Rings, Extensions, and Cohomology 9781003071815, 9781138402058, 9780824792411, 0824792416, 1138402052

Table of contents :
Cover ......Page 1
Half Title ......Page 2
Title Page ......Page 8
Copyright Page ......Page 9
Preface ......Page 10
Table of Contents ......Page 12
Contributors ......Page 14
Daniel Zelinsky: An Appreciation......Page 16
The Centralizer on H-Separable Skew Group Rings......Page 22
Contributions of PI Theory to Azumaya Algebras......Page 30
Cocycles and Right Crossed Products ......Page 40
Engel-Type Theorems for Lie Color Algebras......Page 52
Constructing Maximal Commutative Subalgebras of Matrix Rings ......Page 56
Galois Extensions Over Local Number Rings......Page 62
Infinite Extensions of Simple Modules over Semisimple Lie Algebras ......Page 88
Smoothing Coherent Torsion-Free Sheaves......Page 108
Projective Covers and Quasi-Isomorphisms......Page 114
On Dihedral Algebras and Conjugate Splittings ......Page 128
On H-Skew Polynomial Rings and Galois Extensions......Page 134
Separability and the Jones Polynomial ......Page 144
A Note on Grӫbner Bases and Reduced Ideals......Page 160
Bicomplexes and Galois Cohomology ......Page 164
Adjoining Idempotents ......Page 176
Separable Polynomials and Weak Henselizations ......Page 186
Faithful Representations of Lie Algebras Over Power Series......Page 202
Idealizers of Fractal Ideals in Free Group Algebras ......Page 216
Elements of Trace Zero in Central Simple Algebras......Page 226
Canonical Modules and Factorality of Symmetric Algebras......Page 234
Splitting Properties of Extensions of the Wedderburn Principal Theorem ......Page 244
Index ......Page 262

Citation preview

rings, extensions, and cohomology

PURE AND APPLIED M ATH EM ATICS A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed University o f Delaware Newark, Delaware

Earl J. Taft Rutgers University New Brunswick, New Jersey

CHAIRMEN OF THE EDITORIAL BOARD S. Kobayashi University o f California, Berkeley Berkeley, California

Edwin Hewitt University o f Washington Seattle, Washington

EDITORIAL BOARD M. S. Baouendi University o f California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute o f Technology Marvin Marcus University o f California, Santa Barbara W. S. Massey Yale University Anil Nerode Cornell University

Donald Passman University o f Wisconsin—Madison Fred S. Roberts Rutgers University Gian-Carlo Rota Massachusetts Institute o f Technology David L. Russell Virginia Polytechnic Institute and State University Walter Schempp Universitdt Siegen Mark Tepty University o f Wisconsin—Milwaukee

