of Amfor this relative situation, namely the simplicial subcomplex Tm of parts J of £ suchthat m — nj e S. Thus, one can see that the dimension of Hi(Tm) is exactly thenumber of degree m elements in a minimal set of £ -homogeneous generators of thez-th-order syzygy module in the above minimal resolution of the S-module k[S].

Now, for a fixed m € S, one has a key long exact sequence of type

. . . -> Hi+l(Qm) -^Ki^ ffi(Am) -> Hi(Qm) -+#<_! ̂ ...

where H.(Qm) and K, are appropriated vector spaces of the following nature. First,H.(Qm) is the homology of a complex associated to the vertex m of a graph QQwith coloured edges constructed from the knowledge of Q, which has C as colour setand which is called the Apery graph. The vertex set of QQ consists of the elementsm of type q + nj where q & Q and I C C. Edges of colour n £ C join a vertexm1 to another m whenever m — m' = n. The complex associated to m has as z'-thchain space the one freely generated by the subsets / C C of cardinality i + I suchthat TO — n/ E. Q, the boundary map being the projection of the usual simplicialboundary. Second, the spaces K. are much more difficult to describe and we avoidthe details. However, they can be computed in successive steps in two different andcomplementary ways. One, in terms of new graphs of exactly the same type thanQQ but with other concrete sets instead of Q. Another in terms of homologies oftype H.(Tmi) where the elements m' are of type TO — n/ with / C C. See [8] for thedetails and some applications.

An extra consequence of the construction of the complexes Tm is that one hasa way to characterize the depth r of the ring k[S}. Recall that the three integers r,d and e associated to a positive semigroup are such that r < d < e. The integersd and e are easily obtained from 5. To obtain r, in [8] it is proved that if TOis an integer with 1 < r$ < d, then the inequality r > TO is equivalent to the factHe-ro(Tm) = 0 for every m & S. In particular, for ro = d one gets a characterizationof the Cohen-Macaulay property by the property

He-d(Tm) = 0

for every m € S, which, for the simplicial case e = d means that all the complexesTm are connected. From this, it is easy to recover the well known characterizationdue to Goto [12] with asserts that, for simplicial semigroups, the Cohen-Macaulayproperty is equivalent to the property

TO € G(S),n, n' e £,n ^ ri ,m + n e S,m + n' 6 S1 => TO e S.


Other characterizations of the Cohen-Macaulay property for nonsimplicial cases aregiven in [23] and [20].

A standard application which illustrates the use of the technique of the abovelong exact sequences is to the case of simplicial Cohen-Macaulay semigroups (i.e.those for which r = d = e). Since, in that case, the complexes Tm are connected,one can deduce that Ki = 0 for every i, so that one gets

for every m and i. Thus, the minimal resolution for simplicial Cohen-Macaulaysemigroups can be derived from a unique combinatorial object, the Apery graph.

For the general case, there are other ways to derive free resolutions for R from aunique combinatorial object. Namely, as shown in [2] this can be done either fromthe Taylor or from the hull complexes, H and E respectively.

Let us explain how this works. For it, consider for each of the above cellularcomplexes an associated complex of ^-modules given as follows. The z-th orderchains are the elements of the free A-module generated by the i-dimensional facesof the considered cellular complex, and the boundary map is given on any such face# by

y^e(H,H') —— H'V a"'

where the sum ranges over all the faces H' of the considered cellular complex,e(H,H') € {0,1,—!} denotes the incidence index for the cellular complex, and&H •, a/H are the elements defined above. Recall that, from the definition of regularcellular complexes, the incidence index satisfies the properties e(H, H') — 0 unlessH' is a facet of H, so the above sum is extended only to facets of H in the cellularcomplex.

Because of the properties of the Taylor and hull complexes, one has that whatone actually gets are free A-module resolutions of k[M] = k[Nh + L}. Moreover, theresolution, which is Z^-graded by construction, is in fact invariant by the action ofthe lattice L induced from its action on II. This means, that each one of the chainof A-modules is in fact also a free .AfLj-module, where A[L] is the algebra of thegroup L on the coefficient ring A. Notice that one has A[L] = k[Nh x L], so thefirst projection Nft x L — > N^, seen as an N^-module homomorphism, gives riseto a surjective A-linear map A[L] — > A. Now, by extending scalars via the mapA [L] -— > A and taking into account that

k[M] ®A[L] A = k[S] = R,

one gets a complex which is in fact S-graded and exact. Thus, according to theconsidered cellular complex, one gets two S'-graded resolutions of R, which arerespectively called the Taylor and the hull resolution.

Both resolutions are, in general, far from being minimal; however, if the data TTQis generic, then the hull resolution is (isomorphic) to the minimal one. However, theyare interesting and useful since they keep the action of the lattice L. Notice that,as commented before, the hull complex is equal to the Scarf complex for the genericcase, so, in that situation the Scarf complex can be directly used instead of the hullone for constructing the above resolution, which is minimal besides. Nevertheless,we remark that the generic case is combinatorially characterized by the fact that



the simplicial complexes Am which are not connected have connected componentswhich are full simplices. This is a strong assumption from the combinatorial pointof view, so, in general, if one wants to know about the minimal resolution the onlyavailable description (by the moment) is that discussed before, based on the studyof the simplicial complexes Am.

In the general (non generic) situation the hull resolution has, nevertheless, an-other nice property, as in fact it is a finite one, i.e. the free ^-modules involved areor finite rank and the number of them is finite. This is a non obvious statementwhich follows from the fact that the Graver basis is finite. See [2] for details.

8 MULTINUMERICAL SEMIGROUPS

In practice, the toric data TTQ (of a semigroup with a given generator set) is oftengiven in arithmetical terms. In fact, the group G(S) being finitely generated, it isnothing but, up to isomorphism, one of type

Zd x Z/giZ x ... x Z/giZ

for convenient integers d,l,gi,.. • ,gi-Thus, if such an isomorphism is, a priori, considered, then TTQ becomes equivalent

to the specification of the coordinate (d + /)-tuples (in the above product group) ofthe generators HI ,..., n^ of S. A semigroup given by such a specification is called amultinumerical semigroup. For the simplest case d = 1 and / = 0, they are usuallyreferred to as numerical semigroups in the literature.

What one would need, therefore, is to study toric varieties within arithmeticsfrom multinumerical semigroups. This means to deduce the behaviour and geomet-rical properties of those varieties from arithmetic properties of the (d + Z)-tuples ofintegers or modular integers given by the semigroup generators.

Such an arithmetical study becomes, nevertheless, difficult and it is an openproblem except in rather few cases. The difficulties arising can be explained if onelooks at the discussion in the above section on how combinatorics are involved in thedevelopment of toric geometry. In fact, using objects such as polytopes or simplicialor cellular complexes avoids having to deal with delicate relations among numbers.

However, mathematically speaking, once that combinatorial methods have grownup and produced nice results, one can hope and try to interpret them in the frame-work of arithmetics. This strategy is used in [8] for affine and projective toric curvesand in [5] for affine and simplicial projective toric surfaces. For the general case,good computational results dealing with equations are also derived by Vigneron in[18]. To show the possibilities of the above strategy, we will discuss, here, suchresults for curves.

An affine toric curve is given by the numerical semigroup S given by a setA of h nonnegative integers. One has r = d = 1 and, since the cone C(S) hasonly one edge, also e = 1. Thus, this case is a simplicial Cohen-Macaulay one.Then, pick a partition of A in a set £ consisting of any single element s € A andas complementary set C the set of the h — 1 remaining elements. Now, considerthe Apery set Q consisting of those integers q & S such that q — s £ 5, andfrom it construct the coloured graph QQ. It is not difficult to translate the graphstructure into arithmetical relations, so that the homologies //j(Am) = Hi(Qm) for


the vertices m of QQ can be derived from such relations. One concludes that theminimal resolution for affine toric varieties can be obtained in complete arithmeticalterms from the set of generators of the given numerical semigroup. See [8] for details.

A projective toric curve of degree s is given by a subsemigroup of N2 generatedby a set A = £ U C, where £ is the set consisting of the two elements (s, 0) and(0, s) and C consists of elements (GI, s — c\),..., (c^_2, s — 0^-2) for different valuesCj with 0 < Cj < s. The semigroup S can be polarized by the function A given by\(c, c') — (c + c')/s, so that 5 defines an embedding of the projective toric curve inp/i-i_ Notice that one has d = e = 2 and that either r = 2 or r = 1, depending onthe projective curve to be or not to be arithmetically Cohen-Macaulay.

Let Si be the numerical semigroup generated by GI, . . . ,Q l_2,s, and for eachc 6 Si denote by /z(c) the least number of the above generators of Si needed toachieve the sum c. Notice that the function fj, satisfies the property /j,(c) < /z(c—s) + lfor every c e S whenever c — s S S. By translating into arithmetics the methodsin [8], in [9] it is shown that the projective toric curve is arithmetically Cohen-Macaulay if and only if one has /i(c) = n(c — s) + 1 for every s e Si such thatc - s e S.

In general, from the knowledge of the function fj, it is easy to find the Apery setQ relative to the above partition A = £ U C as well as the set D consisting of thoseelements m in S such that m — (s, 0) € S, m — (0, s) e S, m — (s, s) ^ S. One canconsider a coloured graph Q-p in an identical way as QQ but replacing Q by D. In[8] it is shown that the vector space Ki in the long exact sequence of the previoussection can be identified with the homology Hi(Dm], where this last homology hasalso an identical construction than that for the case of the set Q. Thus, one deducesthe long exact sequence

. . . -> Hi+1(Qm) -> Hi(Dm) -> tfz(Am) -> Hi(Qm} - > . . . .

The involved homologies as well as the image maps in this exact sequence can begiven in aritmetical terms from the given data s, GI , ..., c^-i- From here, this is so forthe reduced homologies H. (Am), therefore, the minimal resolution of the projectivetoric curve is obtained from arithmetics.

9 APPLICATIONS

The development of toric geometry has provided applications to many problems ingeometry. This is related to the fact that, quite often, toric varieties are objects onwhich one can determine and describe the main ingredients involved in the consid-ered problems. Applications also occur to some problems external to geometry andalgebra, in such a way that, toric geometry is becoming also an interesting topicof applied mathematics. Those external applications are mainly related to appliedcombinatorics or to applied optimization. We will end this paper by illustratingthis situation with two examples of current research.

The first one is the coin exchange problem, a classical problem of appliedcombinatorics. The approach and results are recently obtained by Campillo andRevilla in the paper [9]. Assume one has a coin system with coins of valuesc\ < c-2 < ... < Ch-i- Then, setting s = c^-i one has a projective toric curveZ, namely that of degree s given by the (polarized) subsemigroup of N2 generatedby the elements (0, s), (d, s - c i ) , . . . , (ch-i, s - ch-i) = (s,0).

The exchange problem aims at achieving a value c in the semigroup S\ generatedby the coin values in an appropriate way. One wants, in particular, to achieve thevalue c with the minimum number of coins, namely the integer p,(c) introducedin the above section. The problem can, therefore, be formulated as to give goodways or algorithms to achieve the value c with /z(c) coins in practice. Commentsin the above section show how this problem is mathematically close to that of thedetermination of equations and syzygies for projective toric curves.

Usually considered coin systems have a strong property, namely that for themthe greedy algorithm to achieve the values c with //(c) coins works. The greedyalgorithm achieves a value c e Si by first taking the largest coin Cj such that GJ < cand, then, restart with the value c — Cj and continue in the same way. If, for everyc £ Si, the greedy algorithm uses fj,(c) coins then one says that it works for thesystem. From the discussion at the end of the last section, one deduces that if thegreedy algorithm works then Z should be arithmetically Cohen-Macaulay.

From this, one shows how toric geometry yields an interesting new class of coinsystems with nice properties, namely the Cohen-Macaulay ones, i.e. those suchthat the associated projective toric curve Z is arithmetically Cohen-Macaulay. Forthem, in general, the greedy algorithm to achieve values with a minimal number ofcoins is not available, but one has an alternative new and good algorithm to do so(see [9] for details).

The second application is to integer linear programming, also a classical prob-lem, this time of applied optimization. Integer linear programming is related tomultinumerical subsemigroups of Zd, which, for the sake of simplicity, will be as-sumed to be positive. Let S be such a subsemigroup and assume that it is generatedby the elements HI, . . . , n^ 6 Zd. The integer linear programming problem consistsin finding the optimal solution with non negative integral coordinates to one of type

which minimizes a linear map (the cost map)

Here, the coefficients p^ are real numbers in general.An integer linear program can be seen, therefore, as the specification of type

(TTQ, p) where TTQ is the data of a semigroup and generators as above and p the costfunction. For each m & S the solutions of the integer linear programming problemfor ?n are among the elements in the fiber TTQl(m) and, moreover, among the verticesof the polytope Om.

Now, notice that, once one fixes any monomial ordering on the variables xi, . . ,,Xh (for instance the reverse lexicographic ordering), the cost function gives rise toanother monomial ordering by comparing two monomials, first, by the value of pon the exponents and, second, in case of equal values of p by the previous fixedordering (i.e. the weighted ordering corresponding to the above one).

Then, one can prove that the reduced Grobner basis of the ideal / given by TTQrelative to this new ordering provides a minimal test set for the integer programmingas described in the sequel.

In fact, the reduced Grobner basis is generated by binomials, therefore, it canbe viewed as a subset Up of the lattice L. On the other hand, one has the propertythat if x = (xi,..., Xh) € Nh is in a fiber and 1 <E L (i.e. a feasible solution) thenx — 1 is again a feasible solution whenever x — 1 g Nfe. Then, the set Up is a test setfor the program as it satisfies the following two conditions. First, if x is a feasiblesolution which is not optimal, then there exists 1 € Up such that x — 1 is also afeasible solution. Second, if x is an optimal solution to a program, then x — 1 isnot a feasible solution for every 1 G Up. The condition on the Grobner basis tobe reduced implies that Up is minimal among the subsets satisfying the above twoconditions. Test sets provide nice algorithms, in the obvious way suggested by bothconditions, to solve the integer linear programming problem.

Non reduced Grobner bases provide non minimal test sets. In particular, theset U giving the universal Grobner basis in section 6, which is finite and the unionof all Up for all cost functions, is a test set for all programs when p varies, i.e. itis a data which only depends on TTQ. See [16] and [17] for details. For algorithmsinvolving cases of non positive semigroups see [3], or consider the Lawrence lifting(see for example [22]).

REFERENCES

[1] A. ARAMOVA, J. HERZOG, Free resolution and Koszul homology. J. of Pureand Applied Algebra, 105 (1995), 1-16.

[2] D. BAYER, B. STURMFELS, Cellular resolutions of monomial modules.J.reine angew. Math. (1998) 502, 123-140.

[3] F. DI BIASE, R. URB ANKE, An Algorithm to Calculate the Kernel of CertainPolynomial Ring Homomorphisms, Experimental Mathematics, Vol.4, No. 3(1995), 227-234.

[4] E. BRIALES, A. CAMPILLO, C. MARIJUAN, P. PISON, Minimal Systemsof Generators for Ideals of Semigroups, J. of Pure and Applied Algebra, 124(1998), 7-30.

[5] E. BRIALES, A. CAMPILLO, C. MARIJUAN, P. PISON, Combinatorics ofsyzygies for semigroup algebras. Collet. Math. 49 (1998), 239-256.

[6] E. BRIALES, P. PISON, A.VIGNERON, The regularity of a Toric VarietyPreprint, University of Seville (1999).

[7] A. CAMPILLO, C. MARIJUAN, Higher relations for a numerical semigroup.Sem. Theor. Nombres Bordeaux 3 (1991), 249-260.

[8] A. CAMPILLO, P. GIMENEZ, Syzygies of affine toric varieties Journal ofAlgebra to appear.

[9] A. CAMPILLO, M. REVILLA Coin exchange algorithms and toric projectivecurves. Preprint (1999).

[10] M.P. CAVALIERE and G. NIESI, On monomial curves and Cohen-Macaulaytype Manuscripta Math. 42, (1983), 147-159.


[11] W. FULTON, Introduction to Toric Varieties, Princeton University Press(1993).

[12] S. GOTO, N. SUZUKI, K. WATANABE, On affine semigroup rings, Japan.J. Math. 2(1),(1976), 1-12.

[13] J. HERZOG, Generators and relations of semigroups and semigroup rings,Manuscripta Math. 3, (1970), 175-193.

[14] M. HOCHSTER, Rings of invariant of tori, Cohen-Macaulay rings generatedby monomials, and polytopes. Annals of Math. 96(2), (1972), 318-337.

[15] M. HOCHSTER, Cohen-Macaulay rings, combinatorics and simplicial comple-xes Lect. Notes in Pure and Appl. Math. Dekker 26(1977), 171-223.

[16] S. HOSTEN, B. STURMFELS, GRIN, An Implementation of Grobner Basesfor Integer Programming, In E. Balas and J. Clausen editors, Integer Program-ming and Combinatorial Optimization, LNCS 920, Springer-Verlag, (1995),267-276.

[17] S. HOSTEN, R. THOMAS Grobner basis and integer programming B. Buch-berger and F. Winkler (editors) Grobner Bases and Applications, Lect. NotesSeries 251, London Math. Soc., (1998), 144-158.

[18] P. PISON-CASARES, A. VIGNERON-TENORIO, N-solutions to linear sys-tems over Z. Preprint of University of Sevilla (1998).

[19] P. PISON-CASARES, A. VIGNERON-TENORIO First Syzygies of Toric Vari-eties and Diofantine Equations in Congruence To appear in Comm. in Algebra.

[20] U. SCHAFER, P. SCHENZEL Dualizing complexes of affine semigroup rings,Trans. A.M.S. 322(2),(1990), 561-582.

[21] B. STURMFELS, R. WEISMANTEL, G. ZIEGLER, Grobner basis of lat-tices, corner polyhedra and integer programming. Beitrage zur Algebra undGeometrie36(1995), 281-298.

[22] B. STURMFELS, Grobner Bases and Convex Polytopes, AMS UniversityLectures Series, Vol. 8 (1995).

[23] N.V. TRUNG, L.T. HOA, Affine semigroups and Cohen-Macaulay rings gen-erated by monomials. Trans. A.M.S. 298(1), (1986), 145-167.

[24] A. VIGNERON-TENORIO, Semigroup Ideals and Linear Diophantine Equa-tions. Linear Algebra and its Applications, 295 (1999), 133-144 •


Canonical Forms for Linear Dynamical Systems overCommutative Rings: The Local Case

M. CARRIEGOS, Departamento de Matematicas, Universidad de Leon. 24071-Leon. Spain.E-mail: demmcv@unileon. es

T. SANCHEZ-GIRALDA, Departamento de Algebra, Geometrfa y Topologia, Uni-versidad de Valladolid. 47005-Valladolid. Spain.1E-mail:sanchezgiral@cpd. uva. es

Abstract

In this paper we survey some results on the Brunovsky canonical form for thefeedback equivalence of reachable linear dynamical systems over commutative rings.Besides the general definitions and well known facts we include some very recentresults: A normal form for reachable linear dynamical systems over a local ringR is given. This normal form is a canonical form in the case of 2-dimensionalsystems. Finally, the case of discrete valuation domain R is specially treated:We obtain a complete set of invariants and a canonical form for a reachable n-dimensional linear dynamical system E with the property that the invariant R-modules M-f, Mj% ..., M^ are free except at most one of them.

1 INTRODUCTION

The importance of the feedback action in Control Theory is well known: The mod-ification of a dynamical system to achieve some desired behavior, since for exampleits stabilization, is very important by its applications.

On the other hand, several authors have worked in the last years to extend linearcontrol results to systems defined over a commutative ring R. Morse's paper [17]is one of the main initial works on the control of systems over rings and containsa constructive proof that reachability implies pole assignability for multi-input sys-tems defined over a polynomial ring in a single indeterminate with coefficients in


114 Carriegos and Sanchez-Giralda

a field. Actually, the theory of systems over rings is well developed and of interestas an active and rich area of research. The influence in this area of R.E. Kalmanfrom 1970 has been essential. We would like to call the reader's attention to [20] asa general reference in control theory and [3] and [5] as references in linear controltheory from an algebraic point of view.

When R is a field, the classification of reachable linear dynamical systems forthe feedback action is due to P. A. Brunovsky [4], R.E. Kalman [14], and W.A.Wonham and A.S. Morse [23]. In this case, the feedback equivalence class of areachable linear dynamical system E = (A, B) is characterized by a finite set ofpositive integers k\, ...,ks which are called "the Kronecker indices of E = (A,B)",because they are identical to the classical indices associated with the pencil ofmatrices (zldn — A B)

In the general case the feedback classification problem is still open. Moreoverthe feedback classification problem is called "wild problem". Nevertheless, thereare several results which determine canonical or pseudo-canonical forms for thefeedback action of linear dynamical systems over special rings.

This paper is organized as follows: In section 2 we review the feedback group,the feedback equivalence of linear dynamical systems over a commutative ring R,and the feedback invariant R- modules Np and Mf associated to E, where i is apositive integer lower than the dimension of the system E.

Section 3 is devoted to study well known facts about feedback equivalence forsystems over a field, such as the Brunovsky canonical form. For a commutative ringR, we show that every reachable linear dynamical system over R is equivalent toa Brunovsky canonical form if and only if .R is a field. Consequently, Brunovsky'sclassification is a complete classification of reachable linear dynamical systems overR if and only if .R is a field.

In section 4, we review the notion of change of scalars in a linear dynamicalsystem and we introduce some new results: A normal form is given for reachablelinear dynamical systems over a local ring. This normal form becomes a canonicalform in the case of m-input 2-dimensional reachable linear dynamical systems overa local domain, hence a feedback classification for those systems is given. Thecase of m-input, rx-dimensional reachable linear dynamical systems over a discretevaluation domain is also studied, and we obtain a canonical form for reachablesystems E with the property that all invariant .R-rnodules Mp are free except atmost one of them.

2 LINEAR DYNAMICAL SYSTEMS OVERCOMMUTATIVE RINGS: THE FEEDBACK GROUP

Let R be a commutative ring with identity element. An m-input n-dimensionallinear dynamical system E over R is a pair of matrices (A, B), where A is an n x nmatrix and B is an n x m matrix with elements on R. The system E = (A, B) issaid to be reachable if the columns of the n x mn matrix

A*B= (

which is called the reachability matrix of the system E, generate Rn.

Let X be a p x q matrix with entries in R. We denote by Uj(X) the j-tii


Linear Dynamical Systems over Commutative Rings 115

determinantal ideal of X, that is, the ideal generated by all j x j-minors of thematrix X. Note that the rn-input, n-dimensional system E = (A, B) is reachable ifand only \iUn (A*B) equals R (see [3, Theorem 2.3.] for details).

The feedback group Yn,m is the group, acting on m-input, n -dimensional lineardynamical systems E = (A, B) generated by the following three types of transfor-mations:

1. A i—> A' = PAP~l ; B i—> B' = PB for some invertible matrix P. Thistransformation is consequence of a change of base in Rn, which is called thestate module.

2. A i—> A' = A ; B i—> B' = BQ for some invertible matrix Q. This trans-formation is consequence of a change of base in Rm, which is called the inputmodule.

3. A i—> A1 = A + BF ; B i—> B' = B for some mxn matrix F, which is calleda feedback matrix.

The system E' is feedback equivalent to E if it is obtained from E by an elementof Fn>TO.

Let E = (A, B) be a linear dynamical system of size (m, n) (i.e. m -inputn-dimensional) over R. For i — l,...n we denote by Np the submodule of Rn

generated by the columns of the (n x i- m)-matrix (jE?|AB| • • • \Al~lB^. We denotealso by (S|>1S| • • • \Al~lB] the homomorphism of R -modules

(B\AB\ • • • |Ai-1B) : R*'"1 -> Rn

denned by the matrix (B\AB\ • • • lA^B). We denote by Mp, for each i = 1, ...n,the quotient .R-module Mp — Rn/Np. Then note that by the Cayley-Hamiltontheorem, the jR-modules Mp and Mp are equal for each i = n + l,n + 2,...

Recall that the linear dynamical system E is reachable if and only if Np = Rn.Then note that E is reachable if and only if Mp = 0 for each i = n, n + 1,...

LEMMA 2.1 Let E = (A,B) and E' = (A',B') be two feedback equivalent lineardynamical systems of size (m, n) over a commutative ring R. Then we have thefollowing properties.

1. Np is isomorphic to Np for each 1 < i < n.

2. Mp is isomorphic to Mp for each 1 < i < n.

PROOF.- (See [12, Lemma 2.1.]) D

REMARK 2.2 Note that by the previous Lemma one has that the sets:

and

are two sets of invariants associated to the linear dynamical system £ by the actionof the feedback group. The following example shows that the sets of invariants{./V,p}. ,. , and {Mp\ . ,. , are not sufficient to state the class of £ by the action*• ' l<i<« *• l ' l<i<n J

of the feedback group Fmtn(R).

Let R = TL be the ring of integers and set A =

0 0

0 03 0

and B = 1 00 5

Put A' = and B' = B and consider the linear dynamical systems £ =

We have:

N? = (B\AB) = Im

N? = (B\A'B) = Im

0 5 3 0

1 0 0 00 5 4 0

By [13, Theorem 2.4], the system E is not feedback equivalent to E' because thefollowing congruence

3 = 4-u/i2 (mod5)has no solution lih^TL and u is a unit in TL.

3 CANONICAL FORM FOR SYSTEMS OVER FIELDS

First, we recall a classical result due to P.A. Brunovsky [4].

THEOREM 3.1 Let E be an m-input, n-dimensional reachable linear dynamicalsystem over a field k. Then there exists a finite set of positive integers k\ > k% >... > ks > 0 uniquely determined by E with X/i=i k^ = n such that E is feedbackequivalent to Ec = (Ac, Bc) where Ac and Bc are described below:

/

Ac =

where Ei is the k; x L- matrix

00

0 \0

Es

( °0

0V °

1 0

0 1

0 • • •0 • • •

... o \'•. o

' • - 10 0 /

Linear Dynamical Systems over Commutative Rings

and

Br =

117

/ e,0

0 •62 •

• 0• 0

0 •0 •

• ° \• 0

\ 0 0 ••• es 0 ••• 0 /

where &i is the ki x 1 matrix

ei= ( 0 . . . 0 1 )*

The integers {fc;}*=1 are called the Kronecker indices o/E, and £c = (AC,BC)described above is called the Brunovsky canonical form associated to the Kroneckerindices {ki}*=l.

PROOF .- See [4], [9], [12] or [20]. D

REMARK 3.2 The set of positive integers {ki}1<i<s is a complete set of invari-ants associated to the system E for the feedback action. In fact, two reachablelinear dynamical systems E and E' (over a field k) are feedback equivalent if andonly if they have the same set of Kronecker indices.

REMARK 3.3 The Kalman controllability (reachability) decomposition (see [20,3.3.]) is extremely useful in giving simple proofs or facts about reachability forlinear dynamical systems over a field and provides a clasification of arbitrary lineardynamical systems over a field (see [7, Theorem 1.16.])

The following result shows that the set of increasing non-negative integers

or equivalently the set of decreasing non-negative integers

are two complete sets of invariants associated to E for the feedback equivalence.

PROPOSITION 3.4 Let k be afield and E = (A,B) a reachable linear dynamicalsystem of size (m,n) over R. We put of — dimk(Mp) for 1 < i < n. Then theset of decreasing non-negative integers {crf}l<i<n is a complete set of invariantsof the class of equivalence of E (i. e. E is feedback equivalent to E' if and only ifaf = af for all i = 1,2, .., n).

PROOF.- By Theorem 3.1 we can assume that E = (A, B) is a Brunovsky canonicalform. With the notations of that theorem, we have

/

V o

0 0 \0

E*es 0

Note that E^Ci = 0 for h > ki. Moreover, the intersection of the vector subspaceslm(AhB) and lm(Ah' B) of kn is {0} when h ̂ h'.

If the sequence of Kronecker indices associated to E is {ki, ...,ks} where

KS2 = • • • < KSt = • • • =

then the sequence {dim//?} is

s2s3s

ksskss + sikss + s2

KSS -j~ (feS

kss + • • • + (kst_l - kst_2) s t_i + st

kss + • • • + (kat_1 - fc s t_2) st-i + (kst - fc^J st

(The reader can see [8, Proposition 2.5.] for details of the calculation).Now, we conclude that we can obtain the Kronecker indices {ki}1<i<s from the

invariants {dim A^}1<i<n and since erf = dim^M?) = n — dimNp , it follows that

the set of Kronecker indices {ki}1<i<s of the linear dynamical system E = (A,B)can be obtained from the indices {<rf },,., . This proves that the set of indices{ ^ ) l<i<n ^{erf }l<i<n

are equivalent data to the set of Kronecker indices of the system S,which completes the proof. D

REMARK 3.5 Suppose that R is an arbitrary ring and S = (A, B) is an m-inputn-dimensional reachable linear dynamical system over R. In general, E is not feed-back equivalent to a Brunovsky canonical form Ec = (AC,BC). For example, in [13]it is shown that if R is a principal ideal domain and E is an TO -input, 2-dimensionalreachable linear dynamical system over R, then E is feedback equivalent to a systemE = (A, §} of the form

7 / 0 0 \ s / 1 0 • • • 0A = ( f o j ' s = ( o d . . . owhere <i and / are coprime (i.e. the ideal generated by d and / is the whole ringR).

REMARK 3.6 The above classification of reachable m-input 2-dimensional lineardynamical systems over a principal ideal domain R is generalized in [10] to a


clasification of m-input 2-dimensional reachable linear dynamical systems over thequotient ring R/ (pn) where R a principal ideal domain, p a nonzero prime of R andn a positive integer.

DEFINITION 3.7 Let S = (A,B) be an m-input n-dimensional reachable lineardynamical system over R. We say that S is a Brunovsky system if S is feedbackequivalent to a Brunovsky canonical form in the sense of Theorem 3.1.

The following result characterizes the Brunovsky systems over a ring R whichsatisfies that all finitely generated projective modules over R are free.

THEOREM 3.8 Let R be a commutative ring. Let S = (A,B) be a m-input n-dimensional reachable linear dynamical system over R. IfEisa Brunovsky system,then the R-module Mp is free for all i — 1,2, ...n.

Moreover, if every finitely generated projective R-modules is free, then the systemS is a Brunovsky system if and only if Mp is a free R-module for all i = 1,2, ...n

PROOF.- (See [12, Theorem 3.1.]) D

It is natural to define a Brunovsky ring as a commutative ring such that everyreachable linear dynamical system is equivalent to a Brunovsky canonical form.Note that for a Brunovsky ring the above theorem characterizes the feedback orbitof any reachable linear dynamical system. The following result proves that the classof Brunovsky rings equals the class of fields.

THEOREM 3.9 Let R be a commutative ring. The following are equivalent:

1. R is a Brunovsky ring.

2. R is a field.

PROOF.- Note that (2) => (1) is Brunovsky's Theorem (Theorem 3.1).Conversely assume that R is a Brunovsky ring. First, we claim that R is an

absolutely flat ring (see [1] or [2]). Let a be a finitely generated ideal of R.

a = (ai,...,am).

Consider the (m + l)-input 2-dimensional linear dynamical system £(a) given by

0 0 \ / 1 0 0 ... 0

0 0 0 ... 01 0 0 ... 0

Note that the reachability matrix of

A*B = 1 0 0 . . . 00 ai a-2 . . . ar,

verifies that the determinantal ideal U^ (A*B) is the whole ring R. ConsequentlyS(a) is reachable and therefore a Brunovsky system. Then, by the above Theorem3.8, the .R-module Mf'"' = R2/lm(B) is free.


Consider the finite free resolution of the ^-module M-p

Rm+l ^R2^ ME<-> _ 0

Since Mjp ° is free, it follows by [6] that the determinantal ideals associated tothe matrix B are principal ideals generated by an idempotent element of R. Inparticular, the ideal

Ui (B) = (a1,a2,...,am) = a,is a principal ideal generated by an idempotent and we have proven that everyfinitely generated ideal of R is principal generated by an idempotent. This is equiv-alent to R being an absolutely flat ring (see [1, ex. 27, p. 39])

Now let us prove by contradiction that R is a local ring. In fact, suppose thatthere exist two maximal m and m' ideals of R such that m ^ m'. Let x € m andx £ m' and consider the linear dynamical system over R given by

0 J ' \ 0 x

The system £^x) is reachable, so £(x) is feedback equivalent to a 2-input 2-dimen-sional Brunovsky canonical form. But note that the following two systems are theonly 2-input 2-dimensional Brunovsky canonical forms:

The form £{1,1} associated to the sequence of Kronecker indices

"-1 = "'I ~ 1) "'S = ^4 = ' ' ' — 0,

and given by0 0 \ / 10 0 ) ' \ 0

And the form £{2} associated to the sequence of Kronecker indices

k± = 2, A;2 = &3 = • • • = 0,

and given by0 1 \ / 0 00 0 J ' \ 1 0

An easy calculation shows that the matrix

1 00 x

\

is not equivalent to the matrix1 0

x 0 1neither to the matrix

0 01 0 J '

Then the system £^x^ is neither feedback equivalent to the system £{1,1} nor to thesystem £{2}- This is a contradiction and the ring R is local.

Collect the properties of R; that is, R is local and absolutely flat. This isequivalent to R being a field (see [1, ex. 28, p. 39]) D

4 DEALING WITH THE LOCAL CASE

In this section we study some invariants for linear dynamical systems over a localring. First note that if f : R —* R' is a ring homomorphism and £ = (A, B) is an m-input n-dimensional linear dynamical system over R, where A — (<%) and B = (6^);then one can construct the linear dynamical system /* (E) = ((/(«ij)) , ( / ( & f c z ) ) )over R' . The system /* (E) is called the extension of £ by change of scalars fromR to R' via /. Next we review some properties of extension of scalars in lineardynamical systems:

THEOREM 4.1 Let f : R — > R' be a ring homomorphism and let £ and £' be twolinear dynamical systems over R. Then, we have the following properties.

1. For each i = 1,2, ...n, there is a natural isomorphism

2. If £ and £' are feedback equivalents, then the systems /* (£) and /* (£') arefeedback equivalent.

PROOF.- See [8, Lemma 2.1] D

Now, let p be a prime ideal of R and consider the natural extension of scalarsfrom R to Rf given by

ip : R -> #pr i— > r/1 '

and the natural extension of scalars from R to R$/pRv given by

TTP : R ->

In the sequel, we use the notation Ep = ip (E) and £(p) = TT* (E).

The next definition follows naturally from the extension of scalars in a lineardynamical system (see [8]).

DEFINITION 4.2 Let R be a commutative ring, E and E' two m-input n-dimen-sional reachable linear dynamical systems over R. We say that E and E' are locally(resp. pointwise) feedback equivalent if and only if £p and Ep (resp. E(p) andE'(p),) are feedback equivalent for each prime ideal p of R.

The following is straightforward from Theorem 4.1:

E and E' are feedback equivalent^

E and E' are locally feedback equivalentU

E and E' are pointwise feedback equivalent

At this point, two natural questions arise:

• When do the reverse implications hold?

• Study of feedback equivalence, local feedback equivalence and pointwise feed-back equivalence.

The pointwise feedback relation for reachable linear dynamical systems is treatedin [8]. Now we introduce some original results in order to characterize the feedbackequivalence of reachable linear dynamical systems over a local ring and hence tostudy the local feedback equivalence.

The following proposition provides a normal form for reachable linear dynamicalsystems over a local ring.

PROPOSITION 4.3 Let (R,m) be a local ring and let S = (A,B) be a reach-able m-input, n-dimensional linear dynamical system over R. Then, S is feedbackequivalent to the system E^ = (Ah,Bh) where

Ah =

and in general, for each j = 2,..., s — 1

0R(2)Bh

0A(2)Ah

,Bh =0xl

A? =0 0

and finally0

where all the entries of the matrices X\,..., -X"s_i are in the maximal ideal m of R.Moreover one has the following identities relating the invariant R-modules

[Mf] Ki<n

to the positive integers {Ci}i

• Ifi<s, then one has:

d:mR/m (Mp ®R R/m) =n-

• On the other hand, if s < i < n, then Mp = (0) .

PROOF.- Let us prove the result by induction on n, the dimension of the system S.The case n = 1 is straightforward because every reachable m-input 1-dimensio-

nal linear dynamical system is feedback equivalent to the system

Suppose that the result holds for 1, ...,n — 1, and let us prove the case n.Consider the reachable m-input n-dimensional linear dynamical system £ =

(A, B). Since £ is reachable, it follows that Un(A*B) = Un(B\AB\...\An~lB} = R,


consequently Ui(B) = R, and hence £1 = max{7 : U^(B) = R} > 1. Then by [18,Theorem 12] the matrix B can be transformed into B'

B' - PBQ - (' Mb0

0Y

by means of elementary row and column operations. Moreover, from the maximalityof £1, we have that each entry of Y belongs to the maximal ideal m.

Consequently the system E = (A, B) is feedback equivalent to the system

T2 Idf 0

Therefore S is equivalent to

£' = (A', B') = PAP'1 + PBQ

Y

,PBQ}=

05(2)

0 Idf0 Y

Note that the £1 -input, (n — £1 )-dimensional linear dynamical system

is reachable. Consequently, by the induction hypothesis, there exist invertible ma-trices P(2),Q(2) and a feedback matrix F^ such that

p(2)jB(2)F(2) =

and

Finally consider the invertible block matrices given by

p/= '0 p(2)

p(2)p(2)y

Idand

where

and

F'= a

-iF(2)p(2)B(2)g(2)



Then, we have the equalities

P'B'F' = Ah

andP'B'Q' = Bh =

Idf0

0

for some matrix X\ with all its entries in the maximal ideal of R. Consequently thesystem E is feedback equivalent to E/j. This completes the proof D

REMARK 4.4 Let k be a field and let E = (A, B) be an m-input, n-dimensionallinear dynamical system over k. The above Proposition provides in this case aconstruction of a canonical form for the system E; this canonical form is not theBrunovsky canonical form (See [7, Chapter 1]).

REMARK 4.5 Let E = (A, B) be a reachable linear dynamical system over a localring R. When E = E^, we say that E is in normal form. Let m be the maximal idealof R and consider the linear dynamical system E(m) obtained by natural change ofscalars from R to R/m. Then the matrices B,B^2\ ... are in Hermite normal formand the reachability indices are

(m) = dimR/mm) == dimfi/m (Mp ®fl R/m) .

In fact, the indices ai can easily be calculated from the equalities:

E(m) _ _ ,f f ,. s

COROLLARY 4.6 Let (R,m) be a local domain, let E = (A, B) be a m-input,^-dimensional reachable linear dynamical system over R. Then two reachable lineardynamical systems E and E' are feedback equivalent if and only if the R-modulesMp and Mp are isomorphic.

PROOF.- By Proposition 4.3, we may suppose that the systems E and E' are innormal form, so consider the linear dynamical systems

= \A =

andE' = I A' =

B =

B' —

1 0

0

where x,x' & m.First, suppose that the linear dynamical systems E and E' are feedback equiv-

alent, then the matrices

and

B =

B' =


are equivalent. Hence the -R-modules Mf = R2/lm(B) and Mf' = R/Im(B') areisomorphic.

Conversely suppose that the /Z-modules M-p and Mf are isomorphic. Thereare two cases:

• Case 1: Suppose that x = x' = 0. It follows that E = £' and we are done.

• Case 2: Suppose that x ^ 0 and x' ^ 0. Since R is a domain, it follows that xand x' are not zero-divisors. Consequently the linear maps B : R2 —> R2 andB' : R2 —» R2 determined by the matrices B and B' respectively, are injective.Therefore, by [16, Theorem II.13.(b)], there exists an invertible 2 x 2 matrixP such that

PB = B1.

p= , Pn Pi2Consider the matrix

p=(P21 Pi'2

then we have the following equalities

Pn = 1Pi2x = 0

P21 = 0

p22X = x'.

Hence E and E' are feedback equivalent via the feedback action (P,Q,F)given by

P=(l °V 0 p22

Q = M>x2

^ = ( 0 ) 2 X 2 .

This completes the proof. D

The next result characterizes the feedback equivalence of some reachable lineardynamical systems over a discrete valuation ring. We use the following charac-terization of discrete valuation rings [2, VI.3.6.]: The local domain R is a discretevaluation ring if and only if R is a principal ideal ring (every ideal of R is a principalideal). First we need a Lemma.

LEMMA 4.7 Let (R,m) be a local ring and let S = S/j = (A,B} be an m-input,n-dimensional linear dynamical system in normal form as in Proposition 4-3. Con-sider a positive integer i with 1 Mp -* 0 •

Since Mp is a free finitely generated .R-module, it follows that the determinan-tal ideals associated to the matrix [B\AB\---\Al~lB} are trivial ideals, that isUj (B\AB\ • • • l^-1^) is either (0) or the whole ring R. (See [18]).

On the other hand, a calculation on the determinantal ideals of

shows that:

• If j < £1 + •

• If j > £1 + •

(B\AB\---\Ai~'LB')

then Uj (B\AB\ • • • \Al-lB] = R.

then Uj (B\AB\ • • • lA^B) C m.

Moreover one has that

Hi (X^ C %+...+?i+1 (B\AB\ • • • \Al~lB] ,

therefore U\ (Xi) — (0) and consequently Xi = (0). This completes the proof D

THEOREM 4.8 Let (R,m) be a discrete valuation ring, let E = (A,B) be an m-input, n-dimensional reachable linear dynamical system over R. Suppose that everyfeedback invariant R-module Mp is torsion free except, perhaps, Mp. Then, theordered set of R-modules {Mp,..., Mp} forms a complete set of invariants (up toisomorphism) associated to E for the feedback equivalence.

PROOF.- By the Proposition 4.3, we can suppose that the system E is in normalform, that is,

where

A = 5(2) •y

in general, for each j = 2, ..., s — 1

0 ,B<» =

and finally0

where the matrices Xi,...,Xs-i have all their entries in the maximal ideal m of R.First note that since Mp is torsion-free for all i > 2, it follows by Lemma 4.7

that Xi = (0) for all i > 2.Let T~im and H.n be the basis

j" ;

and


of Rm and Rn respectively for the normal form E = E/i . Now, we construct newbases Jim and "Hn of Rm and Rn respectively such that the system S = E^ relatedto these new bases is

0 0

A?0

.A. i

with X, = (0) for alii > 2 and

( o o \

\where d\\d^ • • • \dj are the elementary divisors associated to the quotient R-moduleMp.

Consider the finite free resolution of the .R-module Mp

(B\AB\--Ai~lB)Rn Mp 0

Since Mp = Rn/lm (B\AB\ • • • Ai~1B) is torsion-free for i > 2, it follows by Lemma4.7 that Xi = (0) for all i > 2, and that Mp is generated by

Moreover, we claim that the /?-module tor(Mf') is in fact generated by

Consider the homomorphism of .R-modules

Mp -»x + Np H-» A(x) + Np '

if x + Np e tor(ME), then A(x) + Np 6 tor(M2E) = 0. Then we have that

ABei + Np i tor(Mf)

for each i = !,...,&. (Note that A(ABei) + Np = A2Bet + Np is an element ofthe basis Un and hence A(ABei) + Np ^ 0). Therefore the .R-module tor(Mf) isgenerated by

{ABei:3+1+N?,...,ABei:2 + Np} .

From the above facts, we have that

B (2)0 Idf 0

xl


128

where

Carriegos and Sanchez-Giralda

and Xi — (0) for each i > 2.Define the free .R-modules

and note that FI is a free direct summand of Rn and F? is a free direct summand ofRm (because both are generated by a subset of a basis of Rn and Rm respectively) .Consider the .R-module homomorphism given by the matrix X* :

F2 X-l F1 •

Since R is a principal ideal domain, it follows that there exists a change of basisQ* and P* and a new basis {j^+i, ...,r/m} of F2 and {ABe£3+1, ..., ABe^} of Fjwhich diagonalize Jf*, that is, the following diagram (with exact rows) conmutes

F2 X-l F1

Q* I _ P' I ,

where di\d^ • • • \d7 are the elementary divisors of X^ and

d, \

Now consider the basis

of Rm and the basis

Hn =

A2Bely..., ,, AsBei, ...,

Then, we claim that the system £ = E/j in these new bases has the desired formE = (A,B\.

Note that the matrix B related to the new basis l~im of Rm and 7in is on theform

B = +1,..., Brjm) =


\ V (o) / /

And note that the matrix A related to the new basis 7im of Rm and Jin is on theform

Now, since X% = (0) it follows that ABe^+i = 0, yl-Be^3+2, ..., ABe^ = 0, andconsequently ABe^+i = 0, ^4_B^3+2, • • - , ABe^2 = 0. On the other hand, for eachi > 3, the matrix Xi is the zero matrix, so AlBeî+i = 0, ..., AzBeî+1 = 0. Hence,the matrix A in the new basis "Hm of /?m and "Hn is of the form

0 0 = A

with Xj = (0) for alH > 2. This completes the proof D

COROLLARY 4.9 Let (R,m) be a discrete valuation domain, let S = (A,B) bean m-input, n-dimensional reachable linear dynamical system over R. Suppose thatevery feedback invariant R-module Mp is torsion free except, perhaps M?. Then,the index io, the R-module Mp and the ordered set of non negative integers

form a complete set of invariants associated to S for the feedback equivalence.

PROOF.- The feedback invariant /^-modules Mf , ..., M^_1 are free, so the feedbackequivalence class of E is characterized by the positive integers < ai >

I )and

l<j<»o-l

by the feedback equivalence class of a (aig_2 ~ îo-i HnPut, (o"j0_i )-dimensionallinear dynamical system 6l°~l (E) (see [7, Theorem 1.11]).

The feedback invariant /^-modules M- of the system 8l°~l (E) are relatedto the feedback invariant /^-modules of E by the isomorphisms:

for all j — 1,2,... (see [7, Theorem 1.11]). Consequently all invariant .R-modulesMJ are free except Ml . Hence, by Theorem 4.6, the feedback equiv-

alence class of §z°~~l (E) is characterized by the invariant .R-modules M- up


to isomorphism. Therefore the feedback equivalence class of E is characterized bythe J?-modules Mp up to isomorphism and the result follows in a straightforwardway. D

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[3] J.W. BREWER, J.W. BUNCE, F.S. VAN VLECK: Linear Systems over Com-mutative Rings. Marcel Dekker (1986).

[4] P.A. BRUNOVSKY: A classification of linear controllable systems. Kyber-netika, 3, 173-187 (1970).

[5] R. BUMBY, E.D. SONTAG, H.J. SUSSMANN, W. VASCONCELOS: Remarkson the Pole-Shifting problem over rings. Journal of Pure and Applied Algebra,20, 113-127 (1981).

[6] A. CAMPILLO-LOPEZ, T. SANCHEZ-GIRALDA: Finitelly generated pro-jective modules and Fitting ideals. Collectanea Mathematica, XXX , 2°, 3-8(1979).

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[8] M. CARRIEGOS, J.A. HERMIDA-ALONSO, T. SANCHEZ-GIRALDA: Thepointwise feedback relation for linear dynamical systems. Linear Algebra andits Applications, 279, 119-134 (1998)

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An introduction to Janet bases and Grobner bases

F. J. CASTRO-JIMENEZ, Dpto. de Algebra, Universidad de Sevilla, Apdo. 1160,E-41080 Sevilla. Spain.E-mail: castro@cica. es

M.A. MORENO-FRIAS, Dpto. de Matematicas, Universidad de Cadiz, Apdo. 40,E-11510 Puerto Real. SpainE-mail: [email protected]

I INTRODUCTION

The results of Buchberger ([7]) on Grobner bases in commutative polynomial ringshave been generalized by several authors (see for example [6], [11] for early treat-ments) to the case of some rings of linear differential operators.

Independently, the work of Riquier [43] and Janet [21, 22] on the algebraic ap-proach to the systems of partial differential equations was discovered by Pommaret[40, 37] (see also [36],[42]). Since then, these works and the ideas behind them havebeen thoroughly explored, generalized and firmly established within the frameworkof effective methods for the resolution of systems of partial differential equations(see for example [44], [24], [15, 16], [46, 47], [48], [27], [41]).

Despite this, as far as we know, there is still no systematic comparison betweenthe Janet bases (called by him completely integrable systems) and Grobner basesapproach. Most references in the literature accept that both of them are "essentiallyequivalent".

This is the task we undertake in the present work. From this point of view, onecan regard this paper as a "survey" on the subject. Concretely, we show that, undercertain hypotheses, when the linear differential equations have their coefficients ina field, every completely integrable system is a Grobner basis and conversely (see4.1.2, 4.1.3, 4.2.5, 4.2.6). This is particularly useful in the case of rings of differentialoperators with constant coefficients. This being a commutative polynomial ring, wethink we can regard Janet bases as a precedent of Grobner basis (in the commutative


134 Castro-Jimenez and Moreno-Frias

case).Of course, we cannot directly apply the results of Janet to the case of differential

equations with coefficients in a ring. This generalization does exists for Grobnerbases theory, in the case of "general" rings of differential operators (see for example[6], [11, 12, 13], [23], [26], [20], [45]).

Just a few words about references. Although the list is not exhaustive (onGrobner bases theory in the differential case), we have included firsthand, the ref-erences used in writing this paper and then some other references that we consideruseful to the reader as a complement of the point of view we have developed here.

The authors have greatly benefited from the work of J.-F. Pommaret, F. Schwarzand V.P. Gerdt while writing this article. We also wish to thank J.M. Tornero fora careful reading of the manuscript.

2 MONOMIALS

Let k be a field. Let us denote by M.(X) the set of monic monomials of thecommutative polynomial ring k[Jf] = k [x i , . . . , xn}. If a = (ai,..., an) € N" wewill write Xa instead of a;"1 • • •<". In ([21]; (1920)) Janet gives a proof of theso-called "Dickson's lemma" and, as a consequence, he proves ([21], pp. 69-70) thefollowing two lemmas:

LEMMA 2.0.1 Let I be an infinite subset of M(X). Then there exists a finitesubset F C I such that for all Xa € I there exists X@ & F such that Xa is divisiblebyX?.

LEMMA 2.0.2 Let Si C 82 C • • • C S^ C • • • be an increasing sequence of subsetsin A4(X) such that for all i, each monomial in <Si+i \ Si is not divisible by amonomial in Si. Then this sequence is finite.

2.1 Janet modules

DEFINITION 2.1.1 We say that a subset J of M(X) is a Janet module if eitherJ — 0 or each multiple of a monomial in J lies in J:

\/Xa <= J,V/? e N" we have Xa+fi e J.

REMARK 2.1.2 Let (f> : M(X)-^~Nn be the canonical map <j)(Xa) = ( o (J}.

DEFINITION 2.1.3 Let J ^ 0 be a Janet module. A finite subset B of J is saidto be a basis of J if each monomial in J is divisible by some monomial in B.

PROPOSITION 2.1.4 Each Janet module has a basis.

PROF.- Apply 2.0.1.

Janet Bases and Grobner Bases 135

2.2 Multiplicative variables. Classes

Here we will give the definition (Janet, [21], p. 75-76) of multiplicative (and non-multiplicative) variables and the definition of the class of a monomial.

DEFINITION 2.2.1 Let T be a finite subset of M(X) and Xa € f',1. xn is said to be a multiplicative variable with respect to (or simply, for) Xa inJ- if for all X@ G J- we have /3n < an.2. Xj, 1 < j' < n — I , is said to be a multiplicative variable with respect to (orsimply, for) Xa in J- if the following condition holds: for all X@ & J- with /3n =an, • • • ,/3j+i — otj+i, we have /3j < otj. We denote by mult(Xa,f) the set ofmultiplicative variables with respect to Xa in T. The variables x, £ mult(Xa,F)are called non-multiplicative variables for Xa in T.

DEFINITION 2.2.2 Let Xa be a monomial of the finite set T C M(X). We callclass of Xa in F, noted by Ca,F, the set

Ca,r = {Xa+f> Each variable in X13 belongs to muh(Xa,F)}

Classes corresponding to different monomials are disjoint.

DEFINITION 2.2.3 ([21], p. 79) Let F be a finite subset of M(X) and denote byJ the Janet module generated by F. The set F is said to be complete if the followingholds: For all Xa € J- and for all Xi £ mult(Xa, F) there exists X@ € J- such thatXiXa eC0,f.

Let F be a complete subset of M.(X). Then for each Xa £ J- and for eachXi £ mult(Xa, F) the only X13 6 T such that XiXa € Cp^ verifies that (an,..., ai)is less than (/3n,... ,/?i) w.r.t the lexicographical order (see [21], p. 85).

3 COMPLETELY INTEGRABLE SYSTEMS. JANET BASES

Janet considers in [21] the degree lexicographical order (denoted by <^eK or -<):

( a < \p\a <deg p ̂ I or

( a| = \/3\ and ( « „ , - • • ,0^ <lex (/3n, • • • , / ? : ) ,

where <iex is the lexicographical order.Let k be a field. We denote by k(.X') (resp. k((X))) the quotient field of the

polynomial ring k[X] (resp. of the formal power series ring k[[A"]] = k[ [x i , . . . , xn]]).In this section we will consider the rings of linear differential operators k[9], Qn(k) =k(X)[di,. ..,dn] and Q~n(k) = k((X))[<9i, ...,dn}. We denote by ft any of thesethree rings and by A any of the corresponding fields k,k(J5C), k((X)). Let A/" be aleft 72.-module.

Consider a system of (not necessarily homogeneous) linear differential equations:

(5): Pl ( « ) = / ! , . . . , Pr(u)=fr

where Pi g Tl, fi 6 J\f and the unknown u belonging to A/". Rewrite the equationPJ(U) — fj as

136 Castro- Jimenez and Moreno-Frias

with aa(j),a/3 <E A. The element aa^da^\u] (resp. ^2/3^a(j)a^(u} + /j) is

called the first (resp. second) member of this equation. We will identify da(u) withda and with a. When fj = 0 and if no confusion is possible we will identify theequation Pj(u) = 0 with the linear differential operator Pj.

DEFINITION 3.1.4 ([21], p.105) Let S be a system as above. We say that S isin canonical form (with respect to -<) if the following conditions hold:1) aa(j) = 1 f°r allJ-2) The first members of any two equations are distinct.

DEFINITION 3.1.5 ([21], p. 106) Given a system S in canonical form and f theset of its first members, we call principal derivative (with respect to S) each mono-mial da in the Janet module generated by J- . We call parametric derivatives theremaining ones.

DEFINITION 3.1.6 Let E = aada(u) = £^a a^(u)+f be a linear differentialequation with a7 G A, f G A/" and aa ^ 0. We call support of E the set supp(E) ={7 6 N" | a7 ^ 0}. We call a the privileged exponent of E (with respect to -<) andwe denote it by expx(S) (or exp(E) for short, if no confusion is possible).

DEFINITION 3.1.7 Let S be a system as above and denote by f the set of thefirst members of S . Let E be an element of S . We call multiplicative (resp. non-multiplicative) variable of E (in S) any of the multiplicative (resp. non-multiplica-tive) variables of the first member of E (in J-). The class of E will be the class ofits first member in f (see 2.2).

DEFINITION 3.1.8 The system S is complete if F is complete (see 2.2.3).

Let 5 = {Ei, . . . ,Er} be a complete system of linear homogeneous partial dif-ferential equations and suppose the Ei are in canonical form. Let us reproduce thedefinition of Janet ([21], p. 107): Si, par derivations et combinaisons, on ne penttirer de (S) aucune relation entre les seules derivees parametriques (et les variablesindependantes) , on dira que le systeme est completement integrable.

Denote by I the left ideal (in TV) generated by S and write

DEFINITION 3.1.9 ([21], p. 107) The complete system S is said to be completelyintegrable if the only element in I with support in Nn \ A(S) is the zero element.

DEFINITION 3.1.10 Let I be a left ideal ofR, generated by a finite homogeneoussystem S. The system S is called a Janet basis (of I) if S is completely integrable.

4 JANET BASES AND GROBNER BASES

4.1 Homogeneous systems

The theory of Grobner bases developed by Buchberger [7] for commutative poly-nomial rings has been generalized to ideals in rings of differential operators and

in particular to ideals in ft (see [12, 13], [23], [26], [20], [45]). If / is a left idealof ft we denote by Exp(/) the set of privileged exponents exp(P) for P in / (see3.1.6). A finite subset {Pi,...,Pr} C I is said to be a Grobner basis of / if

Given a - (a 1 , . . . , ar) E (N")r we define the partition {Ai , . . . , Ar, A} of Nn

associated to a as follows:

i-l r

Fori = l , . . . , r ; A, = (a* + Nn) \ ({J A,-); A = N" \ ((J A<).j=l i=l

If E_ = (Ei, • • • ,Er] E TV we call partition associated to E_ the partition associatedto (exp(£1),...,exp(.Er)).

THEOREM 4.1.1 (Division theoremj/n 11). Consider (El,---,Er) E TV withPi ji 0, i = 1, • • • ,r. Let {Ai, • • • , Ar, A} be the associated partition of N". Then,for all E & ft, there exists a unique (Qi, • • • , Qr, R) € TV+l such that:

1- E = E[=i QiEi + R.2. If R ^ 0, each monomial of R (in the variables di, • • • ,dn) lies in A.

3- If Qi 7^ 0, each monomial cda of Qi (with c E A) satisfies a + exp(Ei) C A,.

In fact, this division theorem (and its proof) is explicit in Janet's work ([21],pp. 100 and 106) when the set {Ei,..., Er} is complete and in canonical form.PROOF.- The proof is analogous to the commutative polynomial ring case (see forexample ([1], p. 28) because the coefficients of the differential operators belong tothe field A and because Leibnitz's rule implies that for all a E A and a E N™, daa-ada is a differential operator of degree less or equal than |a — 1 = ai + • • • + an — 1.

THEOREM 4.1.2 Let I be a left ideal of Tl and B = {E±,... ,Er} C /. If B isa Janet basis of I then B is a Grobner basis of I, with respect to the monomialordering -< on N™.

PROOF.- Write A = A(S) = Uj=i(exP(£j) + N")- Let -P be in / and supposeexp(P) ^ A. By the division theorem in Ti (see above) there exists R E Ti withsupp(.R) C N" \ A, such that P - R G / and exp(P) = exp(E). So R ^ 0. But thisis impossible by the hypothesis (see definitions 3.1.9, 3.1.10).

We say that a differential operator P 6 ft is monic if the coefficient of itsprivileged monomial is 1.

PROPOSITION 4.1.3 Let B = {El}..., Er} be a Grobner basis of a left ideal I offt. Suppose exp(-Ej) ^ exp(Ej), for i ^ j, Ei is monic for all i and B is complete.Then B is a Janet basis of I.

PROOF.- Let R be a non zero element of / with support contained in N™ \ A(S).Then exp(R) E Exp(7) = A(£?) which is a contradiction.

THEOREM 4.1.4 (Criterion for complete integrability). Let

S = {E1,E2,---,Er}

be a subset of monic elements in Ti. Suppose that for all i and for all non-multiplicative variables d^ for Ei (in S) we have d^Ei = ^j=i ^^ -Ej such thatthe only variables in each monomial (in the variables di , • • • , dn) of A^ are multi-plicative variables for Ej, Vj = 1, 2, • • • , r. Then we have:

1. For all HeU,

where the only variables of each monomial in Qi are multiplicative variablesfor Ei in S. Here (Ei, E2, • • • , Er) is the left ideal (ofR.) generated by the Ei.