LECTURE NOTES IN PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

N. Jacobson, Exceptional Lie Algebras L.-A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis /. Satake, Classification Theory of Semi-Simple Algebraic Groups F. Hirzebruch, W. D. Newm ann, and S. S. Koh, Differentiable Manifolds and Quadratic Forms /. Chavel, Riemannian Symmetric Spaces of Rank One R. B. Burckel, Characterization of C(X) Among Its Subalgebras B. R. McDonald, A. R. Magid, and K. C. Smith, Ring Theory: Proceedings of the Oklahoma Conference Y.-T. Siu, Techniques of Extension on Analytic Objects S. R. Caradus, W. £ Pfaffenberger, and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces E. O. Roxin, P.-T. Liu, and R. L. Sternberg, Differential Games and Control Theory M. Orzech and C. Small, The Brauer Group of Commutative Rings S. Thornier, Topology and Its Applications J. M. Lopez and K. A. Ross, Sidon Sets W. W. Comfort and S. Negrepontis, Continuous Pseudometrics K. McKennon and J. M. Robertson, Locally Convex Spaces M. Carme/i and S. Malin, Representations of the Rotation and LorentzGroups: An In­ troduction G. B. Seligman, Rational Methods in Lie Algebras D. G. de Figueiredo, Functional Analysis: Proceedings of the Brazilian Mathem atical Society Symposium L. Cesari, R. Kannan, and J. D. Schuur, Nonlinear Functional Analysis and Differential Equations: Proceedings of the Michigan State University Conference J. J. Schaffer, Geometry of Spheres in Normed Spaces K. Yano and M. Kon, Anti-Invariant Submanifolds W. V. Vasconce/os, The Rings of Dimension Two R. £ Chandler, Hausdorff Compactifications S. P. Franklin and B. V. S. Thomas, Topology: Proceedings of the Memphis State University Conference S. K. Jain, Ring Theory: Proceedings of the Ohio University Conference B. R. M cDonald and R. A. Morris, Ring Theory II: Proceedings of the Second Oklahoma Conference R. B. Mura and A. Rhemtulla, Orderable Groups J. R. Graef, Stability of Dynamical Systems: Theory and Applications H.-C. Wang, Homogeneous Branch Algebras £ 0. Roxin, P.-T. Liu, and R. L. Sternberg, Differential Games and Control Theory II R. D. Porter, Introduction to Fibre Bundles M. Altm an, Contractors and Contractor Directions Theory and Applications J. S. Golan, Decomposition and Dimension in Module Categories G. Fairweather, Finite Element Galerkin Methods for Differential Equations J. D. Sally, Numbers of Generators of Ideals in Local Rings S. S. Miller, Complex Analysis: Proceedings of the S .U .N .Y . Brockport Conference R. Gordon, Representation Theory of Algebras: Proceedings of the Philadelphia Conference M. Goto and F. D. Grosshans, Semisimple Lie Algebras A. /. Arruda, N. C. A. da Costa, and R. Chuaqui, Mathematical Logic: Proceedings of the First Brazilian Conference F. Van Oystaeyen, Ring Theory: Proceedings of the 1 9 7 7 Antwerp Conference F. Van Oystaeyen and A. Verschoren, Reflectors and Localization: Application to Sheaf Theory M. Satyanarayana, Positively Ordered Semigroups D. L Russell, M athematics of Finite-Dimensional Control Systems P.-T. Liu and E. Roxin, Differential Games and Control Theory III: Proceedings of the Third Kingston Conference, Part A A. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices J. Cigler, V. Losert, and P. Michor, Banach Modules and Functors on Categories of Banach Spaces

47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97.

P.~T. Liu and J. G. Sutinen, Control Theory in M athematical Economics: Proceedings of the Third Kingston Conference, Part B C. Byrnes, Partial Differential Equations and Geometry G. K/ambauer, Problems and Propositions in Analysis J. Knopfmacher, Analytic Arithmetic of Algebraic Function Fields F. Van Oystaeyen, Ring Theory: Proceedings of the 1 9 7 8 Antw erp Conference B. Kadem, Binary Time Series J. Barros-Neto and R. A. Artino, Hypoelliptic Boundary-Value Problems R. L. Sternberg, A. J. Kalinowski, and J. S. Papadakis, Nonlinear Partial Differential Equations in Engineering and Applied Science B. R. McDonald, Ring Theory and Algebra III: Proceedngs of the Third Oklahoma Conference J. S. Golan, Structure Sheaves Over a Noncommutative Ring T. V. Narayana, J. G. Williams, andR. M. M athsen, Combinatorics, Representation Theory and Statistical Methods in Groups: YOUNG DAY Proceedings T. A . Burton, Modeling and Differential Equations in Biology K. H. Kim and F. W. Roush, Introduction to Mathem atical Consensus Theory J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces O. A . Nielson, Direct Integral Theory J. £ Smith, G. O. Kenny, and R. N. Ball, Ordered Groups: Proceedings of the Boise State Conference J.Cronin, Mathem atics of Cell Electrophysiology J.W. Brewer, Power Series Over Commutative Rings P.K. Kamthan and M. Gupta, Sequence Spaces and Series T. G. McLaughlin, Regressive Sets and the Theory of Isols T. L. Herdman, S. M. Rankin III, andH . W. Stech, Integral and Functional Differential Equations R. Draper, Commutative Algebra: Analytic Methods W. G. M cK ay and J. Patera, Tables of Dimensions, Indices, and Branching Rules for Rep­ resentations of Simple Lie Algebras R. L. Devaney and Z. H. Nitecki, Classical Mechanics and Dynamical Systems J.Van Geel, Places and Valuations in Noncommutative Ring Theory C. Faith, Injective Modules and Injective Quotient Rings A . Fiacco, Mathematical Programming with Data Perturbations I P. Schultz, C. Praeger, andR. Sullivan, Algebraic Structures and Applications: Proceedings of the First Western Australian Conference on Algebra L Bican, T. K ep ka, a n d P. N em ec, Rings, M o du les, and Preradicals D. C. Kay and M. Breen, Convexity and Related Combinatorial Geometry: Proceedings of the Second University of Oklahoma Conference P. Fletcher and W. F. Lindgren, Quasi-Uniform Spaces C.-C. Yang, Factorization Theory of Meromorphic Functions O. Taussky, Ternary Quadratic Forms and Norms S. P. Singh and J. H. Burry, Nonlinear Analysis and Applications K. B. Hannsgen, T. L. Herdman, H. W. Stech, and R. L. Wheeler, Volterra and Functional Differential Equations N. L. Johnson, M. J. Ka/laher, and C. T. Long, Finite Geometries: Proceedings of a Con­ ference in Honor of T. G. Ostrom G. /. Zapata, Functional Analysis, Holomorphy, and Approximation Theory S. Greco and G. Valla, Commutative Algebra: Proceedings of the Trento Conference A . V. Fiacco, Mathematical Programming with Data Perturbations II J.-B. Hiriart-Urruty, W. Oettli, and J. Stoer, Optimization: Theory and Algorithms A. Figa Ta/amanca and M. A. Picardello, Harmonic Analysis on Free Groups M. Harada, Factor Categories with Applications to Direct Decomposition of Modules V. I. Istratescu, Strict Convexity and Complex Strict Convexity V. Lakshmikantham, Trends in Theory and Practice of Nonlinear Differential Equations H. L. Manocha and J. B. Srivastava, Algebra and Its Applications D. V. Chudnovsky and G. V. Chudnovsky, Classical and Quantum Models and Arithmetic Problems J. W. Long/ey, Least Squares Computations Using Orthogonalization Methods L. P. de Alcantara, Mathematical Logic and Formal Systems C. £ Aull, Rings of Continuous Functions R. Chuaqui, Analysis, Geometry, and Probability L. Fuchs and L. Sa/ce, Modules Over Valuation Domains

98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115.

116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140.