2. S is completely integrable.

PROOF.- We can suppose exp(Er) -< exp(£V-i) - < ; • • • - < exp(J5i). The hypothesisimplies that S is complete, exp(dkEi) = exp(A^ '*' Ec^k^) for a unique integerc(fc,z) - - - >can be written as Gi = G\ + Hi where G\ ' is the sum of the monomials of Gwith only multiplicative variables for Ei in S. In particular HI = 0.

We have

i-l i=l i=2

Let us denote 6 = (61, • • • , 5n) = max {exp(HiEi), i = 1, • • • , r} and

ZG = max{z exp(HiEi) = S}.

We call (6,io) the characteristic exponent of X^r=i H j E j -We will consider on N™ x {1, • • • , r} the well ordering defined as follows:

(6,i0) < (S',i'0) «=> or{6 = 6' and ZQ < i'0.

Then we can write

where a € A, exp(a91ai •••d^Eio)=5 and e x p ( £ : i o ) -< 8.

Suppose dk is a non- multiplicative variable for Et0, then by hypothesis we canwrite

HiQEio = oSf1 • • • d^~l • • • d^(dkElo) + HioEio =

where the only variables in each monomial of A^ are multiplicative variables forEJ. Then rewrite

= HioEioi=2

wherefor j ^ io

Now we will compute the characteristic exponent of this new expression:

1. For i0 + 1 < j < r we have exp(#j£j) = exp(a5"1 • • • d%k~l • • • dÂÊj +HjEj) ^ max{exp(a<9? 1 • • • d^k~l ••• dÂÊj), expÊj)}. We have first

j ) -< 6, because of the definition of io, and then

• • • • • • i = al} • • • , ak - , • • • , a

( a i , - - - ,ak - 1, • • • , « „ ) +exp(dkEio) = S.

So, exp^'jEj) -< S for iQ + 1 < j < r.

2. ~

and then exp(H'iQEi0) -< S.

3. For 1 < j < io — 1 we have

jEj) = exp(ad^ • • • d^1 • • • dÂÊj + H^) -<

• • • • • • ,-

The choice of j implies that exp(HjEj) X S and, on the other hand, we have

So, the characteristic exponent (6', i'0) of J^ . H'Êj is less than (6, io) w.r.t thewell ordering <], which implies the assertions of the theorem.

REMARK 4.1.5 As a consequence of this theorem, Janet develops a finite proce-dure constructing a completely integrable system (i.e. a Janet basis) of a left ideal I0/72., starting from an arbitrary system of generators of I. This algorithm should becompared to Buchberger's algorithm computing Grobner bases. Janet's procedure isas follows: a) We can suppose the starting system Si = {Ei, . , . , Er} to be completeand in canonical form (see 2.2.3,3.1.4). b) For each i = 1, . . . ,r and each k suchthat dk is non-multiplicative for Ei, write (see 4.1.1) dÊi — ̂ r=i ^ki ^j ~^~ îk

where

1. Each monomial in A^ (in d\, . . . , dn) is formed only by multiplicative vari-ables for EJ in Si .

2. e x p ( ^ J ) < exp(afcEi) for j = 1, . . . ,r.

3. The support of Rî is contained in N" \ A(5i).

c) If all the Rki are zero, then S is completely integrable (see 4-1 -4)- d) V thereexists Rki 7^ 0 then we consider the new system S% = SiU {Rki} and we restart.

This procedure is finite. Indeed, let Si, i = 1, 2, . . . be the sequence of systemsobtained applying Janet's procedure. Write Fi = (exp(J5) | E £ Si} C Nn. By 2.0.2this sequence is stationary and the procedure is finite.

4.2 Non-homogeneous systems

In this section we will explain how to extend the results of 3 and 4 to systems oflinear non-homogeneous differential equations.

Let 5 be a system of linear, non necessarily homogeneous, differential equations

where Pi 6 72, /j € A/" and the unknown u belonging to A/". We denote by Sh thehomogeneous system PI(U) = • • • = Pr(u) = 0 associated to S.

We will denote by Ei the equation Pi(u) = fi (or Pi(u) — fi =0).We identify the equation Pi(u) = fi (i.e. the equation Ei) with the couple

(Pi, /i) S 72 © TV and we consider the 72.-sub-module M of 72. © M generated by

DEFINITION 4.2.1 Let S = {Ei,---,Er} = {(Pi,fi),---,(PrJr)} be a completesystem in canonical form. Let M be the 72.- sub-module ofR.® M generated by S.The system S is said to be completely integrable if the following holds:1) If(OJ) e M then f = 0.2) If (P, f) € M and P ^ 0 then the support of P is not contained in N™ \ A(S^).

DEFINITION 4.2.2 Let M be the U-sub-module ofU®N generated byS = {(-Pi, /i), • • • , (Pr> f r ) } • We call S a Janet basis of M if S is completely inte-grable.

Denote by £ the (left) 7?.-module 72. © J\f and by -n\ : £ — > 72. the canonicalprojection.

As in 72 we have in £ the notions of privileged exponent and Grobner basis andwe have a division theorem in £.

We still denote by exp : £ \ ({0} ©A/") — > N" the map expx(P, /) = exp(P).

THEOREM 4.2.3 (Division theorem in E). Consider (El,---,ET) € £r with Et =(Pi, f i ) and Pi ^ 0, i = 1, • • • ,r. Let {Ai, • • • , Ar, A} be the associated partition ofN™. Then, for all E = (P, /) € £, there exists a unique (Qi, • • • ,Qr, ( R , d ) ) 6 Tlrx£such that:

2. If R ^ 0, each monomial of Ti (in the variables di, • • • , dn) lies in A.

3. If Qi ^ 0, each monomial cda of Qi (with c € A), satisfies a + exp(Ei) C

PROOF.- Analogous to the proof of 4.1.1. We first write P = Y?i=\ QiPi + Rthen g = f- EL: Qi(fi).DEFINITION 4.2.4 Let M be a H- sub-module of£. A finite subset

of M is said to be a Grobner basis of M, with respect to -<, if the following twoconditions hold:

1. {Pi, • • • ,Pm} is a Grobner basis o/7Ti(M) with respect to -<.

2. For all g £ M, if (0, g) € M then g = 0.

THEOREM 4.2.5 Let M be an U-sub-module of £ = U ® M and suppose B ={(Pi, /i), • • • , (Pr, /r)} C M is a Janet basis of M . Then B is a Grobner basis ofM, with respect to -<.

PROOF.- Analogous to the proof of 4.1.2, using the division theorem 4.2.3.

PROPOSITION 4.2.6 Let B = {El = ( P ^ f r ) , . .. ,Er = (Pr,fr)} be a Grobnerbasis of a left TL- sub-module M ofR.@J\f. Suppose exp(Pi) ^ exp(P7-) for i ^ j, Eiis monic for all i and B is complete. Then B is a Janet basis of M.

PROOF.- Suppose (P, /) e M and supp(P) c N" \ A(5fe). The family {Pi, . . . , Pr}is a Grobner basis of the ideal 7Ti(M) and then A(5/l) = Exp(?ri(M)). So, exp(P) 6A(Sh) and then P = 0.

THEOREM 4.2.7 (Criterion for complete integr ability). Let

S = {E1,E2,---,Er}be a subset of monic elements in £. Suppose that for all i and for all non-multi-plicative variables dk for Ei (in S) we have d^Ei = X^=i Ak^Ej where the onlyvariables in each monomial (in the variables d\, 82, • • • , dn) of A^ are multiplicativevariables for Ej, Vj = 1, 2, • • • , r. Then we have:

1. For all H e £,

1=1where the only variables of each monomial in Qi are multiplicative variablesfor Ei in S. Here (Ei, E2, • • • , Er) is the left Tl-module generated by the E^.

2. S is completely integrable.

PROOF.- Analogous to the proof of 4.1.4.


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Invariants of Coalgebras

J. CUADRA, Dpto. Algebra y Analisis Matematico, Universidad de Almerfa.04120-Almen'a. Spain.

F. VAN OYSTAEYEN, Department of Mathematics, University of Antwerp (UIA).B-2610 Wilrijk (Antwerp). Belgium.

1 INTRODUCTION

In this paper we give a semi-survey of the known results for certain invariants ofcoalgebras, in particular the Picard group and the Brauer group. We also includesome proofs and enriched the theory with some new results and observations, inparticular Theorem 3.2.9 and consequents, Corollary 4.1.13 and Examples 4.1.14,Theorem 4.2.5 and Corollary 4.2.6, Proposition 4.3.10 and Corollaries 4.3.11, 4.3.12.

After some preliminaries in Section 2 we survey the results on the Picard groupin relation with Morita-Takeuchi theory in Section 3. Not so many Picard groupshave been explicitly calculated. This is possible sometimes when the calculationmay be reduced to dealing with automorphisms, more correctly to outer automor-phisms; this is the topic of Section 3.2. Good cases are obtained for basic coalgebras(Theorem 3.2.5) and pointed coalgebras (Corollary 3.2.7). The coalgebra case some-times reduces to a calculation for the dual algebra in case the coalgebra has finitedimensional coradical (Theorem 3.2.9).

We review the general theory for the Brauer group in Section 4.1, again involvingMorita-Takeuchi theory, and we add new examples to the theory. The Brauer groupof a cocommutative coalgebra may be calculated from the Brauer group of theirreducible components. Several questions about the torsioness of the Brauer groupof irreducible coalgebras were pointed out in [25]. In particular, it was conjecturedthat the Brauer group of an irreducible coalgebra is torsion. We solve this questionin Corollary 4.2.6. In fact, we proved that the Brauer group of an irreduciblecoalgebra embeds in the Brauer group of the dual algebra.


148 Cuadra and Van Oystaeyen

Finally we describe some subgroups of the Brauer group which are versions ofthe Schur subgroup and the projective Schur subgroup of the Brauer group of a ring.These have good properties mainly because they define torsion groups whereas theBrauer group itself is not necessarily torsion.

We prove that the Schur group of a cocommutative coalgebra over field of charac-teristic positive is trivial. Also for irreducible coreflexive coalgebras with separablecoradical, these subgroups may be computed from the ones corresponding to thecoradical.

2 PRELIMINARIES

Throughout k is a fixed ground field and M-k denotes the category of k- vectorspaces. All coalgebras, vector spaces and unadorned (g>, Horn, etc., are over k.

Coalgebras and Comodules (See [1], [21]): For a coalgebra C, A and e denote thecomultiplication and the counit, respectively. The category of right C-comodules isdenoted by Mc; for X in M.G , px is the comodule structure map. For X, Y € M.c ,Com^c(X, Y) is the space of right C-comodule maps from X to Y. Similarly, c M.denotes the category of left C-comodules. If D is another coalgebra, then X isa (D, <7)-bicomodule if X e Mc via px, X e DM via xP and (1 <g> px)xP =

Cotensor product : Let M be a right <7-comodule and N a left C-comodule withstructure maps PM and p^ respectively. The cotensor product MOCN is the kernelof the map

The functors MO^— and —D^N are left exact and preserve direct sums. If cMrjand oNs are bicomodules, then MO^N is a (C, £?)-bicomodule with comodulestructures induced by those of M and N.

Let X 6 Mc , we say that X is finitely cogenerated if there is a finite dimensionalvector space W such that X is isomorphic to a subcomodule of W ® C. W ® C isnothing else but the direct sum of C dim(W) times. Sometimes we will write C^instead W ® C with n = dim(W). X is free if X is isomorphic to C^ for someset I. X is said to be a cogenerator if for any comodule M e M.c , M <— > X^\ forsome set /, as comodules. X is injective if the functor XDC— is exact.

Morita-Takeuchi Theory (See [23]): A comodule X € M.c is called quasi-finiteif Com-c(Y, X) is finite dimensional for any finite dimensional comodule Y e M.c ' .Quasi-finite comodules can be characterized by looking at the socle, cf. [23, Prop.4.5].

PROPOSITION 2.0.1 For a comodule X, the following assertions are equivalent:i) M is quasi-finite.ii) soc(M) is isomorphic to (Big/iS,- where {5i}ig/ is a complete set of re-

presentatives of isomorphism classes of simple comodules and Hi are finite cardinalnumbers.

Invariants of Coalgebras 149

For a quasi-finite comodule X e Mc and any Y € Mc the co-hom functor isdefined by

where {Y\} is a directed system of finite dimensional subcomodules of Y such thatY = lim— Y\. When Y = X then h^c(X, X) is denoted by e_c(X) and it becomes

A

a coalgebra, called the co-endomorphism coalgebra.

A Morita-Takeuchi context (M-T context) (C, D,cPo, oQc, /,5) consists ofcoalgebras C, D, bicomodules cPo, oQc, and bicolinear maps / : C —> POrjQ andg : D —> QO/^P satisfying the following commutative diagrams:

P ——^—> PODD Q —————> QUO

in/CP

\ /

Cacp —\

10g

/-> poDQacp

/niThe context is said to be strict if both / and g are injective (equivalently,

isomorphism).

The following result, due to Takeuchi, characterizes the equivalences betweentwo categories of comodules, [23, Prop. 2.5, Th. 3.5]:

THEOREM 2.0.2 LetC,D be coalgebras.a) If F : Mc —> M is a left exact linear functor that preserves direct sums,

then there exists a (C,D)-bicomodule M such that F(-) = — O^M.b) Let M be a (C,D)-bicomodule. The following assertions are equivalent:

i) The functor — ̂ cM.:Mc-+MD is an equivalence,ii) M is a quasi-finite injective cogenerator as a right D-comodule and

e__£>(M) = C as coalgebras.Hi) There is a strict M-T context (C, D,cPo, oQc, / ,<?)•

When the conditions hold, N — h-o(M, D) is a (D,C)-bicomodule and theinverse equivalence is given by — OrjN : MD —> Mc. The coalgebras C and D arecalled Morita-Takeuchi equivalent coalgebras.

Morita-Lin Theory (See [15]): There is another Morita theory for coalgebras,due to Lin. The category of right comodules is embedded in the category of leftC"*-modules and using the classical Morita theory for rings one characterizes theequivalences in the category of comodules. A right (7-comodule M is said to be aningenerator if it is a finitely cogenerated injective cogenerator.

We say that C is strongly equivalent to D if Mc is equivalent to MD via/ : Mc -> MD, g : MD -> Mc, f ( C ) is an ingenerator in MD and g(D) isan ingenerator in M.c'. If both coalgebras have finite dimensional coradical, thenstrongly equivalent is the same as equivalent. Here is the theorem characterizingstrong equivalences:

THEOREM 2.0.3 Let C and D be coalgebras.a) If Mc is strongly equivalent to M,D via f : M.c — > MD and g : M.D — > Jv[° ,

then there are ingenerators P € M° and Q € MD such that C'-M. is equivalent toD*M via

F(-) = D.P£.®C'-: c-M^ D-M,G(-) = c-Q*D.®D'-: D-M-* c-M.

Moreover, f and g are naturally isomorphic to F and G respectively.b) U c*M is equivalent to rj*M. via functors F : C'-M- — > D*-M, G :

D'M^ C'M such thatF(Mc) C MD andG(MD] C Mc , then Mc is stronglyequivalent to MD .

The cocenter (See [25]): Let C be a coalgebra. If we view C as a right Ce-comodule (Ce = C°p 0 C), then C is quasi- finite. The coendomorphism coalgebrahas the following universal property:

i) e-C" (C) is a cocommutative coalgebra with a surjective coalgebra map ld :C — > e-ce(C) which is cocentral, i.e., for all c € C,

ii) For any cocentral coalgebra map / : C — > D there exists a unique coalgebramap g : e_ce(C<) — * D such that / = gld. In particular, an injective coalgebra mapinduces a coalgebra map from e_c=(C f) to e_£>e(D).

e_ce(C) is denoted by Z(C) and it is called the cocenter of C. Let C be acocommutative coalgebra, a coalgebra D is said to be a C-coalgebra if D is acoalgebra together with a cocentral coalgebra map e : D — > C, called C-counit. Amap of C-coalgebras is a coalgebra map which respects the C-counits.

Let D,E be C-coalgebras with C-counits CD^E respectively, and M a (D,E)-bicomodule. M is said to be a bicomodule over C if the following diagram is com-mutative.

M —— f^^M®E <8>6g. M0C

PD

CD <8> 1 T

where T is the twist map. Let F : .M0 —-> A^B be an equivalence of categories. Wehave seen that F is of the form F(-) = -DDM for a (£>,.E)-bicomodule M. Wesay that F is an equivalence over C if M is a bicomodule over C. In this case, wesay that D and E are Morita-Takeuchi equivalent coalgebras over C.

3 THE PICARD GROUP

From Morita-Takeuchi theory we retain that every autoequivalence of M.c is givenby a (C, C')-bicomodule M, which is called an invertible bicomodule. The Picardgroup of C, denoted by Pic(C), was introduced in [24] by taking isomorphismclasses of invertible bicomodules and cotensor product over C. In that paper, an

exact sequence connecting the group of coalgebra automorphisms of C, Aut(C), andPic(C) was given. This sequence yields that the group of outer automorphisms,Out(C), embeds into Pic(C}. In this section we find coalgebras such that Out(C)is the whole Pic(C). So, for these coalgebras the computation of Pic(C) reducesto the computation of automorphisms of C. This result allows us to compute newexamples of Picard group of coalgebras.

3.1 Definitions and properties

In this subsection we recall from [24] the definition of the Picard group of a coalgebraand some of its properties. We refer to the reader to [24] for the proofs.

DEFINITION 3.1.1 A (C,C)-bicomodule M is called invertible if the functor— OCM : M.c —* M° defines a Morita-Takeuchi equivalence. This is equiva-lent to the existence of a (C, C)-bicomodule N and the bicomodule isomorphismsMDCN ^ C and NncM ^ C.

DEFINITION 3.1.2 The Picard group of C, denoted by Pic(C), is defined asthe set of all bicomodule isomorphism classes [M] of invertible (C,C)-bicomodules.Pic(C) becomes a group with the multiplication induced by the cotensor product,that is, for [M], [N] £ Pic(C), [M}[N] - (MUCN\. The identity element is [C]and(M}-1 = [h_c(M,C)}.

Let M be a (C, C')-bicomodule with right and left structure maps pM and MPrespectively, ld : C —* Z(C) the universal cocentral map from C in its cocenter andT ; M ®C —> C* <S> M the twist map. The set

Picent(C) = {[M] e Pic(C) : r(ld ® l)pM = (1 <8> ld) uP\

is a subgroup of Pic(C) called Picent of C.

Remark: Let Co be the coradical of C. If Co is finite dimensional, then everyautoequivalence in Mc induces an autoequivalence in c*M by Theorem 2.0.3. Inthis case Pic(C) C Pic(C*) via the map [M] i—» [M*]. When C is finite dimensionalthis map is surjective and both groups coincide.

PROPOSITION 3.1.3 Let C,D be Morita-Takeuchi equivalent coalgebras, thenPic(C) ^Pic(D).

This proposition shows that Pic(~) is an invariant up to the M-T equivalencerelation.

Let Aut(C) denote the group of automorphisms of the coalgebra C. An au-tomorphism / 6 Aut(C) is said to be inner if there is a unit u & C* such thatf ( c ) = S(c) U(ci)c2u~1(cs) f°r all c& C. We denote by Inn(C) the group of innerautomorphisms of C. Inn(C) is a normal subgroup of Aut(C) and the factor groupOut(C) = Aut(C)/Inn(C) is called the group of outer automorphisms of C.

Let M be a (C, C*)-bicomodule and /,g 6 Aut(C). We denote by f M g thebicomodule constructed in the following way: as a vector space /Mg = M and

, fMgP = ( f ®


THEOREM 3.1.4 There is an exact sequence

where w(/) = [ /Cj] for all / £ Aut(C). Hence, u> induces a monomorphism fromOut(C) to Pic(C}.

The following result is used in the proof of the further results.

PROPOSITION 3.1.5 Let [M], [N] e Pic(C). Then, M ^ N as right comodulesif and only if there exists f 6 Aut(C) such that N = /Mi as bicomodules.

3.2 The Aut-Pic property

Now we investigate when the map w : Aut(C) — > Pic(C) is surjective. In thiscase Pic(C) = Out(C), and thus we reduce the computation of Pic(C) to thecomputation of the automorphisms of the coalgebra C.

DEFINITION 3.2.1 A coalgebra C has the Aut-Pic property ifuj of Theorem 3.1.4is surjective.

The first example of coalgebra with the Aut-Pic property was given in [24, Th.2.10]:

PROPOSITION 3.2.2 Let C be a cocommutative coalgebra, then C has the Aut-Pic property. Consequently, Pic(C) = Aut(C).

Next we study more general coalgebras with the Aut-Pic property.

PROPOSITION 3.2.3 Let C be a coalgebra such that every right injective como-dule is free. Then C has Aut-Pic.

PROOF: Let M be an invertible (C, C)-bicomodule with inverse N, then M, N arequasi-finite injective cogenerators as right comodules. By the hypothesis, we knowthat M = C(n) and N = (7(m) for some n, m > 1 as right comodules. Now,

C ^ MucN ^ C(n]ocN ^ N(n*> ^ c(nm\

as right comodules. Since C has the IBN property, cf. [17, Prop. 4.1], then nm = 1and so M = C as right comodules. From Proposition 3.1.5, it follows that thereexists / e Aut(C) such that M = $Ci as bicomodules. Hence [M] 6 Im(u>). I

More examples of coalgebras with Aut-Pic were given in [8, Prop. 2.5]; thesederive from the consideration of matrix coalgebras.

PROPOSITION 3.2.4 Let C be a coalgebra with Aut-Pic. Then the matrix coal-gebra over C, Mc(n, C), has Aut-Pic for all n > 1.

We recall from [5] that a coalgebra is basic if every simple subcoalgebra is thedual of a division algebra over k. The following theorem shows that basic coalgebrasalso have the Aut-Pic property.


THEOREM 3.2.5 Let C be a basic coalgebra, then C has Aut-Pic.

PROOF: We first prove that for basic coalgebras the isotypic component of a simpleright coideal only contains that coideal. Let / be a non-zero simple right coideal ofC and J its isotypic component. It is well-known that J is a simple subcoalgebra,cf [13]. By the hypothesis, there is a finite dimensional division algebra D over ksuch that J = D* as coalgebras. I is a right coideal of J, so that 1^ is a right idealof D. Since D is a division algebra it must be /-1 = {0} and hence I = J.

We know that C is a quasi-finite injective cogenerator. Using Proposition 2.0.1we have that C = ®aer E(Sa)^na'i where {Sa} is a representative family of rightsimples comodules and na are finite cardinal numbers. Every 5a is isomorphic toa right simple coideal Ia of C . Taking into account that the isotypic componentof a coideal only contains that coideal, it must be na = 1 for all a <E F, and so

Now, let [M] £ Pic(C), then M is a right quasi-finite injective cogenerator.Since M is quasi-finite, again by Proposition 2.0.1, soc(M) = ®agr Sa a , andsince M is an injective cogenerator, then M = (J)aer E(Sa)^ma^ where ma > 1for all a € T. Set P = 0Q6r E(Sa)(m°-^, then M ^ C ® P as right comodules.Let N be the inverse of M, then C = MacN ^ (C'ncN) 0 (POCN) as rightcomodules and so soc(C) = soc(N) 0 soc(POGN). Writting N ^ ®Q€r E(Sa)(ta\from the above isomorphism it follows that ta < 1 for all a e F. But, since Nis a cogenerator, ta > 1, therefore ta — 1 for all a £ F. Thus, N = C as rightcomodules and from C = NOCM, we have that M = C as right comodules. Now,by Proposition 3.1.5, there is / e Aut(C) such that M = jC\ as bicomodules.Hence [M] e Im(w). I

Cocommutative coalgebras are examples of basic coalgebras, so we rediscover[24, Th. 2.10].

COROLLARY 3.2.6 Every cocommutative coalgebra has the Aut-Pic property.

Noting that pointed coalgebras are basic coalgebras we have:

COROLLARY 3.2.7 Every pointed coalgebra has Aut-Pic.

Thus we have found a big family of coalgebras having the Aut-Pic property.Using this result we are able to give new examples of Picard groups of coalgebras.

EXAMPLE 3.2.81.- Let C be the Sweedler coalgebra, i.e., the vector space generated by the set

{dn,sn '• n € IN} with comultiplication and counit given by

= gn= gn <S>sn + sn ®gn+i e(sn)=Q

C is a pointed coalgebra, and by the above corollary, Pic(C) = Out(C). We showthat Pic(C) is trivial. Before computing Aut(C), it is easy to observe:

For i,j € IN with j ^ i + 1 the (gi, <?j)-primitive elements are A(<?j — <jj) forX & k. For all n 6 W, the (gn, gra+i)-primitive elements are angn + (3nsn — angn+iwith an, /3n G k.

Let / £ Aut(C), then / maps group-like elements to group-like elements, (<?i, <7j)~primitives to (/(<?i), /(;?j))-primitives. Since / is bijective, we may conclude that /is of the form:

with on £ k and /% £ k* for all i £ IN. We next check that every automorphism isinner and hence Pic(C) is trivial. Let / be as above, we define u, v : C — » k by:

u(gi) = 1 v ( g i ) = I

It is not hard to prove by induction that u is a unit with inverse v and that

2.- Let r G IV such that r > 2 and C the vector space generated by the setj, ..., a£, 6^, ..., b£ : n £ IV} with comultiplication and counit defined by:

A(<) = oj, ® 4 e(4) - 1

for alH = 1, ..., r and n € IV. C is a pointed coalgebra and then Pic(C) = Out(C).A similar argument to the above one yields: Pic(C) = Sr, the symmetric group ofr letters.

3.- We consider the same vector space as above but with comultiplication andcounit given by:

= 0

for all i = 1, ...,r — 1 and n £ IV. C is a pointed coalgebra and hence Pic(C) =Out(C). Using a similar reasoning as in 1, it can be computed that Pic(C) = 2Zr,the cyclic group of order r.

Suppose that the coradical of C is finite dimensional. If C has the Aut-Picproperty, does it follow that C* the Aut-Pic property? The following result answersthis question.

THEOREM 3.2.9 Let C be a coalgebra with coradical of finite dimension.i) If C has Aut-Pic and Pic(C) = Pic(C*) via [M] -> [M*], then C* has the

Aut-Pic property.ii) If C* has the Aut-Pic property, then C has the Aut-Pic property.

PROOF: i) Let [M] £ Pic(C*), since Pic(C) ^ Pic(C*), then there is a (C, C}-bicomodule JV such that M = N* as (C* , C**)-bimodules. As C has the Aut-Picproperty, we can find g £ Aut(C) such that N = 3Ci. Then N* ^ g,C{ and thusM* £* g.Ct as (C*,C'*)-bimodules.

ii) Let [M] € Pic(C), then [M*] e Pic(C*} and as C* has the Aut-Pic property,there is an automorphism / in C* such that M* = fC^. Let $ : C* —> M* theisomorphism of right C"*-modules. We put n* = $(e) and we define * : M —> C,m i—» E(m)(n*>m(o)}m(i)- Then ^ is a map of right C-comodules which verifiesthat <]/* = $. Thus, \& is an isomorphism of right (7-comodules and hence there isg 6 Aut(C) such that M = 5Ci as bicomodules. So, C* has the Aut-Pic property.

COROLLARY 3.2.10 Suppose that C is a finite dimensional coalgebra. Then, Chas Aut-Pic if and only if C* has Aut-Pic.

COROLLARY 3.2.11 Let C be a cocommutative coalgebra with finite dimensionalcoradical. Then, C* has the Aut-Pic property. Moreover, if C is coreflexive, then

PROOF: We know that J = CQ is the Jacobson radical of C* and C* / J = CQ. SinceCQ is finite dimensional, we have that C* is a semilocal ring and hence Picent(C*)is trivial. Now, [4, Prop. 1.5] claims that a commutative ring has the Aut-Picproperty if and only if its Picent is trivial. Hence, C* has the Aut-Pic property andso Pic(C*) ^ Aut(C*).

If C is in addition coreflexive, by [22, Prop. 7.1], the map from Aut(C) toAut(C*), f ^ f* is an isomorphism. Thus, Pic(C) £* Aut(C) ^ Aut(C*} ^Pic(C*). I

EXAMPLE 3.2.12Let C be the power divided coalgebra, that is, the fc-vector space generated by

the set {co,ci,C2, ....} with comultiplication and counit given by:

A(cn) = £)"=! c% 0 cn^i, e(cn) = S0<n-

for all n e W where S means the Kronecker symbol. It is well-known that C*is isomorphic to the power series ring fe[[x]] and that C is a coreflexive coalgebra.Using the above proposition Pic(C) £* Aut(C) ^ Aut(k[[x}}) ^ Pic(k[[x]]).

When a coalgebra C has the Aut-Pic property, the Picent of C is also described interms of automorphisms. Set Autz(c)(C~) and Innz(c}(C] for the groups of Z(C}-automorphisms and Z(C*)-inner automorphisms, respectively and Outz(c)(C) =Autz(c)(C}/InnZ(C](C}.

PROPOSITION 3.2.13 If C has the Aut-Pic property, then

Picent(C) ^ Outz(C)(C).

PROOF: Let [M] € Picent(C) C Pic(C). By hypothesis there exists / e Aut(C)such that M = SC\ as (C, C)-bicomodules. Since [M] e Picent(C), r(ld ® !)(/ igi1)A = (1 ® ld)A, that is, for c G (7, we have:

Applying e <g> 1, l d f ( c ) = ld(c) and this just means that / is a map of Z(C)-coalgebras. Hence Picent(C] = Outz(c)(C). I

To finish this section, we observe that in the cocommutative or cosemisimplecase the Picent is trivial. This shows lights differences with the ring case. In thecocommutative case, this difference arises from the decomposition of any cocommu-tative coalgebra as direct sum of irreducible subcoalgebras. We will see in the nextsection that this fact is also very important for the Brauer group of a cocommutativecoalgebra.

PROPOSITION 3.2.14 Let C be a cosemisimple or cocommutative coalgebra,then Picent(C) is trivial.

PROOF: From [24, Lem. 2.12] we get that Picent(C) ^ Hi€l Picent(Ci) when{Ci} is a family of subcoalgebras such that C = ® i6/Ci- Suppose that C iscosemisimple, using this remark, it is enough to prove that Picent(C) is trivialwhen C is a simple coalgebra. But if C is simple, C is finite dimensional andPicent(C) £* Picent(C*). Now, as C* is simple, by [9, p. 375], Picent(C*) = {!}.Hence, Picent(C) = {!}.

As C is cocommutative, C has. Aut-Pic and by the above proposition

Picent(C) ^ Autc(C) = {!}.

4 THE BRAUER GROUP OF A COCOMMUTATIVE COALGEBRA

4.1 Definitions and properties

We recall from [25] the construction of the Brauer group for a cocommutative coal-gebra and its most important properties. Proofs of the results may be looked up in[25].

In this section, C is a cocommutative coalgebra. Let I? be a C-coalgebra withC-counit denoted by e. The universal property of the cocenter implies that there isa unique coalgebra map TJ : Z(D) —> C such that e = r j l d .

DEFINITION 4.1.1 D is said to be cocentral if the map r] is an isomorphism ofcoalgebras.

Putting De — DOcDop, D may be viewed as a right or left De-bicomodule.

DEFINITION 4.1.2 D is coseparable over C or C-coseparable if it verifies one ofthe following equivalent conditions:

i) There is a map TT : DOCD —> D such that ?rA = Irj.ii) D is injective as a right De-bicomodule.

DEFINITION 4.1.3 A C-coalgebra D is called Azumaya over C or a C-Azumayacoalgebra if it is C-coseparable and cocentral.

The following proposition lists some properties of Azumaya coalgebras.


PROPOSITION 4.1.4 LetD,E be Azumaya coalgebras over C . Then,i) Dop and DOCE are Azumaya coalgebras.ii) If P is a quasi-finite injective cogenerator, then e_cr(P) is an Azumaya coal-

gebra.in) If C1 is a cocommutative coalgebra and f : C' — > C is a map of coalgebras,

then Da^C' is an Azumaya coalgebra over C".

Let D be a coalgebra and Z(D) its cocenter. Since Z(D) = e_£>e(D), D maybe viewed as an (Z(D), De)-bicomodule. Let /, g denote the canonical maps

/ : Z(D) -* DDD,h^D.(D,De), g : De -+ h_D*(D,D*)Uz(D]D.

Then (Z(D), De, D, hDe(D,De), f , g ) forms a M-T context. Azumaya coalgebrascan be characterized in terms of Morita-Takeuchi equivalences.

THEOREM 4.1.5 Let D be a C -coalgebra. The following are equivalent:i) D is an Azumaya coalgebra.ii) The above M-T context is strict.Hi) D is a quasi- finite injective cogenerator as left C -module and ec-(D) = De.

Denote by B(C) the set of the isomorphism classes of Azumaya C-coalgebras.An equivalence relation is introduced in B(C) as follows: if E, F £ B(C), thenE is equivalent to F, denoted by E ~ F , if there exist two quasi-finite injectivecogenerators M, N in M° such that

^ FDce_c(N),

as C-coalgebras.

THEOREM 4.1.6 The quotient set B(C)/ ~, denoted by Br(C), is an abeliangroup with the multiplication [E][F] = [EO^F], unit element [C] and for [E] theinverse is [E°p\. The group Br(C) is called the Brauer group of the cocommutativecoalgebra C.

If T] : C' — > C is a map of cocommutative coalgebras, then 77 induces a grouphomomorphism 77, : Br(C) -> Br(C') given by rj*([E]) = [EUCC'} for all [E] €Br(C). Thus we have defined a contravariant functor from the category of cocom-mutative coalgebras to the category of abelian groups.

The following proposition claims that the equivalence relation in B(C) is indeedthe Morita-Takeuchi equivalence relation.

PROPOSITION 4.1.7 [D] = [E] € Br(C) if and only if D and E are Morita-Takeuchi equivalent coalgebras over C .

If the coalgebra C is of finite dimension, then the study of the Brauer group ofC is equivalent to the study of the Brauer group of the dual algebra C*.


PROPOSITION 4.1.8 Let C be a coalgebra of finite dimension and D a C-coal-gebra. Then,

i) D is an Azumaya coalgebra if and only if D* is an Azumaya algebra over thedual algebra C* .

ii) The map (— )* : Br(C] — > Br(C*), [D] t— > [D*} is a group isomorphism.

The following results study the behaviour of the functor Br(-) with respect totaking direct sums.

LEMMA 4.1.9 Suppose that there is a family of co commutative coalgebras {Ci}j6/such that C = ®ie/ Ci. Then,

i) Every C -coalgebra D is of the form ®i6/ A where Di = DOCC^ is a C;-coalgebra. Conversely, if the Di's are Ci- coalgebras, then D = (Big/ A is a C-coalgebra and Di = Docd.

ii) Let D,E be C -coalgebras. Then D is Morita-Takeuchi equivalent over C toE if and only if Di is Morita-Takeuchi equivalent over Ci to Di for all i £ I.

in) A C -coalgebra D is C- Azumaya if and only if all the Di are Ci- Azumaya.

THEOREM 4.1.10 Let C = ®ieICit then Br(C] ^Hi€l Br(Ci).

PROOF: In light of the above lemma, we may define a map

13 :

[D] i— > IIig/[A] with Di = DO^Ci. This map is well-defined in view of the abovelemma and because the relation in B(C] is a M-T equivalence relation. Suppose thatnie/[A] = 1 € Hie/ Br(Ci). Then, from Proposition 4.1.7, it follows that A is M-T equivalent to Ci for all i e /. Now Lemma 4.1.9 entails that D is M-T equivalentto C and then [D] = [1] in Br(C). Lemma 4.1.9 also yields that j3 is surjective.That J3 is a group homomorphism is deduced from the fact (DOcE}i = DiO^Eifor alH € / and Azumaya coalgebras D, E. I

Remark: This decomposition has two important consequences:

1) In general Br(C) is not torsion. Let <$ be the rational number field and Cthe group like coalgebra C = @n^j^ <?• It is well-known that for any n 6 IV thereis [An] (E Br(Q) of order n. The coalgebra A = ®n6^ A*n is C-Azumaya, and [A]does not have finite order in Br(C}.

2) To compute the Brauer group of a cocommutative coalgebra it is enough tocompute the Brauer group of irreducible coalgebras.

In view of 2) we may assume that C is cocommutative irreducible and studythe Brauer group in this case. For an irreducible C, the following appeared in [25,Prop. 4.10]:

LEMMA 4.1.11 Let C be an irreducible coalgebra, and D an Azumaya C -coalgebrawith C-counit e.

i) D is irreducible and D* is Azumaya over C* .ii) D is finitely cogenerated free as right C-comodule via e.iii) The map (—)* : Br(C) — > Br(C*), [D] H-> [D*] is a group homomorphism.


If the coalgebra C is in addition coreflexive, it was proved in [25, Prop. 4.12]that the map (-)* : Br(C) -» Br(C*} is injective. Moreover, in this case, if C0denotes the coradical of C, then induced map z* : Br(C) — > Br(Co) for the inclusionmap i : CQ — > C is injective. In [26] a cohomological interpretation of the Brauergroup allowed to prove that both these maps are isomorphisms.

PROPOSITION 4.1.12 Let C be a cocommutative coreflexive irreducible coalge-bra. Then, the maps i* : Br(C) -> Br(C0) and (-)* : Br(C) -> Br(C*) areisomorphisms.

COROLLARY 4.1.13 If C is a cocommutative coreflexive coalgebra, then

PROOF: We write C as direct sum of irreducible coalgebras C = ©i€/ Ci. EveryCi is irreducible and coreflexive. From the foregoing proposition it follows thatBr(Ci) = Br(Cio) where CJQ denotes the coradical of Ci. Now,

Br(C] = Br(Ci) = Br(Ci0) ~ 5 r ( C i 0 ) ^ Br(C0),iel iel i€l

since C0 = @ieICiQ. I

With these results we can provide new examples of Brauer groups of coalgebras.

EXAMPLE 4.1.141.- The Brauer group of a coreflexive coalgebra over the real number field 1R is

isomorphic to a direct product of copies of Zj2- Let C be a cocommutative core-flexive coalgebra, it admits a decomposition as direct sum of irreducible coreflexivecoalgebras C = ®ie/C;. From Theorem 4.1.10, Br(C) ^ Y[iel Br(Ci). For everyirreducible subcoalgebra its coradical is isomorphic to JR or (F, the complex numberfield. Using that Br(]R) ^ %2, Br((H) is trivial and Br(Ci) ^ Br(Ci0) by theabove proposition, Br(Ci) is trivial or isomorphic to -2^2 for all i 6 /.

2.- If C is a coreflexive coalgebra over either an algebraically closed field or afinite field, then Br(C) is trivial. It is proved using a similar argument as aboveand noting that the Brauer group of either an algebraically closed field or a finitefield is trivial.

3.- Let V be a finite dimensional vector space over a field of characteristic zeroand C the symmetric algebra over V. From [21, Prop. 11.0.10], [14, Th. 3.4.3], Cis connected and coreflexive. From Corollary 4.1.13, Br(C) == Br(k).

4.- Let i be a Lie algebra over a field of characteristic cero and C its universalenveloping algebra U(L). By [21, Prop. 11.0.9], [14, Th. 3.4.3], C is connected andcoreflexive. Then, Br(C) ^ Br(k).

5.- Let A be a commutative noetherian algebra. From [14, Th. 3.3.3], the finitedual coalgebra A° is a coreflexive coalgebra. Using 4.1.13, Br(A°) = Br(A^). If inaddition A is pointed, then A° is pointed and so Br(A°] = FJj Br(k) for a suitableset /.



Several questions were pointed out in [25].

1.- Is the Brauer group of an irreducible coalgebra torsion? It is tempting toconjecture that for a cocommutative coalgebra, the non-torsion aspect comes fromthe product decomposition but that for each irreducible subcoalgebra the Brauergroup is a torsion group.

2.- For an irreducible coalgebra C, is the map (-)* : Br(C) —> Br(C*) injective?When is it an isomorphism?

In the coming subsection we prove that these questions have affirmative answers.We finish this subsection with a version of the Auslander's problem for coalgebras.

The Auslander's Problem for Coalgebras

Let C be a cocommutative irreducible coalgebra. When is the map z* : Br(C) —»Br(Co) an isomorphism? We know that it is true if C is coreflexive. We conjecturethat this map is always an isomorphism.

4.2 Torsioness in the Brauer group

In this section we proved that the map (—)* : Br(C) —> Br(C*) is injective. SinceBr(C) embeds in a torsion group, Br(C) is torsion. Thus we solve positively thequestions presented before. In order to do this we need some preliminaries.

Let C be a coalgebra with dual algebra C*. Every right C-comodule may beviewed as a left C"*-module (see [21],[1]). The left C"*-modules arising from a rightcomodule are called rational C*-modules and the full subcategory of rational leftC*-modules is isomorphic to M.c.

Let M be a left C*-module and m G M. We say that the element m is rationalif there are families {mi}f=1 C M and {ci}™=1 C C such that

• in =

The set of rational elements of M, Rat(M), is a submodule of M. The functorRat(-) : c'M —»• c*M, M i-» Rat(M) is a left exact preradical (see [16, page371], [12]). It is well known that there is a bijective correspondence between leftexact preradicals, hereditary pretorsion classes and left linear topologies (see [20,VI, Prop. 4.2]). The left linear topology associated to Rat(-) is the class f of allcofinite closed left ideals of C*. We remember that a left coideal / of C* is calledcofinite if C* /I is finite dimensional as vector space. / is called closed if there existsa subspace W of C such that I = W^^ '. Hence / is cofinite closed if and only ifthere is a finite dimensional subspace W of C such that I = W^c \ The followingwas observed in [16, page 371].

PROPOSITION 4.2.1 Let C be a coalgebra and C* its dual algebra.i) The class J- of all cofinite closed left coideals of C* is a linear topology. That

is:a) If I e J- and J is any left ideal of C* such that I C J, then J € J-.


b) If /, J € JF, then I n J £ JF.cj /// 6 JF ana7 c* £ C*, i/ien (/ : c*) e JF.

nj If M is a left C*-module, Rat(M) = {m € M : annc- (TO) e JF}.mj Aic = {M 6 C..M : Rat(M) = M}.

Observe that if C is an infinite dimensional coalgebra, {0} ^ JF. In the sequel wewill assume this. J- is also symmetric, that is, any ideal in JF contains a two-sidedideal that belongs to J-.

PROPOSITION 4.2.2 Let C be a coalgebra, C* its dual algebra and M & C,M.Then, Rat(M) = {m £ M : annc.(C* • m) £ JF}.

PROOF: Let m £ M such that annc-(C* • m) £ T . Since annc*(C* • m) Cannc'(m), it follows that annc- (TO) £ JF. By the above proposition, m € Rat(M).

Conversely, let TO £ Rat(M), then annc»(m) 6 JF. Let {i>*}i€/ be a basisof annc'(m). We extend this to form a basis of (7*. We only have to add afinite number of elements {y*, ...,y£} because annc*(m) is cofinite. We claim thatannc*(C* • m) = n™=i(annC" (m) : y*).

Let d* £ annc'(C* • m), then 0 = d* • (y* • TO) = (d* * y*) • TO for all z =l,...,n, where * denotes the product in C* . Hence d* * y* £ annc* (TO), that is,d* £ (annc-(m) : j/*) for all z = l,...,n. Thus, d* £ P|"=1(annc.(m,) : T/*).Suppose now that e* £ Pl™_1(annC7. (TTI) : y*) and let c* be in C*. We can writec* = ELi M, + £?=! Ay* with A,, ft e fc. Then,

r n

e* • (c* • TO) = V^ Aje* • (v*. • TO) + Y^ A(e* * y*) • m = 0,i=l i=l

since Vji , e* * y* £ annc* (TO.). Hence e* G annc* (C* • m).Since annc*(m) S JF; (annc* '• y*) 6 -^ f°r all * = 1) • • • > n and so annc*(C* •

(m) : y*) e JF. I

LEMMA 4.2.3 Le^ C be a cocommutative coalgebra, D a C-coalgebra with C-counite, and suppose that D is finitely cogenerated free as right C-comodule via e. If I isa cofinite closed ideal in C* , then D*e*(I) is a cofinite closed two-sided ideal in D* .

PROOF: Since D is finitely cogenerated free as right C-comodule via e, there is afinite dimensional vector space W such that D = W <g> C as right C-comodules.Let h : D — > W ® C be this isomorphism. Let d £ D, and suppose that h(d) =Y?J=IWJ®CJ with Wj e W,Cj e C, then pw®c(h(d)) = (h®l)pD(d) = (/i<8>e)A(cZ).This yields the equality:

Let / be a cofinite closed ideal in C* , then there is a finite dimensional subspace V ofC such that / = V^ . J is a two-sided ideal of C* because C* is commutative. By [21,

Prop. 1.4.3], V is a subcoalgebra of C. We are going to prove that h*(W* 0 V^) =D*e*(V -1).

Let (p e h*(W* 0 I/-1), then there is ̂ e VF* 0 V1- such that p - h*(ip). Wecan write i/j = Y^i=i wl ® & with w* £ W7* and ̂ € V1- for all z = 1, ..., n. Then,

Set (i* = h*(w* 0ec) for i = l,...,n. Then,

= E"=i E(d)« ® £c-> h(d(i)))(ipi, e(d(2)))

= E"=i EJ=i E(C.,.)

Hence, ^ = ̂ =1 dje*(^) e ̂ e^^1). Thus, /i*(W* ® I/-1) C D*eBy linear spaces arguments, it is easy to check that

h*(W* ® F-1) = /i*((W ® V)^) = h'l(W ® F)1.

We check that D*e*(V-L) C h~l(W ® V}^ . Let x* e D*e*(VjL], then there isd* e -D*,Vi € yx,i = l,... ,n, such that z* = E"=i d*e*(Vi)- Let d^ h~l(W ®V),then /i(d) e W <g> I/. Set /i(d) = Yfj=\ wj ® ci with wi € '̂ cj e ^- Since ^ is a

subcoalgebra of C and Cj e V, then Cj(2) € V for all j = 1, ..., s.

== E"=i EJ=i

We conclude that /i-^W7 0 K)1 = ^(V^D*. The space /j-^W 0 F) is finitedimensional because W 0 V is so. Thus we have proved that D*e*(I) is a cofmiteclosed two-sided ideal. I

LEMMA 4.2.4 Let C be a cocommutative coalgebra, D,E C-coalgebras and Ma (E, D)-bicomodule. M is a (E,D)-bicomodule over C if and only if M* is a(D* ,E*)-bimodule centralized by C* .

PROOF: Let erj : D — > C and e^ : E — > C the C-counits of D and £ respectively.D* and S* are C"*-algebras via e*D and ej; respectively. M* is a (D*, S*)-bimodulevia,

, m - e , m =(m) (m)

for all TO e M,tf* G Z?*,e* e S* and m* e M*.

If M is a bicomodule over C, then Y^(m)m(o) ® £^(^(-1)) = £(m) m(o) ®ez?(m(i)) f°r aH rn £ M. Given c* e C*,m* £ M* and TO e M, we have,

{ej;(c*)-m*,m} = £(m){e|;(c*), TO (_I))(TO*, TO(O)}= E(m){CVs(m(-l)))(m*>m(0)}

= E(m)(C*> eD(m( l)))(T O*>m(0)}

= £(m){e£>(c*)>m(l))(m*>TO(0)}

= (m* •e*D(c*),m).

Thus e^c*) • m* = m* • e^(c*) for all c* e C*, m* e M*. That is, M* is centralizedby C*.

Suppose now that M* is centralized by C*, then f*E(c*) • m* = m* • e*D(c*) forall c* e C*,m" e M*. So,

(m) (m)

for any m € M.We can view M ® C embedded in (M* <g> C*)* via the map A : M <g> C -+

(M* OC'*)*,(A(m(8)c) !m* ® c*) = (TO*,TO}{C*,C) for all m* e M*,c* € C*. Form e M,

i))) ,ra*®c*} =E (m)(m*,TO (0)}{c*,£E(m (_1))}

) TO(0) ® £D(m ( 1 ))), m* ig) c*},

for all 77i* € M* and c* £ C*. Since A is injective, we obtain

(m) (m)

for all TO € M. This just means that M is a bicomodule over C. I

THEOREM 4.2.5 Let C be cocommutative irreducible coalgebra. The group ho-momorphism ( — )* : Br(C] — > Br(C*), [A] H^ [A*] is injective.

PROOF: Let A be an Azumaya C-coalgebra such that [^4*] = [C*] in Br(C*}. Theequivalence relation in Br(C*) is the Morita equivalence relation over C*, see [19,Ex. 2.19]. So that, A* and C* are Morita equivalent algebras over C*. We haveequivalences over C1*,

Fc>M-+ —— - A'M.

GMorita's theorem yields the existence of a (A*, C*)-bimodule P and a (C*, A*)-

bimodule Q such that:

Since F and G are equivalences over C* , then P and Q are centralized by C* (see[3, page 57]). Let e : A — > C be the C-counit of A, then A* is a C"*-algebra via theinjective map e* : C* — > A*.

We see that F(Aic) C At A. Let M £ Mc arbitrary. By Proposition 4.2.1 ii),iii)we are going to check that ann^*(x} is cofinite and closed for all x 6 F(M). Letx be in F ( M ) , then x = Yî-\Pi ® mi f°r Pi € P, TOJ e M. From Proposition 4.2.1ii), iii), annc> (mi) is cofinite and closed for all i = 1, ...,n. I = P)r=i annc*(ci) is acofinite closed ideal of C*. We write / = e*(I)A*. Since ^4 is an Azumaya coalgebraover (7, by Proposition 4.1.11, A is finitely cogenerated free as right C-comodule.By Lemma 4.2.3, / is a closed two-sided ideal of A* . Let y € /, then there is aj e A*and c* e 7 such that y = ]CJli aje*(cj)-

m n m n m n^ — ̂ ^ — ̂

y ' x =J — 1 Z— 1

where in the second equality we have used that P is centralized by C* , and in thelast equality that c* vanishes all the m^s. Hence / is a cofinite closed ideal in A* suchthat I • x = {0}. Then / C anriA->(x), and we conclude that annA,(x] is cofiniteand closed. So we have proved that F(M) & MA, and thus, F(M°) C MA.

We now check that G(MA) C Mc '. Let N e MA and a; e G(N), then x =Yî=i 1i ®ni with qtj e Q,rii € ^V for all i = 1, ...,n. Since .AT € Ai"4, ann_4.(A* • nj)are cofinite closed two-sided ideals by Proposition 4.2.2. / = HILi annA*^* -Hi) isa cofinite closed two-sided ideal of A*. Since A* is an Azumaya C*-algebra, thereis a bijective correspondence between ideals of C* and two-sided ideals of A* givenby (see [19, Cor. 2.11]):

C* -* A*, JH-> JA*, A* ^C*, A-î fne*(C*) .

Let / = /ne*(C*). Let J be an ideal of C* such that e*(J) = /. Then, given y e J,e*(y) € / and,

n n n

ft (g) nj = P gj£* (y) (8)^ = ^(81 (e* (y) • n;) = 0,

where we have used that Q is centralized by C* and that e*(j/) 6 /. Hence itvanishes all the o^s. So we have that J C annc*(x). We see that J is cofinite andclosed.

Since / is cofinite and closed, there is a finite dimensional subspace W of Asuch that / = WÂ'\ We claim that J = e(W)^c"> . Let 9? e e(W)^c"> C C*and 10 e W, then (e*^),™} = (</?,e(iy)} = 0. So e*(<^>) € /ne*(C7*) = 7 and then<f e J. Conversely, if </? € J, £*(</?) € /. Let v e ^(W7), then there is w & W suchthat v = e(w). Now, (<p,v) = ((p,e(w)} — (e*(if>),w) = 0. e(W) is finite dimensionalbecause W is so. Hence J is cofinite and closed. It follows that annc* (x) is cofiniteand closed. Thus G(N) e Mc ', and so G(MC) C Al-4.

Finally, F(MC) C AÂ and G(MA) C yWc. By Theorem 2.0.3 b), C is stronglyequivalent to A via F' = F\Mc and G' = G\MA. In view of Theorem 2.0.2 a),

let M be a (C, A)-bicomodule such that F'(-) = -DCM. From Theorem 2.0.3a), A*M*j. ®c* - = P ®c* -• Then, M* ^ P as (A*, C*)-bimodules. Since Pis centralized by C* , M* is centralized by C* . Lemma 4.2.4 yields that M is a(C, j4)-bicomodule over C and so F' is a Morita-Takeuchi equivalence over C. Fromthe Proposition 4.1.7, [A] = [C] in Br(C). Thus we have proved that the map(-)* : Br(C) -> Br(C*) is injective. I

It is known that the Brauer group of a commutative ring is torsion, see [19, Th.12.9]. Then Br(C*) is a torsion group. By the above theorem we have that:

COROLLARY 4.2.6 The Brauer group of a cocommutative irreducible coalgebrais a torsion group.

4.3 Subgroups of the Brauer group

In the classical theory of the Brauer group of a commutative ring important sub-groups related to group theory appear, the Schur and the projective Schur sub-groups. It is natural to ask which are the corresponding subgroups to these in theBrauer group of a coalgebra. A Schur and a projective Schur subgroup can bedefined for cocommutative coalgebras by using crossed coproducts. The interestof these subgroups is that Azumaya coalgebras represented by crossed coproductshave a nice structure which comes from the good structure of the crossed coprod-ucts. For example, the Brauer group of a coalgebra may not be torsion, howeverthe Schur subgroup is a torsion group.

Let C be a cocommutative coalgebra and G a finite group with identity elemente. We regard the Hopf algebra H = (kG)* with basis {pg : g S G} dual to the basisof kG; that is pg(h) = Sg h V<?, h € G. The comultiplication and counit are definedby:

Let a : C — > (kG)* <8> (kG)* be a linear map expressed in the following way:

Vc G C,

with ax>y e C* for all x,y £ G. Let C XQ (kG)* be the vector space C <S> (kG)*and we define a map:

Aa(c Xph) = Eaa,a-ift(c3)ci X Pa. ® C2 Xpa-i/i.

C Xa (kG)* is said to be a crossed coproduct if AQ is coassocitative and £c®£(kG)*is a counit.

LEMMA 4.3.1 C XQ (kG)* is a crossed coproduct if and only if the followingconditions hold:

(CU) Normal cocycle conditionag<e(c) = ae,g(c) = e(c) \/g^G,c&C.

(C) Cocycle condition

Vs, r, q e G, c € G.

This construction is a particular case of a more general construction where (kG)*is replaced by an arbitrary Hopf algebra H and H coacts weakly on C. All the detailsof this and an interesting connection of crossed coproducts with Galois coextensionmay be found in [10].

A map a : C — > (kG)* ® (kG)* verifying (C) and (CU) is said to be a cocycle. Ifthe cocycle a is convolution invertible we say that C Xa (kG)* is a twisted groupcoalgebra. The map ea = (1 <8> e) : C Xia (kG)* — > C makes C XQ (kG)* intoa G-coalgebra. The following proposition lists some properties of twisted groupcoalgebras, see [6].

PROPOSITION 4.3.2 Let C,D be cocommutative coalgebras, G,H finite groupsand a : C — > (kG)*®(kG)*, /3 : C —> (kH)* ® (kH)* convolution invertible cocycles.Then,

i) aop : C -> (kG)*°P ® (kG)*0? £* (kG°P)* <g> (kG0?)* defined by aop(c) =Sz yeG ay,x(c)Px ®py is a convolution invertible cocycle.

ii) (C XQ (kG)*)op ^ C Xa°P (kGop)* as C-coalgebras.Hi) The map a x /? : C — » (kG x H)* <g> (/cG x H" )* defined & 2 / a x / 3 = ( l < g > T < g >

l)(a <g>/3)A, where r is the twist map, is a convolution invertible cocycle.iv) (C Xa (kG)*)Oc(C Xp (kH)*) ^ C XQX/3 (kG x H)* as C-coalgebras.v) If f : D — > C is a coalgebra map, then the map a : D — > (kG)* <g> (kG)* given

by a(d) = a f ( d ) is a convolution invertible cocycle.vi) (C Xa (kG)*)OcD ^ D X5 (kG)* as D-coalgebras.vii) a* : kG ® kG — > C* is a normalized cocycle and a*(kG ® kG) C U(C*).viii) (C X\a (kG)*)* ^ C* *Q. kG as C* -algebras.ix) C XQ (kG)* is C- co separable if and only if [G]"1 6 k.

From now on, unless otherwise stated, Ji is a class of groups closed under finiteproducts and taking opposites.

DEFINITION 4.3.3 Let A be a C-coalgebra. We say that A is a projective SchurC -coalgebra relative to TL (H-PSC) if A is C-Azumaya and there exists a twistedgroup coalgebra C XQ (kG)* with G € H and an injective C-coalgebra map i : A — >C XQ (kG)* . When a is trivial, i.e., a(c) = J^ yeG£(c)px ®py, A is called SchurC-coalgebra relative to H (H-SC).

PROPOSITION 4.3.4 The set PSH(C) = {[A] 6 Br(C) : A is H - PSC} isa subgroup of Br(C). This subgroup is called the H-proyective Schur subgroup ofC.

PROOF: Let [A], [5] e PSH(C), then there are twisted group coalgebras C Xa

(kG)* , C x/3 (kH)* with G, H 6 H. and injective G-coalgebra maps i : A —> C XQ(kG)* and j : B -> C Xp (kH)* . The map J°P : Bop -> (G Xp (kH)*)op is aninjective G-coalgebra map and (G Xp (kH)*)op ^ C X()°P (kHop)* by Proposi-tion 4.3.2 ii). Using the left exactness of -DCB°P and C XQ (kG)*ac- it followsthat zDl : AOCB°P -> C Xa (kG)*ncBop and lOjop : C Xa (kG)*ncBop ->

C XQ (kG)*OcC Xpop (kHop)* are injective C-coalgebra maps. From Propo-sition 4.3.2 iv), C XQ (kG)*acC X^P (kHopY = C Xaxf3oP (kG x Hop)*and the composition (lDjop)(iOl) is an injective C-coalgebra map from AOCB°P

to C XJQX/3°p (kG x Hop)*. By hypothesis on H, G x Hop e U and hence[A][B]°P = {ADCB°P} e PSH(C). I

COROLLARY 4.3.5 The set SH(C) = {[A] e Br(C) : A is U- SC} is asubgroup of Br(C) called T-i-Schur subgroup of C .

If H is the class of finite groups then we denote PSH(C) and SH(C) simplyby PS(C) and S(C). PS(C) is called the projective Schur subgroup of Br(C) andS(C) the Schur subgroup.

Let [D] e PSU(C] with injective C-coalgebra map i : D -> C XQ (kG)* whereC Xa (kG)* is C-coseparable. The set of this classes, denoted by PS^(C), is asubgroup of PS(C) since the cotensor product of two C-coseparable coalgebras isagain C-coseparable. All the above definition hold for

PROPOSITION 4.3.6 Let rj : D —=> C be a map of cocommutative coalgebras, thent] induces group homomorphisms 77* : PSW(C) — > PSH(D) and 77*, : 5W(C) — *S^(D). Hence, P5W(— ) and 5W(— ) are contravariant functors from the categoryof cocommutative coalgebras to abelian groups.