P. Fischer and W. R. Smith, Chaos, Fractals, and Dynamics W. B. Powell and C. Tsinakis, Ordered Algebraic Structures G. M. Rassias and T. M. Rassias, Differential Geometry, Calculus of Variations, and Their Applications /?.-£. Hoffmann and K. H. Hofmann, Continuous Lattices and Their Applications J. H. Lightbourne III and S. M. Rankin Hi, Physical M athem atics and Nonlinear Partial Differential Equations C. A. Baker and L, M. Batten, Finite Geometries J. W. Brewer, J. W. Bunce, and F. S. Van Vleck, Linear Systems Over Commutative Rings C. McCrory and T. Shifrin, Geometry and Topology: Manifolds, Varieties, and Knots D. W. Kueker, E. G. K. Lopez-Escobar, and C. H. Smith, M athematical Logic and Theoretical Computer Science B,-L. Lin and S. Simons, Nonlinear and Convex Analysis: Proceedings in Honor of Ky Fan S. J. Lee, Operator Methods for Optimal Control Problems V. Lakshmikantham, Nonlinear Analysis and Applications S. F. McCormick, Multigrid Methods: Theory, Applications, and Supercomputing M . C. Tangora, Computers in Algebra D. V. Chudnovsky and G. V. Chudnovsky, Search Theory: Some Recent Developments D. V. Chudnovsky and R. D. Jenks, Computer Algebra M. C. Tangora, Computers in Geometry and Topology P. Nelson, V. Faber, T. A. Manteuffei, D. L. Seth, and A. B. White, Jr., Transport Theory, Invariant Imbedding, and Integral Equations: Proceedings in Honor of G. M. Wing's 65th Birthday P. Clement, S. bvernizzi, E. Mitidieri, and I. I. Vrabie, Semigroup Theory and Applications J. Vinuesa, Orthogonal Polynomials and Their Applications: Proceedings of the International Congress C. M. Dafermos, G. Ladas, and G. Papanicolaou, Differential Equations: Proceedings of the EQUADIFF Conference E O. Roxin, Modern Optimal Control: A Conference in Honor of Solomon Lefschetz and Joseph P. Lasalle J. C. Diaz, Mathematics for Large Scale Computing P. S. Milojevic, Nonlinear Functional Analysis C. Sadosky, Analysis and Partial Differential Equations: A Collection of Papers Dedicated to Mischa Cotlar R. M. Shortt, General Topology and Applications: Proceedings of the 1 9 8 8 Northeast Conference R. Wong, Asymptotic and Computational Analysis: Conference in Honor of Frank W . J. Olver's 65th Birthday D. V. Chudnovsky and R. D. Jenks, C om puters in M a th e m a tic s W.D. Wallis, H. Shen, W. Wei, and L. Zhu, Combinatorial Designs and Applications S. Elaydi, Differential Equations: Stability and Control G. Chen, E B. Lee, W. Liftman, and L. Markus, Distributed Parameter Control Systems: New Trends and Applications W. N. Everitt, Inequalities: Fifty Years On from Hardy, Littlewood and Polya H. G. Kaper and M. Garbey, Asymptotic Analysis and the Numerical Solution of Partial Differential Equations O. Arino, D. E. Axelrod, and M. Kimmel, Mathematical Population Dynamics: Proceedings of the Second International Conference S. Coen, Geometry and Complex Variables J. A. Goldstein, F. Kappel, and W. Schappacher, Differential Equations with Applications in Biology, Physics, and Engineering S. J. Andim a, R. Kopperman, P. R. Misra, J. Z. Reichman, and A. R. Todd, General Topology and Applications P Clement, E Mitidieri, B. de Pagter, Semigroup Theory and Evolution Equations: The Second International Conference K. Jarosz, Function Spaces J. M. Bayod, N. De Grande-De Kimpe, and J. M artinez-M aurica, p-adic Functional Analysis G. A. Anastassiou, Approximation Theory: Proceedings of the Sixth Southeastern Approx­ imation Theorists Annual Conference R. S. Rees, Graphs, Matrices, and Designs: Festschrift in Honor of Norman J. Pullman G. Abrams, J. Haefner, and K. M. Rangaswamy, Methods in Module Theory

14 1 . 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160.

G. L. Mullen and P. J.-S. Shiue, Finite Fields, Coding Theory, and Advances in Communications and Computing M . C. Joshi and A. V. Balakrishnan, Mathem atical Theory of Control: Proceedings of the Inter­ national Conference G. Komatsu and Y. Sakane, Complex Geometry: Proceedings of the Osaka International Con­ ference /. J. Bakelman, Geometric Analysis and Nonlinear Partial Differential Equations T. M abuchi and S. M ukai, Einstein Metrics and Yan g-M ills Connections: Proceedings of the 27 th Taniguchi International Symposium L. Fuchs and R. Gobel, Abelian Groups: Proceedings of the 1991 Curacao Conference A. D. Pollington and W. Moran, Number Theory with an Emphasis on the M arkoff Spectrum G. Dore, A. Favini, £. Obrecht, and A. Venni, Differential Equations in Banach Spaces T. West, Continuum Theory and Dynamical Systems K. D. Bierstedt, A. Pietsch, W. Ruess, and D. Vogt, Functional Analysis K. G. Fischer, P. Loustaunau, J. Shapiro, E L Green, and D. Farkas, Computational Algebra K. D. Elworthy, W. N. Everitt, and E B. Lee, Differential Equations, Dynamical Systems, and Control Science P.-J. Cahen, D. L. Costa, M. Fontana, and S .-E Kabbaj, Commutative Ring Theory S. C. Cooper and W. J. Thron, Continued Fractions and Orthogonal Functions: Theory and Applications P. Clement and G. Lumer, Evolution Equations, Control Theory, and Biomathematics M. Gyllenberg and L. Persson, Analysis, Algebra, and Computers in Mathem atical Research: Proceedings of the Twenty-First Nordic Congress of Mathematicians W. O. Bray, P. S. Milojevic, and C. V. Stanojevic, Fourier Analysis: Analytic andGeometric Aspects J. Bergen and S. Montgomery, Advances in Hopf Algebras A. R. Magid, Rings, Extensions, and Cohomology N. H. Pavel, Optimal Control of Differential Equations Additional Volumes in Preparation