PROOF: We know that there is a group homomorphism 77* : Br(C) —+ Br(D),[A] H-> [AOCD]. We show that the restriction of 77* to PSH(C) has its imagecontained in PSH(D). Let [A] & P5W(C), then there is a twisted group coalgebraC X\a (kG)* and an injective C-coalgebra map i : A — > C Xa (kG)* for someG 6 7Y. By the left exactness of — OCD, we have an injective Z?-coalgebra mapiOl : AncD -> C XQ (fcG)*Dc£). Since C xa (fcG)*DcD ^ D Xfe (fcG)* byProposition 4.3.2 vi), it follows that [AUCD] € PSH(D). I

As in the Brauer group, when the coalgebra is finite dimensional, the study ofthe 7i-projective Schur is equivalent to the study of the W-projective Schur subgroupof the dual algebra C* .

PROPOSITION 4.3.7 Suppose thatC is a finite dimensional cocommutative coal-gebra and let A be a C-coalgebra. Then,

i) A is a projective Schur (resp. Schur) C-coalgebra relative to Ti if and only ifA* is a projective Schur (resp. Schur) C* -algebra relative to TL.

n) PSH(C) ^ PSH(C*) (resp. SH(C) ^ Sn(C*)) mapping [A] ^ [A*].

The behaviour of the Ti-projective Schur functors is not so good as the Brauerone as the following proposition shows. However, this result will be useful sometime.

PROPOSITION 4.3.8 Let {Ci}i€l be a family of subcoalgebras of C such thatC = ®i€lCt. Then, PSH(C) -> Tli6/ PSH(d) and SH(C) <-+ Yli&1 ̂ (Q).

PROOF: By Theorem 4.1.10, the map 77 : Br(C) -> Hi€l Br(Ci), [D] >-> Y[i€l[Di]is a group isomorphism. Viewing 77 restricted to P5W(C) we obtain a groupmonomorphism, and its image is contained in Hig/ P5^(Cj). Let D be a Ji-projective Schur coalgebra, reasonig as in Proposition 4.3.6 for each C{ we obtain

that DI = DOcd is a proyective Schur Gj-coalgebra relative to H for all i s /.Therefore ILe/[A] € lL6, ̂ «(GZ). •

The nice structure of the twisted cogroup coalgebra makes the following resultpossible though the Brauer group is not torsion.

PROPOSITION 4.3.9 Let C be a cocommutative coalgebra over a field k, thenis a torsion group. If char (k) — 0 then S (C) is torsion.

PROOF: Let [D] € S^(G), then there exists an injective G-coalgebra map i : D — >C® (kG)* with G G T~i and JGJ" 1 G k. Using the universal property of the cocenterit may be proved that the cocenter of C ® (kG}* is C <g> Z(kG)*, where Z(kG)denotes the center of kG. kG is Azumaya over Z(kG) because \G\~l € k. Then wecan find n 6 IN such that (kG)n ^ Mm(Z(kG)) = Z(kG) ® Mm(k). Denote by Dn

the cotensor product of D n times. By Proposition 4.3.2 vi), there is an injectiveG-coalgebra map

i' : Dn -> C <g> (kGn)* ^ G <g> Mm(Z(kG))* ^ C <g> Z(kG)* <g> Mm(k}* .

We can consider Dn as a subcoalgebra of C ® Z(kG)* <g> Mm(k)* . Since G <g>Z(kG)*®Mm(k}* is an Azumaya G®Z(A:G)*-coalgebra, by [25, Cor. 3.17] there ex-ists a subcoalgebra C' of C®Z(kG)* such that Dn = C' ®Mm(k)* as C-coalgebras.Hence their cocenters are isomorphic, i.e, C = C' and thus Dn = C ® Mm(k)* asG-coalgebras. This means that [L>n] = [C] in Br(C).

If char(k) = 0, then the group coalgebra C <8> (kG)* is always C-coseparable byProposition 4.3.2 ix). Hence 5^(C) = SH(C). I

In the irreducible case, the injectivity of the map (— )* : Br(C) — > Br(C*)entails the following:

PROPOSITION 4.3.10 Lei C be a cocommutative and irreducible coalgebra thenthere are

injective group homomorphisrns PSU(C) -+ PSH(C*), and SH(C) -> SH(C*).

PROOF: Theorem 4.2.5 yields the existence of an injective group homomorphism(-)* : Br(C) -» Br(C*), [A] ̂ [A*]. The restriction of this map to P5W(G) isan injective group homomorphism and its image is contained in P5'W(C'*). Let[A] e PSH(C), then there is a twisted group coalgebra C XQ (kG)* with G <E Hand an injective C*-coalgebra map i : C X (kG)* — > A. Dualizing this map andusing Proposition 4.3.2 viii) we obtain a surjective C*-algebra map from C* *Q* Ginto A*. This fact combined with the fact that A* is G*- Azumaya (by Proposition4.1.11) leads to [A*] € PSH(C*).

For SH(C) we use the same argument taking into account that a* is trivial whena is trivial. I

We may deduce some properties of S(C) from properties of S(C*). The followingresult is an example of this.

COROLLARY 4.3.11 If C is a cocommutative coalgebra over a field with charac-teristic different from zero, then S(C) is trivial.


PROOF: We suppose that C is irreducible, then S(C) ^-> S(C*}. Since k has nonzero characteristic, C* is a fc-algebra of positive characteristic and applying [11,Prop. 1], we have that S(C*) is trivial. Hence S(C) is trivial. If C is not irreduciblethen, since it is cocommutative, there exists a family of irreducible subcoalgebras{Ci}iel such that C = ®ie/C;. From Proposition 4.3.8, S(C) ^-> Y[i€lS(Ci), butS(d) = {0} for all i 6 /, thus S(C) = {0}. I

COROLLARY 4.3.12 Let C be a cocommutative coreflexive irreducible coalgebra.If C0 is separable, then PSH(C) ^ PSH(C0) and SH(C) ^ SH(C0). In particular,ifC is connected, then PSn(C) ^ PSH(k) and SH(C) ^ SH(k}.

PROOF: Proposition 4.1.12 claims that the inclusion map i : CQ —-> C inducesan injective group homomorphism i* : Br(C~) —> fir (Co). Then, the restriction toPSH(C), z*» : PSH(C) -> PSH(C0) is also injective. Since C0 is separable, theMalcev-Wedderburn decomposition for C yields the existence of a coalgebra mapTT : C —> Co which splits the inclusion map i : CQ —> C, i.e., wi = Ic0- By thefunctorial properties of P5W(—), i**?!-** = \PSn^c}- Then z** is surjective andhence an isomorphism. I

EXAMPLE 4.3.13

1.- Let V be a finite dimensional vector space over a field of characteristic zeroand C the symmetric algebra over V. We know that C is connected and coreflexive.By the above corollary, PSH(C) ^ PSH(k), and SH(C) ^ SH(k).

2.- Let L be a Lie algebra over a field of characteristic cero and C its universalenveloping algebra U(L). Since C is connected and coreflexive, PS'W(C') = PS'!~i(k),and SH(C) ^ SH(k).

Acknowledgments

J. Cuadra is grateful to Professors J.R. Garcfa Rozas and B. Torrecillas for theircomments and ideas in the proof of Theorem 4.2.5.

REFERENCES

[1] E. Abe, Hopf Algebras, Cambridge University Press, 1977.

[2] M. Auslander and O. Goldman, The Brauer Group of a Commutative Ring,Trans. Amer. Math. Soc. 97 (1960), 367-409.

[3] H. Bass, Algebraic K-Theory, Benjamin, New-York, 1968.

[4] M. Bolla, Isomorphism between Endomorphism Rings of Progenerators, J. Al-gebra 87 (1984), 261-284.

[5] W. Chin and S. Montgomery, Basic Coalgebras, Collection Modular interfaces(Riverside, CA, 1995), 41-47.



[6] J. Cuadra, J.R. Garci'a Rozas and B. Torrecillas, Subgroups of the BrauerGroup of a Cocommutative Coalgebra. J. of Algebras and Representation The-ory, 3 (2000), 1-18.

[7] J. Cuadra, J.R. Garci'a Rozas, B. Torrecillas and F. Van Oystaeyen, On theBrauer Group of a Cocommutative Coalgebra, to appear in Comm. in Algebra,2001.

[8] J. Cuadra, J.R. Garci'a Rozas and B. Torrecillas, Outer Automorphisms andPicard Groups of Coalgebras, to appear Revue Roumaine de MathematiquesPures et Appliquees, 2000.

[9] C.W. Curtis and I. Reiner, Methods of Representation Theory with A-pplications to Finite Groups and Orders, Vol. II. Wiley, 1987.

[10] S. Dascalescu, S. Raianu and Y.H. Zhang, Finite Hopf-Galois Coextensions,Crossed Coproduct and Duality, J. Algebra 178 (1995), 400-413.

[11] F. DeMeyer and R. Mollin, The Schur Subgroup of a Commutative Ring, J.Pure and Applied Algebra 35 (1985), 117-122.

[12] J. Gomez Torrecillas, Coalgebras and Comodules over a Commutative Ring,Revue Roumaine de Mathematiques Pures et Apliquees 43 (1998), 591-603.

[13] J.A. Green, Locally Finite Representations, J. Algebra 76 (1982), 111-137.

[14] R.G. Heyneman and D.E. Radford, Reflexivity and Coalgebras of Finite Type,J. Algebra 28 (1974), 215-246.

[15] B. I-Peng Lin, Morita's Theorem for Coalgebras, Comm. in Algebra 1 No. 4(1974), 311-344.

[16] B. I-Peng Lin, Semiperfect Coalgebras, J. Algebra 49 (1977), 357-373.

[17] C. Nastasescu, B. Torrecillas and F. Van Oystaeyen, IBN for graded Rings. Toappear in Comm. Algebra.

[18] P. Nelis and F. Van Oystaeyen, The Projective Schur Subgroup of the BrauerGroup and Root Groups of Finite Groups, J. Algebra 137 (1991), 501-518.

[19] M. Orzech and C. Small, The Brauer Group of a Commutative Ring, LectureNotes in Pure and Applied Mathematics 11, Marcel-Dekker, New-York, 1975.

[20] B. Stentrom, Rings of Quotients, Springer-Verlag, 1975.

[21] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.

[22] E.J. Taft, Reflexivity of Algebras and Coalgebras, Amer. J. Math. 94 (1972),1111-1130.

[23] M. Takeuchi, Morita Theorems for Categories of Comodules, J. Fac. Sci. Univ.Tokyo 24 (1977), 629-644.

[24] B. Torrecillas and Y.H. Zhang, The Picard Groups of Coalgebras, Comm. inAlgebra 24 No. 7 (1996), 2235-2247.



[25] B. Torrecillas, F. Van Oystaeyen and Y.H. Zhang, The Brauer Group of aCocommutative Coalgebra, J. Algebra 177 (1995), 536-568.

[26] F. Van Oystaeyen and Y.H. Zhang, Crossed Coproduct Theorem and GaloisCohomology, Israel J. of Mathematics 96 (1996), 579-607.


Multiplication Objects

J. ESCORIZA and B. TORRECILLAS, Dpto. Algebra y Analisis Matematico,Universidad de Almeria. 04120-Almeria. Spain.

l: [email protected] and [email protected]

AbstractA concept of multiplication object in monoidal categories is given and

it is proved that it generalizes that of multiplication module, multiplicationmodule relative to a torsion theory and multiplication graded module. Itis proved that the endomorphism ring of an indecomposable multiplicationobject having the descending chain condition on multiplication subobjects islocal in any braided monoidal category.

1 INTRODUCTION

Multiplication modules over a commutative ring are narrowly related to finitelygenerated, projective or distributive modules, sharing important properties withsome of these families. As for commutative multiplication rings, they are of in-terest in multiplicative ideal theory (see [14]). Hereditary rings and von Neumannregular rings can be studied from the common perspective of being multiplicationrings. By defining the concept of multiplication module relative to a torsion the-ory we can study completely integrally closed domains as multiplication rings withrespect to the canonical torsion theory, i.e., the one induced by the height oneprime ideals, and therefore some divisorial properties can be investigated in thissetting. The graded concept of multiplication ring (a gr-multiplication ring) is use-ful in order to characterize graded rings which are multiplication rings (cf. [9]).Moreover, some arithmetically graded rings which are important in Algebraic Ge-ometry such as Dedekind graded domains or generalized Rees rings are examples ofgr-multiplication rings (cf. [26]). The study of multiplication objects in monoidalcategories provides us with a common framework to consider the concept in the dif-ferent above-mentioned situations. For example, in order to prove that a stronglygraded ring R is a multiplication ring, we can consider the Bade equivalence be-tween the category of graded /^-modules and the category of .Re-modules and then


174 Escoriza and Torrecillas

apply the properties of multiplication objects in these monoidal categories (cf. [9]and [10]).

Monoidal categories were introduced by Benabou and Mac Lane in 1963 (cf. [2]and [16]). They are, between others, responsible for the narrow relationship betweenQuantum Groups and Knot Theory and constitute the adequate framework forHopf algebra representations (see [13]). The concepts of multiplication module andmultiplication module relative to a torsion theory (cf. [5] and [6]) have already beenstudied from a categorical point of view in [8], where the concept of multiplicationobject in a commutative Grothendieck category is established. However, the naturaldefinition of multiplication graded module, also investigated in [8, Section 5], is notthe particularization of the definition of multiplication object in the category gr-_Rof graded right .R-modules and degree preserving homomorphisms. This problemis solved in this work where monoidal categories turn out to be the ideal setting toresearch into all these concepts, including that of multiplication graded modules.The aim of this paper is to study multiplication objects in monoidal categoriesand some particular cases such as the category of right C-comodules, C being acocommutative coalgebra or the category of Doi-Koppinen modules. The paper isorganized as follows: Section 2 is devoted to introduce monoidal categories, givingexamples of them and explaining their elementary properties. In Section 3, somegeneral properties of multiplication objects are obtained. The study is illustratedwith some special cases. In Section 4, we obtain some results on endomorphismsof multiplication objects in braided monoidal categories. We have proved thatthe endomorphism ring of an indecomposable multiplication object which verifiesthe descending chain condition on multiplication subobjects is local in any abelianbraided monoidal category. Throughout this work every category is abelian.

2 MONOIDAL CATEGORIES

A category C is said to be a monoidal category (some authors call this a tensorcategory) if there is a bifunctor <8>c : C x C —» C (called tensor product), an object I(called the unit of the monoidal category) and natural isomorphisms

a{A,B,c} • (A ®c B) ®c C -> A ®c (B ®c C),

1A : I ®c A -> A, rA : A <g>c I -> A,

for each A,B and C in C, verifying the following formulas, which will be brieflydenoted by (1), (2), (3) and (4) and where / : A -> A', g : B -> B1 and h : C -> C"represent morphisms in C.

f ®c (9 ®c h) o a{A>B^} = a{A',B',c'} ° (I ®C g) ®c h1A> o (17 <s>c /) = / o 1A, rAi o (/ ®c \i) = f o rA

(IA ®c IB) ° a{A,i,B} = TA ®c IBa{A,B,C®cD} ° a{A®cB,C,D} = (^A ®C a{B,C,D}) ° a{A,B®cC,D] ° (a{A,B,C} ®C ID)

Property (3) is known as the triangle axiom and property (4) is called the pentagonaxiom. In case the isomorphisms a{A,B,c}>^/i and rA are identities, C is said to

Multiplication Objects 175

be a strict monoidal category. The category defined above will be denoted by(C,®c, / ,a , / , r ) .

It is well-known that in every monoidal category,

(see, for example, [12, Proposition 1.1]). Another important fact for our purposesappears in [13, Proposition XI.2.4], where it is stated that the set of endomorphismsof the unit, End(I), is a commutative monoid for the composition. Moreover, iden-tifying I<S)cI with /, the tensor product coincides, in End(I), with the composition.

Given two monoidal categories C and D, with units / and /' respectively, afunctor F between them is said to be a monoidal functor if there exists an isomor-phism $0 : /' -» FI and a family of isomorphisms $2,A,B : FA ® FB — > F(A ® B),verifying the following equalities, which will be called ful, fu2 and fu3 respectivily:$2,A,B®C ° (Ip A ® $2,B,c)°a{FA,FB,FC} = F(a{A,B,C})°®2,A®B,C ° (®2,A,B <8> IFC)rFA = F(rA) o $2,^,7 ° UFA <8> $o)-W = F(1A) o $2,/,A o (*o ® IFA).

When the functors intervening in an equivalence of categories are monoidalfunctors, the categories are called monoidal equivalent. Notice that the two tensorproducts have been denoted in the same way.

EXAMPLES. We recollect some examples of monoidal categories, most of them well-known.

The category Vect(K) of vector spaces over a field K, with / = K and thetensor product being that of vector spaces. The maps I and r are defined by1(1 ® v) = r(v <S) T) = v for all v 6 V (V being an object in the category). Thecategory Mod-K[G] of representations of G over K or, equivalently, of K[G}-modu\esis a submonoidal category of Vect(K) with the structures g(u <8> v) = gu ® gv andgk = k for all g e G, u, v 6 V and k € K.

Any monoid, regarded as a discrete category is a strict monoidal category, wherethe tensor product is the multiplication.

The category Ab of abelian groups with the usual tensor product and with 2Zas the unit.

The category of right ^-modules, Mod-/?, when R is commutative, with theusual tensor product and R being the unit.

The category of K-algebras with the tensor product of /("-algebras, K being theunit.

The category of all /?-/?-bimodules, R being any ring, with the tensor productover R.

The opposite category of any monoidal category.The category ( B-M,®,K,a,l,r) of left B-modules, B being a bialgebra.Let gr-R be the category consisting of all .R-modules graded by a group G, R

being a commutative ring and morphisms preserving the degree. Then, the categorygr-R is a monoidal category considering R as the unit, defining the gradation forthe tensor product of two /?-modules, M and N, by means of (M <S> N)g, which isthe additive subgroup generated by the elements x ® y with x £ Mh, y & NI suchthat Ih = g (see [19, p. 12]) and with maps a, I and r as in Mod-/?.

If H is a Hopf algebra over a field K, the category of all representations of His monoidal (cf. [4, Example 5.1.4]). Numerous examples, different form the aboveones, can be found in [4] and [28].

A Grothendieck category, i.e., a cocomplete abelian category with exact directlimits and with one generator (see [23, Chapter V]) is called commutative if it has agenerator U such that its endomorphism ring, R = Endc(U), is commutative. Sucha generator is called endocommutative. By the Gabriel-Popescu theorem, thesecategories are just the categories equivalent to quotient categories of a categoryof modules over a commutative ring with respect to a hereditary torsion theory,i.e., the ones equivalent to a category of the form Mod-(R,r), consisting of ther-injective and r-torsion-free /^-modules (cf. [23, Chapter X]). Therefore, theypreserve many properties of categories of modules over commutative rings. Theyhave been studied in [1] and the references there. By the Gabriel-Popescu theorem,there is an equivalence between any Grothendieck category C with generator U andMod-(R, T), where R = Endc(U). Using notations as in Stenstrom's book (cf.[23]), we consider the adjoint functors i : C — > Mod-/? and a : Mod-.fi! — > C.Let T : C -^Mod-R be the functor given by T(C) = Homc(U,C) and the inducedfunctor T' : C — * Mod-(R, T), the latter being an equivalence of categories.

THEOREM 2.1 If C is a commutative Grothendieck category, then there existsa monoidal category of the form (C,<3>c,U,a,l,r), where U is an endocommutativegenerator of C.

Proof. Let T be the torsion theory which arises through the Gabriel-Popescutheorem applied to C. The tensor product making Mod-(R,r) into a monoidalcategory will simply be denoted by <g>. Then, for any objects A,B in C, one definesA <8>e B = G(T'(A) ® T'(B)), where G is the inverse equivalence of categories ofT1 . The natural isomorphism between (A ®c B) ®c C and A ®c (B <S>c C} will bedenoted by O,{A,B,C}- This isomorphism <I{A,B,C} arises from

(A ®c B)®CC = G(T'(A ®c B) ® T'(C)) = G(T'G(T'(A)

and, therefore, it is defined as

where I^M represents the natural isomorphism between T'G(M) and M. Let IA =TIA ° G(lx'(A)) and rA = t]A ° G(rT>(A))i where 77,4 is the natural isomorphismbetween GT'(A) and A. The tensor product of morphisms is defined as / ®c 9 =

Now, we point out that the category Mod-(R,r), R being a commutative ring,is a monoidal category. The proof can be found in [15, Proposition 3.2] or in [11,Proposition 1.2.2]. Say that M ® N = a(M ®R N), that the unit is a(R), IM andTM are the ones induced by the multiplication of M and R. The tensor product ofmorphisms is defined by / ® g = a(f <S>K <?)• Finally, a{LtM,N] is induced by thetensor product <£)R and its associativity. The rest of the proof is easy but tediousand for this reason is omitted (all details can be found in [7]). I

The following result shows that, because of the form in which the tensor producthas been defined, the equivalence T' is a monoidal equivalence.

THEOREM 2.2 Given a commutative Grothendieck category C, equivalent to Mod-(R,r) by means of the functor T', then T' is a monoidal equivalence between C (byusing the tensor product defined above) and Mod-(R,r).

Proof. The isomorphism <&o is IT'(U) and ^2,A,B is ^T^A^T'IB)' Let us checkthat the necessary properties are verified.

With our notation,

0<{FA,FB,FC} =

'(C\> ° aT'(A),T'(B),T'(C)}-

By naturality, we have

F(a{A,B,C}}

(a{AiB,C}) ° ̂

) } ° a{T'(A),T'(B),T'(C)}

and this expression is equal to the above one by simplifying the last two parentheses.Thus ful is verified.

In our case,

T'(rA) o $2>y4j / o (1T,(A} ® $0) = T'(rA) o

By applying T' to the equality rA = r/A o G(rT'(A)), it follows that

T'(rA) = T'(r,A)oT'G(rT,(A}) = ^T>(A) °T'G(rT,(A)).

By naturality, the second member of the last equality is r-j-'(A) ° 4>T' (A)®T> (U) • Bysubstituting,

(rA) O $2ij4i/ o (1T,(A) ® $0) =• rT'(A] ° ̂ T'(A)®T'(U)^T\A)®T'(U) = TT'(A)T'

which proves property fu2. The proof for fu3 is totally analogous and is thereforeomitted. I

3 GENERAL PROPERTIES OF MULTIPLICATION OBJECTS

Throughout this section (C,®c,I,a,l,r) (or simply C) stands for a monoidal cate-gory. The tensor product <S>c will be substituted by ® when no confusion arises.

DEFINITION 3.1 An object X ofC is a multiplication object if every subobject Yof X can be written as Y = l\ (J <8>c -X") = TX (X ®c H] for some subobjects H, J ofI.

In the proof of most results below only the condition using rx is verified, thecondition with lx, being similar. Recall that an object S is called simple if it hasno subobjects except 0 (if the category has a zero object) and itself. It is easily seenthat the unit of every monoidal category and every simple object are examples ofmultiplication objects. An easy computation allows us to obtain the next result.

PROPOSITION 3.2 Let (C,<8,/,a, l , r ) be a monoidal category. Assume that Xis a multiplication object in C and f : X —> Z is a morphism in C. Then, f ( X ) isa multiplication object in C.

REMARK 3.3 Since a quotient object of X is the image of X by the canonicalprojection, we have that every quotient object of a multiplication object is also amultiplication object.

The following theorem shows that multiplication objects are preserved by monoidalequivalences.

THEOREM 3.4 Let F : C —> £> be a monoidal equivalence between two monoi-dal categories. Then, X is a multiplication object in C if and only if F(X) is amultiplication object in TJ.

Proof. Since F is an equivalence of categories, every subobject of F(X) is ofthe form F(Y), where Y is an subobject of X. As X is a multiplication object,Y = lx(J <8> X) = rx(X ig> H), for some subobjects J, H of /. By applying F,we obtain that F(Y) = F(lx)(F(J ® X)) = F(rx)(F(X <g> H ) ) . On the otherhand, F(J®X) = $2 jx(FJ ® FX) = $2 jx ° ($o ® lpx)(J ® FX). Therefore,F(Y) = [ F ( l x ) o $2,j,V o ($0 ® 1FX)](J ® FX). By fu3, F(Y) = 1FX(J ® FX).The other condition is proved similarly. I

REMARK 3.5 From the proof of Theorem 3.4, it is clear that every dense monoi-dal functor preserves multiplication objects.

Let C be a commutative Grothendieck category with an endocommutative generator,U, and R = Endc(U). Let X be an object of C. Given a morphism r G R, sinceC is commutative, it is possible to consider, for X, the morphism induced by themultiplication by r, i.e., r* : T(X) —> T(X), given by r*(a) — a o r for everymorphism a from U in X. Since T is a full functor, there exists an endomorphismr** de X, such that T(r**) = r*, i.e., r** oa = aor for every morphism a : U —> X(since T is faithful, this assignment is unique). By taking into account all theseconsiderations, we gave the next definition in [8]. An object X in C is called amultiplication object if for any subobject Y of X there exists a family of morphisms{ri}i£i such that Y = X]ig/7"i*(-^0 ^or some endocommutative generator U.

Let R be a commutative ring, T a hereditary torsion theory over R and Man -R-module. The functor a : Mod-R —>Mod-(R,r) maps the .R-module M intoits module of quotients with respect to T, the E-module MT (see [23] for furtherinformation). The following result shows that the definitions of multiplication objectin monoidal categories and commutative Grothendieck categories coincide.

THEOREM 3.6 Let C be a commutative Grothendieck category with endocommu-tative generator U and let (C, ®c, U, a, I, r) be the corresponding monoidal category.An object X of C is a multiplication object in C if and only if it is a multiplicationobject in (C,®c,U,a,l,r).

Proof. Let R = Endc(U). Then the Gabriel-Popescu theorem says that C isequivalent to the category Mod-(R,r) via T'. Every object of Mod-(J?, r) is ofthe form a(M) for some M £ Mod-/?. So, a(M) is a multiplication object in themonoidal category Mod-(R,r) if and only if, for every subobject a(N) of a(M),there exists an ideal A of R such that a(N) = r0(M)(a(M) <g> a(A)) = a(M).a(A).By following the same steps as in the proof of [8, Theorem 4.4], this is equivalent tosay that a(M) is a multiplication object in the commutative Grothendieck categoryMod-(R,r). Since T' is an equivalence of commutative Grothendieck categoriesmapping an endocommutative generator to an endocommutative generator, by [8,Proposition 3.6], the object X of C is a multiplication object if and only if T'(X)is in the commutative Grothendieck category Mod-(R,r). On the other hand, T'is also a monoidal equivalence. By Theorem 3.4, X is a multiplication object inthe monoidal category C if and only if T'(X) is in Mod-(.R, T). At the beginning ofthe proof, we noticed that, in Mod-(R,r), the two interpretations of multiplicationobjects are identical, which finishes the proof. I

As examples, we shall characterize multiplication objects in some particular cases.

EXAMPLE 1. Graded R-modules. Recall that if R is a commutative ring, then agraded .R-module M is a multiplication graded module if every graded submodule Ncan be written as N = AM for some ideal A (which can be taken graded) of R. Mul-tiplication graded modules and rings are studied in [8], [9] and [10] but only in somecases the category of graded _R-modules is a commutative Grothendieck category.Now, we show that this concept coincides in any case with that of multiplicationobject in the corresponding monoidal category.

PROPOSITION 3.7 An object M, in the monoidal category gr-R, is a multiplica-tion object if and only if M is a multiplication graded (gr-multiplication) R-module.

Proof. The object M is a multiplication object in gr-R if for every graded sub-module, N, of M (N <gr M), there exists a graded ideal A of R such thatN = rM(M ® A) — 1M(A® M). Since MA = rM(M ® A), this is equivalentto say that for every N <gr M, there exists a graded ideal A of R, such thatN = MA and this means that M is a gr-multiplication .R-module. I

Given a graded /^-module M, the g-suspension of M is the -R-module M(g)with the gradation defined by (M(g})h = Mhg for every h G G. It is well-knownthat the functor Tg from gr-R to itself and defined by TgM = M(g) is a monoidalequivalence of categories. As a consequence of Proposition 3.7 and Theorem 3.4,for any g € G, M is a multiplication graded R-module if and only if M(g) is.

If .R is a ring graded by G, we can consider the group ring R[G] with the followinggradation: (R[G])g = QheG^-gh-1^ (see [20])- Then, Dade's Theorem yields thatthe categories Mod-/? and gr-R[G] are equivalent. Moreover, this equivalence isclearly monoidal. By Theorem 3.4, R is a multiplication ring if and only if the ringR[G] is a gr-multiplication ring.

EXAMPLE 2. Recall from [13, Proposition XV.1.2] that a quasi-bialgebra is a K-algebra A with algebra morphisms A : A —> A <8> ^4 and e : A -+ K such thatthe category of left A-modules (A — Mod, <8>^, K, a, l , r ) is monoidal, where ®K isthe usual tensor product in K and a, / , and r are the usual morphism families. In[21, Theorem 2], the dual concept appears: a semi-coalgebra is a coalgebra withcoalgebra morphisms M : C <8> C —> C and u : K —> C such that the category ofright C-comodules (M°,®Ki K,a,l,r) is monoidal (a,l,r are as usual).

PROPOSITION 3.8 If A is a quasi-bialgebra, then an A-module M is a multi-plication object in (A — Mod,(g>K,K,a,l,r) if and only if M = 0 or M is sim-ple. If C is a semi-coalgebra, then a C-comodule M is a multiplication object in(M. ,®K,K,a,l,r) if and only if M = 0 or M is a simple comodule, i.e., it has nonon-zero proper subcomodules.

Proof. Suppose that M is a multiplication object in A-Mod. Then for any A-submodule N there exist submodules / and /' of K such that N = rj^(M <g># /) =IM(I1 ® M). But the only possibilities for / and /' are 0 and K. Consequently, Mhas to be simple. Conversely, every simple module is clearly a multiplication object.The proof for comodules is analogous. I

EXAMPLE 3. Consider a family of groups (G,)i6jv with G0 = {1} and groupmorphisms pn,m '• Gn x Gm —> Gn+m for any (n,m) 6 IN x IN. We define n®m —

and potn = pnfl = \Gn f°r anY n,m,p € IN (after natural identification), then thecategory (Q, ®, 0,1,1,1) is monoidal, where the class of objects of Q is IN, the setHomg(i,j) is 0 if i ^ j and Gj if i = j and the composition in Homg(i,i) is thegroup operation (see [13, XI.3.2]). Since in this category the only subobject of n isn itself, every object is a multiplication object.

EXAMPLE 4. Multiplication comodules. Throughout this example (C, A,e)denotes a cocommutative coalgebra over a field K with comultiplication A andcounit e. The category of right C-comodules will be denoted by M.c. We followthe notation of [24]. If M e M°, then its structure map will be WM '• M —> M (giC1.The corresponding structure map of M as a left C-comodule will be denoted by w'M.Recall that given two right C-comodules M and N, the cotensor product of M andN, MDCN, is the kernel of the map a = WM®^-N-^M®W'N : M<8>JV -» M®C®N.We write IM,N f°r the injection from MO^N into M <g> N. By the definition ofkernels, there exists a unique morphism 99 : M —> MOCC such that iM,N0f — WM-Moreover, it is well-known that it is an isomorphism. The inverse isomorphismof tp will be denoted by TM- In the same way, one defines the isomorphism IM '•CacM -> M. The isomorphisms aL>M,N '• (LOCM)OCN —> Lnc(MOcN) areinduced by the associativity isomorphisms from (L<8> M) <8> N to L® (M ® N). It iseasily seen that (Mc', Oc, C, a, I, r) is a monoidal category. As a consequence, withthe above notations and by taking into account the cocommutativity, we are ableto give the following definition. Henceforth, for simplicity of notation, we write Dinstead of DC-

DEFINITION 3.9 A right C-comodule M is called a multiplication comodule if forevery C-subcomodule N there exists a right coideal D ofC such that ru(MOD) — Nor, equivalently, MOD =

DEFINITION 3.10 A cocommutative coalgebra C is a multiplication coalgebra ifevery right coideal is a multiplication C-comodule, with its natural structure.

Any cocommutative coalgebra C is clearly a multiplication C-comodule. A rightC-comodule M is called cyclic if it is isomorphic to a quotient of the form C/Ifor some right coideal / of C. From Remark 3.3, every cyclic right comodule is amultiplication comodule. A cosemisimple coalgebra is a coalgebra which is the sum ofits simple subcoalgebras. Then, by Remark 3.3, every cosemisimple cocommutativecoalgebra is a multiplication coalgebra.

REMARK 3.11 There are right comodules which are multiplication objects in Jv{c

but they are not multiplication C*-modules.

Let S = {CQ, GI, ....} and C the divided power coalgebra, that is, C = KS withA(CJ) = Y^J-OCJ ® ci-i and £(ci) = ^o.i- C is obviously a cocommutative coalge-bra and therefore, C is a multiplication C-comodule. It is well-known that C* isisomorphic to the power series ring ^[[X]]. The C*-modules of C are of the form0, C and Y^=oKci for some n e ^+ (cf- I24> P- 44D- Then, the morphism ofC*-modules <f : Mn = Ei=oKci -> (K([X]}/(xn+1)) = M'n given by

i=0 i=0

is clearly an isomorphism. Therefore c*C = IJngW M'n where M'n C M^+1 for alln e IV. But this is not a multiplication C*-module as there is no ideal / of /C[[X]]

TS- r r V"ll

such that C = /^J1./. This proves the statement.

Recall from [25] that a C-comodule M is invertible if there exists another C-comodule M' and isomorphisms of C-comodules / : C —> MOM' and g : C —>M'DM such that

a) (idMng) o WM = ct{M,M',M} ° (l^idM) o w'M.b) (idM>tJf) o WM> = a{M',M,M'} ° (gOidM,) o w^,.The next proposition provides us with new examples of multiplication comod-

ules. We use the above notation.

PROPOSITION 3.12 If M is an invertible C-comodule, then M is a multiplica-tion object in M.c.

Proof. Let L be a subcomodule of M. We shall prove that MOg~1(M'OL) andWM(L) are equal by checking that their images under the monomorphism icoincide. Moreover, (idM^g}(MUg~l(M'OL)) = MD(M'OL).

We have

= a{M,M',M} °= a{M,M',M}

By using b), one proves in the same way that w'M(L) = /~1(LDM')DM. I

Let M € Mc and take m € M. Then, there exists a minimal subcomodule ofM, which is moreover finite-dimensional, containing m (see [18, Theorem 5.1.1]). Itwill be denoted by < m >. This notation will be used in the sequel.

PROPOSITION 3.13 A C-comodule M is a multiplication comodule if and only iffor every m G M there exists a right coideal I of C such that WM(< m >) =

Proof. Suppose that for every m G M, WM(< m >) = MD/m for some rightcoideal Im of C. Let N be a subcomodule of M. Then, WM(< n >) = MD/n

for every n G N. Let us consider / = X)neAf^n- Since the sum of subcomodulesis a subcomodule, / is a right coideal. We have WM(N) = WM(Y^neN < n >) —

) . The necessity is clear. I

EXAMPLE 5. Doi-Koppinen modules. Consider the category A-M(H)C of left-right G-Doi-Koppinen modules and ^-linear (7-colinear homomorphisms, where thetriple G = (H, A, C) is a monoidal Doi-Hopf datum over a commutative ring R (see[3] for definitions and further information). By [3, Proposition 2.1], the category( AM(H)C,®R, R, a, l , r ) is monoidal. Given two Doi-Koppinen modules N C M,we write (N : M) for {r G R;rM C JV}. Then, we have a characterization ofmultiplication object similar to that of multiplication modules.

PROPOSITION 3.14 A Doi-Koppinen module M is a multiplication module ifand only if N = M(N : M) for any Doi-Koppinen submodule N of M .

Proof. By following the remarks of [22, §1], it suffices to prove that (N : M) is aDoi-Koppinen submodule of R. It is clear that it is an ideal of R. Let EA be thecounit of A, a G A and r G (N : M). Then, a.r.M = £A(a).r.M C £A(a)N C N.It follows that (N : M) is a left ^4-module. Let WR be the structure map for thecomodule R, then WR(T) = r ig) 1 for every r G R. Since U>(/V :M) = WR\(N : M),i.e., the restriction of WR to (N : M), (N : M) is also a right C-comodule. Thecompatibility conditions are verified as every element of (N : M) is an element ofR. I

4 ENDOMORPHISMS OF MULTIPLICATION OBJECTS

DEFINITION 4.1 A monoidal category ( C , ( S ) , I , a , l , r ) is called braided if thereexists a family of natural isomorphisms CA,B '• A®B — > B®A verifying the followingaxioms:(Bl) (g <8> /) o CA,B = CA',B' ° (/ ® <?) for a^ morphisms f : A — > A', g : B — > B' .

CA,B®C ° a{A,B,c} = (Is ® CA,C) ° O>{B,A,C} ° (CA,B ® lc)-{A,c,B}(S3) ac,A,B ° CA®B,C ° aA,B,c = (CA,C ® IB) ° aA,c,B ° ( I A ® CB,C)-

When the braided category is strict as a monoidal category and, moreover,CA,I — CI,A = IA f°r aH objects A of C, the category C is a strict braided category.

LEMMA 4.2 // (C,®,I,a,r,l,c) is a strict braided category, then rx(X ® A) =lx(A ® X] for every object X of C and every subobject A of I.

Proof. Notice that cx,i(X ® A) = CX,A(X ® A). In fact, let IA be the inclusionmorphism from A into /. By axiom (Bl), (IA ® ^x] ° CX,A = cx,i ° (ix ® M)- Byapplying both members to X®A, we obtain [(iA®^x)°cx,A\(X®A) — cx,A(X®A)and [cx,i o (lx ® iA)\(X ® A) = cx,i(X ® A).

Now, we have rx(X®A) = (lx°cx j)(X®A) = lX(cXj(X®A)) = lx(cx A(X®A))=lx(A®X). I

DEFINITION 4.3 An object X of a category C is called totally invariant if f(Y) CY for every endomorphism f of X and for every subobject Y of X.

PROPOSITION 4.4 If X is a multiplication object in a braided monoidal categoryC, then X is totally invariant.

Proof. It is well-known that every braided monoidal category is equivalent to astrict braided monoidal category. Moreover, since the properties of being totallyinvariant and being a multiplication object are preserved by monoidal equivalences,it is enough to prove the result for a strict category C. Let Y < X and let / 6Endc(X). Since X is a multiplication object, Y = rx(X ® A) for some A < I, Ibeing the unit of C. By Proposition 3.2, f ( X ) = rx(X <g> B) for some B < I. Byapplying / to the first equality, f ( Y ) = (/ o rx)(X ®A)=-rxo(f® li)(X ® A) =r x ( f ( X ) ® A). By substituting the second equality, f ( Y ) = rx(rx(X ® B) ® A) =rx((rx ® li)(X ®B)®A) = (rx o rx)((X ® B) ® A] = (rx o rx)(X <g> (B ® A ) ) .By using the fact that C is braided and strict, f ( Y ) = (rx o rx)((X ® A) ® B) =rX(rx(X ®A)®B)= rx(Y ® B) = rY(Y ®B)CB.I

PROPOSITION 4.5 Let (C,®,I,a,l,r,c) be a braided monoidal category and letX be a multiplication object. Then, « / / , < ? € Endc(X), (f o g)(Y) = (g o f ) ( Y ) forevery subobject Y of X.

Proof. It is clear that it suffices to prove this for a strict category. Thus wesuppose that C is strict. Firstly, we deal with the case Y = X. By Proposition3.2, f ( X ) = rx(X ® A) and g(X) = rx(X ® B) for some subobjects A,B of/. By applying g to the first equality, we have g ( f ( X ) ) = (g o rx)(X g) A) =rxo(g® lx)(X ®A) = rx(g(X) ® A) = rx(rX(X <S> B) ® A). Since C is strictand braided, (g o f ) ( X ) = r\((X ® A) ® B). In a similar way, we have f ( g ( X ) ) =f(rx(X®B)) =rx(f(X}®B) = r'2x((X®A)®B). Let us now consider the generalcase Y < X. Since X is a multiplication object, Y = rx(X ® C) for some C < I.Then, f(Y)=rx(f(X)®C}. Thus g ( f ( Y ) ) = rx((go f ) ( X ) ® C } . By the previousstep, g ( f ( Y ) ) = rx((f o g)(X) ® C) = f ( g ( Y ) ) . I

PROPOSITION 4.6 Let X be a multiplication object in a braided monoidal cat-egory C and let (f € Endc(X). Then, ip is an isomorphism if and only if it is anepimorphism.

Proof. The first step of the proof consists in proving that 9? is an epimorphism if andonly if 931 y : Y —> Y is an epimorphism for every subobject Y < X. Notice that byProposition 4.4, <p\Y is defined from Y to Y. Suppose that 99 is an epimorphism andlet Y < X. Since X is a multiplication object, Y = rx(X ® A) for some A < I. Byapplying <p, we have <p(Y) = <porx(X®A) = rx°(p®li)(X®A) = rx(f(X)®A}.Since ip is an epimorphism, <p(X) = X. Therefore, if>(Y) = rx(X <g> A) = Y. Theother implication is clear. Now we shall prove that if 9? is an epimorphism, then it isa monomorphism. In fact, assume that 9? is an epimorphism. By the first step, wehave 99(0) = 0 and <f>(Y) = Y ^ 0, for Y 7^ 0. It follows that (p is a monomorphism.

The next result is an analogue of Fitting's Lemma.

PROPOSITION 4.7 Let X be a multiplication object in a braided monoidal cate-gory C and let 9? G Endc(X) such that <pn(X) — </3™+1(JQ for some positive integern. Then, X = Ker <pn ®

Proof. The result follows by Proposition 3.2 and a standard reasoning. I

REMARK 4.8 The condition stated in Proposition 4- 7 is obtained when X has thedescending chain condition over multiplication objects because it suffices to considerthe chain of multiplication objects X 3 f ( X ) D • • • .

Recall that an object X is called indecomposable if X = X\ ® X% (X\,X-2 beingsubobjects of X) implies X\ = 0 or X% = 0.

COROLLARY 4.9 Let X be an indecomposable multiplication object in a braidedmonoidal category and let is an automorphism, so is (pn and hence, <p is not nilpotent. Now,suppose that </? is not nilpotent. By Proposition 4.7, X = fn(X)®Ker fn. Since Xis indecomposable, either Ker tpn = 0 or (pn(X) = 0. If tpn(X) = 0, then ipn = 0 andip is nilpotent, a contradiction. Thus Ker <pn = 0 and hence, <pn is a monomorphism.As a consequence, f is a monomorphism and therefore, X = (fn(X). It follows thatipn, and therefore f is an epimorphism. I

PROPOSITION 4.10 Suppose that X is a multiplication object in a braided mo-noidal category C and is an epimorphism.

3. Lp is an automorphism.

Proof. Suppose that <p is a monomorphism. Then, 9?" is a monomorphism as well.By hypothesis, f n ( X ) = <pn(<p(X)). It follows that X = <f(X) and therefore <p isan epimorphism.

Now, assume that (f is an epimorphism. Since X is a multiplication object,Ker <f — TX (X <g> A) for some A < I. By applying f, we have

0 = (p o rx(X <g> A) — rx(v(X} ® A) = rx(X ® A) — Ker p.

Thus (f is a monomorphism. By Proposition 4.6, under these circumstances, 2 isequivalent to 3. I

THEOREM 4.11 If X is an indecomposable multiplication object in a braidedmonoidal category that verifies D. C. C. on multiplication subobjects, then the ringEnd(X) is local.

Proof. By using Corollary 4.9, the theorem is proved as in the module case. I

Acknowledgements. Both authors have been supported by grant PB98-1005from DGES and PAI FQM 0211. The authors are grateful to the referee for hissuggestions and comments.

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[2] Benabou, J., Categories avec multiplication, C. R. Acad. Sci. Paris, Ser. IMath., 256, (1963) 1887-1890.

[3] Caenepeel, S., Van Oystaeyen, F. and Borong Zhou, Making the Category ofDoi-Hopf Modules into a Braided Monoidal Category, Algebras and Represen-tation Theory, 1 (1), (1998) 75-96.

[4] Chari, V. and Pressley, A., A guide to Quantum Groups, Cambridge UniversityPress, Cambridge, (1994).

[5] Escoriza, J. and Torrecillas, B., Multiplication modules relative to torsion the-ories. Comm. in Algebra, 23 (11), (1995) 4315-4331.

[6] Escoriza, J. and Torrecillas, B., Relative Multiplication and Distributive Mod-ules, Comment. Math. Univ. Carolinae, 38 (2), (1997) 205-221.

[7] Escoriza, J., Objetos multiplicacion (Ph. D. thesis), Serv. Publicaciones Univ.Almeria, Almeria, 1997.

[8] Escoriza, J. and Torrecillas, B., Multiplication Objects in Commutative Gro-thendieck Categories, Comm. in Algebra, 26 (6), (1998) 1867-1883.

[9] Escoriza, J. and Torrecillas, B., Multiplication rings and graded rings, Comm.in Algebra, 27 (12), (1999).

[10] Escoriza, J. and Torrecillas, B., Multiplication graded rings, Algebra and Num-ber Theory, Marcel Dekker, (2000) 127-136.

[11] Jeremfas Lopez, A., El grupo fundamental relativo. Teoria de Galois y local-izacion, Alxebra 56, Santiago de Compostela, 1991.


[12] Joyal, A., Braided Tensor Categories, Advances in Math., 102, (1993) 20-78.

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Krull-Schmidt Theorem and SemilocalEndomorphism Rings

A. FACCHINI, Dipartimento di Matematica Pura e Applicata, Universita diPadova. Via Belzoni 7, 35131 Padova, Italy.E-mail: facchini@math. unipd. it

Abstract

We survey some very recent results on the Krull-Schmidt Theorem. Wedescribe the properties of some classes of modules whose endomorphism ringis semilocal and their direct sum decompositions. In particular, we presentresults obtained by D. Herbera, the author and R. Wiegand about the KQof a semilocal ring [FH1 1999, W 1999], an example due to G. Puninski of auniserial module that is not quasismall [PI 1999], and some results proved byCorisello, Barioli, Herbera, Raggi, Rfos and the author about homogeneoussemilocal rings and the Krull-Schmidt Theorem for modules whose endomor-phism ring is homogeneous semilocal [CF 1999, BFRR 1999].

Throughout, ring means associative ring with identity 1^0. If .R is a ring, wedenote the Jacobson radical of R and the ring of n x n matrices over R by J(R)and Mn(R) respectively. All modules are unital right modules unless otherwisespecified. Semigroups are additive and commutative and have a zero element.

1 SEMILOCAL RINGS AND MODULES WHOSE ENDOMORPHISMRING IS SEMILOCAL

A ring R is semilocal if R/J(R) is semisimple artinian. By the Wedderburn-ArtinTheorem, this means that R/J(R) is isomorphic to a finite direct product of ringsof matrices over division rings. Since a ring is semisimple artinian if and only if itis right artinian and its Jacobson radical is zero, we have that a ring R is semilocalif and only if R/J(R) is right artinian, if and only if R/J(R) is left artinian.


188 Facchini

EXAMPLES 1.1 (1) A commutative ring is semilocal if and only if it has onlyfinitely many maximal ideals, because a commutative ring is semisimple artinian ifand only if it is a finite direct product of fields.

(2) Every right (or left) artinian ring is semilocal.(3) Every local ring is semilocal.(4) If R is semilocal, the ring Mn(R) of n x n matrices over R is semilocal for

every positive integer n.(5) Direct product of finitely many semilocal rings is semilocal.(6) Every homomorphic image of a semilocal ring is semilocal.(7) If R is semilocal, the ring eRe is semilocal for every nonzero idempotent

e&R.

For these examples and the other basic properties of semilocal rings presented inthis section see [F 1998, Ch. 1]. Here is a property of semilocal rings that has beenobserved only recently by Pere Ara (for a proof see [BFRR 1999, Theorem 2.2]):

PROPOSITION 1.2 Let e be an idempotent element of a ring R and suppose thatboth the rings eRe and (1 — e)R(l — e) are semilocal. Then R is semilocal.

Being semilocal is a finiteness condition on the ring. For instance, if R is asemilocal ring, then every set of orthogonal idempotents of R is finite; there areonly finitely many simple _R-modules up to isomorphism; and there is only a finitenumber of finitely generated indecomposable projective /^-modules up to isomor-phism [FS 1975, Theorem 9]. Moreover, as we shall see in Theorem 1.3, semilocalrings are exactly the rings with finite dual Goldie dimension. Since the notion ofdual Goldie dimension is not quite standard, we shall briefly recall it here.

Let L be a modular lattice with a smallest element 0 and a greatest element 1.A finite subset {xi\i£l}ofL\ {0} is join-independent if Xj A (Vi=y xi) = 0 f°r

every j e /. An arbitrary subset of L\ {0} is join-independent if all its finite subsetsare join-independent. A modular lattice with 0 and 1 is said to be of infinite Goldiedimension if it contains an infinite join-independent subset. Otherwise it is said tobe of finite Goldie dimension. In this case it is possible to prove that { card X \ Xis a join-independent subset of L \ {0} }, the set of the cardinalities of all join-independent subsets of L \ {0}, has a greatest element, which is a non-negativeinteger dim L, called the Goldie dimension of L (for a proof see, for instance, [F 1998,Theorem 2.36]). Thus every modular lattice with 0 and 1 has a Goldie dimension,which is either a nonnegative integer or oo. For a module AS over any ring S, thelattice Ij(As) of all submodules of AS is a modular lattice with 0 and 1, and theGoldie dimension of L(As) is called the Goldie dimension dim AS of the moduleAS. If L is a modular lattice with 0 and 1, its dual lattice L*, that is, the set Lwith the opposite order, is a modular lattice with 0 and 1. The Goldie dimensiondimL(^4s)* of the dual lattice of i(As) is called the dual Goldie dimension codimAsof the module AS- For a proof of the following result see, for example, [F 1998,Proposition 2.43].

THEOREM 1.3 A ring R is semilocal if and only if the right R-module RR hasfinite dual Goldie dimension, if and only if the left R-module xR has finite dualGoldie dimension. In this case, codlm(RR) = codim(/jl?) =


Semilocal Endormorphism Rings 189

Note that in this statement dlm(R/J(R)) is simply the number of direct summandsin any decomposition of the semisimple module R/J(R) as a direct sum of simplemodules.

We shall be mainly interested in modules whose endomorphism ring is semilocal.Hence, let 5 be an arbitrary ring and let AS be a right S-module, and suppose thatEnd(As) is semilocal. Again, this is a finiteness condition on the module AS- Forinstance, if AS = ®AeA A\ is a decomposition of AS as a direct sum of nonzeromodules A\ and End(/is) is semilocal, then the index set A must be necessar-ily finite. In the following lemma we have collected three important properties ofmodules with a semilocal endomorphism ring. Recall that two direct sum decom-positions AS = ©AeA ^A = ® €M BH of a module AS are said to be isomorphicif there is a one-to-one correspondence tp: A —> M such that A\ = -B^A) for everyA G A .

LEMMA 1.4 Let As,Bs,Cs be modules over an arbitrary ring S and supposeEnd(As) semilocal. The following properties hold:

(a): (Cancellation property) If A © B = A © C, then B = C.(b): (n-th root property) If An = Bn for some positive integer n, then A = B.(c): The module AS has only finitely many direct sum decompositions up to

isomorphism.

By Proposition 1.2, if AS and BS are modules with a semilocal endomorphismring, the module AS ® BS has a semilocal endomorphism ring.

If AS is a module and its endomorphism ring End(.As) is semilocal, thencodim(End(j4s)) is finite by Theorem 1.3, codim(End(As)) = n say. The nextresult, which is due to Dolors Herbera and the author and whose statement has notappeared in the literature yet, says that if AS is isomorphic to a direct summandof a finite direct sum B\ © B^ © . . . © Bm of 5-modules, then AS is isomorphic toa direct summand of a direct sum of at most n of the 5-modules BI, 82,..., Bm.

THEOREM 1.5 Let AS be a module over a ring S and suppose codim(End(^4s)) =n < oo. Let Bi,B2,... ,Bm be S-modules such that AS is isomorphic to a directsummand of 0™^ Bi. Then there exists a subset a of {1, 2 , . . . , m} of cardinalitycard a < n such that AS is isomorphic to a direct summand of @-€a Bj.

EXAMPLE 1.6 The endomorphism ring of any artinian module is semilocal. Thiswas proved by Camps and Dicks [CD 1993, Corollary 6] answering a question posedby Pere Menal [M 1988, Question 16]. Recall that a module-finite algebra over acommutative ring A: is a ring R with a homomorphism of k into the center of Rsuch that R is a finitely generated fc-module. If R is a module-finite algebra overa semilocal noetherian commutative ring, then R is isomorphic to the endomor-phism ring End(As) of an artinian cyclic right module AS over a suitable ring S[FHVL 1995, Corollary 1.3]. This allowed us to show that Krull-Schmidt fails forartinian modules [FHVL 1995]. The problem was the following. If a module Mis both artinian and noetherian, then the (classical) Krull-Schmidt Theorem statesthat the decomposition of M into a direct sum of indecomposable modules is uniqueup to isomorphism. On the other hand, Krull-Schmidt fails for finitely generatedmodules over certain subrings of Z® • • - ® Z [L 1983]. Realizing every module-finitealgebra over a semilocal noetherian commutative ring as the endomorphism ring of

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a suitable artinian right module, we constructed examples of artinian modules forwhich Krull-Schmidt fails [FHVL 1995]. Further examples of artinian modules withparticular direct sum decompositions were later constructed by Yakovlev [Y 1998]and Pimenov-Yakovlev [PY 1998]. We shall come back to the problem of construct-ing artinian modules with particular direct sum decompositions at the end of thenext section.

EXAMPLES 1.7 A number of modules whose endomorphism ring is semilocalwere discovered by Herbera and Shamsuddin [HS 1995]. For instance, they showedthat:

(1) If AS is a module over an arbitrary ring 5 and both dim(As) and codim(As)are finite, then End(^4s) is semilocal and codim(End(As)) < dim(j4s)+codim(A,s).This is the case of serial modules of finite Goldie dimension, that is, direct sumsof finitely many uniserial modules (see Section 3), or, more generally, direct sumsof finitely many biuniform modules (a module is said to be biuniform if both itsGoldie dimension and its dual Goldie dimension are equal to 1).

(2) If AS is a module over a ring 5, the Goldie dimension dim(.As) is finiteand every injective endomorphism of AS is bijective, then End(^4s) is semilocaland codim(End(yls)) < dim(As). For example, artinian modules satisfy theseconditions.

(3) Dually, if AS is a module over a ring 5, its dual Goldie dimension codim(.As)is finite and every surjective endomorphism of AS is bijective, then End(As) issemilocal and codim(End(As)) < codim(As). This happens for noetherian modulesof finite dual Goldie dimension over any ring S, in particular, noetherian modulesover semilocal rings.

2 K0 OF A SEMILOCAL RING

Let AS be a module over an arbitrary ring 5 and suppose R = End(^is) semilocal.If we want to study the direct sum decompositions of AS, it is natural to considerthe full subcategory &dd(As) of Mod-5 whose objects are the modules isomorphicto direct summands of direct sums Ag of finitely many copies of Ag. For example,add(5s) is the full subcategory proj-5 of Mod-5 whose objects are all finitely gener-ated projective right 5-modules. The categories add(As) and proj-7? are naturallyequivalent via the equivalences Homs(As, —): a,dd(As) —> proj-JS and —<S>p.A: proj-R —> add(Ag), and in these equivalences the module AS correspond to the moduleRR. Hence, studying the direct sum decompositions in the category add(Ag) iscompletely equivalent to studying the direct sum decompositions of finitely gener-ated projective right modules over the semilocal ring R = End(^4g). In particular,studying the direct sum decompositions of AS is equivalent to studying the directsum decompositions of the module RR.

The set of isomorphism classes of finitely generated projective right modulesover a ring R can be given the structure of a commutative semigroup with 0 inthe following way. For every finitely generated projective right R-module PR, let[PR] denote the isomorphism class of PR, that is, the class of all right _R-modulesisomorphic to PR. Let V(R) denote the set of all isomorphism classes of finitelygenerated projective right ^-modules (V(R) is actually a set). Define an additionin VR by [PR] + [QR] = [PR ® QR\- Then VR is a commutative semigroup whose

zero is the isomorphism class of the zero module. Here we have defined V(R)using right modules. But the contravariant functor Hom/j(—, R)'- proj-R —> /2-projis a duality between the full subcategory pmj-R of Mod-R whose objects are allfinitely generated projective right .R-modules and the full subcategory .R-proj ofR-Mod whose objects are all finitely generated projective left .R-modules. Henceif we define V(R) using left modules instead of right modules, we obtain the samesemigroup V(R) up to isomorphism.

If PR is a finitely generated projective right module over a semilocal ring R, thenthe endomorphism ring of PR is semilocal by Examples (4) and (7) of 1.1, so thatPR cancels from direct sums (Lemma 1.4(a)). Thus the commutative semigroupV(R) has the cancellation property, i.e., it is contained in an abelian group. LetKQ(R) denote the smallest abelian group that contains V(R), that is,

Ko(R) = { [PR] — [QR] 1 PR,QR finitely generated projective right .R-modules}.

Here [PR] - [QR] = [P'R] - \Q'R} if and only if [PR] + [Q'R] = [P'R] + [QR] in V(R), thatis, if and only if PR® Q'R = PR®QR- Usually one must construct the group Ko(R)considering not the isomorphism classes [PR] of finitely generated projective modulesPR, but the stable isomorphism classes [PR]S, that is, the classes [PR]S of all right-R-modules QR such that there exists a finitely generated projective module XR withPR®XR = QR®XR. But as R is semilocal, the finitely generated projective moduleXR cancels from direct sums, so that two finitely generated projective /^-modulesPR and QR are stably isomorphic if and only if they are isomorphic.

The V and the KQ we are considering can be viewed as functors

V: Rings —> CSemigrps and KQ: Rings —> Ab

from the category Rings of associative rings with identity to the category CSemi-grps of commutative semigroups with zero or Ab of abelian groups respectively.If <f. R —> S is a ring morphism, V((p):V(R) —> V(S) is the semigroup morphismdefined by V(<p)([PR\) = [P ®R S] for every finitely generated projective -R-modulePR. Similarly for KQ(<f): K0(R) -> K0(S).

In passing from V(S) to Ko(S) a lot of information is lost. For example, if S isa ring and TT: S —> S/J(S) denotes the canonical projection of S onto S modulo itsJacobson radical J(S), it can be proved that V(n):V(S) —> V(S/J(S)) is alwaysinjective [FH2 1999, Proposition 2.11]. In particular, if R is a semilocal ring, thering R/J(R) is semisimple artinian, so that each finitely generated R/ J(.R)-moduleis a finite direct sum of simple modules. Thus V(R/ J(R)) is the free commutativesemigroup having the isomorphism classes of simple R/J(.R)-modules as a free setof generators, that is, V(R/J(R)) = Nn, where n is the number of simple R/J(R)-modules up to isomorphism. It follows that KO(TT): Ko(R) —» Ko(R/J(R)) is aninjective group homomorphism of Ko(R) into the free abelian group Ko(R/J(R}) =Zre. Thus Ko(R) also is a free abelian group, i.e., Ko(R) = Zm for some m < n.This shows that the KQ of any semilocal ring is simply a finitely generated freeabelian group, and this structure is too poor to describe what we were interestedin, that is, the direct sum decompositions of RR. If we want to study the directsum decompositions of RR in Mod-_R (recall that this is equivalent to describing thedirect sum decompositions of AS in Mod-S1), we must know how it is possible towrite the element [RR] of V(R) as a sum of elements of V(R). Now V(R) C K0(R),

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so that if we do not want to lose information by passing to Ko(R) we must keeptrack of the subsemigroup V(R) of Ko(R) and its element [RR]. To do this weshall follow Goodearl [G 1991, Chapter 15] considering on Ko(R) the structure ofpre-ordered abelian group with order-unit.

If G is an abelian group and < is a relation on the set G that is reflexive,transitive and translation-invariant (i.e., for any a, b, c € G, a 0 of a pre-ordered abelian group G is anorder-unit if for any a & G there exists n € N such that a < rat. The pre-order <on G is completely determined by the positive cone of the pre-ordered abelian group(G, <): if we associate to every translation-invariant pre-order < on G its positivecone G+ = { a g G | 0 < a } , we obtain a one-to-one correspondence betweenthe set of translation-invariant pre-orders on G and the set of subsemigroups of G(Recall that subsemigroups of G contain OQ). A pre-ordered abelian group withorder-unit (G, <,«) consists of a translation-invariant pre-order < on an abeliangroup G and an order-unit u. Pre-ordered abelian groups with order-unit form acategory. For instance, the abelian group Ko(R) with the pre-order whose positivecone is Ko(R) + = { [PR] PR a finitely generated projective right -R-module} andwith the order-unit [RR] is a pre-ordered abelian group with order-unit, and KQturns out to be a functor from the category of associative rings with identity intothe category of pre-ordered abelian groups with order-unit.

Let's go back to the previous example, in which R is a semilocal ring, TT: R —>R/ J(R) denotes the canonical projection of R onto the semisimple artinian ringR/J(R), the mappings

V(TT): V(R) -» V(R/J(R)) and KO(TT): K0(R) - KQ(R/J(R))

are injective, the semigroup V(R/J(R)) is isomorphic to Nn, where n is the numberof simple J?/J(/?)-modules up to isomorphism, and Ko(R/J(R)) = Zn. The compo-nentwise order on Zn is the pre-order on Z™ whose positive cone is N™, that is, thepartial order defined by (a\,..., an) < (61 , . . . , bn) if a, < 6$ for every i = 1,... ,n.Thus the pre-ordered abelian group with order-unit (Ko(R/J(R)), <, [R/J(R)]) isisomorphic to the partially ordered abelian group with order-unit (Zn, <,u) for asuitable order-unit u in (Z™, <). The order-units of Z™ with respect to the compo-nentwise order < are the n-tuples u = (ui,U2, • • • ,un) € Zn with Ui > 0 for everyi = 1 , 2 , . . . , n. It can be proved [FH1 1999, Lemma 2.2] that the injective mappingKQ(TT): Ko(R) —> Ko(R/J(R)) is an embedding of partially ordered abelian groupswith order-unit, that is, KO(T^)([RR]) = [R/J(R)] and for every x,y & Ko(R), x < yin K0(R) if and only if KQ(n)(x) < K0(ir)(y) in K0(R/J(R)). This shows that

THEOREM 2.1 Let R be a semilocal ring and TT: R —> R/J(R) the canonical pro-jection. Then the pre-ordered abelian group with order-unit (Ko(R), <, [^?fi]) isisomorphic to a subgroup of (Ko(R/J(R)), <, [ R / J ( R ) ] ) via the embedding of pre-ordered groups with order-unit KQ(TT): Ko(R) —> Ko(R/J(R)). Moreover, if n isthe number of simple right R-modules up to isomorphism, the pre-ordered abeliangroup with order-unit (Ko(R/J(R)), <, [R/J(R)]) is isomorphic to (Z™, <,u) for asuitable order-unit u in ( Z ™ , < ) . In particular, (Ko(R),<) is a partially orderedgroup, i.e., its pre-order relation < is also symmetric.

Conversely,

THEOREM 2.2 [FH1 1999, Theorem 6.3] Let G be a partially ordered subgroupof (Zn, <) and u an order-unit of Zn such that u 6 G. Let f : G —> Zn denote theembedding. Then there exist a semilocal hereditary ring R and two isomorphisms ofpartially ordered groups with order-unit g:G^> Ko(R) and h:Zn —> Ko(R/J(R})such that the diagram

G -^ Z"g [ [ h

K0(R) K-^] K0(R/J(R))

commutes.

A very interesting variant of this theorem in the setting of commutative noethe-rian rings has been proved recently by Roger Wiegand [W 1999]. Recall that if Mis a module over a commutative ring k and M* = Honifc(M, k) denotes the dualmodule, the module M is called reflexive if the canonical mapping M —> M** intothe bidual is an isomorphism. Wiegand has proved that

THEOREM 2.3 [W 1999, Theorem 4.1] Let G be a partially ordered subgroup of(Zn, <) and u an order-unit of Zn such that u G G. Let f:G —> Z™ denote theembedding. Then there exist a semilocal ring R that is the endomorphism ring of afinitely generated reflexive module M^ over a commutative noetherian local uniquefactorization domain k of Krull dimension 2 and two isomorphisms of partiallyordered groups with order-unit g: G —> Ko(R) and h: Zn —> Ko(R/J(R)) such thatthe diagram

G -^ Z™9l [h

K0(R) K-^} K0(R/J(R}}commutes.

Wiegand's result is particularly interesting for at least two reasons. On the onehand, the endomorphism ring R of a finitely generated module Ak over a com-mutative noetherian semilocal ring k is a semilocal ring (Example 1.7(3)). AsKQ(R}+ = V(R), the partially ordered abelian group with order-unit Ko(R) con-tains the description of all direct sum decompositions of Ak, so that Theorems 2.1and 2.3 describe all possible direct sum decompositions of any finitely generatedmodule over a commutative noetherian semilocal ring. On the other hand, theendomorphism ring of a finitely generated module over a commutative noetheriansemilocal ring k is a module-finite algebra over k, and we have already remarked inExample 1.6 that every module-finite algebra over a semilocal noetherian commuta-tive ring is isomorphic to the endomorphism ring End(^4s) of an artinian cyclic rightmodule AS over a suitable ring S [FHVL 1995, Corollary 1.3]. Thus Theorems 2.1and 2.3 describe all possible direct sum decompositions of artinian modules.

The method expounded in this section allows us to solve the problem of theexistence of modules with particular direct sum decompositions whenever we areconsidering finitely generated modules over commutative noetherian semilocal rings,or artinian modules over arbitrary rings, or projective modules over semilocal rings,

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etc. The idea is always reducing the problem of the existence of a direct sumdecomposition to an arithmetical problem about elements and subgroups (G, <,u)of a partially ordered abelian group with order-unit (Z™, <,u). We shall make someexamples.

EXAMPLE 2.4 Do there exist two indecomposable finitely generated nonzero mod-ules A and B over a commutative noetherian semilocal ring R such that A2 = J33 ?

If the answer is affirmative, there exist a partially ordered subgroup (G, <,u)of a partially ordered abelian group with order-unit (Z n ,< ,u) and two positiveelements a, 6 £ G such that u = 2a = 36 and such that a and b cannot be written asa sum of two positive elements in G. Here and in the rest of this section when wesay "a positive element" of G we mean an element of GnN™ whose components arenot all zero. Since the equation 2x = 3y has only the solutions x = 3t, y — 2t in Z(where t ranges in Z), it follows that the only positive elements a, b € Zn such that2a = 36 are a = 3c, b = 2c with c ranging in the positive elements of (Zn, <, u). Butfor any subgroup G of Z", if a, 6 6 G, then c = 3c — 2c = a — b 6 G, so that botha = c + c + c and b = c + c are sums of positive elements of G. This contradictionshows that the answer is negative.

We would have obtained the same answer considering artinian indecomposablemodules over arbitrary rings or projective indecomposable modules over semilo-cal rings, instead of indecomposable finitely generated modules over commutativenoetherian semilocal rings.

EXAMPLE 2.5 Do there exist three indecomposable pairwise nonisomorphic fini-tely generated modules A, B, C over a commutative noetherian semilocal ring Rsuch that A2 = B&C?

We are looking for a subgroup (G, <, u) of (Zn, <, u) and three distinct positiveelements a, b, c <G G such that u = la = b + c and such that a, 6, c cannot bewritten as a sum of two positive elements in G. It is easy to find a solution ofu = 2a = 6 + c inZ 2 , for instance u = (2, 2 ) ,a= (1,1), 6= (2,0),c = (0,2). Now letG be the subgroup of Z2 generated by u, a, b, c, so that G = { (x, y) 6 Z2 | x + y iseven }. Since there are no elements in G strictly contained between (0,0) and (1,1) ,no elements strictly contained between (0,0) and (2,0), and no elements strictlycontained between (0,0) and (0,2), the elements a, b and c of G cannot be writtenas a sum of two positive elements in G. This shows that the answer to our questionis positive

We would have obtained the same positive answer considering artinian indecom-posable modules over arbitrary rings or projective indecomposable modules oversemilocal rings.

EXAMPLE 2.6 Do there exist three indecomposable pairwise nonisomorphic fini-tely generated modules A, B, C over a commutative noetherian semilocal ring Rsuch that A3 - B2 ® C ?

We must look for a subgroup (G, <, w) of (Zn, <,u) and three distinct positiveelements a, b, c € G such that u = 3a — 26+ c and such that a, b, c cannot be writtenas a sum of two positive elements in G. It is easy to find a solution of u = 3a — 26+cin Z2, for instance u = (3,6), a = (1, 2), 6 = (0,3),c = (3,0). If G is the subgroupof Z2 generated by u, a, b, c, then G = { (x, y) & Z2 | x + y is divisible by 3 }. Thereare no elements of G strictly contained between (0,0) and (1,2), strictly contained

between (0,0) and (0,3), or strictly contained between (0,0) and (3,0). Thereforea, 6 and c cannot be written as a sum of two positive elements in G. This showsthat the answer is again positive.

We would have obtained the same positive answer considering artinian indecom-posable modules over arbitrary rings, etc.

EXAMPLE 2.7 Looking at the previous two Examples 2.5 and 2.6, it is natural toask: Do there exist three pairwise nonisomorphic finitely generated modules A, B, Cover a commutative noetherian semilocal ring R such that A2 = B © C and A3 =B2®C?

If the answer is affirmative, there exist a subgroup (G, <,u) of (Zn,<,u) andthree distinct elements a, b, c <5 G such that 2a = b + c and 3a = 26 + c. But then inthe abelian group G we would have a — 3a — 2a = (26+c) —(6+c) = b, contradictionbecause a ̂ b. The contradiction shows that the answer is negative in this case.

As we have already remarked, this technique describes all possible direct sumdecompositions of artinian modules. Compare this method with the examplesof artinian modules for which Krull-Schmidt fails due to Facchini-Herbera-Levy-Vamos [FHVL 1995], Yakovlev [Y 1998] and Pimenov-Yakovlev [PY 1998] men-tioned in Example 1.6.

3 UNISERIAL MODULES

A further class of modules whose endomorphism ring is semilocal is the class ofuniserial modules. Let 5 be a ring. An 5-module Us is said to be uniserial if thelattice L(As) of all its submodules is linearly ordered, that is, for any submodulesM and N of A we have M C TV or TV C M.

If A and B are two uniserial modules, we say that A and B belong to thesame monogeny class if there exist a monomorphism A —» B and a monomorphismB —» A. In this case we write [A]m = [B]m. We say that A and B belong to thesame epigeny class, and write [A]e = [B]e, if there are an epimorphism A —» Band an epimorphism B —> A. Belonging to the same monogeny class and the sameepigeny class are two equivalence relations in the class of all uniserial modules. Thenext proposition shows why monogeny classes and epigeny classes are important inthe context of uniserial modules.

PROPOSITION 3.1 Let A and B be uniform modules over a ring S. Then A = Bif and only if [A]m = [B]rn and [A}e = [B]e.

A module is serial if it is a direct sum of uniserial modules. In particular,a module is serial of finite Goldie dimension if and only if it is a direct sum offinitely many uniserial modules. The theorem that follows describes when twoserial modules of finite Goldie dimension are isomorphic.

THEOREM 3.2 (Weak Krull-Schmidt Theorem for uniserial modules, [F 1996])Let AI, ..., An, BI, ... ,Bt be nonzero uniserial modules over a ring S. Then A\ ©. . . © An = BI © . . . ® Bt if and only if n = t and there are two permutations a, T of{1,2,. . . ,n} such that [Ai}m = [Ba^)}m and [Ai]e = [-BT(i)]e for every i = 1,2, . . . ,n.

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This theorem considers direct sums of finite families of uniserial modules. Whatcan be said about direct sums of arbitrary (possibly infinite) families of uniserialmodules? That is, if { Ai \ i 6 / } and { Bj \ j <G J } are two families of uniserialmodules, when can we say that ®i€/ Ai = ®j€j Bjl This problem was studied in[DF 1997], where we proved the next two results. They show that one and a half ofthe two implications of Theorem 3.2 hold for arbitrary families of uniserial modulesas well.

THEOREM 3.3 // {Ai \ i € /} and {Bj j 6 J} are two families of uniserialmodules over a ring S and there exist two bijections <r, T: I —> J such that [Ai]m =(Ba(i)}m and (Ai e - [BT^)}e for every i € /, then ©ie/ Ai = 0jeJ Bj.

Conversely,

THEOREM 3.4 If {Ai \ i 6 /} and {Bj j e J} are two families of nonzerouniserial modules over a ring S and (J)i6/ f/j = ©,-gj Vj, then there is a bijectiona: I —> J such that [Ui}m = [V^.(j)]m for every i 6 /.

We hoped that an analog of Theorem 3.4 could hold for epigeny class too,but this is false, as has been proved very recently by G. Puninski [PI 1999]. Hiscounterexample is the following.

Recall that a chain ring is a ring R such that both the modules RR and fiR areuniserial. A chain domain R, that is, an integral domain that is also a chain ring,is called a nearly simple chain domain if it has exactly three two-sided ideals, whichmust be necessarily the ideals 0 C J(R) C R. For an example of a nearly simplechain domain see [BBT 1990, §6.5]. Right ideals, left ideals and finitely presenteduniserial modules of nearly simple chain domains have very particular behaviors.For instance,

PROPOSITION 3.5 Suppose R is a nearly simple chain domain, and a and b arenonzero noninvertible elements of R. Then:

(1) the left ideal aR and the right ideal Rb are incomparable;(2) aR + Rb=J(R);(3) the right R-modules R/aR and R/bR are isomorphic.

The proof of this proposition, due to Puninski, can be found in Lemma 4.1,Proposition 6.2 and Corollary 4.3 of [PI 1999]. Property (1) is easy. First of all,Puninski proves that the Jacobson radical J = J(R) of a nearly simple chain domainR is not finitely generated. To see this, note that J ^ 0 in a nearly simple chaindomain. If J is finitely generated, then J2 ^ J by Nakayama's Lemma. Now J2 ^ 0because R is a domain. Thus J2 is a two-sided ideal different from 0, R and J,contradiction. Thus J is not finitely generated. If a and b are nonzero noninvertibleelements of R, and the left ideal aR and the right ideal Rb are comparable, forinstance, if Rb C aR, then J = RbR C aR, so that J = aR is finitely generated,contradiction. Property (2) is the key property that allows Puninski to constructthe counterexample. His proof is based on techniques of model theory. Property (3)says that over a nearly simple chain domain R there are only two indecomposablefinitely presented right modules up to isomorphism, RR and R/aR, where a is anynonzero noninvertible element of R. Thus every finitely presented right module over

a nearly simple chain domain R is isomorphic to R^ ® (R/aR)m for two uniquelydetermined nonnegative integers n and m.

Then Puninski shows that there exists a noninvertible element r € R such thatfln>i -^r™ T^ 0- For tmsi ^e chooses an arbitrary nonzero noninvertible elements <E~R. As Rs <2 sR by Proposition 3.5(1), there is t e R such that is ^ sR. Thusthere exists a noninvertible element r £ R with isr = s. Then s = tsr = t2sr2 =i3sr3 = . . .enn>i^«.

Now that this element r has been chosen, Puninski defines a uniserial moduleUK via generators and relations as follows: let UK be the right J?-module witha countable set of generators xi,X2,xs, • • • subject to the relations x^r = 0 andxn+ir = xn for every n > 1. Clearly UK is uniserial. Set VK = R/rR. It is possibleto show that

[/ft®v f l*o)-v^o), (i)where V^ °' denotes the direct sum of countably many copies of VR. Since VRis cyclic and UR is not cyclic, there is no epimorphism VR — > UR. In particular,[I/fi e ^ [UR]B. Thus the epigeny classes that appear in the two direct sum decom-positions UR ® V^ = V^ , i.e., one copy of [UR\e and countably many copiesof [Vftje in the decomposition on the left and countably many copies of [VR],, inthe decomposition on the right, are different. Thus the problem of describing whentwo direct sums of uniserial modules ®ie/ Ai and ®,-ej Bj are isomorphic, that is,finding a generalization of theorem 3.2 for arbitrary families of uniserial modules,is still open.

With his example Puninski solves a problem of [DF 1997, p. Ill], showing thatthere exist uniserial modules that are not quasismall (also see [F 1998, Problem 15on p. 269]; here a module A is said to be quasi-small if for every set { Bi i € / }of .R- modules such that A is isomorphic to a direct summand of ®ie/ B^, there isa finite subset F C / such that A is isomorphic to a direct summand of Q)ieF Bi).Because of isomorphism (1), Puninski's module UR is also an example of a pure-projective uniserial module over a chain domain that is not finitely presented. Inparticular, not every pure-projective module over a chain domain is a direct sum offinitely presented modules.

Puninski's module UR also solves a further problem. In [DF 1998, Proposi-tion 2.6] Nguyen Viet Dung and the author showed that if V is a uniserial moduleover an arbitrary ring R, I is a, non-empty index set and V^ = A ® B, then eitherA or B must contain a direct summand isomorphic to V. In [DF 1998, p. 99] weasked whether under these hypotheses both A and B must contain a direct sum-mand isomorphic to V. Puninski's decomposition V^°) = U © V^°) shows that forA = U and B — V^°) the answer is negative.

In another wonderful paper [P2 1999], Puninski considers uniserial modules overprime coherent nearly simple chain rings that are not domains. Over such a ringR he is able to construct a pure-projective module that is not a direct sum ofindecomposable modules. This shows that not every direct summand of a serialmodule over a chain ring is serial. Thus he answers the question posed in [F 1998,Problem 10 on p. 268]. It is not known yet whether every direct summand of a serialmodule of finite Goldie dimension is serial [F 1998, Problem 9 on p. 268]. Puninski'sexample of a pure-projective /^-module that is not a direct sum of indecomposablesanswers a further question [F 1998, Problem 11 on p. 269], because it is an example

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of a pure-projective module over a chain ring that is not serial.For an example of a prime coherent nearly simple chain ring R that is not a

domain see [D 1994] and [P2 1999, §8].

4 HOMOGENEOUS SEMILOCAL RINGS AND MODULES WHOSEENDOMORPHISM RING IS HOMOGENEOUS SEMILOCAL

Now we consider the class of modules whose endomorphism ring is a homoge-neous semilocal ring. We shall present some results that appear in [CF 1999] and[BFRR 1999], A ring R is homogeneous semilocal if R/ J(R) is simple artinian, thatis, if P/J(P) is isomorphic to the ring Mn(D] ofnxn matrices with entries in somedivision ring D for some positive integer n. For instance, if 5 is a local ring andn is a nonnegative integer, the ring Mn(S) of n x n matrices with entries in 5 is ahomogeneous semilocal ring. A commutative ring is homogeneous semilocal if andonly if it is local. Every homomorphic image of a homogeneous semilocal ring ishomogeneous semilocal. If R is a right noetherian ring and P is a right localizableprime ideal of R, the localization of R with respect to P is a homogeneous semilocalring. Further examples of homogeneous semilocal rings are given in [CF 1999, §§4and 5].

A number of properties of local rings extend to homogeneous semilocal rings.For example,

PROPOSITION 4.1 In a homogeneous semilocal ring R the Jacobson radicalJ(R) is the unique maximal proper two-sided ideal of R, that is, J(R) contains allproper two-sided ideals of R. Conversely, if a semilocal ring R has a unique maximalproper two-sided ideal, then R is homogeneous semilocal.

Obviously, every homogeneous semilocal ring R has a unique simple right moduleup to isomorphism.

THEOREM 4.2 Let R be a homogeneous semilocal ring. Then:(a) There exists a unique indecomposable finitely generated projective R-module

P up to isomorphism.(b) Every projective R-module is isomorphic to a direct sum pW for some

set X.(c) // X and Y are sets, then pW and p(y) are isomorphic if and only if X

and Y have the same cardinality.

We already know that semilocal rings are exactly the rings of finite dual Goldiedimension (Theorem 1.3). Thus it is possible to associate to each homogeneoussemilocal ring R its dual Goldie dimension codim(P), which is a positive integer.It is also possible to attach to each homogeneous semilocal ring R a second nu-merical invariant, the index of R, denoted index(-R). It is defined as follows. ByTheorem 4.2, the finitely generated projective module RR, where PL is a homoge-neous semilocal ring, is isomorphic to a finite direct sum P* of copies of the uniqueindecomposable projective .R-module P, and the positive integer t is uniquely de-termined. This integer t is called the index of the homogeneous semilocal ring P,denoted index(P).



PROPOSITION 4.3 Let R be a homogeneous semilocal ring and let n, t be positiveintegers. Then:

(a) codim(-R) = n if and only if R/J(R) = Mn(D) for some division ring D;(b) index(/2) = t if and only if R = Mf(S) for some homogeneous semilocal

ring S with no nontrivial idempotents.

PROPOSITION 4.4 Let R be a homogeneous semilocal ring and let m be a positiveinteger. Then:

(a) codim(Mm(/?)) = m • codim(R);(b) index(Mm(.R)) = m • index(.R).

The invariant index(jR) always divides the codimension codim(.R) for every ho-mogeneous semilocal ring R, as the following proposition shows.

PROPOSITION 4.5 If P is the indecomposable projective module over a homo-geneous semilocal ring R, then

codim(.R#) = index(l?) • codim(Pfl).

Conversely, for any pair of positive integers t and n with t that divides n, thereexists a homogeneous semilocal ring R with t = 'mdex(R) and n = codim(R)[CF 1999, Example 5.1]. Let us consider the "extreme cases" in the equality ofProposition 4.5:

THEOREM 4.6 Let R be a homogeneous semilocal ring. Then(a) codim(_R) = 1 if and only if R is a local ring;(b) index(.R) = 1 if and only if R has no nontrivial idempotents;(c) codim(/?) = index(.R) if and only if R is semiperfect, if and only if R =

Mn(S) for some positive integer n and some local ring S.

As homogeneous semilocal rings generalize local rings and the Krull-SchmidtTheorem concerns modules whose endomorphism ring is local, it is natural to askwhether the Krull-Schmidt Theorem holds for modules whose endomorphism ringis homogeneous semilocal. The answer is given in our last theorem.

THEOREM 4.7 (Krull-Schmidt Theorem for direct sums of modules with homo-geneous semilocal endomorphism rings, [BFRR 1999]) Let MR be a module overa ring R. Suppose that MR = MI © . . . © Mt = NI © . . . © Nm are two directsum decompositions of MR into indecomposable direct summands and that all theendomorphism rings End(Mi) and End(Arj) are all homogeneous semilocal. Thent = m and there is a permutation IT of {1, 2, . . . ,t} such that Mi = Na^ for every

It is important to note that in order to get the uniqueness of decompositions, itis not sufficient that in the decompositions MR = MI © . . . © Mt = NI © . . . © Nmonly one decomposition MI ffi . . . ® Mt is into indecomposable direct summandsMi whose endomorphism rings End(Mj) are homogeneous semilocal. For instance,for any pair of integers n > I and s > I there exist a ring R and an /^-moduleAR with two non-isomorphic direct sum decompositions into indecomposable direct


200 Facchini

summands AR = PI ® . . . ® Pn = Qs and the End(Pi)'s homogeneous semilocal forevery i = l,...,n ([FH2 1999, Lemma 5.5] and [BFRR 1999, Example 3.4]).

We conclude with a generalization of Theorem 4.7 to arbitrary, possibly infinite,direct sums. It is due to Dolors Herbera and the author. Recall that a moduleAR over a ring R is said to be small if for every family { Bi i € / } of ^-modulesand any homomorphism f. AR — > 0i6/ Bi, there is a finite subset F C / such that

(Di€^ Bi. For instance, finitely generated modules are small.

THEOREM 4.8 Let MR be a module over a ring R. Suppose that MR = ® ig/ Mi=• (J) -6 j NJ are two direct sum decompositions of MR into indecomposable smalldirect summands and that all the endomorphism rings End(Af,) and End(JVj) arehomogeneous semilocal. Then there is a one-to-one correspondence tp: I —* J suchthat Mi = ^<p(i) for svery i G /.

REFERENCES

[BFRR 1999] F. Barioli, A. Facchini, F. Raggi and J. Rios, Krull-Schmidt Theoremand homogeneous semilocal rings, preprint, 1999.

[BBT 1990] C. Bessenrodt, H. H. Brungs and G. Torner, "Right chain rings, Part1", Schriftenreihe des Fachbereichs Math. 181, Universitat Duisburg, 1990.

[CD 1993] R. Camps and W. Dicks, On semilocal rings, Israel J. Math. 81 (1993),203-211.

[CF 1999] R. Corisello and A. Facchini, Homogeneous semilocal rings, preprint,1999.

[D 1994] N. Dubrovin, "The rational closure of group rings in left ordered groups",Schriftenreihe des Fachbereichs Math., Universitat Duisburg, 1994.

[DF 1997] N. V. Dung and A. Facchini, Weak Krull-Schmidt for infinite direct sumsofuniserial modules, J. Algebra 193 (1997), 102-121.

[DF 1998] N. V. Dung and A. Facchini, Direct summands of serial modules, J. PureAppl. Algebra 133 (1998), 93-106.

[F 1996] A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math.Soc. 348 (1996), 4561-4575.

[F 1998] A. Facchini, "Module Theory. Endomorphism rings and direct sum decom-positions in some classes of modules" , Birkhauser Verlag, Basel, 1998.

[FH1 1999] A. Facchini and D. Herbera, KQ of a semilocal ring, to appear in J.Algebra (1999).

[FH2 1999] A. Facchini and D. Herbera, Projective modules over semilocal rings,to appear in the Proc. of the International Conference on Algebra and itsApplications, Athens (Ohio), March 25-28, 1999.

[FHVL 1995] A. Facchini, D. Herbera, L. S. Levy and P. Vamos, Krull-Schmidtfails for artinian modules, Proc. Amer. Math. Soc. 123 (1995), 3587-3592.


[FS 1975] K. R. Fuller and W. A. Shutters, Projective modules over non-com-mutative semilocal rings, Tohoku Math. J. 27 (1975), 303-311.

[G 1991] K. R. Goodearl, "Von Neumann regular rings", Krieger Publishing Com-pany, Malabar, 1991.

[HS 1995] D. Herbera and A. Shamsuddin, Modules with semi-local endomorphismring, Proc. Amer. Math. Soc. 123 (1995), 3593-3600.

[L 1983] L. S. Levy, Krull-Schmidt uniqueness fails dramatically over subrings ofZ®Z®---®Z, Rocky Mountain J. Math. 13 (1983), 659-678.

[M 1988] P. Menal, Cancellation modules over regular rings, in "Proc. GranadaRing Theory Conference", Lecture Notes in Math. 1328, Springer, Berlin,1988, pp. 187-208.

[PY 1998] K. I. Pimenov and A. V. Yakovlev, Artinian modules over one matrixring, to appear in the Proc. of the Euroconference Infinite Length Modules,Bielefeld, September 7-11, 1998.

[PI 1999] G. Puninski, Some model theory over a nearly simple uniserial domainand decompositions of serial modules, preprint, 1999.

[P2 1999] G. Puninski, Some model theory over an exceptional uniserial ring withapplications to decompositions of serial modules, preprint, 1999.

[W 1999] R. Wiegand, Direct-sum decompositions over local rings, preprint, 1999.

[Y 1998] A. V. Yakovlev, On direct sum decompositions of Artinian modules, Alge-bra i Analiz 10 (1998), 229-238.


On Suslin's Stability Theorem for R[XI, ... ,xm]

J. GAGO-VARGAS, Departamento de Algebra, Universidad de Sevilla, Aptdo.1160, 41080-Sevilla. Spain.Ei-m.a,il:[email protected]

Abstract

Let R be a euclidean domain. Suslin's Stability Theorem asserts that anymatrix in SLn(R[xi, . . . ,xm]), with n > 3,m > 1, can be expressed as prod-uct of elementary matrices. We give an algorithmic proof of this theorem,following [9], where the case R a field is treated. Using Grobner bases, wecompute generators of maximal ideals in R[x\ , . . . , xm] and give a normaliza-tion of unimodular vectors. As a corollary, we obtain a constructive proof ofQuillen-Suslin Theorem for R[XI, . . . ,xm}.

1 INTRODUCTION

In this paper, R will denote a euclidean domain, k a field, R[x] will denote the poly-nomial ring R[XI, . . . , xm and analogously fc[x] will be k[xi, . . . , xm}. We assumethat we can factor any element a 6 R and that for any given prime element p G Rwe can factor in (R/(p})[x\. This assumption is needed because we have to find aprimary decomposition of an ideal in R[XI, . . . , xm}.

DEFINITION 1.1 For any ring S, an n x n elementary matrix Eij(a) over S is amatrix of the form I + a • e i j , where i ^ j , a € S and &ij is the nxn matrix whose(i, j) component is 1 and all other components are zero.

DEFINITION 1.2 A square matrix A over a ring S is called realizable if A canbe written as a product of elementary matrices.

DEFINITION 1.3 A vector


204 Gago-Vargas

is called unimodular if (i>i, . . . , vn) = S.Umn(Sf) is the set of unimodular vector with entries in S.

Suslin's Stability Theorem can be expressed as

SLn(R[x]) = En(R[x}) for all n > 3 (2)

where En(R[x\) is defined as the subgroup of SLn(R[x\) generated by the elemen-tary matrices.

This theorem was first proved by A. A. Suslin in [11], and an algorithmic prooffor R = k appeared in [9]. The theorem fails when n = 2, as P.M. Cohn showed in[4], and there exists an algorithm that determines when a given matrix in 6X2 (-^M)is realizable, and in that case, expresses it as product of elementary matrices (see[10]).

In this paper, we extend the algorithmic proof of [9] to R\x\. There exist twomain difficulties. First, given a proper ideal in R[x], we have to give an effectivemethod to build a set of generators of a maximal ideal that contains it. We reducethe problem to the case fc[x], solved in [7]. Second, there exist steps in the processwhere it is needed that a certain polynomial be monic in one of the variables. WhenR = k we can use the Noether Normalization Theorem. For euclidean domains ingeneral we do not have such a tool, but for unimodular vectors it is possible. Wegive a constructive version of a theorem in [12] that allows us to make such a changeof variables. Section 1 is devoted to solve these technical questions.

In Section 2 we adapt the proof in [9] through the previous lemmas, pointing outthe differences. In the same way as in [9], an algorithmic proof of the Quillen-SuslinTheorem for R[x can be deduced.

All required computations can be carried out using Grobner bases, as describedin [1]. So, this paper may be considered a contribution to the computational side ofalgebraic K-Theory, like [6]. As pointed out in [10], these algorithms have potentialapplications to signal processing.

2 CONSTRUCTIONS IN R[x]

2.1 Maximal ideals

THEOREM 2.1 Let k be a field, I = {/1; . . . , fr) C /c[x] a proper ideal. Then it ispossible to compute a set of generators of a maximal ideal M. that contains I.

PROOF.- We follow [7, Proposition 1]. Denote by k an algebraic closure of k.Because / is not the whole ring, we can find a common zero a = (a1; . . . , am) € km

of polynomials /i, . . . , fr. Let M — {g 6 fc[x]|<?(a) = 0}. It is clear that / C M.Let

J = (Ai(o:i), A2(o;i,o:2), • • • , Am(zi, . . . ,o:m)), (3)

where each Aj <E k[xi, . . . , Zj] is the minimal polynomial of <ij over

Then J C M., and J is maximal, because ki = fcj_i[o:j]/Aj. Then, J = M.. D

Suslin's Stability Theorem for R[x,,.. .jcj 205

The same result for the ring R[x] will be reduced to a polynomial ring withcoefficients in a field. So we can assume that R is not a field. The followingpropositions are corollaries of results in [5].

PROPOSITION 2.2 Let I = ( / i , . . . , fr) C R[x] be a proper ideal such that I nR = (s), with s ^ 0. Then it is possible to compute a maximal ideal that containsI.

PROOF.- Let p £ R be a prime element that divides s, and s = ps'. Then theideal J = (p, j\,..., fr) is not the whole ring. If J = R[x] then there would existao, a i , . . . , ar £ R[x] such that 1 = a^p + Y^l=i aih- Multiplying each member bys', we get s' & I n R, which is a contradiction.^

Then we consider the ideal J = ( f i , . . . , fr) in R/(p)[x}. It is a proper ideal,and by Theorem 2.1, we can find a maximal ideal M — ( A i , . . . , \m) C (R/(p))[x\that contains J. Let A, £ R[x], i = 1,... ,m be any lifting of A;. Then the idealM = (p, \i,..., \m) contains J and is maximal, because R[x]/M — ( R / ( p ) ) [ x } / M ,which is a field. D

PROPOSITION 2.3 Let I = {/i,. . . , f r ) C R[x] be a proper ideal such that I nR = (0). Then we can find a maximal ideal that contains I.

PROOF.- By [5, Proposition 8.2], there exists a d £ R - {0} such that

I = (l,d)n(lQ(R)[x}nR[x]). (4)where Q(R) is the quotient field of R. If the ideal (/, d) is not the whole ring,we are under the hypothesis of Proposition 2.2. Otherwise, we have that / =IQ(R)[x] n R[x\. The ideal IQ(R)[x] is not the whole ring, because InR= (0).Then we can compute a maximal ideal MI C Q(R)[x] such that IQ(R)[x C MI-

Let /i = MI fl.R[x]. /i contains the ideal /, and is not equal to the whole ring,because MI is proper. Let

Mi = { A i ( x i ) , A 2 ( x 1 , x 2 ) , . . . , A m ( x i , . . . , x m ) } , (5)

with the polynomials Aj built as in 2.1. Then there exist a^ £ R,i = 1,... ,m suchthat \i = a,i\i € R[x}. Let J\ = ( A 1 ; . . . , Xm} C R[x]. It is clear that Ji C /i, and/x = JlQ(R)[x] n R[x}. The construction of the ideal /i, following [1, pp. 239-241]or [5, cor. 3.8], is done by computing a Grobner basis of Ji, and taking s £ R equalto the least common multiple of the leading coefficients of the polynomial in thatbasis. We are taking x\ < x-2 < ... < xm. We can assume that the generators ofJi are in the Grobner basis. Then

/i = JiQ(R)[x\r\R[x] = JiRsnR[x] = (Ji,st- l)R[x,t]n R[x], (6)

where Rs = 5""1/?, S = {sk} is the localization of R at s. We obtain g\,... ,gn €R[x\ such that /i = ( J i , f i > i , . . . ,#„).

For each i = 1 , . . . ,n, there exist djj-(x, t) £ R[x, t], j = 0,1,... ,m such that3i(x) = ai0(x,t)(st - 1) + EJLia«(x^)^j(x)- If we put t = 1/s, then ^(x) =Y^jLi aij(xi l/s)^j(x)- So each ffi(x) can be expressed as a linear combination ofAI, . . . , Am through polynomials with coefficients of the form a/sk, that belong toR*.

206 Gago-Vargas

Let p 6 R be a prime element that does not divide s. Then s is a unit in thefield R/(p). Let 99 : Rs —> R/(p) be the morphism defined by (p(a/sk) — as~k,and extend it to the polynomial rings in the natural way. Then, if we considerthe images of the polynomials Qi in R/(p)[x], we get that <?j G ( A i , . . . , A T O ) foreach i = 1, . . . ,n. Then every generator of (Ii,p) is in the ideal (Ji,p). The otherinclusion is trivial, and we conclude that (Ji,p) = (Ii,p).

Let us see that (J\,p) is not the whole ring. We can consider the ideal J\ in(R/(p))\x\ generated by the classes Aj, i = 1,... ,m . As p does not divide theleading coefficients of the polynomials that generate J, the polynomials A^ are nonconstant. Moreover, AJ(ZI , . . . , £ $ ) = a;^ + ..., so there exists a common roota = ( a i , . . . ,am), with each cti in an algebraic closure of R/(p). Then, J\ is notthe whole ring, and we have the same conclusion for (Ji,p). We can then applyProposition 2.2 to (/i,p), and the proof is complete. D

2.2 Ideal of principal coefficients

DEFINITION 2.4 Let I be an ideal in R[t}. Iff = a0+ait+.. .+adtd, with ad ^ 0,we say that a^ is the principal coefficient of f. We denote it by a^ = coef(f).

Let J be the set of principal coefficients of polynomials in the ideal /. Then J is anideal in R, that we call c o e f ( I ) . We want to compute a set of generators of c o e f ( I ) .In the ring R[x, t] we take an elimination order in the variable t. For example,xi < 2:2 < • • • < Xm < t. Let G — {<?i (x, t),..., gs(x, t ) } be a Grobner basis of theideal / with respect to this order.

LEMMA 2.5 The ideal coef(I) is generated by the principal coefficients of the poly-nomials <?i(x, t ) , i = 1 , . . . , s. Moreover, given h(x) G J we can compute /(x, t) G /such that c o e f ( f ( x , t ) ) = h(x).

PROOF.- The leading term of each gi(x,t) has the form a;(x)idi. Let h(x) € J. Thenthere exists /(x, t) & I such that coef(f) — h(x). The leading term of /(x, t) has theform h(x)td. Because G is a Grobner basis, there exist hi(x,t),..., hs(x,t) G -R[x][t]such that htd = X^=i hi(x, t)ai(x)tdi. Putting t — 1, we have that h(x) can beexpressed as a linear combination of the polynomials aj(x).

Now, let h(x) £ J. By the previous paragraph, we can compute polynomialshi(x) € R[x] such that h(x) = j^s

i=1 /i i(x)a i(x). Let f(x,t) = ^s=1 hi(x)gi(x,t).

Because G is a Grobner basis, the principal coefficient of /(x, t) is h(x). D

2.3 Normalization of unimodular vectors

Given an ideal / in R[x], it is not possible, in general, to give a change of variablesthat allows us to put a generator as a monic polynomial in one of the variables.However, if the ideal generates the whole ring, we can get a normalization. It isa result that appears in [2] and [12]. We give the constructive version for R[x],following [12, Sections 9, 10].

PROPOSITION 2.6 Let I be an ideal in R[x], R' = R[x}/I, and f1J2 £ #[x]with (/i + /, /a + I) — R'• Then we can find h € R[x] such that (/i + ^1/2) + / isnot a zero divisor in R'.

Suslin's Stability Theorem for /?[*„. . .jcj 207

PROOF.- Through [5], we can compute the prime ideals associated to / in R[x], andcompare among them to extract the maximal sets by inclusion. The proof followsas in [12, Proposition 9.2]. D

COROLLARY 2.7 Let n be a natural number and b = (bi, . . . ,6n)* e Umn(/?[x])a unimodular vector. Then there exists an upper triangular matrix B e En(R\x\)with ones on the principal diagonal such that B-b = (di, . , . , cJn)*, with {eJj, . . . , dn}a regular sequence.

PROOF.- This is the proof of [12, Corollary 9.3], noting that in each step we usea ring of the form Rk — R[x]/Ik for some ideal Ik in R[x], where we can applyProposition 2.6. D

The next place where we have to give a construction is in [12, Lemma 10.5],

LEMMA 2.8 Let I be an ideal in R[x], with htR^(I) > 2. Then there exists aninvertible change of variables xj , . . . , xm <-> j/i, . . . , j/m of the ring R[x] such that Icontains a polynomial monic in the variable y\ .

PROOF.- Let J be the ideal in RQ = R[XI, . . . , xm_i consisting of the principalcoefficients of / with respect to xm. By Lemma 2.5, we can compute a set ofgenerators for J. Following [12, Lemma 10.5], we obtain a g £ J, and it is possibleto extract / G / such that the principal coefficient of / with respect to xm be g.The proof then follows the same steps as in [12, Lemma 10.5]. D

The last result is

LEMMA 2.9 Let n > 3, and b e Umn(.R[x]). Then there exist B € En(R[x]) anda change of variables xi, . . . , xm <-> yi , . . . , ym such that B • b = (cj), where c\ is amonic polynomial in the variable ym.

PROOF.- [12, Lemma 10.6]. D

3 APPLICATIONS TO K-THEORY

The previous results can be applied to give algorithmic proofs of Suslin's StabilityTheorem and the Quillen-Suslin Theorem over the ring R[x\. For that, we followthe proof in [9] for rings k[x]. We give only the specific details.

The proof that En is normal in SLn, for n > 3, ([9, Section 2]) remains equal,because the only algorithmic process that we need is to find elements in R[x] thatexpress 1 as a linear combination of the components of a unimodular vector. Thiscan be accomplished through a Grobner basis.

As in [9, Section 3], we apply induction over the number of variables. LetR' = Rxi,...,xm-i, X = xm.

THEOREM 3.1 Let A € SLn(R'[X}). If AM e En(R'M[X}} for every maximalideal M in R' , then A & En(R'(X}).

PROOF.- Let ai = (0, . . . ,0) G Rm~l and pi G R a prime element. Then MI =(PI,XI, . . . ,x m _i) is a maximal ideal in R' . With the same technique as in theproof of [9, Theorem 3.1], we find an element r\ £ MI- By Propositions 2.2 and

208 Gago-Vargas

2.3, we can compute a maximal ideal M2 that contains r\. We apply again theprocess over M2, and find r2 £ M2. In general, given ri, . . . ,r,-_i 6 R', suchthat (TI, . . . , TJ-I) ^ (1), we can compute a maximal ideal Mj that contains them,and find TJ £ Jvij. After a finite number of steps, we reach a number / such that(TI, . . . ,n) = R'. This condition is verificable through Grobner bases in R'. Therest of the proof remains identical. D

The reduction to SLy, is done through the Elementary Column Property: for n > 2,the group En(R[x\) acts transitively on the set Umn(/?[x]). We review the resultsused in the proof.

The first one is [9, Lemma 4.2]. It remains unaltered, because if /i,/2 e R'[X]and r & R' is their resultant, we can find 51,32 € R'[X] such that r = figi +([3, 13]). A similar proof to Theorem 3.1 can be applied to [9, Theorem 4.3].

THEOREM 3.2 Let

e Umn (£[*]) (7)

a unimodular vector with v$X) monic in X. Then there exist BI € SL,2(R'[X])and B2 € En(R'[X$ such that B1B2 • v(X) = v(0).

PROOF.- Let a = (0, . . . , 0 ) € Rm"1 and pi e .R a prime element. Let MI =(PI,XI, . . . , xm_i) , that is a maximal ideal in /?', and fcj = R'/Mi the residualfield. Following the proof of [9, Theorem 4.3], we find an element r\ € R' that doesnot belong to MI. Applying Propositions 2.2 and 2.3, we can find a maximal idealM.-2 that contains TI. In the same way, we get an element r<2 £ M2- In general, givenri, . . . , TJ_I € 7?' we can find a maximal ideal Mj such that TI, . . . , TJ_I € .A/fj, andTJ ^ MJ. The process follows just like in the original proof of [9]. D

In the proof of the Elementary Column Property ([9, Theorem 4.5]), we have toinclude a slight variation.

THEOREM 3.3 For n > 3, the group En(R[x]) acts transitively on the set

PROOF.- For m = 0, we apply the euclidean division algorihtm over R. By induc-tion, we may assume the theorem for the ring R' = R[XI, • • • ,xm_i]. Let X — xmand

v(X) = : e Umn(R'[X}). (8)

By Lemma 2.9, there exist a change of variables and a matrix B £ En(R'[X])such that Bv(X) = w(F), with wi(Y) monic in Y. By Theorem 3.2 we can findBI e SL2(R'[Y}) and B2 e -Bn(^'[^]) such that B^ • w(7) = w(0) e J?'. By theinduction hypothesis, it is possible to compute B' € En(R'} with B' • w(0) = en.Following the proof of [9, Theorem 4.5], we get that w(F) = (B^1B") • en, with

Suslin's Stability Theorem for R[x,,.. .,r J 209

B^B" e En(R'[Y}). Then, v(X) = (B-1B^1B") • en undoing the change ofvariables, and B^B^B" e En(R'\X}}. D

In the realization algorithm for matrices of the form

SL3(fl[x]) (9)

we first apply Lemma 2.9 to assume that p is monic in xm, because we only addthe product by a matrix in £?s(.R[x]). All the theorems in [9, Section 5] are valid,and this completes the process.

As in [9], the Elementary Column Property implies the Unimodular ColumnProperty: for n > 2, the group GLn(R[x\) acts transitively on the set Umn(.R[x]).In [8] it is shown that this property is equivalent to the Quillen-Suslin Theorem in.R[x], so we get an algorithmic proof of this theorem for R[x].

REFERENCES

[I] W.W. Adams and P. Loustaunau, An Introduction to Grobner Bases, GraduateStudies in Math. Vol. 3, (Amer. Math. Soc., Providence, RI, 1994).

[2] H. Bass, Liberation des modules projectifs sur certains anneaux de polynomes,Seminaire Bourbaki 1973/74, Expose 448, Lecture Notes in Math, vol. 431, 228-254 (Springer-Verlag, 1975).

[3] D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms, Undergradu-ate Texts, in Math., (Springer-Verlag, New York, 1992).

[4] P.M. Cohn, On the structure of the GL^ of a ring, Inst. Hautes Etudes Sci.Publ. Math. 30 (1966) 365-413.

[5] P. Gianni, B. Trager and G. Zacharias, Grobner bases and primary decomposi-tion of polynomial ideals, J. Symb. Comp. 6 (1988) 149-167.

[6] R. Laubenbacher and C. Woodburn, An algorithm for the Quillen-Suslin theo-rem for monoid rings, J. of Pure and Applied Algebra 117-118 (1997) 395-429.

[7] D. Lazard, Solving Zero-dimensional Algebraic Systems, J. Symb. Comp. 13(1992) 117-131.

[8] A. Logar and B. Sturmfels, Algorithms for the Quillen-Suslin Theorem, J. Al-gebra 145 (1992) 231-239.

[9] H. Park and C. Woodburn, An Algorithmic Proof of Suslin's Stability Theoremfor Polynomial Rings, J. Algebral78 (1995) 227-298.

[10] H. Park, A Realization Algorithm for 5L2(-R[xi,... ,xm}) over the EuclideanDomain, SIAM J. on Matrix Analysis and App. 21 n. 1 (1999) 178-184.

[II] A. A. Suslin, On the structure of the special linear group over polynomial rings,Math. USSR Izv. 11 (1977) 221-238.


210 Gago-Vargas

[12] L.N. Vaserstein and A.A. Suslin, Serre's Problem on projective modules overpolynomial rings and algebraic K-Theory, Izv. Akad. Nauk SSSR Ser. Mat. 40(1976) 993-1054.

[13] C. Woodburn, An Algorithm for Suslin's Stability Theorem, Ph.D. Thesis, NewMexico State University, 1994.


Characterization of Rings Using Socle-Fine andRadical-Fine Notions

C. M. GONZALEZ, Departamento de Algebra, Geometrfa y Topologfa. Universi-dad de Malaga. Apartado 59, 29080 Malaga. Spain.

A. IDELHADJ and A. YAHYA, Universite Abdelmalek Essaad. Departement deMathematiques. Faculte des Sciences de Tetouan. B.P. 2121. Tetouan, Morocco.

Abstract

A class C of left modules is said to be socle-fine if for every pair M, N inC: M = N <=> soc(M) = soc(N). By Duality, we say that C is radical-fine iffor every pair M, N in C: M ^ N <S> -^fm — ̂ HTM- In this note we wil1

characterize using the socle-fine and radical-fine notion the following rings:Perfect rings, self-injective rings, PF-rings, QF-rings, SV-rings, V-rings andsemi-primitive rings.

1 INTRODUCTION

The rings considered in this paper will be associative rings with an identity element.Unless otherwise mentioned all the modules considered will be left unitary modules.The notion of socle-fine class has been introduced by A. Idelhadj and A. Kaidi in[7]. A class C of modules is said to be socle-fine if for every M, N in C we havethat M ^ N if and only if soc(M) ^ soc(N).

In [8] A. Idelhadj and A. Kaidi give the dual of the notion of socle-fine. Thisdual notion is called radical-fine. A class D of modules is said to be radical-fineif for every pair M, N in D we have that M = N if and only if r a d M ) =One of the interesting problems that can be posed is that of characterizing ringsusing these notions. In this way A. Idelhadj, A. Kaidi, D. M. Barquero, C. M.Gonzalez and A. Yahya have proved important results that can be found in [9], [8],[7] and [11]. The socle of a module M, denoted soc(M), is the sum of all the simplesub-modules of M. The radical of M, denoted rad(M), is the sum of all the small

212 Gonzalez et al.

sub-modules of M. The injective cover of a module M will be denoted by E(M). IfM has a projective cover we will denote it by P(M). Mod-P denotes the categoryof all right P-modules. A sub-module N of a module M, is large in M, denotedby N < M, and M is an essential extension of N if N n X ^ 0 for every non zerosubmodule X of M. By Duality a sub-module N of M is small in -M, denoted byN <C M, if X + iV ^ M for all proper sub-modules X of M. For notations andterminology we shall follow [2]. This paper is concerned with characterizations ofcertain rings using the socle-fine and radical-fine notions. In section 2 of this paperwe prove that any ring R which contains a non infinite set of orthogonal (non zero)idempotents is right perfect if and only if the class CE of injective hulls of cyclic leftmodules is socle-fine. If R has a T-nilpotent radical, then R is right perfect if andonly if the class T of modules of the form ra^M\, where M is quasi-projective, issocle-fine. Consequently, an arbitrary ring R is right perfect if and only if F and CEare socle-fine. Also we prove that a ring R is self-injective with large socle if andonly if the class fP U J-PE of finitely generated projective P-modules and theirinjective hulls is socle-fine. Consequently R is pseudo-frobenius if and only if R hasa finitely generated socle and J-P U J-PE is socle-fine. An artinian QF-Z ring isQF if and only if the class fP of finitely generated projective modules is socle-fine.Also we show that R is a QP-ring if and only if the class P U PE of projectiveP-modules and their injective hulls is socle-fine. In section 3 of this paper we provethat an qfd-nng is an SV-rmg if and only if the class of modules of finite Goldiedimension is socle-fine. An arbitrary ring R is an SV-ring if and only if the class ofinjective modules and finite-dimensional modules is socle-fine. In section 4 of thiswork we show by duality that a ring R is a left V-ring if and only if the class ofdivisible P-modules is radical-fine. Also we prove that R is semi-primitive if andonly if the class of .R-modules of the form ^, where P is projective and K is smallin P, is radical-fine. If R is perfect then R is quasi-frobenius if and only if the classT U Xp of injective modules and their projective covers is radical-fine. Finally weprove that for a perfect ring, the direct product of any family of projective modulesis projective if and only if the class of all direct products of projective PL-modules isradical-fine. Before we state these results we have to introduce some more notationswhich are useful throughout this work.

1- If R is an arbitrary ring and we have a class C of R -modules, we denote byCE the class of injective hulls of elements in C. If each module of C has a projectivecover, we denote by Cp the class of projective covers of modules in C.

2- If R is any ring we shall denote :

(Mod-P)(Mod-P) fC

<00 = {M e Mod-P I Gdim(M) < 00}— {M e Mod-P | Gdim(M) < n}, where n e N

Class of cyclic P-modulesClass of injective P-modulesClass of projective P-modulesClass of finitely generated projective P-modules

with Gdim(-M) is the Goldie dimension of M.

P

Socle-Fine and Radical-Fine Notions 213

2 PERFECT RINGS AND PSEUDO-FROBENIUS RINGS

A ring R is said to be right perfect if every right /?-module has a projective cover.We say that R is semi-perfect if every cyclic right (or left) .R-module has a projectivecover. We recall that an ideal / is left T-nilpotent provided that for every sequence(<2i)j6M* of elements of / there exists n G N such that anan-i....ai = 0. A ringR is said to be left pseudo frobenius (PF) if it is a left injective cogenerator. In[12] it is shown that a ring R is left PF if R is left injective, semi-local with largesocle. Recall that R is quasi frobenius if R is artinian injective. In [14] Osofsky hasintroduced left pseudo frobenius rings as a generalization of quasi-frobenius rings.For examples of PF-rings which are not QF, see [14] and [13].

PROPOSITION 2.1 If R does not contain an infinite set of orthogonal (non zero)idempotents, then the following conditions are equivalent:

(i) R is right perfect.( i i ) Cs is socle-fine.

PROOF.- (i) =>• (ii) Let M be a non zero element of Cg. Then M is of theform E(C) for some non zero cyclic left module C. Since R is right perfect, by[4, Theorem 22-29] every non zero left module has non zero socle. It follows thatsoc(C') is essential in C and hence M is isomorphic to E( 0 5,), where (J) 5$ is

iel i€l

the socle of C. Therefore CE is socle-fine.(ii) =3- (i) Let M be a left ^-module. If soc(M) = 0, then consider an arbitrary

element x of M. Therefore soc(Rx) = 0 and hence soc(E(Rx)) = 0. Since E(Rx)and 0 are two elements of CE with the same socle E(Rx) — 0, Rx = 0 and thenx = 0. Since x is arbitrary in M, M = 0. By [4, Theorem 22-29] R is right perfect.

PROPOSITION 2.2 For any ring R with left T-nilpotent radical J, the followingassertions are equivalent:

(i) R is right perfect.(ii) R is left (or right) semi-perfect.(Hi) The class F of left R-modules of the form r&l^fM\, whenever M is quasi-

projective, is socle-fine.

PROOF.- (i) =^> (ii) Obvious.(ii) => (in) Since R is semi-perfect, every module of the form rad

J/fM) is semi-

simple as an 4,-module and by [2, Corollary 2-12] ra^M} ^s serni-simple as an R-module, and we have soc(^gn?y) — r ad(Af) • ̂ follows that F is socle-fine.

(Hi) => (i) We have soc(-y) = soc(soc(-j)), and since rad(soc(-j)) = 0 we canwrite soc(4) = soc(—*° ,&•<•<)• Since R and soc(4) are quasi-projective, then 4v J ' vrad(soc(-j)) ' ^ J' n f J > J

and —^° J.]}.. are two elements of F with the same socle. Then 4 = soc(4). andrad(soc(j)) J v J"

hence ^ is semi-simple. Since J is left T-nilpotent, then it follows that R is rightperfect. I

By the above propositions we can give the following characterization of perfectrings.

214 Gonzalez et al.

THEOREM 2.3 For any ring R the following assertions are equivalent:1) R is right perfect.2) F and CE are socle-fine.

PROOF.- 1)=^2) Follows from Proposition 2.1 and Proposition 2.2.2)=>1) Since CE is socle-fine, from [4, Proposition 22-10A] it follows that J is

left T-nilpotent and since F is also socle-fine it follows that R is right perfect. |

LEMMA 2.4 A left R-module P is finitely generated and protective if and only iffor some R-module P' and non zero positive integers n, there is an R-isomorphismP0P' ^ R<-n\

PROOF.- See [2, Corollary 17.3] ,

THEOREM 2.5 For any ring R the following conditions are equivalent.1- R is self-injective with large socle.2- FP U FPE is socle-fine.

PROOF.- 1=»2) Let P be a finitely generated left projective .R-module. Then bythe previous lemma there is a module N and a non zero positive integer n such thatP ® N = PJ"). Since R is injective, P is also injective. Thus JPP coincides withJ~PE- On the other hand if PI is a sub-module of P such that PI Pisoc(P) = 0 thensoc(Pj) =0. Hence PI nsoc(fl(™>) = 0. Since soc(.R(™>) is large in P>), then P! = 0.It follows that soc(P) is large in P. Therefore P is an injective hull of soc(P). If Qis another element of FP, then we have also Q = E(soc(Q)}. If soc(P) = soc(<5),then clearly P = Q and hence FP U J-PE is socle-fine.

2=>1) Conversely, remark that PC and E(R) are two elements of J-P U J-Pe,with the same socle, so P = E(R) and hence PC is left injective. Let now I be aleft ideal of R such that / n soc(R) = 0, then s o c ( E ( I } ) = 0. On the other handE(I) is a direct summand of PC . It follows that E(I) is projective and finitelygenerated. Hence E ( I ) and 0 are two elements of FPuFPE, with the same socle,then E ( I ) = 0 and hence 7 = 0. Therefore soc(R) is large in R. (

COROLLARY 2.6 For any ring R with finitely generated socle, the following con-ditions are equivalent.

1- R is a pseudo-frobenius ring.2- FV U FPE is socle-fine.

PROOF.- See [4, Proposition 24-32] and Theorem 2.5. (

LEMMA 2.7 Let M be a left finitely generated R-module. If the injective hull ofM is projective then it is also finitely generated.

PROOF.- See [12, Lemma 13-6-6, P: 356]. ,

PROPOSITION 2.8 If R is a QF-S artinian ring, then R is QF if and only ifthe class J-P of finitely generated projective R-modules is socle-fine.

PROOF.- Let P be an element of TV. Since R is QF, P is injective and we willhave P ^ E(P). It follows that TV is socle-fine.

Conversely, let P be a finitely generated projective module. Since R is QF-Zartinian, E(P) is also projective. It follows from Lemma 2-7 that E(P) is againfinitely generated. Hence P and E(P) are two elements of TV with the same socle,so P is isomorphic to its injective hull. Therefore every finitely generated projectivemodule is injective. In particular R is injective. Hence R is QF. |

THEOREM 2.9 For any ring R, the following assertions are equivalent:(i) R is quasi-frobenius.(ii) V U VE is socle-fine.

PROOF.- (i)=>(ii) Over a QF ring, every projective module is injective with largesocle. Hence V U VE is socle-fine.