rings, extensions, and cohomology proceedings of the conference on the occasion of the retirement of Daniel Zelinsky

edited by Andy R. Magid The University o f Oklahoma Norman, Oklahoma

CRC Press Taylor & Frands Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 First issued in hardback 2017 © 1994 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN 13: 978-1-138-40205-8 (hbk) ISBN 13: 978-0-8247-9241-1 (pbk) This book contains information obtained from authentic and highly regarded sources. Reason­ able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity o f all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, M A 01923, 978-750-8400. CCC is a not-for-profit organiza­ tion that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system o f payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Library of Congress Cataloging-in-Publication Data Rings, extensions, and cohomology : proceedings o f the conference on the occasion o f the retirement o f Daniel Zelinsky / edited by Andy R. Magid. p. cm. — (Lecture notes in pure and applied mathematics; v. 159) "Papers delivered at a Conference on Rings, Extensions, and Cohomology, held at Northwestern University, August 29 and 30, 1993"—Galley. Includes bibliographical references and index. ISBN 0-8247-9241-6 1. Ring extensions (Algebra)—Congresses. 2. Galois theory—Congresses. 3. Homology theory—Congresses. I. Zelinsky, Daniel. II. Magid, Andy R. III. Conference on Rings, Extensions, and Cohomology (1993 : Northwestern University, Evanston, 111.) IV. Series. QA247.R5752 1994 512\4—dc20 94-11105 CIP

Preface

The papers in this volume were delivered at a conference on Rings, Extensions, and Cohomology, held at Northwestern University, 1993, on the occasion of the retirement of Daniel Zelinsky. In a few cases, the contributors were unable to attend the conference in person and submitted their papers for this volume only. The subjects covered range across commutative and noncommutative ring theory, especially separability and Galois theory, including Lie algebra and module theory: topics which have been the core of the mathematical interests of Daniel Zelinsky, as well as his students, and their students, who are represented here. The papers of T. Kambayashi, A. Magid, and A. Simis et al deal primarily with commutative ring theory, while those of S. Amitsur, D. Haile, M. Rosset and S. Rosset, and A. Rosenmann are primarily in noncommutative ring theory, and W. Brown’s is in both. A. Fauntleroy’s paper is on algebraic geometry, extending some commutative algebra methods to sheaves. The papers of L. Childs, H. Kreimer, and T. McKenzie are in the general area of separability and Galois theory for com­ mutative rings, and those of R. Alfaro and G. Szeto, M. Beattie, and S. Ikehata and G. Szeto deal with generalizations of Galois theory. J. Wehlen’ s paper consid­ ers certain noncommutative algebras over commutative rings. L. Kadison’s paper studies an important case of noncommutative separability. The papers of J. Bergen and R Grzeszczuk, R. Dahlberg, and G. Nelson are on aspects of Lie theory and that of M. Goddard on module theory. Doctoral students of Daniel Zelinsky contributing papers to this volume are William C. Brown, T. Kambayashi, Andy Magid, and Shmuel Rosset. Doctoral students of doctoral students of Daniel Zelinsky contributing papers are Amnon Rosenmann, student of Shmuel Rosset, and Randall Dahlberg and Gray don Nelson, students of Andy Magid. All the conference attendees noted the vitality of the subject, not only among the veteran scholars who spoke but also among the new and recent PhDs who presented papers, a vitality matched by that of the conference honoree. On behalf of all the contributors, we dedicate this volume to our friend and teacher, Daniel Zelinsky.

Andy Magid 111

Contents

Preface

iii

Contributors

vii

Daniel Zelinsky: An Appreciation Andy R. Magid

ix

The Centralizer on //-Separable Skew Group Rings Ricardo Alfaro and George Szeto

1

Contributions of PI Theory to Azumaya Algebras S. A. Amitsur

9

Cocycles and Right Crossed Products M. Beattie

19

Engel-type Theorems for Lie Color Algebras Jeffery Bergen and Piotr Grzeszczuk

31

Constructing Maximal Commutative Subalgebras of Matrix Rings William C. Brown

35

Galois Extensions over Local Number Rings Lindsay N. Childs

41

Infinite Extensions of Simple Modules over Semisimple Lie Algebras Randall P. Dahlberg

67

Smoothing Coherent Torsion-free Sheaves Amassa Fauntleroy

87

Projective Covers and Quasi-Isomorphisms Mark A. Goddard

93

On Dihedral Algebras and Conjugate Splittings Darrell E. Haile

107

On H-Skew Polynomial Rings and Galois Extensions Shuichi Ikehata and George Szeto

113

vi

Preface

Separability and the Jones Polynomial Lars Kadison

123

A Note on Grdbner Bases and Reduced Ideals T. Kambayashi

139

Bicomplexes and Galois Cohomology H. F. Kreimer

143

Adjoining Idempotents Andy R. Magid

155

Separable Polynomials and Weak Henselizations Thomas McKenzie

165

Faithful Representations of Lie Algebras over Power Series Graydon Nelson

181

Idealizers o f Fractal Ideals in Free Group Algebras Amnon Rosenmann

195

Elements o f Trace Zero in Central Simple Algebras Myriam Rosset and Shmuel Rosset

205

Canonical Modules and Factorality of Symmetric Algebras Aron Simis, Bemd Ulrich, and Wolmer V. Vasconcelos

213

Splitting Properties o f Extensions of the Wedderbum Principal Theorem Joseph A. Wehlen

223

Index

241

Contributors

RICARDO ALFARO

University of Michigan-Flint, Flint, Michigan

S. A. AMITSUR Hebrew University, Jerusalem, Israel M. BEATTIE Mount Allison University, Sackville, New Brunswick, Canada JEFFREY BERGEN DePaul University, Chicago, Illinois WILLIAM C. BROWN

Michigan State University, East Lansing, Michigan

LINDSAY N. CHILDS York

State University of New York at Albany, Albany, New

RANDALL P. DAHLBERG

Allegheny College, Meadville, Pennsylvania

AMASSA FAUNTLEROY North Carolina State University, Raleigh, North Carolina MARK A. GODDARD The University of Akron, Akron, Ohio PIOTR GRZESZCZUK Poland

University of Warsaw, Bialystok Division, Bialystok,

DARRELL E. HAILE Indiana University, Bloomington, Indiana SHUICHIIKEHATA

Okayama University, Okayama, Japan

LARS KADISON Roskilde University, Roskilde, Denmark T. KAMBAYASHI

Tokyo Denki University, Hatoyama, Saitama-ken, Japan

H. F. KREIMER Florida State University, Tallahassee, Florida ANDY R. MAGID University of Oklahoma, Norman, Oklahoma THOMAS McKENZIE

Bradley University, Peoria, Illinois

GRAYDON NELSON University of Oklahoma, Norman, Oklahoma AMNON ROSENMANN University of Essen, Essen, Germany

Contributors MYRIAM ROSSET Bar-Ilan University, Ramat Gan, Israel SHMUEL ROSSET Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel ARON SIMIS Universidade Federal da Bahia, Salvador, Bahia, Brazil GEORGE SZETO Bradley University, Peoria, Illinois BERND ULRICH Michigan State University, East Lansing, Michigan WOLMER V. VASCONCELOS

Rutgers University, New Brunswick, New Jersey

JOSEPH A. WEHLEN Computer Sciences Corporation, Integrated Systems Division, Moorestown, New Jersey