(ii)=>(i) Let P be an arbitrary projective P-module. Remark that P and E(P)are two elements of V U VE with the same socles. It follows that P is isomorphicto its injective hull. Therefore, R is QF. I

3 RINGS WHOSE CLASS OF FINITE-DIMENSIONAL MODULESIS SOCLE-FINE

A module M has finite Goldie dimension provided E(M) is a finite direct sum ofindecomposable sub-modules. In the literature, a module of finite Goldie dimensionis sometimes called a finite-dimensional module. If M is a module of finite Goldiedimension, there exists a non-negative integer n such that E(M) is a direct sum ofn indecomposable sub-modules. Moreover, by [6, Lemma 4-12] any other decom-position of E(M) into a direct sum of indecomposable sub-modules has exactely nsummands. Thus n is uniquely determined by M, and is called the Goldie dimen-sion of M (denoted by Gdim(M)). A ring is called qfd-ring provided every cyclicmodule has finite Goldie dimension. The class of qfd-rmg contains all rings withleft Krull dimension. So, in particular, every left noetherian ring is left qfd. Forfurther properties of qfd rings, see [1]. We recall that a ring R is called a left 1^-ringif every simple left Pi-module is injective. In [15] Villamayor has proved that R is a,left V-r'mg if and only if every left ideal of R is an intersection of maximal ideals.Hence the Jacobson radical of each F-ring is zero. A ring R is a right semi-artinianif every non zero right PL-module has non zero socle. We recall that P is a rightSV-fmg if it is a right semi-artinian V-fmg. SV-rings form a special class of Von-Neumann regular rings [3]. In [3] Baccela has proved that R is a right SV-r'mg ifand only if every non zero PL-module has non zero injective sub-module. We recallthat a ring PL is semi-primitive provided that the intersection of its primitive idealsequals zero. In [10] Jacobson has proved that PL is semi-primitive if it has a zeroJacobson radical. Consequently each V-ring is semi-primitive.

PROPOSITION 3.1 Let R be a qfd-ring. Then the following conditions are equiv-alent:

1- R is a right SV -ring.2- (Mod-R)<00 is socle fine.

216 Gonzalez et al.

PROOF.- 1=>2) Let M be a module such that Gdim(M) = n. If n = 0 thenM = 0. Suppose that n ^ 0, hence there are many finitely indecomposable modulesEi,...,En such that E(M) = E\ ® ... ® En. Since R is semi-artinian, for everyi € {!,...,n}, soc(E)i ^ 0. Then for each i, there exists a simple module Si

nsuch that ^ = E(Si). It follows that soc(M) = g) Sj. Since /? is a V-ring then

i=l

soc(M) ^ ^ £'(5',). Hence M ^ soc(M) =* £(M). Therefore over an 5^-ringi=l

every module with finite Goldie dimension is injective and semi-simple. It followsthat (Mod-R)<00 is socle-fine.

2=>1) Conversely, let M be an arbitrary non zero .R-module and let x be a nonzero element of M. Since R is a qfd-fmg then the cyclic /2-module x/? has finiteGoldie dimension. Hence there are many indecomposable R-modules Ei,...,Ensuch that E(xR) = EI ® ... ® En. So, Gdim(>#) = Gdim(E(xR)) = n, hencexR and E(xR) belong to (Mod-R)<00. Since soc(xR) — soc(E(xR)), it follows thatxR = E(xR). Therefore xR is a non zero injective sub-module of M. Hence by [3,Theorem 2-7] R is an SV-ring. (

PROPOSITION 3.2 For any ring R, the following statements are equivalent:1- R is a right SV-ring.2- JU (Mod-R)<00 is socle fine.

PROOF.- 1=>2) By the above Proposition, over an SV-ring every module withfinite Goldie dimension is injective. On the other hand, over a semi-artinian ringevery injective module is of the form E( @ Si), for some set / and some family

ie/(5j)j6/ of simple modules. It follows that J U (Mod-.R)<00 is socle-fine.

2=>1) Conversely, let 5 be an arbitrary simple right .R-module,

Gdim(S') = Gdim(E(S)) = 1,

then E(S) and 5 are two elements of J U (Mod-/?)<00. Since they have the samesocle, 5 = E(S) and hence R is a right V-ring. Let now M be an arbitrary R-module. If soc(M) = 0 then soc(E(M)) - 0. Since 0 and E(M) belong to I, itfollows that E(M] = 0 and hence M = 0. Hence R is semi-artinian. (

PROPOSITION 3.3 For any left noetherian ring R, the following statements areequivalent:

(i) R is left artinian.(ii) The class (J)<2 of injective R-modules of Goldie dimension < 2, is socle-

fine.

PROOF.- (i) => (ii) Over any artinian ring, for every nonzero element M of (T)<2there exists a simple module 5 such that M = E(S). Therefore (2")<2 is socle-fine.

(M) =£> (i) Let M be an arbitrary injective left .R-module. By [16, Theorem4-4] there exists a family (Ei)i^i of injective indecomposable modules such thatM = 0 Ei. Suppose that there exists i0 6 / such that soc(Eia) = 0. Since

ieiGdim(£'j0) = 1, it follows that 0 and Ei0 are elements of (2T}<2 with the same socle.

So Eio = 0 which is not possible, because Eio is indecomposable. Hence for everyi £ I,soc(Ei) ^ 0, so for every i 6 / there exists a simple module Si such thatEi ^ E(Si). Therefore M =* 0 £(.%). It follows by [16, Theorem 4-5] that R is

i€l

left artinian. I

PROPOSITION 3.4 For any left noetherian ring R, the following statements areequivalent:

(i) R is semi-simple.(M) The class (QI)<2 of quasi-injective R-modules of Goldie dimension < 2, is

socle-fine.

PROOF.- (i) => (ii) Obvious.(M) => (z) Since every injective module is quasi-injective, (I)<2 is socle-fine. It

follows by the previous Proposition that R is left artinian. Thus it is sufficientto prove that R has a zero radical. Consider an arbitrary simple left /?-moduleS. We have Gdim(S) = Gdim(£(5)) = 1. Since S and E(S) are quasi-injectivethey belong to (QJ)<2- On the other hand soc(E(S)) = soc(S), so it follows thatS ^ E(S). Therefore R is a left V-ring and from [15, Theorem 2-1] every leftfl-module has a zero radical, in particular rad(/?) = 0. Hence R is semi-simple. |

PROPOSITION 3.5 Let R be a noetherian domain with Krull dimension 1. Thenany class C of direct sums of injective R-modules, with the same finite Goldie di-mension, is socle-fine.

PROOF.- Let M be an element of C. Then M is injective and hence there exists afamily (/?i)ig/ of indecomposable injective modules such that M = 0 Ei, where

i€l

|/| < oo. Since R is commutative noetherian, every indecomposable injective R-module is of the form E(-), where p is a prime ideal of R. So, the fact that the Krulldimension of R is 1 implies that p is either maximal or zero ideal of R. It follows thatevery indecomposable injective /?-module is either of the form E(R) or of the formE(S), where S is simple. Thus there is a family (Si)iei1 of simple modules such thatM = 0 E(Si) 0 E(R)(Iz\ where /i and /2 are contained in /. Let N be another

ie/i

element of (7, then in the same way as before N ^ 0 E(Tj] 0 E(R)(-J2\ fori€J\

some family (Tj}jej1 of simple modules and some set J%. If soc(M) = soc(N) thenTj. Then there is a bijection a : /i — > Jj such that |/i| = |Ji|Sj =

- i j€Ji

and for every i e /j, 5j = T^;). Since M and AT are two elements of C then theyhave the same finite Goldie dimension. So, Gdim(M) = Gdim(N) implies thatI/! | + |/2 1 = [J: + (J2| and hence |/2 = |J2|. It follows that E(R^ ^ E(R)(J21Therefore M = N. ,

COROLLARY 3.6 Over a noetherian domain with Krull dimension 1, any classof finitely generated injective modules with same Goldie dimension is socle-fine.

218 Gonzalez et al.

4 RADICAL-FINE CHARACTERIZATION OF RINGS

An element r of R is said to be a right zero divisor if there exists a non zero elements of R such that sr = 0. A left /^-module E is said to be divisible if E = rE,whenever r is an element of R which is not a right zero divisor. A Z -module M isdivisible if and only if it is injective.

LEMMA 4.1 [16, Proposition 2-6]. Let R be an arbitrary ring. Then everyinjective left R-module is divisible.

PROOF.- Let E be an injective _R-module, let e e E, and let r be an element of Rwhich is not a right zero divisor. Consider the next diagram:

0 -> Rr --> Rf l /gE

where / is defined by /(sr) = se (s e R). Note that, because r is not a rightzero divisor, if sr = 0 then s = 0 and hence /(sr) = 0, so / is well-defined R-homomorphism. Hence there exists a homomorphism g : R —> E which agrees with/ on Rr. Thus

e = f ( r ) =5(0 = r f f ( l )

which shows that E is divisible. I

LEMMA 4.2 [16, Lemma 2-4]. Let E be a divisible R-module and let E' be asubmodule of E. Then -j^ is also a divisible R-module.

THEOREM 4.3 For any ring R, the following conditions are equivalent:1) R is a left V-ring.2) The class TJ of divisible R-modules is radical-fine.

PROOF.- 1=>2) This is obvious, because every module over a V-ring has a zeroradical (see [15, Theorem 2-1]).

2=>1) Let M be an arbitrary left .R-module. By Lemma 4.1, E(M) is divisible.E(M)It follows from Lemma 4.2 that 'ls a'so divisible. Then TaMgM))

E(M) are two elements of D such that

E(M] ^ rad(B(M))

Hence E(M] = r f ) ) , it follows that rad(£(M)) = 0 and hence rad(M) = 0.Since M is arbitrary, it follows from [15, Theorem 2-1] that R is a left V-ring. t

LEMMA 4.4 If <p : M —+ N is an epimorphism and if Ker(y) C rad(M), thenrad(JV) = <Xrad(M)).

PROOF.- See [2, Proposition 9-15]. (

THEOREM 4.5 For any ring R, the following properties are equivalent:1- R is semi-primitive.

J..,.. -r,!,. f___ __

is radical-fine.2- The class S of R-modules of the form •£, whenever P is projective and K <C P,

PROOF.- Let M be an element of <S, then there is a projective module P and asmall submodule K of P such that M — -^ . Consider the application

x H-> x + K

Since K is small in P, then tp is a minimal epimorphism and by lemma 4-4rad(-j^) = ra ^+ — . As P is projective rad(P) = J.P where J is the Jacobsonradical of R. Since R is semi-primitive J = 0 and hence rad(P) = O.It follows thatM has zero radical. Therefore S is radical-fine.

2=^1) Conversely, rad( -j) = 0 implies that -j = ^, % . . Since 0 and J are two

small submodules of R, both R and j are elements of S such that ra

ad(3)--*Vi f V > ^ 4 - _____

rad(fl)

It follows that R = -y and hence J — 0. Therefore R is semi-primitive.

LEMMA 4.6 [2]. If M is a module which has a projective cover P(M). ThenM ~ P(M)

rad(M) ~ rad(P(M)) '

The next theorem is the dual version of theorem 2-9.

THEOREM 4.7 For a left perfect ring R, the following statements are equivalent:1- R is QF.2- T U Tp is radical-fine.

PROOF.- If R is QF, then every injective module / coincides with its projectivecover P(/). Since R is artinian, for every projective Pi-module P, we have rad(P) <CP. It follows by [8, Theorem 4-2-1, P: 49] that / U IP is radical-fine. Reciprocally,let M be an injective P-module. Since PL is left perfect, M have a projective coverP(M). By lemma 4-6 we will have ^fa ^ TJ(

(™(]M}} • Since M and P(M) are two

elements of / U IP, it follows that M = P(M) and hence every injective P-moduleis projective. Therefore, by the Faith-Walker theorem [5] R is QF. |

REMARK 4.8 // the direct product of any family of projective R-modules is alsoprojective then necessarily R is perfect. However, the converse is not true ( see [4,Theorem 22.3'IB P: 168] ). If we let "P^ be the class of all direct product of projectiveR-modules, then we have the next proposition.

PROPOSITION 4.9 For a perfect ring R, the following conditions are equivalent:1- The direct product of any family of projective R-modules is projective.2- PK is radical-fine.

220 Gonzalez et al.

PROOF.- l=>2) Follows from [8, Proposition 5-2-1, P: 73].2=>1) Let (Pi)i£i be an arbitrary family of projective ^-modules. Putting M =

R Pi. Since R is perfect then M has a projective cover P(M). By the lemma 4-6_

• Since M and P(M) are two elements of ̂ then M ̂ P(M).It follows that Y\ Pi is projective.

i el

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About Bernstein Algebras

S. GONZALEZ and C. MARTINEZ, Dpto. de Matematicas, U. Oviedo. C/ CalvoSotelo s/n. 33007 Oviedo. Spain.E-m&il:[email protected] and [email protected]

AbstractThe aim of this paper is to present an overview of some known results and

recent advances on the structure of Bernstein algebras and compare the situ-ation for Bernstein algebras with the general situation for fctfeorder Bernsteinalgebras.

1 PRELIMINARY RESULTS

A very good review of nonassociative algebras that arise in relation to Genetics canbe found in [57]. We will refer the reader to it to understand how nonassociativealgebras can be used in Genetics.

In this paper we will concentrate on some particular classes of those algebras,the so called Bernstein (nth-order Bernstein) algebras. We will try to give a gen-eral overview of the known facts about their structure and the relations with othernonassociative classes such as Jordan algebras, and other algebras relevant in Ge-netics.

Since an exhaustive reference to all articles related to this subject that appearedin the last fifteen years would exceed the admissible length for this article, we willmainly refer to those papers that have been written inside of our research group.

Almost all algebras that appear in Genetics are commutative nonassociativealgebras. Those with a genetic meaning are real algebras, but in order to do analgebraic study of them if has become usual to consider algebras over an arbitraryfield, that in our case this will always be of characteristic different from 2.

In general, algebras that appear in this context have a scalar matrix represen-tation, that is, our algebras will be baric algebras.

DEFINITION 1.I An algebra A over a field F is called baric if it admits a nonzeroalgebra homomorphism uj : A —» F, that is called weight function.


224 Gonzalez and Martinez

In nonassociative algebras we can define powers of an element in several ways. Ifx is an element of a commutative nonassociative algebra we will consider principalpowers of the element x defined by x, x2,..., xt+l = xlx and plenary powersx, x^l, . . . denned by x^+V = x®xW.

If the algebra A has genetic meaning and an element x £ A represents a popu-lation, each element x1 of the sequence of principal powers represents a populationobtained mating back the previous population xl~l with the original populationx (this is usually done in a laboratory) and the sequence of plenary powers con-tains the sucessive generations obtained by random mating within the population,beginning with x (and that appears in nature).

If {ai, 02, • • • , an} is a basis of A, it can be proved that there exists a polynomialin principal powers that annihilates all elements of A ([62]),

f ( x ) =xr + 9lXr-1 + ••• + <9r_iz

where 0, is a homogeneous polynomial of degree i in the coordinates x, of a genericelement x = X)™=i %iai-

We refer to the polynomial /(x) as rank polynomial.

DEFINITION 1.2 A baric algebra A with rank polynomial f ( x ) is a train alge-bra of rank r if the coefficients &i of f ( x ) are functions ofu>(x), where u representsthe weight function.

Working in an adequate extension of F, f ( x ) splits into linear factors

/(x) = x(x — AIU>(X)) (X — \-2fjJ (x}) • • • (x — AT ._iw(x))

and the elements AI , A2, . . . A r _ j are called principal train roots of A.

DEFINITION 1.3 A baric algebra with weight function w is a special train al-gebra if N = KeruJ is nil-potent (that is, Nm = (0) for some m) and all principalpower subalgebras Nl = Nl~lN of N are ideals of A.

Every special train algebra is a train algebra.Genetic algebras form a class of algebras between special train and train algebras.

DEFINITION 1.4 A commutative finite dimensional algebra A is genetic (in thesense of Gonshor [14]) if the algebra has a canonical basis {ao, ai, . . . ,an} withstructure constants A^ (a^j = Y^k=o^ijkO<k) satisfying A0oo = 1, ^ojk — 0 fork < j and A^ = 0 for i,j>Q and k < max(i,j).

This definition turned out to be equivalent to the definition given by Schafer ofgenetic algebras.

DEFINITION 1.5 A baric algebra A with weight function w is a genetic algebra(in the sense of Schafer [59]) if given an arbitrary elementT = aI+f(RXl, . . . , RXn)in the transformation algebraT(A), the characteristic function \XI — T\ is a function

Here, as usual, the transformation algebra T(A) denotes the algebra generatedby the identity operator 7 and all right multiplications Rx : A — > A, x e A.

Bernstein Algebras 225

DEFINITION 1.6 A baric algebra A with weight function ut is called a Bernsteinalgebra (resp. nth- order Bernstein algebraj if the plenary powers of any ele-ment x & A satisfy x^ = w(x)2x^ (resp. a>+2l = (w(x))2"xl™+1]).

When a Bernstein (resp. fc'^-order Bernstein) algebra has genetic meaning,elements of weight 1 (those that represent populations) satisfy x'3l = x^ (resp.z[fc+2] _ x{k+i}^ th^ jS) equilibrium is reached after one generation (resp. k gener-ations) of random mating within the population.

In what follows, A will denote a Bernstein (or fc^-order Bernstein) algebra, ujwill represent its weight function and N will be used to denote the barideal of A,N = Keruj.

2 IDEMPOTENTS

The existence of an idempotent in an algebra A leads in general to a (Peirce)decomposition, so it has always algebraic interest. But idempotents also have ge-netic significance. If an element e2 — e e A represents a population, then geneticequilibrium has been achieved after one generation of random mating within thepopulation x.

So idempotents play an important role in the study of algebras that appear inGenetics, but their existence is not guaranteed. Clearly, if A is a baric algebra ande2 = e is an idempotent in A, then w(e) = 0 or 1.

In the case of .A a Bernstein (resp. fc^-order Bernstein) algebra there alwaysexist idempotent elements. Indeed, if x s A satisfies w(x) = 1, then e = x2

(resp. e = zlfc+1!) is an idempotent element and w(e) = 1. Hence we have a firstdecomposition A = Fe + Keruj.

It is well known ([62]) that if A is a Bernstein algebra then Kerui = Ue® Ze,where Ue = {u e Keru>\2eu = u] and Ze — {z 6 Kerui\ez = 0}.

Products of elements of Ue and Ze satisfy the following relations:

(B.I) UeUe C Ze, UeZe C Ue and ZeZe C Ue.

(B.2) For every u € Ue, z e Ze we have U3 = 0 = u(uz) — (uz)2 = uz2.

The set of idempotent elements of A is I (A) = {e+u+u2 \u£Ue}. Furthermore,if / = e + u + u2 is another idempotent element, then Uf = {x + lux \ x e Ue},Zf = {z- 2uz - 2u2z z e Ze}.

Consequently dimUe = dimUf and dimZe = dimZf are independent of theparticular idempotent element. This allows us to define the type of the Bernsteinalgebra A as the couple (1 + dimUe,dimZe). In this way type(A) is an invariantthat plays an important role in the classification results.

If A denotes a /ct/l-order Bernstein algebra, then A = Fe © Keruj and Keru> =Ue + Ze, where Ue = {x e Kerw\1ex = x}, Ze = {z e Keru\R™z = 0}.

But it is not known how to get all idempotent elements from a given e2 = e inA, as it is known in the Bernstein case.


226 Gonzalez and Martfnez

It was proved in [21] that the dimension of Ue (and so the dimension of Ze) doesnot depend on the particular idempotent element e. Notice that for k > 2 and e, ftwo idempotent elements, we do not know how to express the elements of Uf (resp.Zf) in terms of the elements of Ue (resp. Ze), even if we know the expression of /in terms of e.

In [33], [34] and [35] idempotent elements of second order Bernstein algebras werestudied. In this case, given an idempotent element e, the set {(e + u + u2)2 \ u <E Ue}is contained in I (A), but in general both sets do not coincide. It happens in someparticular cases.

Even in the case k = 2 relations satisfied by products of elements in Ue and Zeare more complicated and difficult to use. For instance U2 C Ze is still valid, butnothing is known about UeZe and Z2.

Also the relations in (B.2) are no longer valid. In the case k = 2 we have insteadthe following relations for arbitrary elements u € Ue, z £ Ze

(B.3) e(uz) + u(ez) e Ue, z2(ez] e Ue, ez2 + 2(ez)2 e Ue,

eu3 + u(eu2) e C/e, eu3 + u(eu2) =0, u3 + 2u(eu2) e Ze,

uz 2 = 2e(uz2 + 4(uz)(ez)) + 2u(ez2 + 2(ez)2).

Coming back to Bernstein algebras, some particular classes can be considered.

DEFINITION 2.1 a) A Bernstein algebra A is nuclear if A = A2, or equivalently ,U2 = Ze for an arbitrary idempotent e.

b) A Bernstein algebra A is called exclusive (or exceptional,) if U2 = 0 forsome idempotent element. Then U2 = 0 for every idempotent e.

c) A Bernstein algebra is called normal or conservative if x2y = w(x)xy forevery x,y G A. Such algebras satisfy Z2 + UeZe = 0 and this gives a characterizationof normality.

Consequently, the three definitions above do not depend on the particular idem-potent e that we consider in the algebra. We know that dimU2 and dim(UeZe + Z2)are invariants of the algebra. This is not the case with dimZ2 and dimU3, whichcan change when we take different idempotents.

A Bernstein algebra is called orthogonal if Uf = 0 for some idempotent elemente. When U3 = 0 holds for every idempotent element, the Bernstein algebra A iscalled totally orthogonal. It can be proved that A is totally orthogonal if and onlyif for some idempotent element, we have U3 = 0 qnd U2U2 = 0 (and so the relationis satisfied for every idempotent).

In [30] it was proved that UQ = {u € Ue \ uUe = 0} is an ideal of the Bernsteinalgebra A, which is invariant under derivations and does not depend of the particularidempotent element e. Indeed, UQ = Hee/M) ^e' ^ *he algebra A is nuclear, then

= 0. This ideal has been very helpful in the study of Bernstein algebras.

3 RELATIONS WITH OTHER CLASSES OF ALGEBRAS

Initially a baric algebra may have more than one weight function. However it isknown (see [62]) that if Keruj is nil, then the weight function is unique.

If A is & Bernstein algebra then N = Kerui is not necessarily nil, understandingthat a (nonassociative) algebra is nil if some principal power of every element iszero. For instance, A = Fe + Fu + Fz with products e2 — e, leu = u, ez = 0,u2 = Q,uz = z2 = u is a, Bernstein algebra with N = Fu + Fz, and zn = u forevery n > 2.

However every element x e Kerut in a Bernstein algebra satisfies (x2)2 = 0 andit can easily be seen that this suffices to prove the uniqueness of the weight function.

This is also the case for fc^-order Bernstein algebras, since elements in Kerujsatisfy x^+^ = 0.

Since the ideal N = Kerui of a Bernstein algebra is not nil, clearly it is notnilpotent in general. Of course this is the case if A is a Jordan-Bernstein algebra,that is, a Bernstein algebra that satisfies also the Jordan identity x2(yx) = (x2y)x,Vx, y e A.

In the case of Jordan-Bernstein algebras it is not necessary to distinguish be-tween principal and plenary powers, since every Jordan algebra is power-associative,that is, the subalgebra generated by one element is associative. Of course the con-verse is not true and there are commutative power-associative algebras that are notJordan. In the case of Bernstein algebras we have

THEOREM 3.1 (see [30]) Let A = Fe + Ue + Ze be a Bernstein algebra. Thenthe following statements for A are equivalent

1. A is power-associative,2. A is a Jordan algebra,3. Z2 =0 and (uz)z = 0 Vu S Ue, z e Ze,4- Z2 — 0 for every idempotent element f2 — f € I (A).

Furthermore, it is proved that the above mentioned ideal UQ satisfies thatis a Jordan-Bernstein algebra. In every Jordan-Bernstein algebra the identity x3 = 0for elements of N is satisfied. The converse is not true, as is shown by the algebraB — Fe + Fu + Fz with e2 = e, eu = ^u, ez = 0, u2 = 0 = uz, z2 — u, that is notJordan, although x3 = 0 for all x e N = Fu + Fz.

The previous fact becomes a useful tool. If UQ = 0 in a Bernstein algebra A, thenA is Jordan (so genetic). If UQ ^ 0, then the quotient A/Uo is a Jordan-Bernsteinalgebra of smaller dimension.

This is used in [1] to prove that, under the assumption chF ^ 2,3,5 the squareof the barideal iV2 is nilpotent (the algebra is not assumed to be finite-dimensional).

The proof uses two well known facts:i) If J is a special Jordan algebra without elements of additive order < 2n and

xn = 0, then J is solvable ([60]),ii) If J is a solvable Jordan algebra, then J2 is nilpotent ([55]).

However Theorem 3.1 is no longer true if we consider A^-order Bernstein alge-bras, k > 1.

Power associative and Jordan second order Bernstein algebras are studied in[35]. Using the counterexample by Suttles of a 5-dimensional nil algebra, solvablebut not nilpotent, we construct an example of a power-associative second orderBernstein algebra that is not Jordan.

Nuclear Bernstein algebras have more significance from a genetic point of view.It is known that the barideal of a finite dimensional nuclear Bernstein algebra isnilpotent.

Using the ideal UQ it is easy to prove that x4 = 0 for all x e N. Clearly(Keru>)n = 0 if n = dirnA and the product of arbitrary 2n — 1 elements of Kerw iszero. Griskhov conjectured that if the nuclear Bernstein algebra A is generated byr elements, then (Keru)2r+2 = 0. It can be shown that a nuclear Bernstein algebracan be generated by r elements if and only if dimUe — dimU3, < r.

We introduce (see [32], [50]) the notion of free nuclear Jordan-Bernstein (and freeJordan-Bernstein) algebra which allows us to prove the above mentioned conjectureand which has been useful in the study of these algebras.

In [4] the above bound is improved, proving that if A is a nuclear Bernsteinalgebra generated by r elements then (Keruj)r+'i = 0 if r is even and (Kerui)r+3 = 0if r is odd.

Nilpotency of the barideal of a Bernstein algebra has been studied in severalpapers. Since it is known that the powers of the barideal N are again ideals of theBernstein algebra A, nilpotency of N suffices to assure that the algebra A is genetic.For instance, in [56] Engel conditions are studied. If we consider a Bernstein algebraA = Fe + Ue + Ze, the operator Lu : Kerui —> Keruj for u G Ue is always nilpotent.Indeed, it satisfies L3 = 0. However Lz : Kerui —> Kerui, z € Ze does not need to benilpotent. Bayara, Micali and Outtara prove that a Bernstein algebra that satisfies(Lz |j/)3 = 0 for every z € Ze is genetic, that is, the barideal Kerw is nilpotent.

In [12] some relations between Bernstein (resp. fct/l-order Bernstein) algebrasand train algebras are studied.

It is known that if A is a train algebra of rank 3 then the following three state-ments are equivalent.

1. A is Jordan,2. A is power-associative,3. The train equation of A is either x3 — w(x)x2 or x3 — 2u>(x)x2 + w(x)2x.

In [12] it is proved that there are no Bernstein algebras of rank 3 with trainidentity x3 = w(x)2x. Indeed, if the baric algebra A satisfies the train identityx3 = w(x)2x, then to be Jordan, to be power-associative and to be Bernstein areequivalent conditions for the algebra. Furthermore, any of the three equivalentconditions above is equivalent to the train identity x2 = w(x)x in A.

It is known (see [61]) that a Bernstein algebra A is a train algebra of rank 3 ifand only if it is Jordan. In such case the train equation of A is x3 = w(x)x2. In[12] the following generalization (if chF = 0) is proved:

THEOREM 3.2 IfchF = 0, A = Fe + U + Z is a Bernstein algebra the followingconditions are equivalent.

1. A is a train algebra of rank less than or equal to k, k > 4.

2. The train equation of A is

where \(k} =^(^( k ~ ^ + k ~ 1 ) for i = I , 2, . . . , k - 2 and \{k} =V l z J \ l J

3. zk~1 = 0 = z ( z - • • ( z ( z u ) } • • •)) for every u e Ue, z e Ze.

k-l

4- zk~l = 0 for every z € Zf and for every f e I (A).

This characterization resembles the one obtained for Jordan-Bernstein algebrasin [30] and suggests that the condition about products in Z is related to the factof being a train algebra and the characterization for Jordan-Bernstein algebras is aconsequence of the fact that they are exactly the Bernstein algebras of rank 3. Thispoint of view has another confirmation in the following result, also proved in [12].

PROPOSITION 3.1 If A is a Bernstein algebra and k > 4 then (Keru)1*-1 nC/o is an ideal of A, that is invariant under derivations and the quotient algebraA/((Kerio}k~l n C/Q) is a train algebra of rank less than or equal to k.

In the same paper some relations between train algebras and second order Bern-stein algebras are studied. It is proved that a second order Bernstein algebra A cannot satisfy a train identity of degree less than 4. If it is a train algebra of rank 4,then the train equation is x4 = ui(x)x3 and if it is a train algebra of rank 5, thenthe train equation is either xb = ui(x)x4 or x5 — |w(o;)x4 — ~uj(x23x

As we have already mentioned, power-associativity and Jordan conditions arenot equivalent in a second order Bernstein algebra (as it is the case in Bernsteinalgebras). However it is proved in [12] that a second order Bernstein algebra A isJordan if and only if it is both power-associative and a train algebra of rank 4.

Notice again the difference with the situation in Bernstein algebras. Secondorder Bernstein Jordan algebras are not characterized as the second order Bernsteinalgebras that are train algebras of rank 4, since there are second order Bernsteinalgebras satisfying z4 = w(x)x3 that are not power-associative.

4 BERNSTEIN PROBLEM

A real algebra A has genetic realization if it has a basis {ai , . . . , an} and a multiplica-tion table didj = Y^ik=i lijko-k such that the structure constants satisfy 0 < 7^ < 1for all i,j,k and X)fe=i 7ijfc = 1-

Such a basis is called a natural basis or stochastic basis and algebras with geneticrealization are the ones that can have genetic significance. The class of algebras

with genetic realization is very big and includes the class of baric algebras, sincethe map u : A —> ffi, w(£iai + • • • + £raan) = £1 + ••• + £„ is a weight function.

In the 20's S. Bernstein ([5],[6]) studied a quadratic evolutionary operator ^that maps the simplex of genetic frequency distributions

An =

into itself and represents the passage of one generation to the next. The operator *is called a Bernstein (or stationary) operator if it satisfies 'I'2 = $, which indicatesthat the population is in equilibrium after one generation.

Associated to every Bernstein operator \& we have a real Bernstein algebra ob-tained by considering the real vector space A over a basis {ap, a j , . . . , an} with themultiplication xy = ^ ( t y ( x + y) — ̂ (x) — $(y) for elements x,y & An and in generalxy = w(x)w(y)xy, if x - u(x)x, y - u(y)y, x,y e Are.

In this way the study of Bernstein operators becomes equivalent to the classi-fication of real Bernstein algebras having a stochastic basis (that is, with geneticrealization). This problem, known as the Bernstein problem has been for years oneof the main problems in this area.

This problem was studied by Lyubich (see [46]) who gives an explicit descriptionof Bernstein operators in the regular and exceptional cases. This implies that theproblem (already solved by Bernstein for dimensions 2 and 3) was totally solved forBernstein algebras of dimension < 4.

In [22] the Bernstein problem in dimension 5 was considered. Since a Bernsteinalgebra of type (2,n — 2) or (n — 1,1) is either exceptional or regular, only type(3,2) has to be considered. It was proved that (up to isomorphism) there are sixnonregular, nonexceptional Bernstein algebras of type (3,2).

The Bernstein problem for type (n — 2,2) , dimension 6 and type (3, n — 3) wasstudied in [36], [37] and [38] respectively. Each time that the dimension of theBernstein algebra increases a little, the difficulty of the problem increases a lot.Due to space limits we do not enter in details of proofs, that are quite technical.

In [40] we proved that a nuclear Bernstein algebra having a stochastic basis isregular. In this way we answer in a positive way a conjecture posed by Lyubich andgive an important step to the final solution of the Bernstein problem, that has beengiven in [39], where the explicit form of all nonregular, nonexceptional stationaryoperators can be found.

5 AUTOMORPHISMS AND DERIVATIONS

Another way to approach the structure of Bernstein algebras is via the group ofautomorphisms and the Lie algebra of derivations.

Since we would like a classification up to isomorphism of Bernstein algebras, itseems natural to attempt a characterization of isomorphisms that allows to treatthem in a simpler way. So in [49] the following theorem was proved

THEOREM 5.1 If A and A' are Bernstein algebras and 9 : A —> A' is a bijec-tive linear map, then 9 is an isomorphism of Bernstein algebras if and only if thefollowing two conditions are satisfied:

i) An element e € A is an idem/potent if and only if 0(e) € A' is an idempotent,ii) xy = 0 in A if and only if 9(x)9(y) = 0 in A'.

Since a general classification, up to isomorphism, has not been possible exceptfor small dimension ([8],[23]), some other classification attempts have been made. In[16] the homotope algebra of a Bernstein algebra (according to the general definitiongiven by Teddy for nonassociative algebras) was studied. It was proved that thehomotope of a Bernstein algebra by an element a is again a Bernstein algebra,whenever w(a) ^= 0. However a natural notion of isotopy does not exist (see [26]).In [19], taking into account relations between an algebra A and some homotope,the notion of quasiisomorphism (weaker than isomorphism) is defined and studied.

If A = Fe + Ue + Ze is a Bernstein algebra, then a derivation D of A is char-acterized by a triple (u, / ,<?), where u = D(e) € Ue, f : Ue —> Ue, g : Ze —» Ze arelinear maps that satisfy the following three conditions:

1. g(uu') = f(u)u' + u f ( u ' ) ,2. f ( u z ) = f ( u } z + ug(z) + 2(uu)z,3. f ( z z ' ) = g(z}z' + zg(z') - 1((uz)z' + (uz')z)for every u,u' 6 Ue, z,z' 6 Ze.

So dim Der(A) < r + r2 + s2, where type(A) = (r + 1, s). The upper bound canbe reached exactly for trivial Bernstein algebras, that is, algebras with (fcerw)2 = 0.

Clearly, there is also a natural lower bound of dim Der(A), that is 0. And againthis bound can be reached, as was proved in [31] where some examples of Bernsteinalgebras having zero derivation algebra are given. In the same paper some necessaryconditions for a Bernstein algebra to have zero derivation algebra were obtained,but sufficient conditions were not found. The fact that the derivation algebra ofa Bernstein algebra may be zero or have the maximal admissible dimension showsthat the structure of Bernstein algebras changes in a very big range and justifiesthat, up to the moment, there is no structure theory for those algebras.

Inner derivations of Bernstein algebras have been studied in [17]. A derivationis inner if it lies in the Lie transformation algebra generated by the (left and) rightmultiplications. If A is a Jordan algebra this Lie transformation algebra is knownto be C(A) = R(A) + [R(A),R(A)}.

Jacobson proved that if A is a semisimple associative (Lie, alternative or Jordan)algebra over a field _F (ch F =£ 2) then all derivations of A are inner.

Schenkman proved that every nilpotent Lie algebra over F has a derivationwhich is not inner.

In [48] inner derivations of a Jordan-Bernstein algebra are studied. In this casethe Lie derivation algebra is a 2-graded algebra with its homogeneous componentof degree 1 isomorphic (as vector space) to Ue and consisting of inner derivations.In particular, a Jordan-Bernstein algebra always has nonzero derivation algebra. Inmany cases, for instance if UZ = 0 or if U2 = 0, derivations that are not inner areconstructed in an explicit way.

We conjectured (the problem is still open) that in a Jordan-Bernstein alge-bras there always exist derivations that are not inner. It is proved that a Jordan-Bernstein algebra can be embedded as ideal of codimension 1 in another Jordan-Bernstein algebra that has non-inner derivations.

In [13] derivations in a second order Bernstein algebras are studied. In this case(and in general for a fct/l-order Bernstein algebra) every derivation is still defineduniquely by a triple ( u , f , g ) , but relations that must be satisfied are sensibly morecomplicated. Lower and upper bounds of dim Der(A) are found and an explicitform of inner derivations in power-associative and Jordan second order Bernsteinalgebras is given.

6 SOME OTHER ASPECTS

A standard way to approach a structure is via the study of the substructures. Inthis direction, the knowledge of minimal and maximal subalgebras (ideals) of aBernstein algebra becomes interesting. In [18] one-dimensional subalgebras andminimal subalgebras of a Bernstein algebra have been studied.

It is also a usual problem to study conditions under which an isomorphism of thelattice of subalgebras (resp. ideals) of two algebras A and A' implies an isomorphismbetween those algebras and also to study properties of an algebra that are preservedunder a lattice isomorphism.

Bertrand [7] proved that if A, A' are Bernstein algebras and 6 : JC(A) —> C(A'} isa lattice isomorphism, then dimA = dimA' and dimS* = dimS' for every subalgebraS < A. Furthermore, if type(^l) = (1 + r, 0) or (1,1), then A and A' are isomorphic.

In [9] it was proved that the existence of a lattice isomorphism 9 : C(A) —> C-(A')implies (if type(A) ^ (1,1)) the existence of an idempotent e2 = e € A such that0(< e >) =< e' >, with e'2 = e' e A'.

In general a lattice isomorphism 6 : C(A) —» C(A') does not satisfy 9(Kerui) =Kerui'. If the barideal of A is not mapped into the barideal of A', then the set of one-dimensional subalgebras of A generated by one idempotent element is not applied inthe analogous set of subalgebras of A'. The existence of such isomorphism imposesstrict restrictions on the structure of the Bernstein algebra A. Indeed, in [51] it wasproved that the existence of a lattice isomorphism 9 : £(A) —> C(A') that does notpreserve the nucleus implies that both algebras A and A' are exclusive and type(^4)= type(j4') = (1 + r, s), with s = 0 or 1. Furthermore, if type(A) ^ (1,1), then9(A2) = A'2. In general the isomorphism of the algebras A and A' does not followfrom the existence of the isomorphism 0, but it happens under some additionalassumptions given in the paper.

If two Bernstein algebras A and A' have isomorphic lattices of subalgebras wemay always assume a lattice isomorphism 9 : C(A) —> C(A') that applies the baridealof A to the barideal of A', as it is proved in [52].

The type of a Bernstein algebra and some properties, like the fact of beingexclusive, normal, genetic, Jordan or totally orthogonal are preserved under a latticeisomorphism.

In [58] Bernstein algebras having a lattice of subalgebras that is distributive(resp. complemented) are characterized. Although the isomorphism of the algebrasdoes not follow from the isomorphism of the lattice of subalgebras, it is proved in[58] that (under the assumption dimU > 1 and existence of square roots in the fieldF) the existence of a lattice isomorphism implies the existence of an automorphismif) : F —» F and a semilinear map <j>, <j) : A2 —> A'2 (of automorphism if>) betweenthe square of the algebras A and A'.

A similar study based on the lattice of ideals has been made.In [53] Bernstein algebras having a linear lattice of ideals are considered. Only

two algebras satisfy this condition:1. A-i =Fe i+Fui ,2. A2 = Fe + Fu + Fu2,

where products are given in the obvious way.

In [54] Bernstein algebras with a distributive lattice of ideals are studied andcharacterized. It is proved that a Jordan (resp. nuclear) Bernstein algebra has adistributive lattice of ideals if and only if it is either normal or exclusive. Using thisfact we can prove that a Bernstein algebra that has a distributive lattice of idealsis trivial or its lattice is linear or it is exclusive. So the problem has been reducedto the exclusive case that can be totally characterized. If it is not trivial then dimZ = 1. The following result can be proved:

THEOREM 6.1 Given a vector space U, a nonzero endornorphism f G Endp(U)with coinciding minimal and characteristic polynomials and a fixed element u\ G Uthere exists an exclusive Bernstein algebra B = B(<p, U, u\) with a distributive latticeof ideals, B = Fe + U + Fz such that z2 = u\ and U. Furthermore,B(ip,U,ui] ^B(^',U',u{) if and only if dim U =dim U', ]</? - A/| = [</ - A/| andp.m.v(ui) =p.m.lpi(u'l).

Some properties, like the fact of being trivial, normal or nuclear, can be char-acterized via the lattice of ideals, but in general nonisomorphic Bernstein algebrasmay have isomorphic lattices of ideals and it is not known if exclusivity is preservedby isomorphisms between the lattice of ideals.

Identities

Polynomial identities are also important tools in the structure theory of algebras.

If A is an arbitrary algebra over the field F, we can associate an ideal T(A) toA that is an ideal of F < X >, the free nonassociative algebra over an infinite set ofgenerators, that consists of all identities of A, that is, polynomials f(x\,... ,xn) GF < X > such that f(a\,..., an) = 0 V a i , . . . , an 6 A.

In [2] generators of the ideal of identities satisfied by regular Bernstein algebrasand exclusive Bernstein algebras are found.

If (A, <jj) is a baric algebra we can consider the baric T-ideal T(A, w) that consistsof all polynomials in R < X > which vanish under substitution of elements a € Awith w(a) = 1

Using the weight function w an identity of a baric algebra (A,ui) can be trans-formed into another identity, involving w, that holds for any values of variables inA. It suffices to replace each variable x by x/u>(x] and multiply by the commondenominator.

In [2] the following result is proved.

THEOREM 6.2 If (A,w) is a Jordan-Bernstein or nuclear algebra, then the idealT(A, (jj) has a finite set of generators.

Superalgebras

The origin of superalgebras lies in Physics and they are, at the present moment,an important part of Mathematics. Their use has become an important tool, as canbe shown by the techniques developed by Kemer to study identities in associativealgebras in relation to the speach property. The problem is to prove the existenceof a finite basis of the ideal of identities of a subvariety.

If V is a variety of algebras, the usual way to define a V-superalgebra is:

DEFINITION 6.1 A 2-graded algebra A — AQ + A1 is a V-superalgebra if theGrassmann envelope algebra G(A) = G(V)o ® AO + G(V)i <g> AI belongs to V.

This definition can not be directly applied in the case of Bernstein algebras,since they do not form a variety. This fact opens several ways to "extend" thenotion of Bernstein algebra to a "natural" notion of Bernstein superalgebra.

One way, that works well if we are interested in the study of identities, is thefollowing.

We may consider a baric algebra (A, uj) as an algebra with an additional multi-plication a * b — aiu(b) that satisfies

i) a * (be) = (a * 6) * c,ii) (a&) * c = (a * c)b = a(b * c) ,iii) a * (6 * c) = (a * b) * c

Conversely, if A is a commutative algebra with an additional multiplication * sat-isfying conditions i) to iii), we can consider u(a) e EndpA denned by u>(a) : x — -> x*a and the linear mapping from A to Endp(A), a — » u>(a). Condition i) assures thatit is a homomorphism and by ii) the image is an associative commutative subalgebraK of the centroid of A, Y(A) = {</> e EndF(A) Va, b 6 A (a<f>)b = a(b<t>) = (ab)(j>}= {(j) € Endp(A) Va & A (j)Rb = Rb4>}- By iii) a; is a homomorphism over K.

DEFINITION 6.2 A generalized baric algebra is a commutative algebra with anadditional multiplication * that satisfies i) to iii) above.

Let M be a variety of generalized baric algebras defined by multilinear identities{fi i € a;}. A Z2-graded generalized baric algebra A is called .A/f-superalgebra if itsGrassmann envelope G(A) belongs to .M. In this way, .M-superalgebras are definedby the identities obtained from {/j} in the usual way followed for superalgebras.

In [3] we prove:

THEOREM 6.3 If (A,w) is a Jordan or nuclear Bernstein algebra, then the idealT(A) is equal to the ideal of identities of the Grassmann envelope of a finitelygenerated baric superalgebra.

Another way of considering superalgebras is to concentrate on the barideal of aBernstein algebra. In this way the notion of (U, Z)-Bernstein algebra appears.

DEFINITION 6.3 Let B = U + Z be a commutative algebra over F that is a directsum of two vector spaces U, Z that satisfy U2 C Z, UZ C U, Z2 C U.

B is called a (U, Z)-Bernstein algebra (or graded Bernstein algebra) if it satisfiesfor all u G [/, z 6 Z

1. u3 = 0,2. u(uz) = u,3. uz2 = 0,4- («2)2 = o,5 (uz)2 = 0.

Notice that if A = Fe + Ue + Ze is a Bernstein algebra, then B = Kerw = Ue + Ze

is a (Ue, Ze)-Bernstein algebra. Conversely, if B is a (U, Z)-Bernstein algebra, thenA = Fe + B is a Bernstein algebra in which U = Ue and Z = Ze. So a gradedBernstein algebra defines uniquely a Bernstein algebra, however a Bernstein algebracan define several graded Bernstein algebras, depending of the chosen idempotentelement (and the associated Peirce decomposition).

Considering graded Bernstein algebras instead of Bernstein algebras, we have avariety of graded algebras and we can define the notion of Bernstein superalgebra.

DEFINITION 6.4 LetB = BQ+Bi be a superalgebra over the field F. LetG(B) =GQ <g> BO + GI ® BI the Grassmann envelope algebra of B. Then B is a Bernsteinsuperalgebra if G(B) is a graded Bernstein algebra.

The notions of Bernstein module and Bernstein supermodule can now be givenin the usual way. In [27] and [28] Bernstein superalgebras of low dimension areclassified, up to isomorphism of superalgebras and up to isomorphism of Bernsteinsuperalgebras.

Bernstein supermodules are studied in [29] and the classification of the faithfulirreducible ones is obtained.

It is possible to have a graded Bernstein algebra that is also Jordan, but whoseassociated Bernstein algebra is not Jordan.

In [45] the speciality of Bernstein Jordan algebras is considered. It is provedthat every graded Jordan Bernstein algebra satisfies the Glennie identity G&. Theauthors of [45] construct a graded Bernstein Jordan algebra whose associated Bern-stein algebra is also Jordan, but does not satisfy G§ and, consequently, is not special.

There are many subjects and different directions that have been followed in thestudy of Bernstein algebras. For instance, in [10] exceptional Bernstein algebrasare associated to some graphs, and properties of the Bernstein algebra are relatedwith properties of the associated graph. As we mentioned at the beginning, our


aim has not been to present an exhaustive overview of all the work in the area andwe have chosen the mentioned topics by a "proximity" reason, since they have beenobtained by people in the same research group. For the same reason we includeonly references of the mentioned topics.

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[1] J. Bernad, A. Iltyakov and C. Martinez, Bernstein representations, Kluwer Ac.Publishers, 39-45, (1994).

[2] J. Bernad, S. Gonzalez, A. Iltyakov and C. Martinez, On identities of baricalgebras and superalgebras, J. of Algebra 197, 385-408, (1997).

[3] J. Bernad, S. Gonzalez, A. Iltyakov and C. Martinez, Polynomial identities ofBernstein algebras of small dimension, J. of Algebra 207, 664-681, (1998).

[4] J. Bernad, S. Gonzalez and C. Martinez, On the nilpotency of the barideal ofa Bernstein algebra, Comm. in Algebra 25(9), 2967-2985, (1997).

[5] S. Bernstein, Principe de stationarite et generalisation de la loi de Mendel,Comptes Rendus Acad. Sci. Paris, 177, 528-531, (1923).

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[7] M. Bertrand, Algebres non-associatives et algebres genetiques, Memorial desSciences Mathematiques, CLXII, Gauthier-Villars Editeur, (1966)

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[10] R. Costa and H. Guzzo Jr., A class of exceptional Bernstein algebras associatedto Graphs, Comm. in Algebra 25 No. 7, 2129-2139, (1997).

[11] M.A Garcia Muniz, Ph.D. thesis, University of Oviedo, Spain (1998).

[12] M.A Garcia Muniz and S. Gonzalez, Baric, Bernstein and Jordan algebras,Comm. in Algebra, 26(3),913-930, (1998).

[13] M.A Garcia Muniz and C. Martinez, Derivations in second order Bernsteinalgebras, Nonassociative Algebra and its Applications, Marcel Dekker, 105-124,(2000).

[14] H. Gonshor, Special train algebras arising in genetics, Proc. Edinburgh Math.Soc. (2) 12, 41-53, (1960).

[15] H. Gonshor, Contributions to genetic algebras, Proc. Edinburgh Math. Soc. (2)17, 289-298, (1971).



[16] S. Gonzalez, Homotope algebra of a Bernstein algebra, Hadronic Mechanics andNonpotential Interactions, Nova Science Publish., New York, 186-200, (1992).

[17] S. Gonzalez, Inner derivations of Bernstein algebras, Linear Algebra and itsApplications, 170, 206-210, (1992).

[18] S. Gonzalez, One dimensional subalgebras of a Bernstein algebra, Algebra andLogic, 30(4), 310-327, (1991).

[19] S. Gonzalez, Quasiisomorphisms of Bernstein algebras, Comm. in Algebra,21(11), 4153-4166, (1992).

[20] S. Gonzalez, A. Grishkov and C. Martinez, A general splitting theorem forBernstein algebras, Comm. in Algebra 26(8), 2529-2542, (1998).

[21] S. Gonzalez, J.C. Gutierrez and C. Martfnez, On Bernstein algebras ofnth-order, Kluwer Ac. Publishers, 158-163, (1994).

[22] S. Gonzalez, J.C. Gutierrez and C. Martinez, The Bernstein problem in dimen-sion 5, J. of Algebra 177, 676-697, (1995).

[23] S. Gonzalez, J.C. Gutierrez and C. Martinez, Classification of Bernstein alge-bras of type (3,n-3), Comm. in Algebra 23(1), 201-213, (1995).

[24] S. Gonzalez, J.C. Gutierrez and C. Martinez, Second order Bernstein algebrasof dimension 4, Linear Algebra and its Applications 233, 243-273, (1996).

[25] S. Gonzalez, J.C. Gutierrez and C. Martinez, On regular Bernstein algebras,Linear Algebra and its Applications 241, 389-400, (1996).

[26] S. Gonzalez and J. Laliena, Bernstein algebras and quantum mutation,Hadronic Mechanics and Nonpotential Interactions, Nova Science Publish.,New York, 201-211, (1992).

[27] S. Gonzalez, C. Lopez-Dfaz and C. Martinez, Bernstein superalgebras of lowdimension, Comm. in Algebra 27(9), 4477-4492, (1999).

[28] S. Gonzalez, C. Lopez-Dfaz and C. Martinez, Bernstein superalgebras of di-mension 4, Nonassociative Algebra and its Applications, Marcel Dekker, Inc ,189-203, (2000).

[29] S. Gonzalez, C. Lopez-Dfaz, C. Martinez and I. Shestakov, Bernstein superal-gebras and their supermodules, J. of Algebra 212, 119-131, (1999).

[30] S. Gonzalez and C. Martinez, Idempotent elements in a Bernstein algebra, J.London Math. Soc.(2) 42, 430-436, (1991).

[31] S. Gonzalez and C. Martfnez, Bernstein algebras with zero derivation algebra,Linear Algebra and its Applications, 191, 235-245, (1993).

[32] S. Gonzalez and C. Martfnez, On Bernstein algebras, Kluwer Ac. Publishers,164-170, (1994).



[33] S. Gonzalez, C. Martinez and P. Vicente, Power-Associative and Jordan 2nd-order Bernstein algebras, Nova Journal of Algebra and Geometry, Vol. 2, No.4, 367-381 (1993).

[34] S. Gonzalez, C. Martinez and P. Vicente, Some special classes of 1nd-orderBernstein algebras, J. of Algebra 167, No. 3, 855-868, (1994).

[35] S. Gonzalez, C. Martinez and P. Vicente, Idempotent elements in a 2nd-orderBernstein algebras, Comm. in Algebra 22 (2), 595-610, (1994).

[36] J.C. Gutierrez, The Bernstein problem for the type (n-2,2), J. of Algebra 181,613-627, (1996).

[37] J.C. Gutierrez, The Bernstein problem in dimension 6, J. of Algebra 185,420-439, (1996).

[38] J.C. Gutierrez, Structure of Bernstein population of type (3,n-S), Linear Alge-bra Appl. 268, 17-32, (1998).

[39] J.C. Gutierrez, Solution of the Bernstein problem in the Non-regular Case, J.of Algebra 223, 109-132, (2000).

[40] J.C. Gutierrez and C. Martinez, Nuclear Bernstein algebras with stochasticbasis, J. of Algebra 217, 300-311, (1999).

[41] H. Guzzo Jr. and P. Vicente, Train algebras of rank n which are Bernstein orPower-associative algebras, Nova J. Math., Game Theory and Algebra 6, No.2-3, 103-112, (1997).

[42] H. Guzzo Jr. and P. Vicente, On Bernstein and train algebras of rank 3 ,Comm. in Algebra 26, No. 7, 2021-2032, (1998).

[43] H. Guzzo Jr. and P. Vicente, Classification of 5-dimensional Power-associative2nd-order Bernstein algebras, Submitted.

[44] P. Holgate, Genetic algebras satisfying Bernstein's stationarityPrinciple, J.London Math. Soc. (2) 9,613-623, (1975).

[45] C. Lopez-Diaz, I. P. Shestakov and S. N. Sverchkov , On speciality of BernsteinJordan algebras, Comm. in Algebra. To appear.

[46] Yu. I. Lyubich, Mathematical structures in population genetics, in "Biomath-ematics", Vol 22, Springer-Verlag, Berlin/Heidelberg, (1992).

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[49] C. Martinez, Isomorphisms of Bernstein algebras, J. of Algebra 160, 419-423,(1993).



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About an Algorithm of T. Oaku

M. I. HARTILLO-HERMOSO, Departamento de Matematicas, Universidad deCadiz. C/ Por-Vera, 54. 11403-Jerez de la Frontera. Cadiz Spain.E-mail: Isabel. hartillo@uca. es

Abstract

We give a new version of an algorithm of T. Oaku computing the globalBernstein polynomial associated to a polynomial / e C[xi, . . . ,xn\. The greatdifference is in the homogenization technique we use.

1 INTRODUCTION

Let An(C) be the Weyl algebra in n variables, that is the algebra of differentialoperators in n variables with coefficients in C[zi, . . . , xn}. This algebra is generatedby the elements Zj, di i = 1, . . . ,n with relations [zj,Zj] = [<9j,<9j] = 0, [di,Xj] =Sij. The elements of >4n(C) can be written in a unique way as a finite sum P =!Cae«" aa(x}da where aa(x) e C[x] with x = (z i , . . . , z n ) and d - (di,...,dn).We shall denote this ring by An, instead of An(C) to simplify the notation.

Let /(z) € C[z] and s a new and consider the ring of polynomials -4n[s]. Thenby the Bernstein theorem (see [3]) there exists a b(s] £ C[s] with b(s) ^ 0 and thereexists a P(s) 6 An[s] such that:

b(k}fk = p(k)(fk+l] v^ez . (1)Polynomials b(s) G C[s] such that there exists a P(s) 6 An[s] verifying the equation(1) form an ideal Bf of C[s]. The Bernstein polynomial associated to / is the monicgenerator of that ideal. We denote this polynomial by 6/(s).

Let MI be the free C[z, s][/~1]-module of rank one and denote its free generatorby /s. We can define an action by An[s] on MI as follows:

242 Hartillo-Hermoso

fk

Let M be the submodule of MI generated by /s, that is M = An[s}fs- We writeJ={P&An[s}\P(f")=0}.

We have that C^sH/"1]/8 ~ M[f~1}. Now we define a structure of C[i]-module, where t is a new variable. Assume g(s) <E C[x, sjf/"1], then we define theaction:

Generally, given a </? = ^Cfc=o ^ktk € C[t] we have that:

Let us point out that this action does not commute with the action of An s}over M[f~1}. That is, if P e An and 9? e C[t] then [P, yj] = 0, but (s + l)t = ts. Ingeneral, we have

Since this action is a bijection over M[/~1], we can consider t~l . This leads todefine a connection over the C[i]-module M[f~1} in the following way:

Let fc e Z. We can define

Then we haveM = M0, M f e cM f c _i ,

On the other hand, the Mk are stable under the action of t. This action is definedfor P(s) e An[s by

Then we have that tMk = Mfc+1. That is,

1} = (J Mfc =

Note that A_d_M^ C Mk-i although this sequence is not stable by this action.dt

Let An+i = C[x,t](d,dt). We define the structure of ^4n+1-module on M[f~1}.For di we define the action as before, due to the formal derivation. We define theaction of dt as the action of A_d_ . Given g(s) e C[x, slf/"1], we have:

dt

d t ( g ( S ) f s ) = -sg(s - I)/s-l

An Algorithm of T. Oaku

For f = X]l=o ak(x)tk (:

243

[x,t] we define:

i= ^Tak(x)g(s- s+k

Let N = An+ifs, it holds

M c N C M[r1} = M[f-1}.

Hence, it is clear that

+ -dt fs = 0.(t ~ f ) f ' = 0,

Even more, these elements generate the annihilator of fs in An+i-

LEMMA 1.1. Let be f & C[x], where x = (z1; . . . ,zn). Then the left ideal

is maximal in An+i-PROOF.- Let P e An+i, P ^ J. Then we must prove that (P) + J = An+i-

We write

Using the generators of J, we can write:

i=0

for some m g N and a^(x) e C[x].Then, for an element in this form, we have:

(P)

If we are in the case ai(x) — 0 for all i ^ 0, we are in a simplest case which willbe solved later. Otherwise, we have that / € C[x]. Then it commutes with dt, andao(x) G C[x], which commutes with t. Hence:

, t] = , t]i=0 . .i=0 . i=0

and, once more as ai(x) e C[x] commutes with t:

We have reduced the degree of dt in our element, and then we have an elementa(x) € (P) + J, with a(x) ^ 0 (this fact comes from P g J). If a(x) e C we have


finished. Let us suppose the degree of a(x) to be greater than 1. Then there is anindex i which verifies that the degree of a(x) with respect to Xj is greater than 1.Using that a(x) <E C[x] commutes with dt as well as with the polynomials of C[x]:

+ 7^Tdt. afc) \=(di' a(x)} = -4^--

Thenda(x] ,„. T , / d a ( x ]—~- e (P) + J and deg ' ^ '

we have finally found an element c(x) € (P) + J which deg(c(x)) = 0, that isc(x) e C and we have finished the proof. D

In An+i we can consider the Kashiwara-Malgrange filtration, which comes fromthe vector subspaces:

Vm(An+1) = P = aw,«,f}xat"dfi% e A•Wi = Oi fz / —

We use this filtration and the above results to calculate the Bernstein polynomialof / .

PROPOSITION 1.2. Lei / e C[x]. T/iere exisi P(s) e An[s] and b(s) e C[s],b(s) ^ 0 swc/i f/iai P(s)/s+1 = b ( s ) f s if and only if there exist Q G V-i(A.n+i) andc(s) e C[s], c(s) ^ 0 swc/i that c(-dtt) -Q e J.PROOF.- Let us consider

P(s)fs+1 = 6(s)/s with P(s)6^n[s] and 6(s) e C[s b(s) £ 0.

Using the action defined before we have

p(-dtt)tfs = b(-dtt)fs ^ (b(-dtt) - p(-dtt}t)fs = o,from which it immediately follows that P(—dtt)t € V-i(An+i)-

Conversely, let us consider c(—dtt) — Q € J with Q € V_i(An+i). Then we canwrite Q = ^T=i Qi(tdt}t\ Take the element

and it is now clear that

D

With these results we have reduced the computation of the Bernstein polyno-mial to finding a special element of the ideal J, this element belongs to Vo(^4n+i).Moreover it is the element of smallest degree because it is polynomical in —dtt.

An Algorithm of T. Oaku 245

2 HOMOGENIZATION OF DIFFERENTIAL OPERATORS

The greatest drawback we find in the algorithm of T. Oaku [10] for the computationof the Bernstein polynomial is the calculation of a Grobner basis with respect toa monomial ordering which considers firstly the Kashiwara-Malgrange filtration.Precisely for this reason, a monomial ordering with this restriction can not be awell ordering, and we can not use the algorithms in the Weyl algebra which parallelBuchberger's one. For a deep development of these results the reader may consult[6]. The goal of computing a basis of this form is obtaining the elements of degree0, that is bf(-dtt).