Daniel Zelinsky: An Appreciation It has fallen to me, on this occasion of his retirement from Northwestern Uni­ versity after almost four and a half decades of service in the Department of Math­ ematics, to offer a few comments on Dan Zelinsky: The Scholar and the Teacher, So Far. Actually, this is going to be a pretty seamless retirement. Dan will be back teaching at Northwestern in the upcoming academic year, as he has every fall since 1949. Nonetheless, retirements are significant milestones, and the obligatory speech at the retirement banquet is a significant responsibility, so before assuming it I will relate a story Dan told me when I was a graduate student. This involves the great French mathematician Jean Dieudonne, who was on the Northwestern faculty when Dan Zelinsky was a young faculty member. Dieudonne had written a one line review for Math Reviews the gist of which was “Lemma 1.1 is wrong and the rest of the paper depends on it” . Dan looked up the paper, and found that while indeed Lemma 1.1 was false in the standard terminology, if one used the terminology as the author had introduced it in Definition 1.0, the lemma, and the rest of the paper, was correct, and at least moderately interesting. Zelinsky pointed this out to Dieudonne, who made some mild comment about perhaps writing a replacement review. Zelinsky was worried that this was insufficient to repair the damage to the author’s career, to which Dieudonne replied “Look, if the man is a real mathematician, then he’ll write other papers, and the review won’t matter. And if he’s not a real mathematician, it doesn’t matter anyway.” I like this story for what it says about Dieudonne, for what it says about Zelinsky, and also for the principle it espouses, which in this context means that at the retirement banquet for a real mathematician, like the present occasion, it doesn’t matter whether one gets the speech right or not. Thus relieved of that reponsibility, I’ll begin with yichus, which is a Jewish word meaning Ph.D. genealogy. American universities began awarding the doctorate in the last third of the nineteenth century. The mathematics professor at Yale at that time was Hubert Anson Hewton, who supervised the thesis of Eliakim Hast­ ings Moore, who taught briefly at Northwestern before going to the University of Chicago, where he supervised the thesis of Leonard Eugene Dickson, who supervised the Chicago thesis of Abraham Adrian Albert, who supervised the 1946 Chicago thesis of Daniel Zelinsky. At each link in this chain, by the way, the son is one of the most distinguished students of the father; in Dan’s case my reference for this is Nathan Jacobson’s 1974 AMS Bulletin obituary of Albert. In fact Daniel Zelinsky’s professional mathematical career began before his Ph.D. with his wartime service in the A M G -C (Applied Mathematics Group, Columbia). According to the personnel list in Saunders MacLane’s historical memoir about this group, Dan was their youngest member. From an address at a banquet in honor of Daniel Zelinsky, Evanston, Illinois, August 29, 1993

Daniel Zelinsky: An Appreciation

Dan Zelinsky’s published work spans four decades and ranges across commutative and noncommutative rings, topological methods in algebra, and cohomology, with Galois theory and Brauer groups being recurring themes. An especially significant chunk is his 10 paper collaboration with Alex Rosenberg from the m id-1950’s to the early 1960’s, including a paper in the famous Dimension Club series in the Nagoya Journal (On the Dimensions of Modules and Algebras VIII), in which they were joined by Samuel Eilenberg. That particular authorial trio - Eilenberg, Rosenberg, and Zelinsky - is not only an euphonious joy to recite, but of course significant contributors to postwar algebra. Dan’s 1953 American Journal paper “Linearly compact modules and algebras” is still regularly cited, as I discovered with a brief search under “ Zelinsky” in the Sci­ ence Citation Index. In fact it is instructive to look at the index of that 1953 volume, where one finds the names Baer, Buck, Borel, Carlitz, Cassels, Conrad, Dieudonne, Eilenberg, Goldhaber, Hartman, Herstein, Kaplansky, Kolchin, MacLane, Mackey, Rosenberg, Rosenlicht, Thrall, and Wintner, before reaching Zelinsky. One won­ ders if any current journal will be able to match that for name recognition 40 years from now. Dan also found time to write a widely used textbook on linear algebra, which allows me to move on to some comments about Daniel Zelinsky as a teacher. It was 50 years ago this fall that Dan began his classroom career as a University of Chicago Mathematics Instructor, which he was again in 1946-47. In fact the Department of Mathematics schedule for that year, a copy of which was retrieved by Paul Sally, reveals that for the autumn quarter Dan Zelinsky taught Math 101c at 8:30 and Math 215c at 9:30 every day except Wednesday, with similar assignments at those days and times for the winter and spring quarters also. After spending the next two academic years at the Institute for Advanced Study, Dan accepted a position at Northwestern University for the academic year 1949-50, and there he has remained ever since. A conservative order of magnitude estimate is that he has taught 300 classes at Northwestern, of which I took one. I’d like to comment on that (possibly unrepresentative) sample. I refer to second year graduate algebra, Math D32, in academic year 1966-67. Dan’s teaching style that year ran to the simple graphic. Here’s an example. Sup­ pose we were studying the theorem that a projective module over a local ring is free. Dan would state the theorem in telegraphic style: T h . R local w/ M . P free