Then T. Oaku gives, in a natural way, an homogenization procedure for dif-ferential operators. The use of this technique preserves the Kashiwara-Malgrangefiltration. The following step is to consider an ideal in .4n[s]. It is generated by theprincipal symbol, for the V-filtration, of the elements of F-degree 0 of /.

This process is very similar to graded ring theory. In [7], we find a generaltechnique which begins with an "admissible filtration" in the Weyl algebra. Thenit introduces an homogenization in the Rees algebra with respect to the Bernsteinfiltration. This leads us to a standard basis in the Weyl algebra which preserves the"admissible filtration".

The Kashiwara-Malgrange filtration is an admissible filtration, that is why weuse the new technique. We give in this section the principal results we use in thecalculation of the Bernstein polynomial.

Any element of An+i is of the form:

P= Y 'W.a./jz"^^^,i/a,/3

We consider the Kashiwara-Malgrange filtration, that is:

Vm = {P € An+i <V,<,,a,/3 =0 if v - fj. > m}.

For P s An+i we define ordy(P) as the minimum m such that P € Vm. We definealso the principal symbol of P as

W(P) =v — n=ordv(P)

Let -< be a monomial well ordering. We define the following ordering in p^2rl+2:

v - n < v' — fj,'

v _ M = ^' _ M'or •{ and

The monomial ordering -<v/ is not a well ordering, but the restrictions to thelevel sets of (a, n, /?, v) such that v — /j, — m are well orderings.

From the definition of this monomial ordering we can define as usual [6] theconcept of Grobner basis for -<v, that we shall call ^-standard basis. The problemis that, in general, given a left ideal / of An+i, a F-standard basis of / is not a

system of generators of /. But given a ^/-standard basis of /, {G1; . . . , G;}, thefamily {ov(Gi), . . . , ov(G;)} is a system of generators of giy(/) [7]. We even knowthat the family (ov(Gi), . . . , <7y(G;)} is a Grobner basis of giy(7) for the wellordering -<.

To compute a F-standard basis we must use homogenization techniques. Thegreat difference we find between [7] and [10] is that the ring in which the Weyl

3ra is embedded in order to homogenizing the operators is

with relations

[x0,Xi] = [x0,t] = {x0,di} = [x0,dt] = j] = [xi,t] = [di,dj} = [di,dt] = 0,

This is a graded algebra, with the degree of the monomial XQXat^d^d^ beingk + \a + n + \(3\ + v. This algebra is isomorphic to the Rees algebra associated tothe Bernstein filtration of An+i-

We define in a natural way the homogenization of the operator P £ An+i. GivenP we denote by ordT(P) its total order

ordT(P) = max{|a + n + \/3\ + v \ aa^^tV ^ 0}

and we define the homogenization of P, denoted by h(P), as:

Now we define a well ordering -<y in the Rees algebra, which will lead us toobtain a V-standard basis:

' k + \a + fj. + \/3\ + v < k' + a' + // + |/3'| + v1

k + \a\ + fj. + \J3\ + v = k' + \a'\ + // + \j3'\ + v1

or { and

Starting from a system of generators {Pi, . . . ,Pr} of an ideal / of An+i, wejust consider the ideal / in the Rees algebra generated by {h(Pi), . . . , h(Pr}}.Using an algorithm which resembles Buchberger's, we obtain a standard basisof / for ~<y. We will denote this basis by Q = {Gi,...,Gs}. We have thatthe new set Q = {G1|Xo=1, . . . ,Gs|Xo=i} is a F-standard basis of / and then&v(G) = {°v(Gi|Xo=i), . . . ,<?v(Gs\x0=i)} is a system of generators of the idealgry (/) . Moreover we have obtained a Grobner basis for -< of the ideal giv (/) (see[7]).

An Algorithm of T. Oaku 247

3 COMPUTATION OF THE BERNSTEIN POLYNOMIAL

First, we fix the notations and the orderings we shall use later.Let -< be a monomial well ordering in N2n+2 such that it is an elimination order

for x and d. For example, we may consider a Ith elimination order of Bayer andStillman [2].

We define the mapping <p:

 gr°v(An+i)£OT<MP)p jf ordy(P) > 0

f - -or .f

Every P & gTy(An+i) can be written in an unique way as a polynomial in —dtt.Let us write in this case P(—dtt). We substitute formally —dtt by s. We write inthis case Q(P) for P(s). Then, we consider the ring homomorphism:

P I — > 0(P).Then © is a ring isomorphism.

THEOREM 3.1. Let be f ( x ) e C[x] where x = (xl,. . . ,xn). Let J be the leftideal of An+i defined by:

Let Q = {GI, . . . ,Gr} be a V -standard basis of J with respect to the ordering pre-viously fixed. Define Q' by

G' = {av(Gi), . . . ,av(Gr)} n CM,] = {Gi, . . . , G'J.

/ / ? P S P f

is a system of generators of the ideal Bf ofC[s].

REMARK 3.2. T. Oaku gave a complete computational solution to the openproblem of calculating the Bernstein polynomial in [10]. His work was based, aswell as ours, on the Grobner basis theory for rings of differential operators but someimportant differences between his algorithm and ours must be emphasized.First of all, Oaku goes from J C An+i to a homogenized ideal Jh C An+i[zo] but,neither is Jh the homogenization we used, nor is ,4ra+i[xo] the Rees algebra of An+iwith respect to the Bernstein filtration.Afterwards, his extremely specific homogenization process leads to the necessityof performing two Grobner basis computations in order to obtain the Bernsteinpolynomial. This is due to the fact that his method for obtaining a Grobner basisof a(^(J)) (an analog for gry(J)) does not allow him to eliminate the variablesxi, ...,xn, and this final step requires then another Grobner basis (now in a poly-nomial ring) to be computed. As it has been stated in theorem 3.1 our algorithm

needs only one Grobner basis computation to calculate the Bernstein polynomial of/.

NOTE 3.3. We have obtained a system of generators of an ideal of C[s], whichis a principal ideal domain. To compute the Bernstein polynomial, we just need tocompute the greatest common divisor of these elements.PROOF.- We shall use proposition 1.2 to prove this result. The mapping 0 justtakes —dtt to s. We do not use the action of 0, and we shall consider the Bernsteinpolynomial b(—dtt) using the fact that 0 is an isomorphism.

We have, using 1.2, that an element g € C[— dtt] belongs to B/ if and only ifthere is an element Q G V^i(An+i) such that g — Q e J. Equivalently, it sufficesthat g <E gry(J)nC[—dtt]. Because if we have an element g — Q g J in the previousway, then ffv(g — Q) = 9 G gr1/(J). Besides oidy(g) = 0 and it is V- homogeneous.Hence g 6 gr^(J) n C[-dti\.

Conversely, if we have an element g € gr^(J) n C[— dtt], then g € C[t,<9t] andg & gr^(J) and therefore g 6 C[— dtt]. We have g e gry(J), so there is an elementF e J such that ay(F) = g, then F = g -Q, where Q e V-i(An+i)-

Then, in order to prove this theorem it suffices to prove that Q((p(Q')) is asystem of generators of gr^(J) n C[— dtt].

We have that the principal symbol ov(CJ) = {ov(Gi), . . . ,ov(Gr)} is astandardbasis of gry (J) for -<. If we choose -< to be an elimination order, then we have thatQ' is a system of generators of gr^(J) n C [ t , d t ] , and these generators result to be^-homogeneous.

Let us see that <p(G') generates gr^(J) nC[— dtt] as an ideal of C[— dtt]. Firstly,we note that the G^ are F- homogeneous and belong to grv(J). Then we findy(G^) € giy(J), because we just multiply at the left side and we obtain an operatorwith V-order 0.

Conversely, if we have H € gry(J)nC[-dtt], (in particular H e gry ( J)nC[i, dt]),then we have a relation:

s

H — y ^HjGj.i=i

The element H is F-homogeneous and its F-order is equal to 0. The elements G'tare ^-homogeneous and they lie in C[t,dt]. We shall denote by m^ its l^-order. Itis clear that we can always choose Hi ^-homogeneous with y-order — m^ and in

We write Hi as:

1 ~if mi > 0

[d~m* if mi < 0.This leads to

where the H[ are F-homogeneous of K-order 0 and belong to C[t,dt}. ThereforeH- 6 C[-dtt]. We have proved that ip(Q') generates gr^( J) n C[-dtt}. D

EXAMPLES 3.4. Using a program in COMMON LISP due to prof. Jose MariaUcha we have computed the algorithm described before. We find the Bernstein

An Algorithm of T. Oaku

polynomial in these cases:

249

x2 + xy3 + y4

x2 + xy2 + y3

(x

x(x + y)(xz •

y ( x -

xy(x + y ) ( x z •

b f ( s )

s+1

(* + !)

|)(s + i)3

REFERENCES

[1] A. Assi, F. J. Castro Jimenez, J. M. Granger. How to calculate the slopes of aP-module. Compositio Mathematica, 104, 107-123, 1996.

[2] D. Bayer and M. Stillman. A theorem on refining division orders by the reverselexicographic order. Duke J. Math. 55, 321-328, 1987.

[3] I. N. Bernstein. The analytic continuation of generalized functions with respectto a parameter. Functional Anal. Appl. 6, 273-285, 1972.

[4] J.-E. Bjork. Rings of Differential Operators. North-Holland, 1979.

[5] A. Borel et al. Algebraic D-modules. Academic Press, Boston, 1987.

[6] F. J. Castro Jimenez. These de Seme cycle Universite Paris VII, 1984.

[7] F. J. Castro Jimenez and L. Narvaez Macarro. Homogenising differential oper-ators. Prepublicaciones de la Facultad de Matematicas, Universidad de Sevilla,36, 1997.

[8] M. Kashiwara. Vanishing cycle sheaves and holonomic system of differentialequations. Lecture Notes in Math. 1016, Springer-Verlag, 134-142, 1983.

[9] B. Malgrange. Le polynome de Bernstein d'une singularite isolee, LectureNotes in Math. 459, Springer-Verlag, 98-119, 1975.



[10] T. Oaku. An algorithm of computing 6-functions. Duke Mathematical Journal,87, vol. 1, 115-132, 1997.


Minimal Injective Resolutions: Old and New

M. P. MALLIAVIN, Universite Paris VI. 10 Rue St. Louis, L'Ile-75004, Paris.France.E-mail: [email protected]

I INTRODUCTION

It is well know that if R is a commutative noetherian ring and if

0 — ->R — >I

is a minimal injective resolution of R than

where E(-) denotes an injective envelope of a fl-module, /^(p) is the cardinal

dimfc

and fc(p) is the residual field at p, i.e. fc(p) = (_R/p)p .If inj dim R < oo and R is commutative then, by H. Bass, each indecomposable

injective modules E ( R / f ) appears exactly in I1 for i = htp. More precisely H. Bassproved that /Mj(p) = 1 if htp = i and /Zj(p) = 0 if not, so 7* is the direct sum ofinjective hulls of R/ip, where p runs in the set of prime ideals of height i in R.

To obtain analogous results for non commutative rings, we need alternativehypotheses.

Let A be a left and right noetherian ring. It is known for a long time (by [13])that the injective dimension of the module AA is equal to the injective dimensionof A -A if the two are finite and in this case we note inj dim(A) = d this commondimension.

One of the main tools in the theory of finite injective dimensional noethe-rian rings is the spectral sequence of Ischebeck (~ 75'): If A is a noetherian and

252 Malliavin

injdim/i = d and if M is a left noetherian A-module, there exists a spectral se-quence

E%q(M) = ExtpA(Extq

A(M,A),A) => mp~q(M)where W>~q(M} = 0 if p ^ q and H°(M) = M.

2 GRADE AND AUSLANDER-GORENSTEIN RINGS

Let A a noetherian ring and M a left .A-module. We name grade of M denotedj ( M ) the natural number or +00 defined as follow

Of course j(Q) = +00.If inj dim A = d < oo, one proves that j^(M) < d for every non zero A-module

M.

DEFINITION 2.1 Let A be a noetherian ring. A left (or right) finitely generatedA-module M satisfies the Auslander condition if for any q > 0 one has j(N) > qfor every A-submodule N of finite type of Extg

A(M,A),

DEFINITION 2.2 The noetherian ring A is said Auslander— Gorenstein of di-mension d if

1. mjdimA = d < oo,

2. every finitely generated right (resp. left) A-module satisfies the Auslandercondition.

EXAMPLES 2.3 Commutative Gorenstein rings; quasi- frobenius artinian rings, forinstance kG for any field and any finite group, any enveloping algebra of finitedimensional Lie algebra, iterated Ore polynomial algebras

k[Xi}[X2;T2,S-2} . . . [Xm;Tm,6m}

where TJ is an automorphism and 6j a Tj-derivation of k[Xi][. . .][Xj',Tj,6j], ([2],[4])-REMARK 2.4 If in 1. we assume gidimA < oo we say that A is Auslander regular.

All the examples above (except perhaps commutative Gorenstein rings and Q.F.-artinian rings) are Auslander regular.

3 COHEN-MACAULAY CONDITION

Recall that if A is an algebra over a field k

GK dim A = supv lim —— ; ———Ign

where V is a dimensional finite subspace of A with I & V and if M is an A-modulefinitely generated i.e. M = AF such that dim^ F < oo, then

_., , . . —— lgdimT/"FGK dim A = lira —— - ————Ign

if A is finitely generated by a finite dimensional space V .

Minimal Injective Resolutions 253

DEFINITION 3.1 Let B be a k-algebra noetherian with GKdimB = w 6 N. Onesays that B is Cohen— Macaulay (CM in short) if for every non zero finitelygenerated B -module one has

GKdimM +j(M) =u.

EXAMPLES 3.2

1. If B is a commutative a/fine k-algebra, B is CM iff B is a equidimensionalCohen- Macaulay ring.

2. If k = C and B is an iterated skew Ore extension, B is CM by [6].

4 RESULTS

DEFINITION 4.1 Let B be a noetherian ring and p a completely prime ideal of B.We denote /^(p, B} the dimension over the skew field F r ( B / f ) of the vector spaceFr(B/p)®B/fExti(B/v,B).

REMARK 4.2 /^j(p,5) ^ 0 iff E(B/y) appears in P, the ith module of minimalinjective resolution of

THEOREM 4.3 (See [8] and the bibliography given there). Let B be a k-algebraover a field k, Auslander-Gorestein satisfying CM, ancJp a completely prime ideal ofB. Let h be the grade of the left B-module B/p. Then /x/,(p, B) ^ 0 andju,(p, B) = 0

This theorem applies to the enveloping algebras of solvable finite dimensionalLie algebras over a field of characteristic 0 or to the Weyl algebra An over a fieldof characteristic 0. Also it applies to many recent algebras, the so called g-skewiterated polynomial algebra over the field C say, ([6]), and when q 6 C is not a rootof unity in C. In fact K. R. Goodearl and E. S. Letzter have proved that each primeideal is completely prime.

As another example there is the C- algebra Oq(MnC), C- algebra of coordinatesof the quantic matrices n x n (q ̂ \/l), so is the quantum algebra of coordinatesof SL(n).

CONJECTURE 4.4 This should be true for all the semi-simple quantum groups fora parameter q not a root of I .

PROBLEM 4.5 If h = j(B/p) does Mp,B) = l? Le- does E(B/*>} appear in theappropriate term of the minimal injective resolution %B one and only one time?This is the case for B — U(£) where £ is a solvable finite dimensional Lie algebraover a field of characteristic 0 and for the Weyl algebra over a field of characteristic0.

I want to present you other cases for this works.Let Uq(M+) be the quantized enveloping algebra of a maximal nilpotent subal-

gebra A/"+ of a dimensional simple complex Lie algebra Q. We assume Q of type A,D, or E or of type B%.


254 Malliavin

For the three first cases Ringel proved that Uq(J\f+) is a g-skew iterated Oreextension and for the last case I proved it using a result of [12]. This result is notknown neither for the general case of a semi-simple Lie algebra nor for other types ofsimple Lie algebra (Bn, n > 3, Cn, F, G). Under the above hypothesis T. Lenaganand K. Goodearl proved that Ucl(Af+), q =£ VT, q € C, is an affine noetherianC-algebra with finite GK-dimension which is Auslander regular, Cohen-Macaulayand (thanks to C. Ringel and to M. Takeuchi) each of its prime ideals is completelyprime.

PROPOSITION 4.6 For Q simple over C of type A2 ~ sl(3,k) or B2 ^ so(5,k},any prime ideal ofUq(J\f+) q ̂ \/I is a completely prime ideal. Iff is one of theseideals with d = j ( B / p ) then /^d(p,T) = 1 and /ij(p,T) = 0 if i ^ d.

PROOF: For A2 this is proved in [9] and it is easy because each (completely) primeideal is generated by a regular normalizing sequence (zi,z2, . . . , zn) with n < 3.Using an old result of [3] there is a [/,j(A/"+)-isomorphism

For 82 the algebra is

B = C[2,ei i3][ei,2,0-i,2][e2,3, 02,3, 02,3]

where z is central 0-1,2(0) = z, 0x2(61,3) = g~2e1]3, 0-2,3(z) = z, cr2,3(ei,2) = <?~2ei> 2 ,02,3(61,3) = <?2ei,3, <S2,3(ei,2) = -g~2ei,3, S2,s(z) = 0, and ^2,3(61,3) = z.

The prime ideals which contain z are generated by a normalizing regular se-quence (z, ui, . . . , un) (n < 4) and the result follows like for A2. There are also theprimes z — a, a G C which are maximal.

The primes p ^ 0 such that p n C[z] = 0 are a little more difficult to handle.It is necessary to apply [10] which said that htp = 1 and there is a Ore set S inUq(N+) and a central element t such that

Then the grade of Uq(J\f+}/'p is one and

- l ~ ~1 l ~1dimFr(-)®ExtlB(~,B) = dim Fr(S~1-) <g> Extl(--, S~1B] = 1.

V P P / V P P /n

PROPOSITION 4.7 For type A2 (resp. B2) if B = Uq(M+), the minimal injectiveresolution gB is

where 1° = Fr(B) and for i > 1 where n = 3 (resp. n = 4)

n — l?i ON Z?i1 — &-[ & 12/2

where

p6Spec(T),htp=i

Minimal Infective Resolutions 255

and E\ is the injective envelope of a sum of critical modules, each of them beingtorsion modulo their annihilator. Moreover 1% = (0) and

where J runs over the family of 2 -sided ideals of finite codimension of B.

In general, there is an antiautomorphism of U+ , u — > ul where u € U+ suchthat e\ = ej, {ei} being the Serre basis of A/"+.

Take V = Cg [N] the restricted dual of U+ where N is the unipotent group withLie N = A/"+. Then V is a left U+ -module under

(xf)(u) = f ( x l u )

each 6j acting locally nilpotently.Applying this to the case A% and B% gives with notations as in Proposition 4.7.

PROPOSITION 4.8 In is isomorphic to the restricted dual of B.

PROOF: It is the same as in [1]. D

PROBLEM 4.9 The same result should be true for every Uq(J\f+) where A/"+ is amaximal nilpotent subalgebra of a semi-simple finite dimensional Lie algebra Q.

REFERENCES

[1] Barou G., Malliavin, M. P.: Sur la resolution injective minimale de 1' algebreenveloppante d'une algebre de Lie resoluble. J. of Pure and Applied Algebra,37, 1985, 1-25.

[2] Bjork, J. E.: The Auslander condition on noetherian rings. Seminaire Dubreil-Malliavin 1987-88, Lecture Notes in Math., 1404, Springer Verlag, 1989, 137-173.

[3] Brown K. A., Levasseur T.: Cohomology of bimodules over enveloping algebras.Math. Z., 189, 1985, 393-413.

[4] Ekstrom E.K.: The Auslander condition on graded and filtered noetherianrings. Seminaire Dubreil-Malliavin 1987-88, Lecture Notes in Math., 1404,Springer Verlag, 1989, 220-245.

[5] Goodearl K.R., Lenagan T. H.: Catenarity in quantum algebras. J. of Pureand Applied Algebra, 111, 1996, 123-142.

[6] Goodearl K.R., Letzter E.S.: Prime ideals in skew and q-skew polynomialsrings. Mem. Amer. Math. Soc., 109, 1994.

[7] Levasseur T., Stafford J. T.: The quantum coordinate ring of the special lineargroup. J. of Pure and Applied Algebra, 86, 1993, 181-186.

[8] Malliavin M. P.: Algebre de Bass et extensions de Ore iterees. Collect. Math.,45, 1994, 85-99.


256 Malliavin

[9] Malliavin M. P.: La catenarite de la partie positive de 1'algebre enveloppantequantified de 1'algebre de Lie simple de type B%. Beitrdge zur Algebra undGeometme, 35, 1994, 73-83.

[10] Ringel C. M.:PBW-basis of quantum groups. J. reine angew. Math., 470, 1996,51-88.

[11] Takeuchi M.: The g-bracket product and quantum enveloping algebras of clas-sical types. J. Math. Soc. Japan, 42, 1990, 605-629.

[12] Zaks A.: Injective dimension of semiprimary rings. J. of Algebra, 13, 1969,73-89.

[13] Zelevinsky A.V.: Parametrization of canonical bases via the generalized de-terminantal calculus. Summer school on Hall Algebras and quantum groups,Hesselberg, August 1999.


Special Divisors of Blowup Algebras

S. E. MOREY, Department of Mathematics, Southwest Texas State University,San Marcos. Texas 78666-4616. USA.Ei-mail:[email protected]

W. V. VASCONCELOS1, Department of Mathematics, Rutgers University, 110Frelinghuysen Road, Piscataway. New Jersey 08854-8019. USA.E-ma,il:[email protected]

Abstract

The nature of the divisors of a Noetherian ring A provides a window to ex-amine its arithmetical and geometric properties. Here we examine the divisorsof Rees algebras of ideals-the so called blowup algebras-from the perspective ofthe operation of shifting. It permits the organization of some previous resultsin a more structured manner and at the same time predicts the occurrenceof several new divisors. Of these, we single out the fundamental divisor thatplays a more basic role than the canonical module.

1 INTRODUCTION

The set of divisors of a commutative Noetherian ring A is the class of rank oneA-modules that satisfy the condition 52 of Serre. It is a notion based on thatof unmixed height 1 ideals of a normal domain. The latter is rich in additionalstructures such as the divisor class group.

The approach we will follow here is to define the divisors of A through theintervention of a finite homornorphism tp : S —> A, with 5 admitting a canonicalmodule ws- The divisors of A are those A-modules arising in the following manner:Suppose dim A = d, dim S = d + n, and let L be a rank 1 A-module. We will call

Martially supported by an NSF grant


258 Morey and Vasconcelos

the canonical module of L, and the set of divisors is made up of all w/,, moreprecisely, of their isomorphism classes, Div(yi). A distinguished divisor of A is itscanonical module

Another divisor is A itself, when it already satisfies the property 52. It is a con-sequence of the theory of the canonical module that Div(A) is independent of thehomomorphism A, a centerpiece of the cohomologytheory of A; (ii) the divisor of Serre IR[It}; and (iii) its dual,

which will play an even more critical role than U>A- It will be labeled the fundamentaldivisor and one of our aims is to explore its role in the arithmetical study of theRees algebra U = R[It}.

We will emphasize the study of graded divisors for the following reason that weillustrate with the canonical module of Ti. Suppose that R is a Noetherian localring and 7 is an ideal of positive height. We shall assume that R has a canonicalmodule U>R. (As a consequence both ~R. — R[It] and Q = grj(R) also have canonicalmodules.) If / G / is a regular element of R, then / will also be a regular elementon uj-fr, so that

From this one has that the a-invariant of Tl is — 1. Thus the graded module(and T>(I) likewise) has a special representation

un = Dit + D2t2 + ••• + Dntn + • • • ,

where IV s are fractionary ideals of R.A basic property of these components — shared by many other graded divisors

in general and all graded divisors when R is an integral domain — is that they definea decreasing filtration D\ D D% D • • •. This peculiar property will be exploitedsystematically. It defines an operation on the set Divh(ft) of such divisors,

S : Divh(ft) H-* Divh(ft)

and a corresponding construction of a graded module over

= L/S(L).

We shall call S the shifting operator of Divh(7£), and one of our goals is to determineconditions for a divisor L to be an end, that is not to lie in the image of S. This isconnected to the notion of prolongation of a divisor, meaning from a given D whatconditions are required to solve the equation S(L) = D, and what is the structureof the set of solutions.

One of the results will show that under general conditions a Cohen-Macaulaydivisor L can hardly be in the image of Sn for n large (determined by dim R) . On

Special Divisors of Blowup Algebras 259

the other hand, there remains the issue of the existence of ends and whether theyare always finite in number (up to isomorphism).

Applying the construction to the module T>(I) yields G(7). The structures ofG(w-j^) and of G(/) as grj(/?)-modules will be used to describe very explicitly inmany cases of interest the canonical modules of 72 and Q, but also will shed lighton when the Cohen-Macaulayness of Q forces 72 to be Cohen-Macaulay.

Another goal is to identify Cohen-Macaulay divisors and to examine their re-lationship to the reduction number of the ideal. Additionally, we are interested infinding the divisors that carry information about the Briangon-Skoda number of /.

This paper is partly a review of the literature on the divisors of a blowup al-gebras but is organized around the new shifting operation and the introduction ofa new kind of divisor, the fundamental divisor of a Rees algebra. Its contents aredistributed as follows. Section 2 outlines our approach to the study of the divisorsof a ring A. The definition and the construction of a pairing of divisors require thatA be a homomorphic image of a Gorenstein ring. In case A is a Rees algebra, theother novel definition is that of the shifting operator

S : Divh(A) ̂ Divh(^).

In the next section we give an exposition of the divisor class group of a normal Reesalgebra. Section 4 is an application of the theory of the shifting operator to describea class of canonical modules of Rees algebras that have a well-packaged form. Fora Cohen-Macaulay Rees algebra R[It] over a Gorenstein ring R, they correspondto ideals for which g i f ( R ) are Gorenstein.

The more novel material here begins with the introduction of the fundamentaldivisor T>(I) of a Rees algebra 72 = R[It] in Section 5. It has the property <S(P(/)) =ui-fc, so in some fashion is a more primitive divisor than the canonical module of 72.The components of T)(I),

are important carriers of information about the Cohen-Macaulayness of 72, includ-ing the reduction number of I. We single out the class of ideals for which D\ ~ WR.We will say that such divisors have the 'expected' form. It will follow from a re-sult of Lipman that ideals of regular local rings always have this property. We willidentify other classes of ideals with similar behavior.

In the brief section 6, we give a calculation of how the Cohen-Macaulayness ofT>(I) also bounds the reduction number of I. Finally, for a Cohen-Macaulay localring (R,m) of dimension d > 0, where OT = (m, /£), we examine several instancesof the exact cohomology sequence

0 ̂ HdM(I^}-i - Hd

M(n}0 -^ Hdm(R) -^ HffVK)-! -+ Hd^(n)0 = 0

which is one of the most significant exact sequences in the cohomology of Reesalgebras. The vanishing of <f> is equivalent to V(I) having the expected form. Inseveral classes of ideals (ideals of near linear type, equimultiple) we identify suchevents.


2 DIVISORS

We shall use various sources for general results on the theory of Cohen-Macaulayrings: [4], [6] for the general theory and [32] for special properties of Rees algebras.Unexplained terminology or notation can be found in [32].

Let R be a Noetherian ring and / C R be an ideal, / = (/i, • . • , /n)- We assumethat R has finite Krull dimension, dim R = d, that its total ring of quotients isArtinian, and height / > 0. If R has a canonical module MR, we can use forthe presentation morphism the standard one, S = R{Ti,...,Tn] — > R [ I t ] , whenws = WRS[—n}.

The semigroup of divisors

To obtain divisors we must supply fodder to the process outlined above. When Ais the ring of polynomials R[t], some divisors are simply extensions of the idealsof R, JR[t\. If A is however a Rees algebra R [ I t ] , there are constructions likeJ(l,t)m C R[t], or more generally (Jit, . . . , Jmtm).

Let us begin with the formal definition of divisors.

DEFINITION 2.1 Let A be a Noetherian ring. An ,4-module L is a divisor if itsatisfies the following conditions:

(i) L satisfies the condition (£2) of Serre;

(ii) If K is the total ring of fractions of A then K ®A L ~ K.

An equivalent way to describe a divisor is: If A has an Artinian total ringof quotients, L is a, module isomorphic to an ideal H C A all of whose primarycomponents have codimension one.

We begin by listing some elementary properties of divisors as defined here. Weemphasize that a module L over the Noetherian ring A has rank 1 if it is torsionfreeand L ®A K ~ K, where K is the total ring of quotients of A.

PROPOSITION 2.2 Let A be a Noetherian ring with an Artinian total ring ofquotients that admits a presentation S — > A as above. The following hold:

(i) For any fractionary ideal L the module

satisfies the condition 5*2 •

(ii) There is a natural isomorphism

D -» Ext

on modules D = U>L.

(iii) // DI and D2 are divisors, the operation

defines a monoid structure on the elements of Div(A) which are principal incodimension one.



(iv) If A satisfies the condition 82 and is Gorenstein in codimension one, then

UJL ~ Hom^(Hom(L, A), A)

is an isomorphism for every rank one module L.

(v) If A is an integrally closed domain then for every divisor L, LoHom /i(L, A) ~A. As a consequence, the isomorphism classes of divisors form a group, theso-called divisor class group of A.

(vi) If A is an integral domain with a canonical module UA, the smallest rationalextension of A that has the property 82 is called its 82 -ification,

0-> A — > A — >C ->0.

(C in the graded case is the so-called Hartshorne-Rao's module of A). It canbe obtained as ^

A = RomA(u>A,LL>A) =

(vii) A and A have the same sets of divisors, Div(A) — Div(A).

Homogeneous divisors and the shifting operation

The homogeneous divisors of 72. = R[It] are those of the form

n>r

The role of the components Dn is its distinguishing feature. We begin with anobservation which will be used in several constructions. Note also its extensionsto more general nitrations such as the integral closure filtration, the Ratliff-Rushfiltration and some symbolic powers.

There are other divisors of a Rees algebra R[It] with the property that for everynonzero divisor / € /, L/ ~ JfR[t] f°r some ideal J C R — that is Lf is 'extended'from R. This is, for instance, the case for all divisors if R is integrally closed.There are also examples of homogeneous ideals in more general cases — and we willconsider one later.

DEFINITION 2.3 For each graded, divisorial ideal of R[It]

n>r

the subideal

n>r

is the shifting of L.


Note that in view of the exact sequence

1]) — > L — > L r f - > 0 , r = inf{n Ln ^ 0},

S(L) is also a divisor, as asserted in the introduction. It guarantees that the gradedmodule over gij(R)

0 -> S(L) — > L — > G(L) -> 0

has positive depth.

The following shows that the shifting operation is well behaved in all integraldomains.

PROPOSITION 2.4 Let R be a Noetherian integral domain with a canonical mod-ule and let I be a nonzero ideal. Then for every homogeneous divisor L of R [ I t ] ,S(L) c L.

PROOF: It will suffice to show that the conductor ideal L : <S(L) has codimensionat least 2. From the construction, It is contained in the conductor L : S(L). Onthe other hand, passing to the field of fractions K of R, we obtain KL = KS(L) =K[t]tr , which shows that the conductor contains a nonzero element x of R. Thismeans that it contains (It,x) which is an ideal of height at least two as ( I t ) is aprime ideal. Since L has the condition (52) of Serre, S(L) C L. D

Prolongation of a divisor

The process of shifting can run backwards under some conditions. Let

L = L1t + L2t2 + •••

be a divisor. By a prolongation we mean another divisor

D = L0t + L^2 + • • • ,

that is S(D) = L. As a prerequisite we must have that ILo C LI but moreconditions are to be met. We consider a case when prolongation can be achieved.

PROPOSITION 2.5 Let R be a Cohen-Macaulay domain with canonical ideal MRand let I be an ideal of height at least 2. Let

L = Lit + L2t2 + ••• , IujRcLiCuR

be a divisor of R[It}. Consider the ideal D defined by

0 -> Lt — > D — > u>Rt -> 0.

Then D is divisorial.

PROOF: As before we are still assuming that locally R is a homomorphic image ofa Gorenstein ring. We may assume R is local of dimension d and write A = R[It]

as a homomorphic image of a graded Gorenstein domain 5 of dimension d+1. Thecondition £2 f°r an ideal such as D is equivalent to

height (ann (Extls(D, u>s))) > 2 + i, i > 0.

Consider the usual Ext sequence

0 — > Hom(wRi,o;s) = 0D,ws) -> Ext l(Lt,ws) -» Ext2(w f l^,ws) = 0,

and the isomorphisms

L^ws i > 1.

For z > 1 the isomorphism just above says that height (ann (Extl(D, ws))) > 2 + z.For i = 1, note that .Ri"1 is already annihilated by It. What is needed is for thecokernel of the map <p to have an annihilator, as /^-module, of height at least 2.Then its annihilator over 5 will be have codimension at least 3, and Ext1(Z),o'5)sitting in the middle of a short sequence of modules with annihilators of codimensionat least 3 would have the same property.

We can test the cokernel by localizing at height 1 primes of R when R[It\ localizesinto Rp[t], a polynomial ring. But in this case Dp ~ Rp[t]. Thus Ext1(£),ws)p = 0and therefore the cokernel of </?p will vanish as well. D

REMARK 2.6 An obvious issue is whether every divisor has a proper prolonga-tion. Another point is whether among the prolongations of a divisor there exists acanonical (minimal) one. The argument above suggests that if LQ and L'Q are thenew components of prolongations of L, then LQ n L'Q may also work out.

It is not difficult to show that if R is a Cohen-Macaulay local ring with acanonical ideal and / is an ideal of positive height, then the fundamental divisor ofR[It] is the unique prolongation of the canonical ideal of R[It .

Let us work out a more general approach to prolongations. Suppose L =Sn>i Lntn is a divisor of R[It] that we seek to prolong into D = 5Zn>i Ln-itn-There are at least two requirements on LQ:

LI C L0

I-Lo C LI,

plus some divisoriality condition on LQ yet to be fully determined. There is howeverat least one situation when it leads to a unique "solution" for LQ.

PROPOSITION 2.7 Suppose that height / > 2 and LI is an ideal with the con-dition 83 of Serre. Then LQ = LI and D is a prolongation of L.

PROOF: Consider the exact sequence

0 -> / — >R — » R/I -> 0,

and apply Honift(-,Li):

0 -> EomR(R/I, Li) = 0 — » EomR(R, LI) — > HomR(/, LI)

Since 7 has height at least two and LI satisfies the condition 82 of Serre, / containstwo elements forming a regular sequence on LI; thus Ext 1

R (R/I, LI) = 0. Thismeans that LI = Hom^/, LI). Now note that LQ C Honifl(7, LQ), which with theother containment shows that LQ = LI.

One can now repeat the proof of Proposition 2.5. Using the sequence

Extzs(Lx,ws) — -> Ext*s(I>,ws) — > Extl

s(L,ws)

one sees that the codimension of the module in the middle must be at least i + 2.(This is the place where we used the condition £3 on LI.) D

The process of shifting cannot be indefinitely applied without changing keyproperties of the divisor. Consider for instance the case of a divisor

L = Rt + L2t2 + --- + Lntn + • • • , Ln+1cLn.

The exact sequence0 -> Ln>2 — > L — > #t -> 0,

shows that if R and L are both Cohen-Macaulay then both <S(L) and G(L) will alsobe Cohen-Macaulay. This follows simply from the standard device of examiningthe relationship between the Cohen-Macaulayness of Tl and g r j ( R ) in terms of thefollowing exact sequences (first paired in [15]):

Q (1)

Q^It- R[It] — > R[It] — > R -> 0, (2)

with the naive isomorphism

It • R[It] ~ / • /?[/£]

playing a pivotal role.

Another general property of Rees algebras that we shall make use of is thefollowing observation of Valla:

PROPOSITION 2.8 Suppose that the residue field of R is infinite. Then

(a) R[It] has a system of parameters of the form

{xi,x2 + ait, £3 + ait, . . . ,xd + ad-i

where {xi, . . . , Xd] is a system of parameters of R and (ai, . . . , a^) is a reduc-tion of I .

(b) (Valla) // / is m-primary then one can choose the a» 's to form an arbitraryminimal reduction of I and set Xi = a,i, Vi.

The following shows that some restriction must be placed on iterated applica-tions of shifting. For simplicity we use that the early components are isomorphic toR but using Cohen-Macaulay ideals of height one would serve the same purpose.

PROPOSITION 2.9 Let R be a Cohen-Macaulay ring and let I be an ideal ofheight at least 2. Suppose

L = Rt + • • • + Rtr + Lr+itr+l + •••

is a Cohen-Macaulay divisor. Then r < height /.

PROOF: We may assume that (R, m) is a local ring of dimension d > 2 and that/ is m-primary. Let J = ( a i , . . . , a < i ) be a minimal reduction of /. Accordingto Proposition 2.8, the elements aj , a2 — ait,..., ad — ad—it, ajt form a system ofparameters for R[It] and therefore will form a regular sequence on the Cohen-Macaulay L. (More properly, on LM where M is the maximal homogeneous idealof R[It}.)

Observe the result of the action of this system of parameters on Rt + • • • + Rtr:after reduction modulo aj , multiplication by a2 — a-\t on R/(ai)t + • • • + R/(ai)tr~1

has the same effect as multiplication by a2 only so that

L/(ai,a2 -ait)L = R/(ai,a2)t + • • • + R/(al,a2)tr~1 + • • • .

Repeating the argument up to the element a^ — ad-it, gives a module

L/(ai,...,ad-ad-it)L = R/(alt..., ad)t + • • • + R/(alt... ,ad)tr~d+1

+higher components.

It is clear that (with r > d) adt is not regular on this module. D

Dimension one

Let R be an integral domain of Krull dimension 1 and let / be a nonzero ideal.Set R = Un>1 Hom#(/ra, /") and / = IR. Note that if UJR is the canonical moduleof R, then UJR = UJR. The module / is a locally free ideal of R and the Reesalgebra 72. = R[It] is the 52~ification of 72. = R[It] (see [26]). The canonical andfundamental divisors of 72. are now easy to describe: ui-j^ = uifttR, and 'D(I) c± uj^.

The other divisors are more difficult to describe, but the following observationis useful. For simplicity suppose that R is a local ring and J = (a) is a minimalreduction of /. The homogeneous divisors of 72. are the fractionary ideals of 72. =R[at] on which {a, at} is a regular sequence. Let J = ZX>i Jntn be such a divisor.We claim that J = JitR. Using the shifting operator repeatedly it will suffice toshow that J2 = aJi. Suppose that bt2 € J; since J2 C Ji, we have the relation withcoefficients in J,

at-bt = a- bt2.Thus bt2 = at • ct, with ct G Jit.

3 DIVISOR CLASS GROUP

The divisors of a Rees algebra 72 — R[It] are easier to control when R is an integrallyclosed domain and / is a normal ideal. We will assume these conditions hold in thissection and give a general description of the divisor class group Cl(72).

There are two approaches to the calculation of Cl(72). One can apply to 72general localization properties that isolate distinguished batches of divisors (see[7]), or make use of specific properties of Rees algebras. The second method willalso be useful for certain algebras of symbolic powers and for the integral closure ofordinary Rees algebras. In the main, our discussion is a merge of several sources:[10], [11], [12], [28], [34] and [35].

THEOREM 3.1 Let R be an integrally closed Noetherian domain and let I be anormal ideal of height at least two. There is an exact sequence of divisor classgroups

0 -> H —> Cl(ft) -£+ Cl(fl) -> 0,where H is a free abelian group generated by the classes [P] defined by the minimalprimes of 772.

PROOF: Since 72 is a graded ring, to construct Cl(72), it will suffice to considerhomogeneous divisors. Let P = X^n>o Pntn be a height one prime of 71. Define afunction from this set of divisors into the group CI(_R) by putting

f ( P ) = [Pol-This map clearly defines a homomorphism of divisor class groups, which we denoteby the same symbol.

The following properties are easy to verify, (i) For each divisorial prime ideal pof R, T(p) = 72 n p72p is a divisorial prime of 'R, and </?([T(p)]) = [p] (and thereforef is surjective). (ii) Another set of valuations for 72 are restrictions of valuations ofK[t], where K is the field of fractions of R; their classes in Cl(72) are clearly linearcombinations of [T(p)]'s. (iii) If for a homogeneous prime P, the height of PQ is atleast two and I <£_ PQ, localizing at PQ yields that P is not divisorial. This meansthat the kernel of tp must indeed by given by certain combinations of [ Q i ] , . . • , [Qs],where {Qi,... ,QS} is the set of associated primes of 772 (note that this ideal isunmixed).

Suppose there is a relation among the [Qjj's. We write it as

ai[Qi] + ••• + ar[Qr] = ar+1[Qr+1} + ••• + as[Qs],

where the a^ 's are non-negative integers. For each Qi denote by q; its component ofdegree 0. The q^ are prime ideals of height at least two. Converted to a relation in72, this means that there exist homogeneous elements / = atm, g = btn of 72, suchthat the ideals

f-f[Q?, 9- H QTi=l i=r+l

have the same value at each valuation of 72.. In particular for a valuation vf definedby T(p) or arising from a valuation of K[t],

In particular a = 6, m = n, so we can cancel / and g, which leads to a contradictionas Oi=i QT nas a nonzero value at the valuation denned by Q\. D

REMARK 3.2 The same description of divisor class groups applies equally to otherfiltrations. For example, if R is a normal domain, / is an ideal of height at leasttwo and the integral closure 72. of 72. is Noetherian, then Cl(72.) has a presentationas above. Similar comments apply to Noetherian algebras of symbolic powers.

COROLLARY 3.3 Let R be a regular local ring and let I be a prime ideal of heighttwo. If S is either the integral closure of Ti = R[It] or the symbolic power algebra72,j = ̂ n>o I tn (assumed Noetherian), then <S is quasi- Gorenstein.

PROOF: By the theorem and remark, S is a Krull domain whose divisor class groupis generated by the classes of the minimal primes of IS. To ascertain that S isquasi-Gorenstein, we localize at I and get in both cases Rj[Ijt], which has a uniquesuch prime, say P. Thus [u>s] = o\P\. But the localization is a Gorenstein ringsince // is generated by a regular sequence of two elements. Thus a = 0, whichshows that ws = tS. D

To identify the set {Qi, . . . , Qs} of minimal primes of 7.72. is usually very difficult.We consider a few cases of interest.

(i) Suppose that I is an unmixed normal ideal of height one. The inclusionR '—> 72 defines an embedding of divisor class groups

C\(R) -^ 0(71), [p] ̂ [T(p)]

which is an isomorphism.

(ii) Let R be a factorial domain and let / be a prime ideal that is generically acomplete intersection and is such that the associated graded ring gr/(.R) isan integral domain (in this case 72 is automatically normal, see the commentbelow). In this case, C\(R) = 0 and C!(72) =

Ideals of linear type

Let / be a (normal) ideal of linear type, that is 72. is the symmetric algebra SR(!)(see [32, p. 138]). Let

H* JU Rn — » / _> 0

be a presentation of /. One can determine the minimal primes of 772 using thematrix tp. For a prime ideal p C R there is an associated prime ideal

Assume that R is universally catenary in order to avoid dwelling into technical-ities. In this case, if in addition S ( I ) is equidimensional, one sees that

height T(p) = 1 if and only if v ( I y ) = height p.

We also observe that unless p is a height one prime itself, / is not free if T(p) is tohave height one.

PROPOSITION 3.4 Let R be a universally catenary Noetherian ring and let I bean ideal such that 5(7) is a domain. Then the set

(T(p) height p > 2 and height T(p) = 1}

is finite. More precisely this set is in bijection with

{p C R 7p not free, p 6 Min(R/It(ip)) and height p = rank(^) - t + 2},

where 1 < t < rank(^).

PROOF: Given a prime p C R, set t = n—z/(7p) . If height p > 2 and if height T(p) =1 then, since rank(V') = n — 1, by the preceding remarks we have

height p = n- I - t + 2 and p D /t(VO \ /t-i(VO-

On the other hand, since 5(7) is a domain, ^(7p) < height p for each nonzeroprime ideal. Therefore, one has height 7t(i/>) > rank(^) — t + 2. It follows thatp G Mm(R/It('4>)). The converse is similar. D

Reducedness and normality

There are very few criteria of normality for Rees algebras outside of general Jacobiantests, which do not take into account the nature of a Rees algebra.

We begin with the following observation:

PROPOSITION 3.5 Let R be a Noetherian normal domain and let I be an idealsuch that gr/(J?) is a reduced ring. Then 72 = R[It] is integrally closed.

PROOF: Consider the extended Rees algebra, 72.e = R[It,t~1]. Since Q = 72,e/(t~1)is reduced it follows that 72e satisfies the (52) and (Ri) conditions of Serre. Thus72e is normal and since 72, — 72e n R[t], 72 is also integrally closed. D

Under some additional conditions on the ideal 7 one obtains a very explicitdescription of the divisor class group of such algebras.

THEOREM 3.6 ([16]) Let (R,m) be a quasi-unmixed local ring and let I be anideal of finite projective dimension. If Q = gr/(7?) is a reduced ring then Q is atorsionfree R/1-algebra. Moreover, if I is a prime ideal then Q is a domain.

PROOF: We may assume that the residue field of R is infinite. Let P be a minimalprime of 772; then P72.p = 772p. We want to relate these primes to the associatedprimes of 7.

Since Q is reduced, 7 is a radical ideal. Let {pi , . . . ,ps} be the set of minimalprimes of 7. For a prime p = pj, set

T(p) = kernel (7e^(72/p7e)p).

Since 7p = p7?p has finite projective dimension, 7?p is a regular local ring andtherefore T(p) will be a prime ideal.

The claim is equivalent to saying that any minimal prime of ITL is one of theT(pi), i.e., that PdR — pi for some i. We suppose otherwise, set q = P(~}R, localizeat q, change notation and assume that P n R = m is the maximal ideal of R. Twoobservations arise: (i) / (£_ m2 and (ii)

dim/? = dimft/P < dimTl/mU = 1(1) < dimR.

By induction on g — height /, we are going to show they are untenable.We use an argument of [20], that treats the more general condition (Rk) of Q.

If g = 0, then 7 = 0 as an ideal of finite projective dimension over a local ring mustcontain regular elements. Let a\,..., an be a set of generators of 7, let Xi,..., Xnbe independent variables and set R = R(Xi,..., Xn), TO = m7? and Q = Q ®R R.Further, set x = £"=i a^, x' = x + 72 <E <51; and 7 = T/(x) C 7? = R/(x). Since7 <£_ m2 and g > 0, x 0 m2 is 7?-regular, being a generic element of 7. By [21, p.130], 7 is an ideal of finite projective dimension of 7?. Finally, x' is a generic elementof Q+. It follows that a; is a superficial element of 7, and G/(x') is still reduced atits minimal primes since for every Q G Proj (G/(x')), (G/(x'))q is the localizationof a polynomial ring over Q. Since a; is a superficial element of 7, gry(7?) is reducedon Proj (gTj(R)). By the induction hypothesis, 1(1) < dim7? — 1 = dim7? — 2.Finally, setting K = R/fh, we have

e(I) = dim Proj (grj(R) <8)^ K) + I = dim Proj (G/(x') ®^ K) + 1 (3)

= dim(G®KK)/(x')>e(I)-l, (4)

to complete the proof. D

EXAMPLE 3.7 Let R = k [ { x , y , z ] ] / ( y 2 - xz), k a field, and let p = (y,x)R.Since 7? is a domain and (y) : x — (y) : x2 it follows that y, x is a d-sequence sothat grp(7?) ~ 5/j/p(p/p2), the symmetric algebra of p/p2. We then obtain thatgrp(7?) = k[z,u,v]/(uv) which is reduced but not a domain. Note however thatproj. dim.^p = oo.

COROLLARY 3.8 Let R be a quasi-unmixed normal domain and let I be an idealsuch that Q — grj(R) is reduced. Denote by {p i , . . . ,ps} the minimal primes of Iand by T(ipi) the corresponding minimal primes of I1<L (uniquely determined by thetheorem). Then Ti = R[It] is a Krull domain whose divisor class group has apresentation

0 -> 77 —* Cl(ft) -£+ Cl(7?) -4 0,where 77 is freely generated by the [T(pj)].

4 THE EXPECTED CANONICAL MODULE

There is an obvious need for standard models for canonical modules of Rees algebras.One that surfaced early is given by the following example.

EXAMPLE 4.1 If the ideal 7 is generated by a regular sequence /1 ( . . . , fg, g > 2,the equations of 71 = R[It] are nice:

In other words, the defining equations of the algebra are generated by the Koszulrelations of the /j's.

Knowing this description of Tl leads immediately to its canonical module ([3])

There are however many other instances of Rees algebras whose canonical mod-ules have this form. It warrants the following:

DEFINITION 4.2 Let R be a Noetherian local ring with a canonical module U>Rand let / be an ideal of positive grade. The canonical module of Tl = R[It] is saidto have the expected form if

for some integer 6.

The connection with the canonical module of gij(R) is provided in the followingresult ([10], [37]).

THEOREM 4.3 Let (R,m) be a Noetherian local ring with a canonical moduleu> — UJR and let I be an ideal of positive grade. Assume that Tt = R[It] is Cohen-Macaulay and set a = —a(Q) for Q = gij(R). Then the following are equivalent:

(a)

(b)

PROOF: Denote by ui-j^ the canonical module 72,. Applying Hom-j^(-, w-^) to thesequences

o -> /ft — >n — >g -> oand

o -> itn — > 7^ — > R -* o,we obtain

and0 -

Suppose that (b) holds. From the last sequence we obtain <^>Itji = w#(l, t)a~1Tl.Feeding this into the previous sequence, given that w^ = tu>It-j^, we obtain thatthe canonical module of Q is as expected.

For the converse it is convenient to express these exact sequences of canonicalmodules in the following manner:

^—^ X—^ X—>n _ . _ \ r) j-n __ _ \ c1 +n __ , _ \ u1 fTi _ n/V- / j "I I /v / _j ''• C/ / j Tl 5

and

E ^—> _ 1n itR- ~ / ^ n * •

From which we get (noting that a > 0)

El = LJEn = Dn_! n > 2En = Dn 1 <n< a

En/Dn = Fn n>a.

It follows that Dn = LO for n < a. For n = a, we have

Ea/Da = A.-1/A, = <"/Da ^ W/IU.

On the other hand, IDn C -Dn+i for all n, gives epimorphisms

UJ/ILJ —» u>/ Da -» w/I(jj.

It follows that Da = 7u>. The remainder of the argument is similar. n

REMARK 4.4 For an ideal I in the situation of this theorem, one has a(gr7(/?)) =a(gr/ ( R f ) ) for any minimal prime p of /. Thus if / is a generic complete intersec-tion, a(grj(R)) = —height /. More generally, a(gTI(R}) = r(/p) — height /.

Gorenstein algebras

The arrangement of the proof above can be used in the proof of the following resultof Ikeda ([18]).

THEOREM 4.5 Let (R, m) be a Noetherian local ring and let I be an ideal of gradeat least 2. If R[It] is Cohen-Macaulay the following conditions are equivalent:

(a) R[It] is Gorenstein.

(b) UR~R and wgr/(fi) ~ gr/(JR)[-2].

PROOF: (a) => (b): Setting K - R[H] and uj-j^ = ]Cn>i Dntn, we have

) +n+l

The embedding

0 -» S P / = u — > P J — > G P / = w -> 0

shows that if w^ ~ 72. then R = DI C U>R and / • w/j C -R; it follows that D\ — R =<JJR since w^ lies in the total ring of quotients of R and grade / > 2. As Dn = In~l

for n > 2, the canonical module of £/ is as asserted.(b) =^ (a): w^ = ujRt(l, t}a~2K ~ 7^ by the hypothesis and Theorem 4.3. D


5 THE FUNDAMENTAL DIVISOR

Our main purpose is to introduce a divisorial ideal associated to a Rees algebraand sketch out some of its applications. It helps to explain old puzzles while at thesame time providing quite direct proofs of earlier results. The reader will note thatit is a mirror image of local cohomology modules of Rees algebras. Its Noetheriancharacter however permits a control of computation that is not always possible withArtinian modules.

Let (R, m) be a Cohen-Macaulay local ring of dimension d, with a canonicalmodule w. Let / = (/i, . . . , fn) be an ideal of positive height. Fix a presentation of72, B = R[Ti, . . . ,Tn] — > 72. We set UB = w ®R B[—n\ as the canonical module ofthe polynomial ring B. The canonical module of 72 is the module

This means that £>(/) can be written as

D2t2

where each Di is an J?-submodule of K, the total ring of fractions of R. Wefix this representation of T>(I) from a given projective resolution of / • 72. and thecomputation of the cohomology.

This divisorial ideal carries more information than the canonical module w-^.Indeed, we can view T>(I) as made up of two parts,

D(I) = Drf + D2t2 + D3t3 + • • • •(0 (")

DI will be called the leading part of 7J>(7), while (ii) is called its canonical part inagreement with the next observation.

A consequence of the proof of Theorem 4.3 is the following relationship betweenthe canonical and fundamental divisors of a Rees algebra.

PROPOSITION 5.1 Let R be a Noetherian ring with a canonical module and letI be an ideal of positive grade. Then the graded components ofujj^ and ofD(I) arerelated in the following manner:

n>2

that is, T>(I) is a prolongation

DEFINITION 5.2 Let R be a Noetherian local ring with a canonical module UIRand let / be an ideal of positive grade. The fundamental divisor of 72. = R[It] issaid to have the expected form if DI ~ U>R.


Let us phrase the condition DI ~ WR in terms of cohomology. We assume thatR is a Cohen-Macaulay local ring of dimension d > 0 and set 7£ = R[It}. From theexact sequence

0 -» in[-l] — > Tl — > R -> 0,we have

0 - HdM(IT^)-i -^ Hd

M(U)0 -^ Hdm(R) -^ H%\IK)^ -^ H%l(U)0 = 0.

PROPOSITION 5.3 Let D be the fundamental divisor of U. Then DI ~ LOR ifand only if f = 0.

One issue is what is the most approachable mapping to examine. Of course themost sensible move thus far has been to prove directly that Hj^(TV)o = 0; we willfollow this route in the last section. Now we must look at the mapping (I) and the Cohen-Macaulayness of R[It]

The next result recasts aspects of the characterization of the Cohen-Macaulay prop-erty of 72 = R[It] in terms other than the vanishing of local cohomology. The fol-lowing is one aspect of the criterion of [19] in the terminology of the fundamentaldivisor.

THEOREM 5.4 (ARITHMETICAL CRITERION) Let (R,m) be a Cohen-Macaulaylocal ring with a canonical module LU and let I be an ideal of positive height. Thefollowing equivalence holds:

_ . „ , , , , ( Q is Cohen-Macaulay and ,_,K, is Conen-Macaulay -4=> < „ (5)y ~ ^ '

Before we give a proof we consider the case of 1-dimensional rings, when theassertions are stronger.

THEOREM 5.5 Let (R,m) be a 1-dimensional Cohen-Macaulay local ring witha canonical module w and let I be an m-primary ideal. The following equivalenceholds:

72. is Cohen-Macaulay 4=4> DI ~ u>. (6)

PROOF: We prove that if DI ~ u>, then / is a principal ideal. The other assertionshave been established before or will be proved in the full theorem.

We may assume that the residue field of R is infinite. Let (a) be a reduction of/ and Ir+1 = alr , where r is the reduction number of / relative to (a). We claimthat r = 0. To this end, consider R[at] whose canonical module is atu>R[at]. Let

+ ••• ~H.omR[at](I-R[It},atuR[at}).

is defined by the relations

D I - I C au>C a?uj

C arw.

The descending chain of fractional ideals of R,

arw: Ir C • • • C auj: /,

implies thatDl = Aw = a rw:/ r ,

where A is some element in the total ring of fractions of R. This equality meansthat

w = Hom/j(/ ra~ rA, w),

and therefore that Ira~r\ c± R, since Hom^(-,w) is self-dualizing on the fractionalideals of R. This means that Ir is a principal ideal and / will also be principal, as/? is a local ring. d

Proof of Theorem 5.4. We consider the long exact sequences of graded B-modules that result from applying the functor Horns ( ' , W B ) to the sequences (1)and (2). We have:

0 — >Ext%-l(H,uB) — >ExtB~l(It-n,ujB} — >Ext%(R,wB) = w (7)

and

0 —— Extl-\n,uB) — ExtnB-l(I • 72,wB) — + Ext^(a,o>B) = wg (8)

• • • — » ExtzB(72,wB) -^ Extl

s(7 • 72, WB) — > 0, i > n.

In the first of these sequences, in degree 0, we have the injection

0 -> DI -^ w — > D -> 0 (9)

that is fixed and that we are going to exploit repeatedly. Suppose that Q is Cohen-Macaulay and DI ~ w. In this case, 93 is an injection of modules with the (62)property that is an isomorphism in codimension 1. Thus n. In theother sequences meanwhile, the mappings 9i are surjections for all i > n. In viewhowever that Ext^(J • 72., WB) ~ Ext^(/t • 72., WB) , as ungraded modules, 0j beingsurjections of isomorphic Noetherian modules must be isomorphisms. This impliesthat Extg(/-72-, WB) ~ Extg(/f -72, WB) as graded modules, which is a contradictionsince one is obtained from the other by a non-trivial shift in the grading.

Conversely, if TL is Cohen-Macaulay, from (1) we have that Q is Cohen-Macau-lay, and from Extg(72.,ws) = 0, we have an isomorphism DI ~ w. D

Divisorial component

We now examine two cases when the behavior of other components of T>(I) impacton its first component.

PROPOSITION 5.6 Let R be a Cohen-Macaulay local ring with a canonical idealuj and let I be an ideal of height > 2. If uj-j^ = X/n>i Lntn has a component Ln thatis a divisorial ideal of R then the fundamental divisor T>(I) has the expected form.

PROOF: By Proposition 2.7, we have that once some Ln = K is divisorial then allthe previous components of any prolongation are equal to K. This gives us DI — K.In addition we have a homomorphism a : K —> u> which is clearly an isomorphismof divisorial ideals of R in codimension one. But this is all that is needed to identifyK and w. D

PROPOSITION 5.7 Let (R, m) be a Cohen-Macaulay local ring with a canonicalideal w, with infinite residue field and let I be an ideal of height g > 2. If DI = D%then DI ~ <jj, that is T>(I) has the expected form.

PROOF: Let a € / be a regular element and choose b £ I satisfying the followingtwo requirements: (i) b is regular on R/(a), and (ii) b is a minimal generator of /and its initial form b* € Qi does not belong to any minimal prime of Q. We claimthat the ideal (a, bt)~R. has height 2. If P is a prime ideal of height 1 containinga,bt, it cannot contain /, since 6*, the image of bt in Tl/ITL = Q does not lie inany minimal prime of Q. This shows that Tip is a localization of a polynomial ringRc[t], and in this case (a,bt) obviously has height 2. We then have that a, bt is aregular sequence on the 7£-module T>(I] which has the property (52). As in theproof of Theorem 5.4, if D\ is not isomorphic to u>, we may assume that in thenatural sequence

cokernel y> is a module of finite length. If DI = D^, since a is regular on/D(/)/6iX'(7), this implies that a is regular on D^/bDi = Di/bDi. But this is acontradiction since DI has depth 1. D

Veronese subrings

A simple application of Theorem 5.4 is to show that a common device, passingfrom a graded algebra to one of its Veronese subrings in order to possibly enhanceCohen-Macaulayness, will not be helpful in the setting of ideals with associatedgraded rings which are already Cohen-Macaulay. This is a well-known fact amongexperts.

Let Ti = R[It] be the Rees algebra of an ideal /, let q > 1 be a positive integerand denote

v—"\

/ ^ '

the q Veronese subring of Ti. Our purpose here is to prove:

THEOREM 5.8 Let R be a Cohen-Macaulay ring and let I be an ideal of positiveheight such that the associated graded ring Q = grj(R) is Cohen-Macaulay. ThenTZ. is Cohen-Macaulay if and only if any Veronese subring Tto is Cohen-Macaulay.

PROOF: Most of the assertions are clear, following from the fact that as an Tio~module, "R. is finitely generated and contains Tlo as a summand. As for the hypothe-ses, if Q is Cohen-Macaulay, the extended Rees algebra A = R[It,t~l] will also beCohen-Macaulay, and the ring A/(t~q) with it. Since the associated graded ringQo of Iq is a direct summand of the latter, Q§ is Cohen-Macaulay.

It will suffice to show that the fundamental divisors of 72. and 7?o,

= Lgt" + L2qt2" + • • • ,

relative to the respective algebras, satisfy D\ ~ Lq.Let WQ denote the canonical module of 72.Q. Let us calculate T>(I] as

s=l

The degrees have been kept track of, permitting us to match the components ofdegree 1, respectively DI on the left and Lq on the right. The remaining assertionwill then follow from Theorem 5.4. D

Symbolic powers

Ideals whose ordinary and symbolic powers coincide provide a clear path to thefundamental divisor.

PROPOSITION 5.9 Let (R,m) be a Cohen-Macaulay local ring with a canonicalmodule u>, and let I be an ideal which is generically a complete intersection. Supposethat for each prime ideal p D /, with height(p//) > 1, £(/p) < height p. Then

PROOF: We claim that the mapping labeled <p above is an isomorphism:

We must show that C = 0. By induction on the dimension of R, we may assumethat C is a module of finite length.

If m is a minimal prime of /, this ideal is a complete intersection. Suppose thenthat / is not m-prirnary. By assumption £(/) < height m, so that height mR > 2.We may thus find a,b e m so that height (a, 6)7?. = 2. Since T>(I) is an (52)-module over 7?, a, b must be a regular sequence on £>(/). In particular, a, b is aregular sequence on DI, which is clearly impossible if C is a nonzero module offinite length. D

The following is an application to the symbolic powers of a prime ideal (see [31]for the Gorenstein case, and [2] for the general case):

COROLLARY 5.10 Let R be a Cohen-Macaulay ring and let p be a prime idealof positive height such that R9 is a regular local ring. Suppose that p(n*> = p" forn>l. Then -R[pt] is Cohen-Macaulay if and only if gr~(K) is Cohen-Macaulay.

PROOF: The condition on the equality of the ordinary and symbolic powers of pimplies the condition on the local analytic spread of p. In turn, this condition ispreserved after we localize R at any prime ideal and complete. D

For these ideals one can weaken the hypothesis that Q be Cohen-Macaulay in anumber of ways. Here is a result from [25]:

THEOREM 5.11 Let R be a Gorenstein local ring of dimension d and let I bean unmixed ideal of codirnension g > I , that is generically a complete intersectionand is such that I^> = In for n > 1 . Then Tl is Cohen-Macaulay if and only if Qsatisfies (Sr) for r = \^}.

A first step in the proof consists in the following calculation ([25]):

PROPOSITION 5.12 Let R be a Gorenstein local ring and let I be an unmixedideal of codirnension g > 1, that is generically a complete intersection and is suchthat /(n) = In for n > 1. Then the canonical module ofTl = R[It] has the expectedform, that is

QUESTION 5.13 Which toric prime ideals p have the property that p("> = pn forn > 1? Particularly interesting are those of codirnension 2 and dimension 4.

Equimultiple ideals

One landmark result in the relationship between % and g-fj(R) was discovered byGoto-Shimoda [8] (later extended in [9]).

THEOREM 5.14 (GOTO-SHIMODA) Let (R, m) be a Cohen-Macaulay ring of di-mension d > 1 with infinite residue field, and let I be an equimultiple ideal ofcodirnension g > 2. Then

„ . „ , , , . f Q is Cohen-Macaulay and . .K. is Cohen-Macaulay <=> < , T. a (10)v '

PROOF: We may assume that R is a complete local ring, and therefore there is acanonical module w. Let J be a minimal reduction of /. Since J is generated by aregular sequence, the Rees algebra 7?_o = R[Jt] is determinantal and its canonicalmodule is (see Example 4.1)

We can calculate T>(I) as

£>(/) = Horn-Fig (in, WQ) = Dit + D^fi

where DI must satisfy the equations

/ • A C w

I9~l-Dl C uI9 •Dl c J • w

Note that since w has (52) and height / > 1, DI can be identified with a subidealof u> and coincides with u> in codimension 1.

Suppose H is Cohen-Macaulay, so that DI ~ u>. But DI C u> and both frac-tionary ideals are (S%) and thus they must coincide since they are equal in codi-mension 1. From the equation I9 • uj C J • w, it follows that I9 is contained inthe annihilator of u>/ J • u>. But this is the canonical module of R/J, and thereforeI3 C J. Since Q is Cohen-Macaulay, by [30] we must have I9 = J • Ia~l.

For the converse, the equations give that D\ = o>, so we may apply Theorem 5.4.n

We are going to reinforce a one-way connection between the reduction numberof an m-primary ideal / and DI (I) in another case.

PROPOSITION 5.15 Let (R,m) be a Cohen-Macaulay local ring of dimensiond > 2, with a canonical module MR, and let I be an m-primary ideal of reductionnumber r(J) < d. //depth grj(J?) > d — 1, then DI(!} ~ U>R.

PROOF: Let J be a minimal reduction such that Ir+1 — JIr . By hypothesis r < d.Consider the associated Sally module

O^IUo — >IK — > S - » 0 , (11)

with ft0 = R[Jt], n = R[Ii\. As observed in [33], the condition depth gi(R) > d- 1means that 5 is a Cohen-Macaulay module. It particular, according to [33], S willthen admit a filtration whose factors are the T^o-modules

In/jr-1[T1,...,Td][-n+l], n<r.

Let B be a presenting Gorenstein ring for HQ, that is, a Gorenstein ring ofthe same dimension as Tlo and a finite homomorphism Tlo- This justmeans that we can define the divisors of 7?-o and of H using B. Dualizing the exactsequence above with B gives the exact sequence

0 - "iK — * w/ft0 —— "s — + Ext^m, B) -> 0. (12)

We make two observations about some terms of the sequence. First, from theexact sequence

0 -> IK0 — > 7^0 — , R/I[Tlt. . . ,Td] -> 0,

and the fact that w-j^ has the expected form u>n(l, i)d~2i7?-o, we get that w^-^j isgenerated in degree > 1. On the other hand, given the factors of the nitration of 5,MS is a Cohen-Macaulay module admitting a filtration whose factors are the duals

where (-)v means the Matlis dual functor. When these are put back into (12),n < r < d, we get that the components of degree 1 of the terms on the left coincide,that is DI(!) = MR, as desired. D


Regular local rings

The following is a rather surprising property discovered by Lipman ([23]; see also[29]):

THEOREM 5.16 // (R, m) is a regular local ring, for any nonzero ideal I thefundamental divisor of R[It] has the expected form.

If / is an m-primary ideal, this statement follows from a basic form of thetheorem of Briancon-Skoda (see a discussion of this theorem and its role in theCohen-Macaulayness of Rees algebras in [1], [2]). One of its consequences is ([23]):

COROLLARY 5.17 Let R be a regular local and let I be an ideal. Then U = R[It]is Cohen-Macaulay if and only if Q = grj(R) is Cohen-Macaulay.

Before proving Theorem 5.16, we first recall some facts of local cohomology.Let A = 0n>0 An be a finitely generated graded algebra over the Noetherian ringR = AQ. Let N be a finitely generated graded ^-module. The cohomology ofcoherent sheaves over Proj (A) is expressed by the following Cech complex. Let/o , . . . , fs be a set of homogeneous elements of A+ such that A+ C \/(/o, • • • ,/«)•The (limit) Koszul complex of the fi is

-» A — > Afi ^ 0).i=0

This is a complex of Z-graded ^-modules. We denote

K ( f 0 , . . . , f s ) ® N = K ( f 0 , . . . , f s ; N ) .

For a given integer n, the Cech complex of the sheaf J\f(n) is the subcomplex

of elements in degree n. Here X = Proj (A) and J\f(n) is the sheaf associated tothe module N[n].

This construction defines the short exact sequence of chain complexes, where Nis viewed as concentrated in dimension zero:

A(n))[-l] — * K(/0, . . . , /.; N) — > N -> 0. (13)n

Since X/i = Spec ((^4/Jo), the Cech complexes give rise to the cohomology ofthe sheafs J\f(n) on the scheme Proj (A). More precisely one has:

THEOREM 5.18 Let N be a finitely generated graded A-module and denote by A/"the corresponding sheaf on X — Proj (A) . Then for all i > I and all integers nthere exists a natural isomorphism of finitely generated R-modules

Moreover for all integers n there exists an exact sequence

0 -> H°A+(N}n -^Nn^ r(X,Af(n)) -^ H\+(N)n -> 0.


Let (R, m) be a Noetherian local ring and let 71 = R[It] be the Rees algebra ofan ideal of positive height. In the sequence (13), set A = 71, let J be an ideal ofR and apply to it the functor ^jf^(-)- Taking the hyper-cohomology (see [36]) ofthe sequence of complexes, one obtains the Sancho de Salas exact sequence ([27];we follow [23] also):

PROPOSITION 5.19 Let 971 = (J,7l+) and let E = Proj (U®R/J). For anyfinitely generated graded A-module N there exists a long exact sequence

Proof of Theorem 5.16. Following [17], let (R, m) be a regular local ring ofdim/? = d and let / be a nonzero ideal. Write 72 = R[It], X = Proj (TV) andE = X XR R/m. The Sancho de Salas sequence, in degree 0, is

— Hdm(R) -£* Hd

E(x,ox) -^ [/4+1(^)]o -+ o.Note that [#4+1(72,)]0 = 0.

On the other hand, according to [22] and [24], for any regular local ring,

from which it follows easily that the mapping <p is an isomorphism.Finally, after noting that O\ — IOx[—l], another application of the Sancho de

Salas sequence to the module / • 72,, in degree — 1, yields the isomorphism

HdE(X, 0X) -(H^1 (171)^.

In other words, we have that DI ~ R, as desired. D

6 COHEN-MACAULAY DIVISORS AND REDUCTION NUMBERS

We want to study the relationships between the algebra R[It] having one of the dis-tinguished divisors we have examined thus far — Serre, canonical and fundamental —and the reduction number of /.

Serre divisor

In this case the relationship is very straightforward according to the following.

PROPOSITION 6.1 Let (R, m) be a Cohen-Macaulay local ring of dimension d >0 and let I be an ideal of positive height. If IR[It] is Cohen-Macaulay then r(7) <

PROOF: Set 71 = R[It], Q = gr/(.R), and 2Jt = (m,72+). We consider the cohomol-ogy exact sequences related to (I) and (2):

HdU = 0 -+ HK — » H R — * H

andH^m = 0 — * H^CK) — > H&(g) — H^l(IU).

Since IR, is Cohen-Macaulay we can find its a-invariant localizing at the totalring of fractions of R, a(IK) = a(K[t}) = -I.

The first of the sequences above says that H^(TVj is concentrated in degree 0,and therefore from the second sequence we obtain a(Q) < 0. Since Q is Cohen-Macaulay, for any minimal reduction J one has a(Q) > rj(/) — 1(1}, to completethe proof. D

Canonical divisor

The basic listing of the properties of divisors (Proposition 2.2) gives a crude inter-pretation of the Cohen-Macaulayness of the canonical module.

PROPOSITION 6.2 Let (R,m) be a Cohen-Macaulay local ring of dimension d >0 and let I be an ideal of positive height. The canonical module w-j^ of *R is Cohen-Macaulay if and only if the 82 -ification R ofRis Cohen-Macaulay.

7 VANISHING OF COHOMOLOGY

In this section we discuss some calculations of the cohomology of Proj (R[It]} withthe aim of detecting the vanishing of E1

M (R) for i < dim R. These groups play asignificant role in predicting the geometric properties of Proj (R) but are very hardto determine explicitly in great generality.

Ideals of linear type

We now treat an interesting general property of ideals of linear type treated byHuckaba and Marley ([14]). Here is their approach to the calculation of <Zd(72.) withsome variations in the case the ring R is not Gorenstein.

THEOREM 7.1 Let (R,m) be a Cohen-Macaulay local ring of dimension d > 0,let I be an ideal positive height and set 72. = R[It\. Then a^CR) < 0 in the following

(i) ([14]) R is Gorenstein and I is of linear type;

(ii) R is an integral domain with a canonical module and I is of linear type;

(iii) R is a Gorenstein integral domain and I is an ideal such that for each primeideal p, z/(/p) < height p + 1, and the canonical module of 72 satisfies thecondition 83 of Serre.

PROOF: (i) Let / = (a1; . . . , an) be a set of generators of / and R. = R[Ti, . . . ,Tn}/Ja presentation. By hypothesis J is generated by forms of degree 1. Since we mayassume that the residue field of R is infinite and J is an ideal of height n — 1,there are n — I forms /i, . . . ,fn-i of J of degree 1 generating a regular sequence.Set A = S/F, where S = R[Ti, . . . ,Tn], F — (/i, . . . , / n_j) , and consider thepresentation 72. = A/L, where L = J/F. Note that since u>s = S[—n] is thecanonical module of 5, UA = S/F[-1] ~ A[-l}.

Let M be the maximal homogeneous ideal of 72. We show that H^(7i)n = 0for n > 0. By local duality (see [4, Section 3.6]), if E is the graded injectiveenvelope of R/m and if for a graded A-module N we set Nv = HomA(N,E), wehave (H^CR^Y = Ext\(Tl,A[-l]). Thus the isomorphisms

mean that it suffices to show that Ext^(72, A[— !])„ = 0 for n < 0.First, observe that from the exact sequence

O ^ L — > A — >K^O,

and the fact that Hom^A, yl[— l])n = 0 for n < 0, we have

E.omA(L, A[-l})n ~ Ext^(72, A[-l])n for n < 0.

Let / G Homs(L, y4[— !])„ and let LI be the set of homogeneous elements of L ofdegree 1. Then /(Li) C A[~l}n+1 = An. If n < 0 then An = 0 and /(Li) = 0.Since I/ is generated by LI, f = 0 in this case. Therefore it suffices to prove thecase n = 0.

At this point we remark that if R is assumed to be just Cohen-Macaulay andup. is its canonical module (which we may assume after completing R), then theargument above already shows that a<i(72.) < 0. It is to ascertain the case n = 0that we must consider additional restrictions on R.

In case (i), we claim that L = 0 : (0 : L). Both ideals are unmixed of heightzero, so it suffices to check equality at the localizations of A at prime ideals p ofheight 0. Since Ap is a Gorenstein Artinian ring the equality holds for any of itsideals. Now observe that (0 : L ) f ( L ) = /((O : L)L) = /(O) = 0, and thereforef ( L ) C 0 : (0 : L) = L. In particular f ( L i ) C A0 n L = 0, as desired.

(ii) This case is very similar. First assume that R is complete. Let K C R bea canonical ideal of R. Then the canonical module of A is U>A — (KS/KFS)[— 1].Notice that ((JJA)U = 0 for n < 0, and that (UA}I = K.

Note that L is a prime ideal and since A is a Cohen-Macaulay ring we musthave L — 0 : (0 : I/), as the right hand side consists of zero divisors and thereforecannot properly contain the minimal prime L of the Cohen-Macaulay ring A. Nowwhen we consider the module Hom^L,^), as above /(Li) must live in degree 1of u A, so it is a subideal H of K such that (0 : L)H = 0. In the original ring 5 thisequality means that

(F : J)H C KFS C FS,and therefore H C F : (F : J) = J. But this is a contradiction since J contains noelements of degree 0.

If R is not complete, we start with the equality L = 0 : (0 : L) and thencomplete. K C R is a canonical ideal and we can proceed as above.

(iii) The condition on the local minimal number of generators of / means thatin a presentation 5 = R[Ti, . . , , Tn] — » 72. the component of degree 1 of the kernelhas height n — I . The ring A is defined as above, A = S/(fi, . . . , fn~i) and A[— 1]is its canonical module. As in case (i), we must show that the module Hom/t(L, A)has no elements in negative degrees.

Observe that 0 :A L <t- L. Indeed localizing at the nonzero elements of R,the ideal F = (/i, • - . , /ra-i) becomes a regular sequence generated by linear formsover a field, and therefore it is a prime ideal of height n — 1. This shows thatFj = Jj, which implies the assertion. Furthermore, since L is a prime ideal, theideal L + 0 :A L has height greater than 0 and thus contains regular elements sinceA is a Cohen-Macaulay ring; this implies that L + OA '• L = L ® 0 \A L. We furthernote that the image of 0 :A L in A/L — Tl is a canonical ideal of K.

For any homomorphism / € Hom^L, .A) (respectively / (E HomÔ :A L,A)),the equality /((O :A L)L) = (0 :A L ) f ( L ) = 0 shows that /(L) C 0 :A (0 :A L) = L,that is / 6 Hom/i(L,L) (respectively / € HomÔ :A L,0 :A L)). Consider theexact sequence

0 -> L®0:AL-Â-^ A/(L <$0:AL) = (A/L)/((L ®Q:A L)/L) = U/un -> 0.

Applying Hom/i(-,J4) we have the short exact sequence

0 -> EomA(A,A) -> Horru(L,L)©HomA(wft,a>ft) -* Ext^C/e/w^,^) -> 0. (14)

We recall that Homû;^, w-^) = 72., the 52-ification of 72. (see [26] for a discus-sion), and that w-j^ = ov,. Moreover, there is a canonical isomorphism

which is the canonical module of both rings Ti/u-j^ ^-> 'R./w-j^. We now use anargument that goes back to Peskine, that under the condition that w-j^ has 63 thenthe ring 7?./w-^ is quasi-Gorenstein. Indeed from the sequences

0

0 -> Hom^C^,^) — > RomA(u-ji, A)

and the identifications HomJ4(7^, A) = w-^, Hom^(w-^, A) = 7£, we have a naturalembedding _

This inclusion is an isomorphism whenever 72. is Cohen-Macaulay. It is thus anisomorphism at each localization at height at most 1 in the support of 72.. Since w-^has the condition 83 by hypothesis the cokernel has positive depth or vanishes. Itfollows easily that it must be zero.

It follows that the modules at the two ends of the exact sequence (14) have noelements in negative degrees, and therefore HomA(L,L) doesn't either. n

We note that this is a calculation at the edge of a very general result on localduality.

Equimultiple ideals

This class of ideals is very amenable to calculation of cohomology. The following isrelevant to the fundamental divisor.

THEOREM 7.2 Let (R,m) be a Cohen-Macaulay local ring of dimension d > 1and let I be an m-primary ideal. Suppose that depth gTj(R) = d — 1 and r(7) < d.Then ad(R[It\) < 0.

PROOF: We make use of the special nature of the Sally module of the ideal /. Wemay assume that the residue field of R is infinite and let J be a minimal reduction.Consider the Sally module S j ( I ) ([31]):

0 -> IR[Jt] — > IR[It] —> S j ( I ) -» 0.

By [13], depth U = d, and therefore from (1) depth IR[It] > d. It follows thatS j ( I ) is a Cohen-Macaulay module (over R[Jt]). Taking local cohomology in thesequence above with respect to the homogeneous maximal ideal A/" of R[Jt\, we havethe exact sequence

0 - Hf(IR(It\) — » H r ( S j ( I ) ) -^ H ( I R [ J t \ ) -^ H l ( I R [ I t } ) - 0.

On the other hand, we have the exact sequence

0 -> IR[Jt] — > R[Jt] — > B = R/I[Ti,. . . , Td] -> 0,

of Cohen-Macaulay modules. From the exact sequence of cohomology

0 -» H*f(B) — H$-l(IR(Jt]) — > H$-l(R(Jt}) -+ 0,

we have that F^+1(/B[Jt])n = 0 for n > 0.Now we appeal to the fact that the Sally module S j ( I ) , being Cohen-Macaulay,

admits a filtration whose factors are the modules

P/ JP~l [Ti ,...,Td][-i+l], i<r = r(7).

As a consequence, H^(Sj(I)) admits a similar filtration of cohomology modulesand we obtain that the a-invariant of S j ( I ) is at most r — d — 1, which shows thata,d(IR[It]) < — 1, as we are assuming that r < d. The cohomology sequence of theSally module also yields the isomorphism

Finally, we consider the cohomology of the sequence

0 -> tIR[It] = I R [ I t ] [ - l ] — -> R[It] —^R->0.

We obtain in degree 0

0 = H*f(IR[It\)-i — » ^r(^[/t])0 -^ ̂ (E) — + Hfr^IRilt])-! -+ 0,

from which we get H%f(R[It])o = 0, because as observed previously

H$-1(IR[It])-1=H*(R).

n

COROLLARY 7.3 Suppose further that Proj (R[Ii\) is Cohen-Macaulay. Thensome Veronese subring of R[It] is Cohen-Macaulay.

Links of primes

In this extended example we calculate the cohomology of the Rees algebra of adirect link of a prime ideal. We will use Theorem 7.4 and other constructions of [5]:

THEOREM 7.4 Let R be a Cohen-Macaulay ring, p a prime ideal of codimensiong, and let z — ( z i , . . . , 2 g ) C p fee o regular sequence. Set J — (z) and I = J:p.Then I is an equimultiple ideal with reduction number one, more precisely,

I2 = JI,

if one of the following conditions hold

(Li) jRp is not a regular local ring;

(L<2) Rf is a regular local ring of dimension at least 2 and two elements in thesequence z lie in the symbolic square p^ 2 ^ .

Let (R, m) be a Cohen-Macaulay local ring of dimension d and let p be a primeideal of height g > 2. Pick a complete intersection ideal J C p of height g and set/ = J : p. We assume in place the conditions of Theorem 7.4 so that /2 = JI. Inparticular this will occur if Rv is not a regular local ring. Finally set 72 = R[It}.

If J? is a Gorenstein ring and p is a Cohen-Macaulay ideal, then / is a Cohen-Macaulay ideal and the algebra 72 is Cohen-Macaulay. To make the calculation tofollow more interesting we will not assume that p is necessarily Cohen-Macaulay.

THEOREM 7.5 For all integers i,n>0, F^(72)rl = 0.

PROOF: Let 72o = R[Jt] and observe that 772o = 172. Consider the exact sequences

0 _ m0 __ n0 _, G' = R/I[Tlj . . . ,T f f] ̂ o,

and0 -> /720[-1] — > 72 — > R -> 0.

We may take local cohomology with respect to the maximal homogeneous idealof 720, which we still denote by 9JI = (m, Jt). Since 720 and R are Cohen-Macaulay,for i < d we have

Since g > 2, it follows from Hlm(IKQ[-l}) ~ Hl

m(R,} that is Hlm(R.)n = 0 for n > 0

and i < d — 1.Another simple inspection shows that /i^hl(/72[-l])0 = H^(R), from which

the remaining assertions follow. D

286 Money and Vasconcelos

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[17] E. Hyry, Coefficient ideals and the Cohen-Macaulay property of Rees algebras,Proc. Amer. Math. Soc., to appear.

[18] S. Ikeda, On the Gorensteinness of Rees algebras over local rings, Nagoya Math.J. 102 (1986), 135-154.

[19] S. Ikeda and N. V. Trung, When is the Rees algebra Cohen-Macaulay?, Comm.Algebra 17 (1989), 2893-2922.

[20] M. Johnson and B. Ulrich, Serre's condition Rk for associated graded rings,Proc. Amer. Math. Soc. 127 (1999), 2619-2624.

[21] I. Kaplansky, Commutative Rings, University of Chicago Press, Chicago, 1974.

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[24] J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem ofBriangon-Skoda about integral closures of ideals, Michigan Math. J. 28 (1981),97-116.

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288 Morcy and Vasconcelos

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Existence of Euler Vector Fields for Curves withBinomial Ideal

A. NUNEZ, Departamento de Algebra, Geometria y Topologfa, Universidad deValladolid. 47005-Valladolid. Spain.E-mail: anunez@agt. uva. es

M.J. PISABARRO, Departamento de Matematicas. Universidad de Leon. 24071-Leon. Spain.E-mail: [email protected]

Abstract

In this paper, we deal with monomial curves with several components.Here, monomial means " binomial ideal" . We study the monomial parametriza-tions of these curves and present an algorithm that characterizes whether theyare tangent to some Euler vector field, returning this field when it is possible.

1 INTRODUCTION

Given elements hi,...,hn € Z, /ij > 0, we can consider the morphism

given by <&(Xi) = thi . It is known that the ideal / = ker$ defines a closed irre-ducible curve in kn, C. This type of curves are called "monomial curves". Theideal / is generated by binomials X— — X—. Conversely, a prime ideal generatedby a set {X~* — X-*\ 1 < i < s}, with some additional conditions, is the idealof polynomials vanishing on a monomial curve. There is no change if we considercurves given by a morphism <&(Xi) = \ithi , with Aj € k; the only diference is thattheir ideal is generated by binomials of the type X— — cX—, c £ k.

Therefore, we have the closely related concepts of "monomial curve" (monomialparametrization) and "binomial ideals" (binomial equations). It would be interes-ting to extend the concept of monomiality to reduced curves in kn. We propose

290 Nunez and Pisabarro

for it the point of view of the implicit equations, that is, a "monomial curve" willbe, essentially, a curve C such that the ideal of C is generated by binomials. Thebinomial ideals have been studied by Eisenbud and Sturmfels in [E-S], and thisstudy allows us to characterize monomial curves in terms of the parametrizationsof its components.

We are interested in studying which of the known properties of the irreduciblemonomial curves are true for monomial curves with several components. In thispaper, we consider that of the existence of Euler tangent fields. An Euler vectorfield in kn is one of the form Y^i=i az^igfr with a* € TL. It is known that if C isan irreducible curve in kn then it is monomial if and only if there exists an Eulerfield of vectors tangent to it. This field, whenever it exists, is closely related to theparametrization of C.

In Section 1 we give the basic definitions and facts for irreducible monomialcurves. In Section 2, we extend the concept of monomial curve to reduced curvesin kn, and study, by using results of [E-S], some properties of its components. InSection 3, we prove (Corollary 4.3) the main result we need in order to find Eulerfields for C:

Denote by Ci,...,Cd the irreducible components of C, and by hj the vector ofexponents of a monomial parametrization of Cj. There exists an Euler vector fieldtangent to C if and only if there exist a £ Z,n and A j , . . . , \d & k such that a^ = \ihjifor all j £ {1,..., d] and i € {1,..., n} such that Xi $. /(Cj).

Finally, in Section 4 we use this result to give an algorithm that returns us thevector of coefficients, a, for an Euler tangent field for C, if it exists.

2 IRREDUCIBLE MONOMIAL CURVES

Throughout this paper, k will be an algebraically closed field of characteristic zero.Let C be an irreducible curve. C is a monomial curve if for 1 o such that C is the image of the map $ : k —> kn givenby <E>(t) — (\ithl,..., Xnthn). In this case, we will say that C admits the monomialparametrization

( Xl =X1th>

: (1)„, _ \ j-hnXn — Ani

Let C be an irreducible monomial curve with parametrization (1) as above. Inthe following, we will write Zc '•= {i € {!,...,n}| A» 7^ 0} and call it the cellassociated to C. If AJ = 0, we will write hi := 0. Finally, if / is any ideal in thepolynomial ring k[X] = k[Xi,...,Xn], we will write Zj := {i 6 {!,...,n}\ Xi ^ /},and call it the cell associated to / (note that the cell associated to C is the cellassociated to the ideal of polynomials vanishing on C).

An ideal / C k[X] is a binomial ideal if there is a set of binomials, {X—* — CjX-;},with Cj g k, generating /.

Let P be a prime ideal of k[X]. We will say that P is combinatorially finite if itdoes not contain binomials of the type X~— c, m — (mi,..., mn), such that TOJ > 0for all i £ Zp.

Vector Fields for Curves with Binomial Ideal 291

An Euler vector field over k[X] is a fc-derivation of the ring k[X] of the form6 = X^=iai^sY~> with oi,...,an € Z. We will call a := (ai,...,an) the vector ofcoefficients of S.

If C is a curve in kn, I := I(C) the ideal of polynomials vanishing on C and 8a fc-derivation of k[X], S is a tangent vector field of C if 5(7) C 7 (i.e., if 5 inducesa fc-derivation of the coordinate ring of C).

The following result is well known: (see for example [G] for the equivalencebetween b) and c); the equivalence between a) and b) can easily be deduced from[E-S])

THEOREM 2.1 Let C be a irreducible curve of kn with ideal I :- I(C). Thefollowing conditions are equivalent:

a) C is monomial.

b) I is binomial and combinatorially finite.

c) There exists a tangent Euler field of C with non-negative coefficients.

3 REDUCED MONOMIAL CURVES

DEFINITION 3.1 Let I be any ideal ofk[X]. We will say that I is combinatoriallyfinite if all its associated primes are combinatorially finite,

DEFINITION 3.2 Let C be a curve in kn. We will say that C is monomial ifI(C) is binomial and combinatorially finite.

Using the Corollaries 1.7., 2.2. and 2.5. in [E-S] we obtain:

THEOREM 3.3 Let k be an algebraically closed field. If I is a binomial ideal ink[X] then every associated prime of I is binomial.

In [E-S]-Theorem 4.1., we have also the following characterization:

THEOREM 3.4 Let k be any field and X C kn an algebraic set. Then, I ( X ) isbinomial if and only if the following three conditions hold:

Cl For each coordinate cell (k*)z := {(PI, ...,pn) € kn\pi ^ 0 <^> i e Z}, the idealof polynomials vanishing on the set X PI (k*)z is binomial.

C2 The family of sets U — {Z C {1,..., n} \X n (k*)z ^ 0} is closed under takingintersections.

C3 If Zi,Z2 6 U and Zi C Z2 then the coordinate projection (k*)Z2 -» (k*)Zl

maps X n (k*)22 onto a subset of X n (k*)Zl .

Let C be a monomial curve in kn and Ci,...,Cd its irreducible components.Then, by using 3.3 and 2.1, we deduce

THEOREM 3.5 Cj is a monomial curve for 1 < j < d.


In this situation, we will denote by (\j,hj,Zj) the elements of a monomialparametrization of Cj, and define Z* := {i € {!,...,n}/ hji = 0}. The followingtheorem follows in a straightforward way from 3.4:

THEOREM 3.6

a) The set U :— {Zi,..., Zj, Z*,..., Z*d} is dosed under intersections.

b) If Zi C j?2 then there exists a e Z>o such that prZl(h_2) = aprZi(h_1) (whereprZi is the projection onto (k*)Zl).

c) If ZI C Z-2 then prz,(h_2) = pr^h^ = 0.

REMARK 3.7 In fact, from 3.4 we have obtained (see [P]) a complete set of com-putable conditions to characterize when a curve with monomial irreducible compo-nents is monomial in the sense of 3.2, but this result is not necessary for the purposeof this paper.

DEFINITION 3.8 Given p,q e (Z U {0})™, we will say that p and q are pro-portional in the intersection, and denote this by p~q, if there exist a, b € Z,(a, b) ̂ (0, 0), such that api = bqi for all i such that pi, qi € Z.

PROPOSITION 3.9 Keeping notations as in 3.6, for every j,k 6 {!,...,d} wehave hj—hk.

PROOF.- There exists s 6 {1, ...,d} such that Zj n Zk is Zs or Z* (3.6, a)). In thefirst case, there exist Oj , ak e Z>o such that p?zs(!ij) — ajPrzs(^«) and Wzs(h.k) =

akpTZa(hs). We can suppose that aj ^ 0, in the other case Wz,(hj) — 0 and we

have the proportionality. Then for all i € Zs, h^ = ^hji and hj~hk. The caseZ n Zk = Z* is obvious.

4 MONOMIAL CURVES AND EULER VECTOR FIELDS

In order to calculate Euler tangent fields for a curve C, we will take into accountthe following:

PROPOSITION 4.1 Let 5 be a k-derivation of k[X}. Then, 6 is tangent to acurve C if and only if 6 is tangent to every irreducible component of C.

PROOF.- Let Ci, ...,Cr be the irreducible components of C. Write / := I(C) andPi := I(Ci). We have / = nPt, hence it is obvious that if 6(Pi) C Pi Vi then6(1) C /. Conversely, suppose 6(1) C / and take / 6 TV For each j ^ i, chooseQJ € Pj such that g0 $ Pi. Then, as 6(f 0 9j) € I C Pi, we have 6(f) f] 9j € Pi

jjti j^tiand therefore 6(f) € Pi. Q

So, let us make more precise part of the proof of 2.1:

THEOREM 4.2 Let C be an irreducible monomial curve, Z the cell associated toC and h the vector of exponents of a monomial parametrization of C. Then, theEuler vector field with coefficient a is tangent to C if and only if the vectors a andh are proportional in the intersection (that is, in the coordinates of Z).


PROOF.- Let / be the ideal of C. As / is binomial there exist a sublattice Lp C Zz

and a group homomorphism p : Lp —> k* such that I —< Xi i $. Z > + <(X~+ - p(rn)X—- | m € Lp} > ([E-S], Corollary 2.5), where m+ is the positive partof m and m = rn+ — m_. Hence, if i ^ Z then the coefficient a, can take any value,because it is irrelevant in order to check whether 6 is tangent to C or not. It is clearthen that we can suppose Z — {!,...,n] and / =< {X—+ - p(m)X—-\ m £ Lp}.In this case, it is easy to deduce from [E-S], Th 2.1, that Lp = {m £ TUl\ m-h = 0},so rank(Lp) = n — 1.

Given m G Lp, write A := {i mi > 0} and B := {i mi < 0}. Then,r\ r\

and this polynomial belongs to / if and only if ~^ieA aimi = SieB aimi

Th. 2.1). Therefore 6(1) C I if and only if a • m = 0 for all m G Lp. Sincerank(Lp) = n — 1, we have that a and h are proportional. rj

We are interested in comparing, for a curve C C kn, the condition of beingmonomial with the condition of existence of an Euler tangent field for C. We canlimit ourselves to curves with monomial irreducible components (see 3.5), C =GI U ... U Cd- If ht is the vector of exponents of a monomial parametrization of Ci,the above results give us:

COROLLARY 4.3 There exists an Euler vector field tangent to C if and only ifthere exists a G 22™ such that Q—hj for all j G {1,..., d}.

From 3.4 we can see that to check monomiality of C, the coefficients of theparametrizations of the components of C are relevant, while 4.3 shows us thatthe existence of an Euler tangent field depends only on the exponents of theseparametrizations. So, for example, we consider the curve C with components

_ i x = t ^ I x = £l =

I(C) is not binomial, because its reduced Groebner basis with respect to the lex-icographic order with x > y is {—4j/x2 + y2 + 3x4}, that is not a set of binomials(see [E-S] Corollary 1.2). However, the field X-^- + 2Y'-^ is tangent to it.

Nevertheless, 3.6 gives us relations between the exponents of parametrizations,so it is natural to ask if it is possible to find an Euler tangent field for any monomialcurve in kn. The answer is no, but in the next section we propose an algorithm tocheck, for a curve with monomial irreducible components, whether the field existsor not and to construct it in the first case. Although it is possible to treat thegeneral case, for the sake of simplicity we will restrict ourselves to the case in whichall the components go through the origin, that is, hij ^ 0 for every j € {!,...,d},i e Zj.

5 ALGORITHM

Take vectors h^,..., hd G (22>o U {0})™ and define Zj := {i hji G Z>o}.

LEMMA 5.1 I f Z j H Z k = 0 for allj,k e {!,..., d}, j + k, then there exists a e Zn

such that a~h,j for all j e {!,..., d}.

PROOF.- Given i g {1, ...,n}, if there exists j € {1, ...,d} with i e Zj, it is uniqueand then we define a-i := hji\ in the other case, aj can take any integer value. rj

From now on, we will suppose that the vectors are proportional in the intersec-tion, that is, hj—hk for every j, k 6 {1, ..., d}.

LEMMA 5.2 // f|f=i •% ^ 9, then there exists a <E Zn such that a~/^ for all

PROOF.- Choose m e fliLi-^i- For j e {!,..., d}, define Aj := j^-, a beingthe lowest common multiple of {hjm j e {l,...,d}}. For each i G {!,..., n}, ifthere exists j G {!,..., d} such that i belongs to Zj we define a* := Xjhji. Thisconstruction is independent of the choice of j since if j, fc are elements of {1, ..., d}with z e Zj n Zfc , as m £ 2j D 2^ and h^ ~hk we have

hki hkm Xj

and Xjhji — Xkhki- If i ^ Uj=i -^j> S^ve ^° ^i any integer value. It is clear that thevector a constructed satisfies the statement. rj

Now, define AI := {j € {!,..., d}\ 1 6 Zj}, and for k e {2, ...,n}, Afe := {j e{!,..., d}| 1 i Z.j,...,k- 1 i Zj,k e Zj}. Writing Zfc := \Jj£Ak Zj and using 5.2,we can construct for each k & {!,..., n} such that Ak ^ 0 a vector ft such thatft2fe = 0 if and only if i <£ Zk and h^^hj for all j & Ak.

LEMMA 5.3 // there exists a e Z™ swc/i t/iai a~ftj- /or a// j e {!,..., d} iftenft f c l~ft f c 2 /or a// & i , &

PROOF.- Choose any k such that Ak ^ 0. Given i,m € -Zfe, there exist j, / 6such that i & Zj and m £ Zi. We have hj~hk, ft;~ft fc, ftj— a, ft;— a and ̂ CZ; C Zk,k£ Zk. Then,

/izfc = ftjj = QJ and ft^ = hlm = am

ftjt ^jfc Ofc ftfc ftife flfc

so

D

ALGORITHM 5.4Input: F = {h_i, ..., ft^}Output: F = 9 or F= {a}, a€Zn

Step 1: Check if there exist j,k & {!,..., d} such that hj^hk. If so, return

Step 2: Construct, for j £ {!,... ,d}, Zj := {i £ {!,..., n}\ hji ^ 0}.Step 3: Check if for all j, k £ {1,..., d}, j ^ k, Zj n Z^ = 0. If so, construct a

as in 5.1 and return F :— {a}.Step 4: Choose j,k £ {!,...,d}, j ^ k, Zj n Zk ^ 0. Realign coordinates in

such a way that 1 6 ZjC\Zk, construct, for k 6 {!,..., n}, Ak and, if Ak ^ 0, h byusing 5.2, as explained before, set F := {hk Ak ^ 0} and apply the algorithm to F.

THEOREM 5.5 Let C C kn be a curve with monomial irreducible componentsand let F = {/i1;..., hd} be the exponents associated to parametrizations of its com-ponents. Apply the previous algorithm to F. Then, if the output is 0, there is noEuler tangent field for C; if the output is {a}, then the Euler field of coefficient ais tangent to C.

PROOF.- Observe that the set {!,..., d} is the disjoint union of the sets Ak, k e{!,...,n}. Hence, if we execute step 4, AI contains at least two elements, so thenew F has less than d elements. Moreover, if a is such that a~/z.fc for k € {1,...,n},Ak =/= 0, from 5.3 we know that also a~/iy for every j £ Ak- This, together with 5.1and 5.2 proves the theorem. Q

For example, consider the curve C = C\ U ... U CV, where

Xi = t

x2 = 0x3 = t C2 = <X4 =t

. x5 = 0

xl = t2x2=Qx3 = 0 C3 = <x4 =0X5 = t

xi =0X2 = t

X3 = t

X4 = 0

X5 = t

1,0,1,1,0) ft2 = (2)0,0,0,l) A3 = (0,1, 1,0,1){1,3,4} 25 = {1,5} Z3 = {2,3,5}

xi —0x2 = Qx3 = 0 C7= <x4 =0X5 = t

Xi = 0X2 = t

^63 —— ̂

x4 =0x5 =0

= t

x4 = 0

= (1,0,0,0,0) As = (0,0,1,0,0) he = (0,0,0,0,1)2, = {1} Z5 - {3} Z6 = {5}

This is a monomial curve, the ideal of C is generated by the binomialsXiX2, XiXs - X%, XiXi — X%, -X3X2 + X$, X2X4, X5X2 — X5X3,+ X^X3, ^4X3 — X%, X^X^}. Using Algorithm 5.4, we construct in step

4 the vectors hl = (2 ,0 ,2 ,2 ,1) and h2 = (0,1,1,0,1), hence the curve C has notangent Euler field, as h1 ^ h2.

REFERENCES

[E-S] D.ElSENBUD & B.STURMFELS, Binomial Ideals, Duke Math. J., Vol. 84, No.1, July 1996.


[G] G.-M. GREUEL, On deformation of curves and a formula of Deligne, In:Algebraic Geometry, La Rdbida 1981 (Eds.: Aroca, J.M. / Buchweitz, R.-O./ Giusti, M. / Merle, M.). Lecture Notes in Math. Vol. 961, pp. 141-168,Springer 1983.

[P] M.J. PlSABARRO, Curvas monorniales con varias componentes, Tesis (inpreparation).


An Amitsur Cohomology Exact Sequence for Involutive Brauer Groups of the Second Kind l

A. SMET and A. VERSCHOREN, Department of Mathematics and ComputerScience, University of Antwerp, RUGA. 2020-Antwerp, Belgium.E-mail: [email protected], [email protected]. be

AbstractIn this note, we construct a version of the Chase-Harrison-Rosenberg se-

quence, linking Amitsur cohomology groups to involutive invariants of thesecond kind.

1 INTRODUCTION

In [16], Saltman proved that any class in ker(Cores : Br(S) —» Br(R)), for any etalequadratic extension S of R, can be represented by an Azumaya algebra endowedwith an involution of the second kind. This fact inspired Parimala and Srinivas in[12] to define a new invariant Br(X, 8) for a scheme X where Y —> X is an etalecover of degree 2 with a non-trivial automorphism 6.

This group is based on sheaves of Azumaya algebras equipped with an involutionof the second kind. In [15] a more explicit construction of this group is given forKrull domains using Orzech's notion of suitable categories, [9].

Some exact sequences where constructed, involving a variant of the Picard group.In [14] a version of the Chase-Harrison-Rosenberg sequence is constructed for invari-ants of the first kind of a Krull domain, starting from the classical Chase-Harrison-Rosenberg sequence as in [2, 4] or the relative version as in [3].

In this paper, we aim (a) to present a brief overview of the construction ofthe invariants of the second kind associated to a divisorially etale extension ofKrull domains; (b) to write out a version of the Chase-Harisson-Rosenberg sequenceinvolving these invariants.

1 Research supported by an F.W.O. research grant.


298 Smet and Verschoren

In the first section we list some generalities on Krull domains and divisoriallattices which will be used throughout this note. We also recall the definitions ofthe Picard and the Brauer group in the framework of suitable categories.

In the following section we recall the definitions of divisorially separable exten-sions and divisorially etale extensions. We define the category of algebras withinvolutions of the second kind, and, in particular, we characterize involutions of thesecond kind on trivial Azumaya algebras, i.e. endomorphism rings of divisorial lat-tices. Finally we construct the involutive Brauer group of the second kind and theinvolutive Picard group of the second kind, both associated to a suitable categoryC C T>(R). Proofs of the results in this section can be found in [15].

In the last section we derive the exact sequence which links these invariants ofthe second kind to Amitsur cohomology groups.

2 GENERALITIES

Throughout this text, R denotes an arbitrary (not necessarily noetherian) Krulldomain with field of fractions K and X(R) C Spec(R) the set of its height oneprime ideals. We then know that Rv is a discrete valuation ring for all p G X(R)and that R = -^p within the field K.

We call an .R-module M divisorial if it is torsionfree, and if the canonical inclu-sion M <^-» dfi(M) is an isomorphism, with

dR(M)= p| Mp

where the intersection is taken within K (&# M. We refer to [8, 9, 19] for furtherdetails on the functor dp..

The modified tensor product of torsionfree /?-modules M and TV is defined to beM®RN = dR(MN), where MN is the image of M®RN in K®R(M®RN). Thisoperation shares many properties with the ordinary tensor product, we again referto [8, 9] for details.

An R- module M is said to be an R- lattice, if it is torsionfree and if we may finda finitely generated R- module N with the property that M C N C K®R M. If R isnoetherian, an R-lattice is thus just a finitely generated torsionfree .R-module. Wedenote by T>(R) the full subcategory of .R— mod consisting of divisorial R-lattices.

Divisorial R-lattices behave in many respects as finitely generated projectiveR-modules, if one replaces the ordinary tensor product by the modified tensorproduct. For example, if E and F are divisorial ^-lattices, then so are E ®R Fand Hom,R(E,F). In particular, for any E e T>(R), we have E* e T)(R) as well,and there is a canonical isomorphism E ®R E* = EndR(E). Let us also point outthat Endfi(E) (&R EndR(F) = EndR(E ®R F), for any pair of divisorial R-latticesE and F. This, and many other properties, may be proved by a straightforwardlocal-global argument, i.e., using the fact that an R-linear map u : E — > F betweendivisorial R-lattices is an isomorphism if and only if for any p e X(R) the inducedmap up : Ep — > Fp is an isomorphism of /2,,-modules. We refer to [9, 19] for detailsand further properties.

As in [9], we define a full subcategory C of T>(R) to be suitable, if it possessesthe following properties:

Exact Sequence for Involutive Brauer Groups 299

1. R&C;

2. if E, F e C, then E §>R F e C and HomR(E, F) 6 C;

3. if E §>R F 6 C for some E e C and F <E £>(#), then ^ e C.

Typical examples include C — ~D(R) and C = P(R), the category of finitelygenerated projective /Z-modules.

A divisorial J?-lattice L is said to be divisorially invertible, if it has rank one,i.e., if K (&R L = K, or equivalently if L ®R L* = -R. For any suitable subcategoryC C T>(R), the group Pic(C) consists of isomorphism classes {I/} of divisoriallyinvertible .R-modules L e C, the group law being induced by the modified tensorproduct. Obviously, {R} is the identity element of Pic(C) and {L}^1 = {L*} forany divisorially invertible fl-module L & C. Typical examples of this constructioninclude Pic(D(R)) = Cl(R), the (divisor) class group of R, cf. [5, 9, 11, 19], andPic(P(R}) = Pic(R), the Picard group of R, cf. [1].

An /^-algebra A is said to be a divisorial Azumaya algebra, if A € 'D(R) and ifthe canonical map

A ®R Aopp -> EndR(A) :a®b^(x^> axb)

is an isomorphism. It is easily verified that this is equivalent to A being a divisorial/^-lattice and Av an Azumaya algebra over Rv for every p 6 J^(/?). From this onededuces that R — Z(A), the center of A, and that EndR(E) is a divisorial Azumayaalgebra for every E G T)(R).

We denote by Az(C) the set of (isomorphism classes of) divisorial Azumayaalgebras, which belong to the suitable subcategory C C T>(R). One defines anequivalence relation ~ on Az(C) by putting A ~ -B, whenever we may find E,F £ Ctogether with an isomorphism

A ®R EndR(E) ^B®R EndR(F).

The set Br(C} of equivalence classes [A] of A e Az(C) may be endowed with a groupstructure, the multiplication being induced by the modified tensor product. It iseasy to verify that the identity element in Br(C) is [Endn(E)} for any E € C andthat [A]~l = [Aopp] for any divisorial Azumaya algebra A. Typical examples includeBr(P(R)) = Br(R), the Brauer group of R, cf. [4, 6, 10, et al], and Br(D(R)) =/3(R), the divisorial Brauer group of R, cf. [9, 11, 19].

3 INVOLUTIVE INVARIANTS OF THE SECOND KIND

Assume the inclusion of Krull domains R c—> 5 to be a Krull morphism, i.e., assume5 to be a divisorial .R-module. The multiplication map

m : S ®R S -» 5 : s ® s' >-» ss'

determines a map /j, : S(&RS —> S. We call 51 divisorially separable over /?, if /z makes5 into a projective 5 ®R 5-module or, equivalently, if there exists an idempotente in 5 §>R S for which p ( e ) — 1 and e(s §> 1 - I §> s) = 0 , for all s 6 5. In theterminology of [3], this notion is just what should be called real a\-separability,where o-i =

An inclusion R °-> S of Krull domains, with 5 divisorial over R, is said to bedivisorially etale2 if S is divisorially separable over R and S 6 TJ>(R).

From now on, we fix an inclusion R °-> 5 of Krull domains, such that S is adivisorially etale quadratic extension of R.

Let a be the canonical /^-automorphism of S and let A be an iS-algebra. Wedenote for any 5-module P by aP the 5-module with underlying additive group Pand with 5-action given by s.p = o~(s)p, for any s € S and p € P. An involution ofthe second kind on A is an anti-automorphism a on A with a2 = id A and a|s = cr.So a is actually an isomorphism A —> CT/lopp.

The modified tensor product (v4,a) <8> (B,(3) of algebras with involution of thesecond kind (A, a), (B,f3) is given by (A <§ B,a §> (3), where (a §> /3)(a § b) =a (a) <S> /3(6). A morphism (A, a) —> (B, /?) of algebras with involution of the secondkind is an algebra morphism u : A —> B with (3 ou = uo a.

For any S'-module P, we denote by Pv the module a(P*). Clearly, f f(P*) S*(CTP)* through the isomorphism 0, defined by d(f)(p) — o~(f(p)) for any / € <r(P*)and p e g.P. Any S-module homomorphism / : P —* Q induces /v : Qv —> Pv. Wehave also that (P § Q)v £^ Pv § Qv for P, Q e £>(S) and that P ^ (Pv)v if P is areflexive S'-module.

If M, AT, L are S-modules and L is divisorially invertible then any homomorphismM ® L —» ./V induces a homomorphism M —* TV (gi Lv, after identifying L (g> Lv with5 through the map / ® / H-> a ( f ( l ) } . In the lemma below, we shall use the samenotation for these two maps.

Just as in [7], we now have:

LEMMA 3.1 Let P be a divisorial S-lattice and let a be an involution of the secondkind on Ends(P). Then there exists a divisorially invertible S-module L, togetherwith a descent map 8 : L —> L over R and an isomorphism h : Pv <8> L —> P suchthat hv = h(l ® 8) and a(u) = h(uy §, l)h~l for u e Ends(P}. The pair (L,6) isunique up to isomorphism and h is unique up to a unit of S, once L is fixed. Theinvolution a is completely determined by the couple (L,6) and the map h.

Since L is a reflexive and divisorially invertible S'-module, we have an isomor-phism

where the last isomorphism is obtained through Morita theory. This isomorphism9 is defined by 6(f § l)(p) = a(f(p))l for alH e L, f e Pv and p e aP. Hence wehave an isomorphism

Ha : P % Pv <8 i A Hom(aP, L) S <rHom(P, aL) -> aHom(P, L),

which defines a nonsingular bilinear map

H:Px,P^L: (p,p') -» Ha(p')(p).2One should actually call these finite divisorially etale extensions, but since no others will be

considered in this text, no ambiguity can arise.

Conversely, any nonsingular bilinear map H : P x aP —> L defines a unique mor-phism h : L <g> Pv —> P. Of course, instead of working with bilinear maps, wemay also view H as a sesquilinear map H : P x P —> L. We then have thatH(sp,s'p') = sa(s')H(p,p') for all p,p' € P,s,s' € S. We may then prove thefollowing:

PROPOSITION 3.2 The sesquilinear morphism H : P x P —* L is e-Hermitian,i.e., H(p,p') = eo(H(p' , p ) ) for allp,p' € P and e 6 S with ecr(e) = 1.

We denote by Si the set of all s € 5 with sa(s) = 1. The above construction alsoadmits a converse:

PROPOSITION 3.3 Let P,L e V(S) with L divisorially invertible and endowedwith a descent map 6, then any nonsingular e-Hermitian morphism H : P x P —> Lwith e £ Si canonically determines an involution a of the second kind on Ends(P}.

We will usually denote a constructed in this way by a//.From the foregoing results, we may conclude that every involution of the second

kind on a trivial Azumaya algebra over 5 is determined by an essentially uniquecouple (L, 6), where L is a divisorially invertible 5-module with descent map 6, anda nonsingular e-Hermitian morphism H with values in L, for some e € Si. We willcall e the type of H.

As in [13] we can now define a variant of the Brauer group associated to Azu-maya algebras with an involution of the second kind. Let C C £>(S) be a suitablesubcategory. We denote by C** the set of couples (E,H), with E e C and H anonsingular e-Hermitian form for some e 6 Si, i.e., a nonsingular sesquilinear mor-phism H : E x E —> S such that H(e,e') = ea(H(e',e)) for every e,e' & E and forsome e e Si. The category C** has a product given by

(E, H) § (E1, H') = (E®E',H® H'},

where (H ® H')(e ® e1, f ® /') = H(e, f)H'(e', /'). It is easily verified that thisproduct is well-defined. If the type of H is s and the type of H' is e', then the typeof H®H' is eg-'. A morphism (E, H) —> (E', H') is an S-linear map / : E —> E' suchthat H(e,e') = H ' ( f ( e ) , f ( e ' ) ) . Denote by Az**(C) the set of isomorphism classesof divisorial Azumaya algebras in C, endowed with an involution of the second kind.In particular, any (E, H) € C** yields a divisorial Azumaya algebra with involutionof the second kind (Ends(E},aH) 6 Az**(C).

We call two divisorial Azumaya algebras (A, a) and (B,/3) with involution ofthe second kind similar, if there exist (E, H) and (F, K) 6 C** such that

(A §> Ends(E), a®aH)^(B® Ends(F), /3 g aK).

The set of equivalence classes [A, a] for this relation is denoted by Br**(C] and iscalled the involutive Brauer group of the second kind of C. If C = 'D(S) then wedenote it by /3**(S/R) and if C = P(S) by Br**(S/R). The multiplication in thisgroup is induced by the modified tensorproduct, the identity element being the class[Ends(E),aH] for any (E, H) e C**. To prove that Br**(C] is actually a group,we need to show any class [^4, a] to have an inverse. We can prove that for any(A, a) 6 Az**(C), we have

[A,a\.[aA,a] = 1.

We will define below an associated group Pic**(C), based on isomorphism classesof couples (L, H) € C** , where L is divisorially invertible. In order to do so, let ustake a brief look at Hermitian forms on divisorially invertible 5-lattices.

First, note that any nonsingular sesquilinear form on S is of the form

Hu : S x S — > S : (x,y) i— > xua(y),

for some u £ S* . Indeed, if H is a sesquilinear form on S, then we have

H(x,y)=xH(l,l)<7(y)

for all x, y € S, so H = Hu with u = H(l, 1). Moreover, for H = Hu to be nonsin-gular, we obviously need u to be invertible. It is now clear that Hu is e-Hermitian, ifwe put £ = ua(u)~1 € 5j. In particular, this implies that all nonsingular sesquilin-ear forms on 5 are e-Hermitian, for some e g S\. In the general case, i.e., for anarbitrary divisorially invertible L € C, the same result holds. This may easily beverified through a straightforward local-global argument, as L is isomorphic to S ateach height one prime of 5.

Let us now consider the set Pic**(C] of isomorphism classes {L,H} of couples(L,H) € C**, where L is divisorially invertible. The multiplication in Pic**(C)is induced by the modified tensor product and we will show that Pic**(C) is agroup, called the involutive Picard group of the second kind of C. If C = P(S) resp.C = T>(S), we will denote this group by Pic**(S/R) resp. Cl**(S/R).

First, let us note that

On the other hand, the inverse of {L,H} & Pic**(C) is the class {aL,aH}. Indeed,H induces an isomorphism aL — > L* , hence an isomorphism

This is a straightforward consequence of the fact that

H!((H x aH)(l ® m, I' g TO')) = <*(e)H(l, m)H(m', I

and

' '

(H ® aH}(l ®m,l'® TO') = cr(e)H(l, l')H(m', TO)

and the fact that both are locally equal to lcr(l')a(u)ucr(m)m'. The previous dis-cussion and the basic properties of the modified tensor product thus show thatPic**(C) is an abelian group, indeed.

4 AMITSUR COHOMOLOGY

Suppose R <-* S is an inclusion of Krull domains then for any covariant functorF from the category of commutative /^-algebras to the category of abelian groups,the groups H'l(S/R,F) are the Amitsur cohomology groups Ker Si/Im (5j_i of thecomplex


where Sn = Y^=i(-l)i+lF(£i} for each positive integer n and e*g®(n+2) mserts a i at the i-th location.

Assume R,R,S to be Krull domains and S to be a divisorially etale quadraticextension of R and let a be the canonical /^-automorphism of S. We define thefunctor /j,ai from the category of torsionfree S-algebras to the category of abeliangroups, by putting

for any commutative torsionfree S-algebra T and with a' the canonical extensionof a to T.

We cannot directly use Cl** in the theorem below, as Cl** is only defined onKrull domains (and Krull morphisms). In order to remedy this, we introduce avariant Cl*s* , which to any S-algebra T € T>(S) associates the group Cl*s*(T) ofisomorphism classes {E, H} such that E is a T-module belonging to 'D(S) and His an isomorphism

the group law being induced by the modified tensor product (as T-modules butwith the intersection taken over primes in 5). It is clear that C7g*(S) = Cl**(S)and that Cl*s*(T) = Cl**(T) when the S-algebra T belongs to T>(S) and T is aKrull domain. Any morphism T — > T' between S-algebras T, T' € T>(S) induces agroup homomorphism Cl*s*(T) — > Cl*s*(T'), defined by sending the class {E,H} to{T' ®T E, HI §T H}. For any S-algebra T e £>(S) we define /3J*(T) in the sameway. We may now prove:

THEOREM 4.1 Assume R,R',S to be Krull domains and S to be a divisoriallyetale quadratic extension of R, where a is the canonical R- automorphism of S.Assume further that R' is divisorially separable and faithfully flat over R and letS' = S ®/j R' . Then there is an exact sequence

, ) -^(3**(S'/S) -H^S'/S, Cl*

')

The Hl(S'/S,F) are the Amitsur cohomology groups and

where the map (3**(S) — * /?£*(S') is defined by sending the class [A, a] to [A ®sS',a®sS'}.

PROOF: In this proof tensorproducts are taken over S, unless otherwise mentioned.Since R'/R is divisorially separable, tensoring over R with S yields that S'/S is alsodivisorially separable. One can also deduce that S' is a divisorially etale quadraticextension of R' with canonical automorphism a' = a ® id. Since R'/R is faithfully


flat we obtain by tensoring with 5q over R that S'^/S^ is faithfully flat for anyq £ X(S). So, we can apply divisorial descent theory, see for example [17, 18], tothe extension 5 <-* 5'.

We shall give an explicit description of the connecting homomorphisms, leavingthe other details to the reader.

(1) Construction of a^. We have to show that H°(S'/S,/v) = Hcr(S), but thisfollows immediately from descent theory.

(2) Construction of a\. Let t be a representative of the class [t] € H1(S'/S, p.a')-Multiplication by t defines a descent datum v : S' 0 S' —> S' (g> S'. By descent thereexists a divisorial 5-module / and a map 77 : / <8> 5' —> S', where

The couple (/, 77) is unique with respect to the commutativity of

. S'&nS I £: T ̂ : Qf \ Qt £: £</

09 1 Q9 ^ J 09 ^

T12

S" ® 5' 5' i 5'

Of course, / e 2? (5) and / is a divisorially invertible S-module, as / <8> S" isa divisorially invertible (over 5) S"-module. Multiplication by er'(t) also defines adescent datum which yields, by unicity of descent, the divisorial S'-module al anda map rj* : al <8> S' — > S' where 77* = 77 o (a §> 1) such that the following diagram iscommutative:

S' § al § 5'

5' ® S1

5' g 5'

' § 5'

It may easily be verified that the following diagram is commutative:

S'®I®aI®S' ——> S' g /g S" ——> 5" g 5'T12 Ti2 -t

4- 4 - 4 -

T23 /(gl-a-'(t) -cr'(t)

/ g „/ g 5' g 5"7T- /̂ g 5' g 5' —r^> S" g 5'

with u = S' ® I ® 77*. Putting L = I ® „!, the outer diagram reduces to:

S" g L g 5" y > 51' 0 g'

"I 1"-4^ -^-

L g 5' g 5' —--> 5' g 5'

where TJ* <S> 5'). Using the map

6 : L § 5' = / §> al ® S' -^ / § 5' —^ 5'

we find that (f> = S' ® 6 and -0 = 9 ® S'. By unicity arguments, it follows thatthe couples ( L , 6 ) and (S, /j,) are isomorphic, where /z : S <g> 5' —* S" is defined bysending s <g> s' to ss'. So there exists an essentially unique isomorphism

H :I®J -> 5

such that the following diagram is commutative:

i®«i®s' —9-^ s1

We define ai([£]) = {/, H} G Cl**(S). It is easy to see that a\ is a homomor-phism of groups and {I,H} depends only upon the class [t] of t in H1(S'/S,/v)-The first fact is proved by using a similar argument as in the construction of theisomorphism H. If c*i([f]) = {I,H} and QH([<']) = {/',#'}, then we obtain thefollowing commutative diagram:

5' § / § /' § 5' -^> 5" g 5'

i®r®s'®s' ——> s1 ® s'Q^)S

where/3 7" ̂ Tf <-> O/ ®^ T ̂ Of ^ Ofv : 1 <g> j (gi 6 —> Y <g) 6 —> o

as before. So it easily follows that a i ( [ t t ' } ) = {I (& I',H ® H'}. For the secondassertion, we have to verify that if [t] = [1], then (/, H) == (S, HI). If £ = s (g> cr'(s),for some s G fj,a>(S'), and if we let

7 7 : 5 ( 8 ) 5 ' — > S ' ' : r ( 8 ) a ; i — > srx,

then we have the following commutative diagram

^ ^ S'lgm ^5' <g> 5 ig. 5' ———> S' <g) 5'

T12

By unicity arguments we find that 7 = 5 and it follows that (/, H) = (5, HI).(3) Construction of f a . For any {/, H} & Cl**(S), define

This map is obviously a well-defined group homomorphism.(4) Construction 0/71. Any [{/,#}] G H°(S'/S,Cl*s*) yields an S' ® S'- iso-

morphism <f : S' (g> / — > 7 <g> 5'. The map / = y^'Vsyi is an S - automorphismof 5' ig> 5' <g> /, so it is multiplication by a unit i G 5* , because / is a divisori-ally invertible S"-module (with the modified tensorproduct taken over 5). Putting7i([{/, H}]) = [t], this yields a well-defined group homomorphism 71. So, we findthat 1 = 71({/, #}{„,/,#}) = ta'(t), hence t G ̂ (S1®3).

(5) Construction 0/0:2-Let P be the 5"-module S' <g> S' , where the action of 5' is defined by s(x <g> y ) =

sx (gi y and let t £ S' <g> 5' <8> 5' represent an element of H2(S'/S, /v)- Defineft : Pt = S' ® P — » P2 = P § 5' by /t(y) = T2s(ty), then /t induces a map

y>t : 5' ® Ends,(P) -» Ends>(P) g 5'

by sending u G Ends,^s,(S' ig) P) to ftuf^1 . Because </pt is a descent datum, thereexists a divisorial Azumaya algebra .A(t) endowed with an isomorphism A(t)®S' ^>Ends,(P}.

It remains to construct an involution on A(t). We have a nonsingular Hermitianform

5' x 5' -> 5 : ( x , y ) H-> Trs,/s(x<T'(y)).

We thus obtain an isomorphism

from which we deduce maps which send u £ End(S') to M' G End(S' ) and u €End(S') to w € End(S'vv] where u' = /j,ufj,"1 and u(f ) = tou',\/t € 5/vv. Becauseof the isomorphism

ev : S' ^ S' : s i— > eus,

where eus is the map defined by e?Js(</?) = cr'(</?(s)), for any s € 5', we also obtaina map

7 : End(S') — > End(S') : u i— > ezJ"1 o u o eU.

Using the fact that u(£) = t/iu/x"1 with t £ S' , we find that w(e?Js)(/z(x)) =cr /(Tr(u(x)(7 /(s))), for all x,s € S' , But we have also that u(evs) = ev^(u)(s)whence u(evs)(/j,(x)) = a'(Tr((xa'(^(u)(s)))) for all x, s € S' . From these twofacts, it follows for all x, s G S' and all u G End(S'}, that

hence

- ' (7(uou)(s))) = Tr(u(v(x})o-'(s)) =

for all x,s G S". So, 7(11 o v) = j(v) o -y(u), in the same way we can prove thatl(tu) = o-(t)j(u) for t G 5, u G End(S'), which means that 7 defines an involution ofthe second type on Ends(S'). From this we can also deduce an involution J3 = 5"<8>7

on Ends'(P)- This involution restricts to an involution on A(t), if we can show thefollowing diagram to be commutative:

0 ———> A(t) ———> Ends'(P) -^ Ends,(P) ® S'

ft

0 ———> A(t) ———> Ends'(P) —~t Ends,(P) § 5'

The upper diagram is obviously commutative, so we have to focus on the lowerdiagram. This problem reduces to the commutativity of

5' § 5' § End(S') -̂ -> S' § End(S') § 5'I I

S'<g>S'®-y Sl'®7i8)S'4~ ~r

S' § 5' § End(S') -̂ -> 5' § End(S') <§ S"

where we may obviously assume

hence

with s,s' £ S',u & End(S') and some Uij : y i—» Cju(a'(ci)y).So, if we want the diagram to be commutative, it suffices to prove that 7(1*^) =

7(u),-j. Using the trace map we find that

We now put a2([t}) — [A(t),/?j. The element [yl(i),/3] is independent of therepresentation t of [i] and we thus obtain a well-defined group homomorphism.

(6) Construction of {3%. Let [A, a] 6 j3**(S'/S), then we have an isomorphism

B : (A®S',a®S')^(EndS'(Q),(3),

where Q e ~D(S) is an S'-module and j3 = an with H : Q x Q — > S' a nonsingulare-Hermitian form for some £ € S^. The map 6 induces an isomorphism

with /?i = S" <g> /3 and /?2 = /3 <8) 5'. Then 93 is induced by some isomorphism/ : (S" § (?) §S'®S' 7 ̂ ̂ ® 5/- Define

s/(Qv ® 5')

by the commutativity of the following diagram:

Ends,$s,(S' g g) —P—+ Ends^s,(Q g 5')I

toft

where t : / t—> /v is the map induced by the transposition map. It is clear that forany u G Ends,^s,(S' ® g), we have (t o <f)(u) = (/v)~1(u <S> l)/v, where u (g) 1 isthe image of u in Ends,^s,(S' ® Q} ® Ends,^s,(I}. It thus follows that if) mapsuv to (/V)" I(M ® l)/v, from which one easily deduces that if> is induced by theisomorphism

n = (/v)~1 • (5' g gv) g , - , 7V —* gv g 5*'One obtains an induced commutative diagram

5,(ggs')

The vertical map

7 : Ends,$s,((S' g g) ®s,&s, /)

is induced by k = g~l(Ha g 5')/, where Ha : Qfits into the commutative diagram

Qv is denned by #, and thus

•ggs"

Since 7 is essentially induced by the isomorphism H, it easily follows that K =(H <8> 5') (g) fj,, for some isomorphism of 5" <g> 5'-modules /z : / —> Jv. Denoteby L : / x / —» 5' ® 5' the associated nonsingular e-Hermitian form, then wemay thus define /?2 by putting ^([A, a]) = {/,!/}. It remains to show that {/, L}represents an element of H1(S'/S, C7g*), but this trivially follows from the fact that^2 = f s f i i which implies 7j ® J3 = 72. One easily verifies that J3^ is a well-definedgroup homomorphism.

(7) Construction of j2- Let {/, #} € H1(S'/S, Cl*s*), then we have an isomor-phism / : /i ® /s —> /2- It follows that

Auts/g,4(7;12 < /3 4 ,

so it is multiplication by a unit t 6 5®4. Putting ii({I,H}} = [i], we obtain awell-defined group homomorphism 72. Because 1 = 72({/,-H"}{cr'/, 77}) = tcr'(t), itfollows that £e/ iC T / (S '® 4) .


REFERENCES

[1] Bass, H., Algebraic K-Theory, W.A. Benjamin, Inc., New York, 1968.

[2] Chase, S.U. and Rosenberg, A., Amitsur cohomology and the Brauer group,Mem. Amer. Math. Soc. 52 (1965) 20-65.

[3] Caenepeel, S. and Verschoren, A., A relative version of the Chase-Harrison-Rosenberg exact sequence, J. Pure Appl. Algebra 41 (1986) 149-168.

[4] DeMeyer, F. and Ingraham E., Separable algebras over commutative rings,Lecture Notes in Math. 181, Springer Verlag, Berlin, 1971.

[5] Possum, R.M., The divisor class group of a Krull domain, Ergebnisse der Math-ematik 74, Springer Verlag, New York, 1973.

[6] Knus, M.A. and Ojanguren, M., Theorie de la Descente et Algebres d'Azumaya,Lecture Notes in Math. 389, Springer Verlag, Berlin, 1974.

[7] Knus, M.A., Parimala, R. and Srinivas, V., Azumaya algebras with involutions,J. Algebra 130 (1990) 65-82.

[8] Lee, H. and Orzech, M., Brauer groups, class groups and maximal orders for aKrull scheme, Canad. J. Math. 34 (1982) 996-1010.

[9] Orzech, M., Brauer groups and class groups for a Krull domain, in: BrauerGroups in Ring Theory and Algebraic Geometry, Lecture Notes in Math. 917,Springer Verlag, Berlin, 1981, 66-90.

[10] Orzech, M. and Small, C., The Brauer Group of a Commutative Ring, M.Dekker, New York, 1975.

[11] Orzech, M. and Verschoren, A., Some remarks on Brauer groups of Krull do-mains, in: Brauer groups in Ring Theory and Algebraic Geometry, LectureNotes in Math. 917, Springer Verlag, Berlin, 1981, 91-94.

[12] Parimala, R, and Srinivas, V., Analogues of the Brauer group for algebras withinvolution, Duke Math. J. 66 (1992) 207-237.

[13] Reyes, M.V. and Verschoren, A., Involutive Brauer groups of a Krull domain,Comm. Algebra 23 (1995) 471-479.

[14] Reyes Sanchez, M.V., Smet, A. and Verschoren A., Involutive invariants of aKrull domain and Amitsur cohomology, in: Rings, Hopf algebras and Brauergroups, Lecture Notes in Pure and Appl. Math. 197 (1998) 239-256.

[15] Reyes, M.V., Smet, A. and Verschoren A., Involutive invariants of the secondkind, Comm. Algebra 27 (1999) 6069-6102.

[16] Saltman, D., Azumaya algebras with involution, J. Algebra 52 (1978) 526-539.

[17] Smet, A. and Verschoren, A., The strong (PDE) condition, Quaestiones Math,to appear.



[18] Smet, A., Involutive invariants of Krull domains, Ph. D. thesis, in preparation.

[19] Van Oystaeyen, F. and Verschoren, A., Relative Invariants of Rings: The com-mutative Theory, Dekker, New York, 1984.


Computation of the Slopes of a D-Module of TypeVr/N

J. M. UCHA-ENRIQUEZ1, Dpto. Algebra. Universidad de Sevilla. Apdo. 1160,E-41080 Sevilla. Spain.E-mail: ucha@ algebra, us.es

Abstract

We provide in this paper a natural generalization of the algorithms of [1]as well as a collection of examples for modules of type T)r/N, where N is asubmodule of T>r. This family of examples is a first step in the study of thebehaviour of the slopes under elementary operations over D-modules (directsum, syzygies,...).

1 INTRODUCTION

Let V be the sheaf of linear differential operators over C™ with holomorphic coeffi-cients.

The purpose of this work is to present an account of the explicit methods withGrobner bases that can be managed to obtain classical invariants of a coherent(left) P-module M. These invariants are called the slopes of M. along a smoothhypersurface of the base space, and they have to do with the irregularity of Ai. Theywere introduced by Laurent in [8] (see the work of Mebkhout [13] for the notion oftranscendental slope). Laurent and Mebkhout proved in [9] that the transcendentalslopes and the algebraic slopes are the same. The analogue in dimension one is thepaper of Malgrange [10] for the perversity the irregularity sheaf. For an introductionto the theory of P-modules see [2], [7] and [12]. For a more effective point of view,see [4] and [16].

In [1] the main theorems are developed to make the slopes computable in the caseof modules of type £>//, where / is a (left) ideal. They prove in a constructive waythe finiteness of the number of slopes, and use a technique of homogenization (with

1 Partially supported by DGESIC PB97-0723.


312 Ucha-Enriquez

respect to the considered filtration) to solve the problem of the infinite processesthat could appear in the calculations.

We provide in this paper a natural generalization of the algorithms of [1] and acollection of examples for modules of type VT/N, where N is a submodule of T>r.This family of examples is a first step in the study of the behaviour of the slopesunder elementary operations over P-modules (direct sum, syzygies,...). We havechosen the simpler Rees Algebra homogenization, as it appears first in [5] (the caseof submodules is explicitly done in [17]).

2 DEFINITIONS

Let A and T> be the rings of differential operators respectively over G[XI, ...,xn] (theWeyl Algebra) and C{TI, ..., xn}. In dimension one, given a differential operatorP(x,dx) = YT=oai(x)dx e C{x}[dx] = V (with am ^ 0), we can consider thelinear map

P : 0/0 — + 0/0,that sends u G O/O to P(u). The following results are very well known:

THEOREM 2.1 P(u) = 0 has a regular-singular point in 0 if and only if

Ker(P) = 0.

THEOREM 2.2 P(u) = 0 has a regular- singular point in 0 if and only if

max{j — val(cij)} = m — val(am).

Therefore, the combinatorial object 'P(P)m

j=0

detects that the equation has a non convergent solution.

EXAMPLE 2.3 For P = x2dx + I the equation P(u) = 0 has a solution u

In any dimension, given a linear form L : Q2 — > Q with coefficients p, q rela-tively prime, you can define the L-order2 of an operator P = J^V mgN2n aa/3^aD^as follows:

1 - a l , aaf3

where Af(P) is the Newton diagram of P. The corresponding graded module grL(.4)has3, in general, different gradations with respect to the linear forms F = (1,0) andV = (0, 1). We will denote K = A, T>.

2Here with respect to the hypersurface x\ = 0.3SodogrL(D) if L + F,V.


Computation of the Slopes of a Z>-Module 313

DEFINITION 2.4 Let P be an element ofU. We will call Newton Polygon of P,denoted by P(P), the convex hull of the set

EXAMPLE 2.5 The Newton polygon ofx2dx + I is

Let us consider now the free *4-module J\.p and a submodule N C ~A.P. Let M beM = AP/N. The study of the slopes of M, as it was done for p = 1 in [1], has todo with the grL(^4)-module gri(M), that is

m>0

where

(m}N

Given a graded ring B = ®j.€Z Bk you can obtain new gradations applying atranslation in one of the components: given k\ & Z,

fcez

that is naturally isomorphic to B with a graded isomorphism of degree k\ . If weconsider Bp, we can grade again using this kind of translations on each component.

DEFINITION 2.6 Given the free module Bp, the gradation of Bp with respect to(ki,...,kp) G IP is the graded module

We will write Bp instead of Bp[0, ...,0]. In a natural way

Bp[kl,...,kp - B[kp}.


314 Ucha-Enriquez

REMARK 2.7 In general there is no graded isomorphism between Bp[ki, ..., kp]and Bp[k(, ...,k'p]. It is enough to take p = 2, ki = k% = 0 and k( ^ k'2.

We will denote by A\^ the ideal

ALM = AnngTL(A}(gTL(M)) = {P € grL(A) P • giL(M) = 0}.

Actually we will distinguish two kinds of slopes:

DEFINITION 2.8 L is a geometric slope of M if the ideal J Aj^ is not bihomo-geneous.

DEFINITION 2.9 L is an algebraic slope of M if the ideal A^ is not bihomoge-neous.

In the case p = 1, if J is an ideal of A then A^ = grL(J). We need hereto take into account the more general object A^ connected to grL(N). The lastobject is computable using homogenization in lir (see [5] and [17] for the details):starting from a set of generators of N you can obtain (via computing L-Grobnerbases respect to certain orders) a set of elements in N whose L-symbols generategTL(N).

DEFINITION 2.10 The L-characteristic variety of M is the variety defined by

REMARK 2.11 The object Aj^ depends on the chosen L-filtration. But the char-acteristic variety is stable, and does not depend on the good filtration (see [8]). Allthe L-filtrations obtained by translations over the components are good (see [3]).

3 FINITENESS OF THE NUMBER OF SLOPES

The case Ap is analogous to the case p = 1: two twin lemmas lead to the finiteness.The proofs are a straightforward rewriting of the proofs of [1] and can be foundagain in [17]. We will write L < L1 to express that the linear form L has lower slopethan L'.

LEMMA 3.1 Let N be a submodule of Ap, L ^ V. There is a linear formL(1), L(1) > L such that, for every L', L(1) > L' > L, one has grL< ' (N) =giv(grL(N)).

Computation of the Slopes of a D-Module 315

LEMMA 3.2 Let N be a submodule of Ap, L ^ F. There is a linear form£(2) ( £(2) -> i gucfr fa^ for every u^ £(2) < L' < L, one has giL> (M) =grL(grF(M)).

In the demonstration of the lemmas appears a way of, given a slope L, obtainingthe next possible slope. It could be summarized as this:

• First, compute an LV-basis, an L-basis G whose L-symbols are a V-basis ofgrL(N). In practice, this is obtained using an order that chooses the exponentwith the greatest V order if the L orders are the same.

• Next, look for the lowest slope L' that appears in the set of all the Newtonpolygons of all the components of the basis G. Then L' is the next possibleslope and there is no slope between L and L'.

Due to compactness, we obtain the finiteness theorem:

THEOREM 3.3 Let N be a submodule of Ap. The number of slopes of M = AP/Nis finite.

The next result completes the set of tools to calculate the slopes, and generalizesthe ideal case:

PROPOSITION 3.4 Let us consider Sp = C[X,£}P bigraded with respect to thetwo gradations F and V. Let W be a submodule of Sp. If W is bihomogeneous sois Anns(SP/W).

PROOF.- Take / — fs + ... + ft G S (with each /j a homogeneous component,s < i < t) with f ( S P ) C W . In particular, (/,0, ...,0),..., (0, ...,0, /) belong to Wand so are all their homogeneous components (/j,0, ...,0),..., (0, ...,0, f i ) . Therefore,each component belongs to the annhilator. D

To finish, we recall that if the radical of an ideal is bihomogeneous, the ideal isbihomogeneous as well. The converse is not true in general.

316 Ucha-Enriquez

4 A WAY OF COMPUTING ALM

As we have said, once we have obtained a possible slope L, we have to computeAL

M. Let us consider M = AP/N and M = gTL(Ap)/grL(N), with

where each Qi = (Qi,i, •••, Qi,P) for 1 < i < t. We can suppose that the symbols ofthe Pj generate gr^(N). We will write

Af = Ann&L(A)(e0], l<j<p,

ifor each GJ — (0, ..., 0, 1 , 0 , ..., 0), 1 < / < p. Of course we have

AnngfL(A)(M) = Q Af.

It is enough then to calculate the Af. The following remarks are important:

• If aiQi + • • • + atQt = (b, 0, ...,0), for example, then

(ai,. . . ,a t) £Syz(Qltj,...,Qptj)

for j — 2, . . . , n. Moreover, b is a product of such a syzygy by the vector(Qi,i,---,QP,i)-

• Let us write

and

With each element (ai, ...,at) € &k y°u obtain an element a in Ak,

(0,..., 0 , , 0 , ...,0),

because (ai, ..., a t) is the intersection of all the correspondent sysygies.

• Thus, if we write

Qt,kwe can assure that

i4b = {a e ffrL(^)| (0, ..., 0,^, 0, ..., 0)

Finally, if one wants to look for geometric slopes, one needs the calculation of theradical of Aj^ (see [6]).

Computation of the Slopes of a Z)-Module 317

5 THE ALGORITHM TO FIND SLOPES

The steps to compute the sets of the slopes of a given D-module M are:INPUT: A family of generators of N.OUTPUT: the slopes {Li,..., Lr} of M or 0 if M is regular.

1. Put L = F.

2. Obtain an LF-basis Pi , . . . ,P s} of TV.

3. Let L' be the lowest4 linear form such that <JL (Pj) is not bihomogeneous forsome Pi, 1 < i < s. If such an L' does not exist, M has no more slopes:grL'(7V) = grv(grL(N)) for every L' ^ L, V because of lemma 3.1. Andtherefore grL(N) is bihomogeneous and so is JA1^.

4. If L1 exists, compute Ajfa. We have that L' is a geometric (resp. algebraic)slope <$=£• \/AM (resp. Aj^) is not bihomogeneous.

5. Go to step 2 changing L by L'.

6 ABOUT THE COMPUTATIONS IN V.

GivenN = (P^.^PJcA*,Ne = <P1 ) . . . ,P r)cZ»',

it is necessary to establish whether the calculations that we make in A give theanalytic slopes as well. The answer is affirmative. In fact, we will prove that a setof generators of , / AnngrL^(Ap/N) is a set of generators oftoo. The results that we need are more general:

LEMMA 6.1 Let A C B be two rings, B flat over A. Let M be an A-module andN be a submodule of M. If Lg = L ®A B, then

MB_M\ _M- - ~ - ~

PROOF.- It follows easily from the exactness of

0 — NB — » MB

Moreover, recall that if A C B, B flat over A, with the notation above one has

(/in/2)B = (/1)Bn(/2)B.The submodule case is analogous.

4If L'(a, b) = pa + qb, we mean the lowest value of ̂

318 Ucha-Enriquez

LEMMA 6.2 Let A C B be two rings, B flat over A. Let N I , N2 be submodules ofAp , and let (Ni)B, (N2)B be their extensions to Bp . Then

PROOF.- Of course, (A1'} ®A B ~ (A ®A B)p ~ Bp. Taking the exact sequence

Ap Ap0

JV2

and applying the tensor product, you obtain

Bp Bp

o _ (Nl n N2)B —>Bp-+ —— e T^y- — o,(jVi)s (-/V2JB

and we obtain what we wanted. D

We can now provide the next proposition:

PROPOSITION 6.3 Let A C B be two rings, B flat over A. and

Ap Bp

where N is a submodule of Ap generated by GI, ..., Gs. Then

(AnnA(M))B = AnnB(MB).

PROOF.- Recall that

AnnA(M) = {a (o,0, ..,0), (0,a,0, ...,0), , ..., (0, .., 0,a) e N},

AnnB(MB)^{b\ (6,0, ..,0), (0,6,0, ...,0), ,..., (0, ..,0,6) e NB}.

In particular, there exist some AI, ..., As with

thus ( A I , . . . , A S ) is in the syzygy module of the fc-th components of the Gj, i —1, ..., s, K = 2, ...,p.

Due to flatness, for k — 2 we have

Lk

h=i

for some /?; & B, 12 = l,...,L2, and some ( f j , ^ , . . . , ^ ) e ^4S that are syzygies ofthe set of the second components for 1 < / < L2. Analogously, ( A j , . . . , A s ) is acombination with coefficients in B of syzygies in A between the fc-th componentsfor k = 3, ...,p:

Lk

Computation of the Slopes of a D-ModuIe 319

Because of the last lemma, ( A i , . . . , A s ) could be obtained as a combination of ele-ments in B multiplied by elements in the intersection of the syzygies between thefc-th components, for k — 2, ...,p, namely,

Therefore we can obtain an element 6 of

GS

j

where each (aj,0, ...,0) is in N, for 1 < j < J.With the same method (0,6, 0, ...,0), ..., (0, ...,0, 6) can be obtained as combina-

tions of elements of the type (0, a^ , 0, ..., 0), ..., (0, ..., 0, o^ ) in N.Finally, as the sets

kIk = {a<=A\ (0,...,^,...,0)eAT},

are in fact ideals, we have

AnnB(MB) = J\ n • • • n Jp =

= (I^B n • • • n (/p)B = (/! n • • • n IP)B = (AnnA(M)}B.n

As in [1] the following result holds, and its demonstration is analogous to the idealcase.

PROPOSITION 6.4 Let N be a submodule of Ap and let N' be the extendedsubmodule in T>N . Let L be a linear form. Then grL(N') = grL(T>)grL(N). Moreprecisely, i/{P1; ...,Pr} is an L-basis of N (and then {<7L(P1), ...,crL(Pr)} generategrL(N}), then it is an L-basis grL(N'} too.

Finally we prove that the generators of the radical of an ideal I = (Pi,..., Pr), I cC[X], generate v/F C C{X} too.

PROPOSITION 6.5 Given an ideal I ofC[X] and Ie the extended ideal in C{X}.Then

(Vl)e =

320 Ucha-Enriquez

PROOF. -• First, we can suppose that / is radical without loss of generality: given any

ideal J, if \/~J = H and H verifies that \/He = He, then

and thus (\/J)e =

• Due to the relationship between the primary decomposition of an ideal inG[X] and its extended one in the localized C[-X"](x) it follows that / C C[X]is radical ==> Ie C C{X](x) is radical.

Using a result of [15], if / is a radical ideal of C[X](x) then \/(Ie) = Ie inC{X}.

n

7 EXAMPLES

Here we present a survey of the examples we have treated to test the algorithm.They are a first approximation of the study of the behaviour of the slopes undernatural operations over D-modules.

7.1 Slopes of O [ l / f ] / O .

In [14] appears a way of presenting O[j}/O as a module of type T>P/N.Given the surface / = 0 with / = xn + yn + zn, if -2 is a root5 of the b-function of/, a set of generators of the submodule N is

P! = (xn + yn + zn,0)P2 = (-l,xn+yn + zn)P3 = (dx,nxn-1)P4 = (dy,nyn-1)P5 = (dz,nzn-1)P6 = (xdx + ydy + zdz + n, 0)P7 = (0, xdx + ydy + zdz + 2n)P8 = (xn-ldy-yn-ldx,0)P9 = (Q,xn-ldy-yn~ldx)PIO = (xn-ldz-zn~ldx,Q}Pn = (O,!"-1^-^-1^)Pi2 = (2/n-192-^-1^,0)Pis = (0,yn-ldz-zn-ldy).

The last set is, in fact, a basis of ./V if one considers an FV-order with an inverselexicographic one. As all the components of every element have Newton polygonswith no slopes, trivially O[j]/O has no slopes with respect to x = 0. By anargument of symmetry, neither has with respect to y = 0 or z = 0.

5 Be careful: This condition does not hold for any n.


Computation of the Slopes of a Z)-Module

7.2 Looking for slopes in a syzygy module

We will treat in this example the ideal6/ generated by

Pi = 2xnydx + xndy + IP2 = y2-xP3 = 2

321

The following elements 81,82,83 generate Syz(Pi, P2, PS):

To obtain an FV basis from this module it is necessary to add a new element:

1 3 1~s4 = (xdx -,nxn~ldy + dx, ~ydx - -<

with 84 = \dy$i + 9XS2- From the set of Newton polygons of all the componentsof every Sj we deduce that there is only one possible slope, namely L = —n thatappears in 82 and 83.

(1,-n)

It is easy to verify7 that the set of the Sj is a L-basis, and therefore their symbols6See [4] for a motivation.7To be more precise, taking an order that chooses first the L-order, after the total grade and

finally a lexicographic order. The exponents with this order are the same.

322 Ucha-Enriquez

generate grL(N). They are

<7 (s3) = (0,-2xn+1^-xnyr,-y,y2)

<7L(s4) = ((U.-j/O

The computation of the annihilator is specially easy because the existence of anelement b such that (6,0,0) S grL(JV) implies that 6 = 0. The annihilator is thenthe zero ideal and thus bihomogeneous. The slope — n that was a true slope of T>/I,has in some sense, "disappeared" in the module of sysygies.

7.3 Slopes and direct sums of ideals

This example uses calculations made in [4] about slopes with respect to a hyper-surface tangent to the support of the direct image of T>e^ by an immersion in C2,and about the slopes of the P-module generated by e»p-i<! . We will take them tostudy what happen with the slopes of a module of the type T>2/(Ii 0/2) where £>//jhas some kind of slope. The motivation of this example comes from the principalideal case: it is easy to prove that in this case the slopes of the new module is theunion of the "old" slopes. Unfortunately, this is not true in the general case. Takeh = (x23x + l ,y) , 72 = (y). For L = -1,

which is a bihomogeneous ideal. But T>/I\ has L as a true slope.Let us consider the module M = T>2/N where N is generated by

• pi = (mxq+ldx + xqydy + mxq + 1,0),

• P4 = (0,qyp+ldy -pxq+ldx - Iqxiydy+pq).

with 2 < p < q and m € C. The respective sets of first and second components ofthe elements in N are ideals /i,/2 with L = — q as a common slope of the "D/Ii,with respect to x — 0.

First we calculate an FF-basis, with an order that (after looking for the greatestF and V) chooses the exponent with greatest dx.FV-base. One needs only to adjoinone element ps = (0,^2) to have a basis, where p$£ is equal to

2 (p + l)xq~lydy - P2qdx).

The Newton polygons of P2, Ps y P4 show no possible slopes. But anyway we finda possible slope to study in pi and ps, namely L = —q.

Computation of the Slopes of a D-Module 323

(2, 1 - q)

The {PJ} are an L-basis too: it is enough to use the order that (after L) considersthe greatest dx. The set of symbols is:

• <7L(p!) = (mxq+l£, + xqyrj + I , 0)

<7L(p5) = (0,p2qxq+1£2 q2xq~ly2r]2 - p2qrj).

It is straightforward to compute the b € grL(L>) such that (6,0), (0,6) <E gon this family of modules. If one follows the algorithm to obtain the annihilator,one will deduce that in this case AM = Ci D C2 where C, is the ideal of the z-th

. Thus JAj^ = \fC{ D \fC^. Here the situation iscomponents of

The intersection is the ideal (y^,yrj,mx'2q+2^ + (1 — m)x9+1£2 — £). Because

f , the slope L has been "conserved".

REFERENCES

[1] Assi A., Castro-Jimenez F.J., Granger J.M. How to calculate the slopes of aP-module. Compositio Mathematica 104, 1996, 107-123.

[2] Bjork, J-E. Rings of Differential Operators. North-Holland, Amsterdam 1979.

[3] Castro Jimenez, F.J. These de Seme cycle. Universite Paris VII, 1984.

324 Ucha-Enriquez

[4] Castro Jimenez F.J., Granger J.M. Explicit calculations in rings of differentialoperators. Prepublicaciones de la Universidad de Seville,, 35. June 1997

[5] Castro-Jimenez F.J., Narvaez-Macarro L. Homogenising differential operators.Prepublicaciones de la Universidad de Sevilla, 36. June 1997

[6] Eisenbud D., Huneke C., Vasconcelos W. Direct methods for primary descom-position. Invent. Math.. 110, 1992, 207-235.

[7] Granger, J.M. and P. Maisonobe. A basic course on differential modules.Travaux en Cours 45, Hermann. Paris 1993.

[8] Laurent, Y. Poly gone de Newton et 6-fonctions pour les modules microd-ifferentiels. Ann. Scient. EC. Norm. Sup. 4e serie, 20, 1987, 391-441.

[9] Laurent, Y. and Z. Mebkhout. Pentes algebriques et pentes analytiques d'unZ>-module. Prepublications de I'lnstitut Fourier 372, 1997.

[10] Malgrange, B. Sur les points singuliers des equations differentielles.L 'Enseignement Mathernatique 20, 1974, 147-176.

[11] Mebkhout, Z. Le theoreme de comparaison entre cohomologies de De Rhamd'une variete algebrique complexe et le theoreme d'existence de Riemann. Pub-lications Mathematiques 69, 1989.

[12] Mebkhout, Z. Le formalisme des six operations de Grothendieck pour lescoherents. Travaux en cours, 35. Hermann, Paris 1989.

[13] Mebkhout, Z. Le theoreme de positivite de 1'irregularite pour les ZXmodules.Grothendieck Festschrift III. Progress in Math. 88, 1990, 84-131.

[14] Oaku, T. Algorithms for the b-function and X>-modules associated with apolynomial. Journal of Pure and Applied Algebra, 117 & 118, 1997, 495-518.

[15] Serre, J.P. Geometric Algebrique et geometric analytique. Ann. Inst. Fourier6, 1955-56, 1-42.

[16] Saito, M., Sturmfels, B. and N. Takayama. Grobner deformations of hyper-geometric differential equations. To appear in Algorithms and Computation inMathematics, 6 (1999).

[17] Ucha Enriquez, Jose Maria. Metodos constructivos en algebras de operadoresdiferenciales . Tesis doctoral. Sevilla, 1999.


Symmetric Closed Categories and Involutive BrauerGroups1

A. VERSCHOREN, Department of Mathematics and Computer Science, Univer-sity of Antwerp, RUGA. 2020-Antwerp, Belgium.E-mail: aver@ruca. ua. ac. be

C. VIDAL, Departamento de Computation, Universidad de La Coruna. La Coruna,Spain.E-mail: eicovima@udc. es

Abstract

The main purpose of this note is to introduce and study the involutiveBrauer group B*(C) of a symmetric closed category C, thus presenting a com-mon framework for the constructions in [18, 21, et al].

1 INTRODUCTION

In [23], it has been proved by Saltman that any 2-torsion element in the Brauergroup of a ring R may be represented by an Azumaya .R-algebra with involution.This inspired Parimala and Srinivas [18] to define a version of the Brauer group ofa scheme X denoted by B*(X) and based on sheaves of Azumaya algebras withinvolution on X.

On the other hand, Orzech introduced in [16] the notion of suitable subcategoryC of T>(R), the category of divisorial lattices over a Krull domain R and defined forthese categories a notion of Brauer group, which presents a common framework forseveral particular constructions of Brauer groups scattered throughout the litera-ture. Finally, Reyes and the first author constructed in [21] an involutive version ofOrzech's set-up and used it to present a more transparent approach to the resultsin [18].

Research supported by an F.W.O. research grant.


326 Verschoren and Vidal

However, all of the previous constructions explicitly used some results fromdescent theory (like the existence of a trace morphism for any Azumaya algebra).Although part of this theory generalizes to more general situations like symmetricclosed categories for example, cf. [4], this does not suffice, however, to actuallyconstruct well-behaving associated Brauer groups.

Another point of view is that used by Vitale in [32], where the Brauer group ofa symmetric monoidal category C is defined as the Picard group of an associatedmonoidal category with unital C-monoids as objects and bimodules as morphisms.Actually, the main goal of [31] was to prove that, when one considers the cate-gory C of quadratic modules over a commutative ring R, Vitale's methods may beslightly modified and still allow for the construction of a Brauer group "a la Vitale",isomorphic to the one introduced by the authors of [21].

In the present note, Vitale's techniques are used to define an involutive Brauergroup for any symmetric closed category C. We show that this construction pos-sesses nice functorial features and generalizes the involutive Brauer groups intro-duced in both [18] and [21].

In the first section, we recollect some definitions and results from the generaltheory of closed categories. We do not include proofs, as these may be foundin [1, 4, 6], for example. The second section is concerned with the introductionof monoids with involution and morphisms between them, as well as nonsingularcompatible sesquilinear forms on bimodules.

The third section is dedicated to the construction for any symmetric closedcategory C of a symmetric monoidal category V(C) and the way we may use thisto define the involutive Brauer group of C. Finally, in the last section, we define aclass of functors between symmetric closed categories which induce a correspondingmorphism between the involutive Brauer groups and show how these allow for theconstruction of an exact sequence connecting the involutive Brauer and Picardgroups.

2 SOME BACKGROUND ON CLOSED CATEGORIESo

Let (C, <8>, K) be any symmetric monoidal category with base object K and com-mutativity isomorphisms TAB '• A<&B —» B<S)A satisfying:

1 TAB =

for any A,B,Ce C.We will say that C is closed if there exists a bifunctor

ROM : Copp xC —>C

such that A®- : C -> C is left adjoint to HOM(A, -) : C -> C, for any A in C. Wewill denote by a A resp. /?£, the unit resp. counit of this adjunction.

In this case, C is also a C-category, which means that

• for any A, B, C £ C, there is a morphism:

: HOM(A,B)®HOM(B,C) -> HOM(A,C);

Symmetric Closed Categories 327

• for any A G C, there exists some

JA:K-+HOM(A,A),

satisfying:

CACD o (cABC®HOM(C, D}} = CABD o (HOM(A, B)®cBCD)cABBo(HOM(A,B)®jB) = cAABo(jA®HOM(A,B}).

Suppose that C is & symmetric closed category. For any object A G C, we saythat (A, rjA, HA) is a monoid in C if r]A : K —> A and HA : A®A —> A are morphismsin C, verifying the usual unity and associativity conditions:

= idA — HA ° (r]A®A)\

Given two monoids (A,T]A,HA) and (B,TJB,HB), a map / : A — * B is said to bea monoid morphism if

/ o r]A = r/B and HB ° (/®/) = / o HA-

The opposite monoid Aopp is defined as (A, rjA, HA ° TAA)- Finally, if A and B aremonoids in C, their monoid product is given by

Let (A,TJA,HA) be a monoid in C. A couple (M, <PM) consisting of an object Mand a morphism fM '• A®M — > M in C is said to be a left A-module if it verifies

fM ° (r)A®M) = idM and fu ° (-^^M) = fM ° (HA®M).

If (M,<PM) and (N,<PN) are two left ^4-modules, a morphism / : M — » N in C iscalled a morphism of left A-modules if it preserves the structure, i.e., if

<fN ° (A®f) = f 0(f>M.

This defines a new category denoted by AC.Right ^-modules may be defined similarly, and yield a category CA. If

(K, TIK,HK) is the trivial monoid in C, then it is clear that j<C ^ C ~ CK-When (A,TJA,HA) and (B,TJB,HB) are two monoids in C and (M,<PM) resp.

(M, 0jvf) is a left yl- module resp. a right B-module, we say that (M, <£M, <^M) is anA5-bimodule if we also have

We denote by ACB the category of >lS-bimodules.

Consider a symmetric closed category C with equalizers and coequalizers. Sup-pose that ( A, TJA, H A) is a monoid in C and that (M, </5jv) and (N,(f>fj) are twoobjects in ^C. We define HOMA(M,N) as the equalizer

"U-MN

HOMA(M, N) ̂ HOM(M, N) ~~ I HOM(M, HOM(A, N)),

where UMN = HOM(tpM,N) and VMN = HOM(M,HOM(A,ipN) oaA(N)).Putting for any (M, <PM] in A.C-

jfc-.K-* HOMA(M, M) = EA(M),

the factorizaction of CIM(K) through the equalizer ZMM, this makes AC into a C-category. If (N,if>fj) and (P,(pp) are two objects in AC, we shall denote by

dMNp '• HOMA(M,N}®HOMA(N,P) -» HOMA(M,P)

the factorization through the equalizer IMR of the morphism

HOM(M,(3N(P) o ([3M(N)®HOM(N,P)})0

o aM(HOM(M,N)®HOM(N,P))o(iMN®iNP)

On the other hand,

(EA(M) = HOMA(M,M),j^,dMMM)

is a monoid in C and (M,j3*M(M)) is a right EU(M)-module, if we denote by j3*Mthe counit of the adjuntion M®— H HOMA(M, -). Moreover

is a left .E,i(M)-module.Finally, if (A,r/A,nA) is any monoid in C and (M, </>M) is a right yl-module, we

define

by (M®A-)(N,(pN) := M<S>AN, where M®AN is the coequalizer in C of the mor-phisms XMJV = M®(fN and T/MJV = 4>M®N , i.e.,

>• M®ANVMN

This last functor factorizes through sC when (B,T]B,fJ>B) is another monoid in Cand (M, y>M> <^M) is an object of sCA.

For any monoid (A, r]A,fj,A) in C and any object M of ^C, there exist morphisms

andVAMA : M®EA(M)M* -> A,

where VMAM is the factorization of duAM through the coequalizer of XMM*2/MM* , and where VAM/I is the factorization of dA^A through the coequalizer ofthe morphisms

We say that M is A-profinite, (resp. an A-generator) if V MAM (resp. VAMA) isan isomorphism. When both conditions are satisfied, we say that M is an A-progenerator. In this case, it follows from [1, (3.3.13)] that

• HOMA(M, -) : AC -ÊA(M) C is & C- equivalence of categories;

• there are mutually inverse C-equivalences

and)- ^ HOMBA(M}(M, -);

• M*®AM ~ EA(M] and M®B/1(M)M* ~ A.

When (A,rjA,Â) = (K,rjK,/j,K), we put VM := V 'MKM and V'M := VKMK

Assume that both (A,T]A,Â} and (B,T)B,^B} are monoids in C and that F HG : ^C — > sC is a C-equivalence. Then it follows from [1, (3.2.4)]) that there existobjects (Q,VQ,(J>Q} € A^B and (L,y>L,<j>L) G s^/i such that

i) <3<S>B- ^ -F and L®^- ~ G;

ii) L®AQ ~ S as BB-bimodules and Q®sL ~ .A as >i/i-bimodules;

iii) HOMA(Q, A) ~ L as SA-bimodules and HOMB(L, B) ~ Q as ^B-bimodules;

iv) HOMA(Q, Q) ̂ B and HOMB(L, L) ~ A as monoids in C;

v) Q is an /i-progenerator and L is a _B-progenerator.

For any monoid (A, TJAÂ) in C, we can define a morphism of monoids

XA : Ae := /1®^°PP ̂ £;(^) = HOM(A, A},

whereXA = HOM(A, ^ A o (/M®^) o (rAA®A)) o a/i(/l®A0^).

When A is a /f-progenerator and x^ is an isomorphism of monoids, we say that Ais an Azumaya monoid. Note that, if (A^TJAÂ) is an Azumaya monoid, we havean equivalence of categories

where MA - HOMA^(A,M), for any Ae-module M.The proof of the following results may be found in [6] :

• if (A, TIAÂ] and (B,T]B,HB) are two Azumaya monoids, then A®B is alsoan Azumaya monoid;

• if P is /C-progenerator, then (E(P),jp,dppp) is an Azumaya monoid;

• if P and Q are both /C-progenerators, there is an isomorphism of monoidsbetween E(P)®E(Q) and E(P®Q).

3 MONOIDS WITH INVOLUTION

From now on, we will suppose that C is a symmetric closed category with equalizersand coequalizers.

Let (A, T]A,HA} be a monoid in C. When a : A —> A is a morphism in C satisfyinga2 = id A, ctorjA = r/A and /J,A ° (ot<S)a) o TAA = & ° HA, we say that the pair (A, a)is a monoid with involution.

Note that there is an obvious way to construct a tensor product (A®B, a<8>/3)for any pair of monoids with involution (A, a) and (P>,/3).

Suppose that (A, a] is a monoid with involution and (M, <^M) is a left A-module.A map h : M®M -> A verifying:

h o (M®(pM] = ^AO (h<S>a) o (M®TAM)

is called a sesquilinear form on M. When h = hoTMM, we say that h is symmetric.Note that when (A, a) = (K, idf;), a sequilinear form h on any object M of C is

just a bilinear map h.Consider another monoid with involution (B,/3) and some(M, </?M, <J>M] m A^B-

Then a sesquilinear form h : M®M —> A on M is said to be /?- compatible if

/i o OM®M) = /i o (M<g></>M) ° (M<S>M<S>/3) o (M®rBM)

As Homc(M®M,A) ~ Homc(M,M*), any sesquilinear form h on M inducesa morphism ha : M ^ M*, by letting /ia = HOM(M,h) o aM(M}. It should beclear that /ia is the unique morphism in C which makes commutative the followingdiagram:

M<S>M

We say that /i is nonsingular, when /ia is an isomorphism.We denote by Q(C) resp. 5(C) the category of pairs (P, h) with P a .fiT-progenera-

tor and h : PcgiP — » A" a nonsingular sequilinear form resp. a nonsingular symmetricsesquilinear form on P. A morphism / : (P, h) — » (Q,h') in these categories is anymorphism / : P — > Q which makes commutative the diagram:

P0P

Any nonsingular sesquilinear form h : M®M —> ^4 on M defines an involutionaft on the monoid EA(M] given by ah = HOM(M,h) o aM(£;^(M)). Here h isdefined as

h = (ha)~l o C?MM/S o TM.EA(M} o (ha®EA(M)).

Note that a^ is the unique endomorphism of EA(M) such that f3*M(M) o (M®a.h) =h.

First of all, if can we show that

(M®ah) o (M®j$f) = idM,

then it will be clear, by the unicity of jfy, that ahoj^ = j f a . But this easily followsfrom the fact that ho (M®j^) = id^-

On the other hand, taking into account the properties of d, the naturality of Tand the definition of a^ and ha, one easily verifies h to be o^-compatible.

4 THE INVOLUTIVE BRAUER GROUP

Following the lines of [32], we introduce a new symmetric monoidal category V(C)as follows. The objects of V(C) are monoids with involution (A, a). A morphismf : (A, a) — > (B,/3) will be a pair / = (M,h) consisting of an A-B-bimodule(M,fMi4>M) and a /3-compatible nonsingular sesquilinear form h : M®M — > Aon M. The composition g o / of / = (M, /IM) : (A, a) — •> (B, j3) and g = (N, h^) :(B,/?)^(C,7) is given by

g o / = (M®BN, hM®BN),

where HM®BN is the factorization over M®sN®M®sN of

h = hM o (4>M®M) o (M®TMB) ° (M®M®hN] o (M®rNM®N).

If (^4, a) is any object in V(C), the identity morphism on (A, a) is (A,^), wherep^4 = ^A ° (A®a). Since one easily verifies that

PA ° (HA® A) = HA ° (HA®U) = HA ° (A®pA)

and

PA o (A® HA) = HA° (^®MA) ° (A®a®a) o (A®TAA)= HA ° (PA®®) o (A®TAA),

it follows that PA is a sesquilinear form. Moreover, since clearly:

PA ° (HA® A) = HA° (A®HA) ° (A®A®a)= HA° (A® HA) ° (A®a2®a)= HA ° (A®HA) o (A<S>a®a) o (A®TAA) o (A®A®a) o (A®rAA)= PA o (A®nA) o (A®A®a) o (A®rAA),

we also have that pA is a-compatible.

If / = (M,hM) : (A, a) -+ (B,(3) and g = (N,hN) : (B,/?) -+ (A, a) aremutually inverse isomorphisms in V(C), we have isomorphisms M®gN ~ ^4 and

~ B of bimodules. This yields a C-equivalence

and, as we have said before, isomorphisms between HOMA(M, — ) ~ N®A~ andHOMs(N, — ) ~ M®B — , as well as monoid isomorphisms between EA(M] ~ B andEB(N] ~ A. In particular, M resp. JV is an /1-progenerator resp. a B-progenerator.

As we have already pointed out that cthM resp. a^N is the unique involutionon EA(M) resp. Es(N) such that /IM resp. /i/y is «hM -compatible resp. cthN-compatible, it follows that a^M = (3 and a^N = a.

We make V(C) into a monoidal category through the bifunctor

-®- : V(C)®V(C) —— V(C),

defined by (A,a)®(B,/3) := (A®B,aAny morphism / = (M, /IM) : (A, a) — > (A', a') and any monoid with involution

(B,/3) define a morphism

f®(B,/3) = (M®B,hM<SlB) :

where

For any monoidal category .4, the Picard group Pic(A) of ^4 is defined to bethe group of invertible elements in the monoid of isomorphism classes of objects inA, with product induced by the inner product in A. As in [32], it thus makes senseto put B*(C) := Pic(V(C})] we call this the involutive Brauer group of C.

Note that if [A, a] e B*(C), there exists (B,/3) in V(C) such that

in V(C). Taking into account ( 1), this implies the existence of (M, /IM) and (N,with M resp. N belonging to A®sC resp. CA®B and h^ resp. /IJY a nonsingularsesquilinear form, such that

and

and/IM oh^ — PA®B and /iw o /IM = UK-

In view of the natural BAopp-bimodule resp. AoppB-bimodule structure of M respN, it is an easy exercise to verify that we have an equivalence

with

and

It easily follows that (A®Aopp,a®aopP) and (K,idK) are isomorphic in V(C) withisomorphism given by (A,h,M ° /iA®yv) and (A,hw o HA^M}- This means, in par-ticular, that the monoids A®Aopp and -E(-A) are isomorphic and also that A is aK-progenerator, i.e., that A is an Azumaya monoid.

EXAMPLE 4.1 When R is a commutative ring and C = jR—mod, a morphism/ : (A, a) —> (B,/3) between two Azumaya algebras with involution is defined tobe morphism of monoids / : A —> B such that (3 o f = f o a, cf. [21]. We canprove, in our set-up, that if / : (A, a) —> (B,/3) is such a morphism with inverseg : (B,{3) —> (A, a), the monoids (A, a) and (B,/3) are also isomorphic in V(C).

If we consider (B,<ps = MB ° (/®-B)>MB) as an object of ^Ce and hB = pA o(g<8)g), it may easily be verified that (B,hB) is an isomorphism in V(C) between(A, a) and (B,/3) with inverse given by (A,hA) defined in an analogous way. Toshow that that hB is /3-compatible, let us point out, for example, that

hB o (B®HB} o (B®B®j3] o (B®TBB)= HA o (^4(gia) o (g®g) o (B®/j>B o (B®(3} o TBB)

?)a o g o /j,B o (5<E)/3) o TBB)o io

o (A®a) o (/j,A o1 o (a®a} o i

The unit element in B*(C) is the class [E(M), ah] for any (M, h] e 5(C). Indeed,we may show the existence of an isomorphisms

(M,h)

in V(C), where h' : M*®M* -> E(M) is given by

ft' - HOM(M, (3M(K)®(ha)-1} o

So /i' is the unique map in Homc(M*®M* , E(M}}, such that the following diagramcommutes:

M®E(M)

The proof of the fact that h' is a nonsingular sesquilinear form is an easy exercisewhich is left to the reader. To finish the proof, we have to show that (M, h) and(M*,/i') are mutually inverse isomorphisms in V(C). For this, we use the fact thatM is K-progenerator, which implies that V'M and VM are isomorphisms. The proofis not difficult but rather technical, so we prefered not to include it here. The onlypoint is to verify the commutativity of the two diagrams:

~^K

M®E(M)M*®M®E(M]M* ——>K®K

andE(M}®E(M)

PF

E(M)

EXAMPLE 4.2 If R is a commutative ring and C — R— mod is the category of R-modules, then it has been proved in [31] that B*(R— mod) is the involutive Brauergroup B*(R) defined by the authors of [21] making use of a "suitable" involutiveversion of Orzech's set up [16] for divisorial lattices over a Krull domain.

On the other hand, if X is a scheme such that 2 e F ( X , Ox), and C = Ox -moddenotes the category of C^-modules over X, it is well known that C is a sym-metric closed category. Following the lines of [31], it is not difficult to see thatB*(OX~ mod) is isomorphic to B*(X), the group defined by the authors of [18].

Other examples, like the relative involutive Brauer group Br*(R,o~) arise in asimilar way as well.

5 FUNCTORIAL BEHAVIOUR

If C and T> are symmetric closed categories with base objects K and K' , respectively,then the functor F : C — » T) is said to be monoidal if there exist morphisms <J>Q :K' -> F(K) and (j)M,N • F(M) <g> F(N] -> F(M ® N) such that:

,L ° ($M,N ® F(L)) = ({>M,N®L ° (F(M) <g> </>N,L), for any M,N and LinC;

• <f>M,K o (F(M) ® 4>0) = idF(M} = <j>K<M ° Oo ® F ( M ) ) , for any M in C;

• F(TMN}0(t>M,N — ̂ W,MOTJ?(M)F(JV)> f°r &I^f ^ , N in C, where T and T' denotethe symmetry isomorphisms in C and T> respectively.

It can be proved [7, (2.2)] that if F : C — > T> is monoidal and <^o and <J)MQ areisomorphisms whenever Q is profinite in C, then:

• if Q is profinite in C, then F(Q) is profinite in T>;

. the morphism F(HOM(Q,M}) -* HOM(F(Q),F(M)} defined as

HOM(F(Q),F((3Q(M) o $Q,HOM(Q,M)) o aF(Q)(F(HOM(Q, M)))

is an isomorphism if Q is profinite in C.

• if F preserves coequalizers, it also preserves progenerators.

PROPOSITION 5.1 Any monoidal functor F : C — > V which preserves coequaliz-ers and satisfies the previous properties induces a group homomorphism

B*(F) :B*(C) ->B*(X>).

PROOF: Let us first point out that for any Azumaya monoid (A,rjA, /J,A) in Cobviously

(F(A),r)F(A - F(r]A) o 4>0, /j,F(A} •= F(^A) o <f>AtA)is an Azumaya monoid in T>. To derive the homomorphism B*(F) : B*(C) — >B*(D), it only remains to show that F maps isomorphic objects in V(C) to isomor-phic objects in V(£>). First note, that if (A, a) € V(C), then (F(A),F(a)) € V(D).This follows from

F(a) o r]F(A) = F(a) o F(r]A) o 00 = F(rjA) ° 4>o =

and

o 4>A^A o (F(a) (g> F(a)) o T'F(A}F(A}.

Next, if (A, a) ~ (B,/3) are isomorphic objects in V(C), with isomorphism givenby (M, /IM)> where (M,<pM,<t>M) 'ls some ylB-bimodule, we have that

is an F(J4)-F(B)-bimodule and also that (F(M),hF^} = F(tiM] ° 4>M,M) definesan isomorphism in V(D) between (F(A),F(a)) and (F (B) , F (/3)) . We leave itas a straightforward exercise to the reader to check that hp(M) 'IS a nonsingularsesquilinear form and that it is F(/3)-compatible.

EXAMPLE 5.2 For any homomorphism R — > S of commutative rings, the hy-potheses of the previous result are obviously satisfied for the induced functor

R-mod — > S'-mod : M H-> 5 ®R S.

We thus obtain a canonical group homomorphism B*(R) — > B*(S).

As in the classical case, we can define an involutive Picard group and obtain anexact sequence relating both groups (Picard and Brauer) under the presence of amonoidal functor between two symmetric closed categories.

The abelian group KQ€ of a category with product (C, _L) is defined as in [2].On the other hand, let fiC denote the category with objects (C, a), where a is aC-automorphism of C 6 C. A morphism (C\,ai) — > (6*2,^2) is just a C-morphism/ : C\ — » C%, fitting into the commutative diagram

We define(Ci.oO -L (C2,a2) = (C1: ± C f

2 )ai -L a2)

and define K\C to be K<£lC modulo the subgroup generated by

[(C, a/3)] -[(C, a)] -[(<?,/?)],

where a and j3 are C-automorphisms of C.Finally, if F : C — > V is a product preserving functor, then <&F denotes the

category with objects (C*i, a, 6*2), where C\ and C2 are objects in C and a : FC\ — *•FC-2 is an isomorphism in T>, and with morphisms

(/ i , /2) :(C1 ,a ,C2)^(Ci,a / ,C2) ,

where the /, : Q — > C '̂ make the diagram

• FC2

Ffi F/2

commutative. We define

(Ci.a.Ca) ± (^i,«',C2) = (d ± C2,a ± a',C2 J_ C2)

and define KI$ to be K^&F modulo the subgroup generated by all

[(C2,/3, C3)].

Recall also, from [2] for example, that a product preserving functor F : C — * T>is said to be cofinal, if given D' € D, we may find D" 6 I? and C1 g C such thatD' ± D" = FC. A full subcategory CQ of C is said to be cofinal, if the canonicalinclusion Co '— > C is a cofinal functor, i.e., if for every C" G Co, we maY nnd C"' 6 Coand C e C such that C" ± C" 9^ C.

From [2] we retain:

PROPOSITION 5.3 With notations as above, we have:

1. if CQ C C is a full, cofinal subcategory of C, then the inclusion induces anisomorphism K\CQ = K\C;

2. if F : C — -> T> is a cofinal, product preserving functor, then there exists aunique homomorphism K\D — » Ki<frF sending the class of (FC,a) to that of(C,a,C). The resulting sequence

KiC -» KiD -» K&F -» K0C -> KQV

is exact.

As we have defined the involutive Brauer group B*(C) as the Picard Group ofthe monoidal category V(C), it should be clear that

B*(C)~K0I(C)

where 1(C) denotes the subcategory of V(C) with objects the invertible Azumayaalgebras with involution.

So, the natural involutive Picard group should be Pic*(C) := Ki(T(C)). Thisconstruction is functorial, because any monoidal functor F : C — > T> satisfying thehypotheses of 5.1, induces a functor

between the associated categories with product. This yields a group homomorphism

Pic*(F) : Pic*(C) -» Pic*(D).

Finally, note that I(F) is cofinal, as for any (B, /?) € I(D) there exists (C, 7) <E V(C)such that

(B,0)®(C,i) ~ ( K ' , i d K I ) ~ ( F ( K ) , i d F ( K ) ) .If we denote by B*(C,T>) the group Ki4>I(F), we thus finally obtain from 5.3:

PROPOSITION 5.4 Any monoidal functor F : C — > T> satisfying the hypotheses0/5.1 induces an exact sequence

Pic*(C} -> Pic*(V) -> B*(C,T>) -> B*(C) -> S*(P)

o/ abelian groups.

EXAMPLE 5.5 If we again consider a commutative ring R and the associatedcategory of P-modules C = -R— mod, we have that

Pic*(C) ~ Pzc*(/?) = Discr(R),

where Discr(R) is Bass' discriminant group, cf. [3, 21], defined to be the group ofisomorphisms classes {L,h}, where L G Pic(R) and h : L®L — > R is a nonsingularbilinear form. Note that, as ha : L — > L* is an isomorphism, then so is h, ash — VL o (L®ha). For any homomorphism of commutative rings / : R — > 5, wethus obtain an exact sequence

1 -» Pic*(#) -> Pzc*(5-) -> Bd*(R,S) -» B*(E) -* 5*(5),

where the injectivity of the map Pic*(R) — > Pic* (5) is easily verified, and wherethe intermediate group Bcl*(R,S) is defined as in [22].

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