proj

Then he would proceed with the proof. One implication here (=^) is obvious and would be so noted with implication symbol and a checkmark. The discussion of the other implication (4=) would begin by entering that symbol in the proof, and then mathematical details would be presented. That particular year, while Dan was doing the details, which were generally all verbal, he would illustrate by sketching to the side on the chalkboard a tight spiral, whose length was dictated by the complexity of the proof. Once achieved, that would be noted by a checkmark after the implication symbol also. Thus finally the chalkboard would display the proof:

Daniel Zelinsky: An Appreciation

Prf.

=> End(5/*) induced by the action of 5 * G on S is a ring isomorphism. Given a ring S and the skew group ring 5 * G , the centralizer of S in S'*G, denoted by A, is the set of all elements in S *G which commute with every element of S'. The group G induces an action on the skew group ring by conjugation; furthermore since A is G-invariant, it induces an action on A. It is easy to see that the fixed ring A G is the center of S *G , denoted by G. The first relation between the H -separability of the skew group ring and the centralizer is given by the following theorem, which has appeared in [Alf], but we give here a different proof using H -systems as defined in [NS75]. The extension A over B is if-separable if there exist some V{ £ C a (B ) and di £ C a®ba (A ), called an if-system, such that £ * M * = 1 ® 1.

T H E O R E M 1. If A is G-Galois over C, then 5 * G is H-separable over S. Proof Let {a t, &,} be a G-Galois basis for A over C. Let Xig =gbi and yig= g ~1 elements of S*G . We claim that {a i ,^ 2 x ig ® Vig} ls an if-system and so 5 * G is if-separable over S. 9

First we need to show that ^ 2 x ig ® Vig 1S in the centralizer of S in S * G ® s S *G : 9

sh I

X i9 ®

\ges

) = )

= ^>2kbi®k 1sk~xh = Y^kbi®k~1sh = [ y^X ja ® yjQJ sh. A€G

hep

But also, Y l aix ig ® Vig = ^ i,g

if-system for 5 * G over S.

k£Q

®

hg

\g&

J

® 9~l = 1 ® 1, hence we obtained an 9

□

In most of the results about H-separable extensions A of a ring R, it has been shown that the double centralizer property plays a very important role; and it is no surprise that the main results of this work depend on the centralizer of the skew group ring satisfying a double centralizer property. In preparation for it, we now give a general proposition for group actions that must be well known but we couldn’t find in the literature.

Centralizer on //-Separable Skew Group Rings

3

P R O P O S IT IO N 1. Let G be a finite group acting faithfully on a ring R. Assume the ac­ tion is G-Galois and assume RG is a commutative ring; then R satisfies the double centralizer property in R *G , (i.e., C b g {C r g (R)) = R), and hence Z (R *G ) = Z (R )G. Proof. Since R G is commutative and the action is G-Galois , then R is an R G- progener­ ator. Hence Hornrg(R ,R ), and so i?*G , is an Azumaya algebra over RG. But also R is separable over RG (see [Alf]), and hence by [OKI87], R satisfies the double centralizer prop­ erty. Furthermore, if T = C b g {R) then we have Z (R *G ) = T G = Z (T )n T G = Z (T )G = ( C a c W n T f = (R n T )G = Z {R )G. □ The group action of G on S induces an action on the centralizer A by conjugation. Although they are different automorphism groups, we can still say that G acts on A , and we will use the same notation for both groups unless it is neccesary to do otherwise. When this action is faithfull, we can form the skew group ring A *G . Since the fixed ring A G coincides with the center G of 5 * G , proposition 1 gives the following corollary. C O R O L L A R Y 1. If A is G-Galois o v e r C , then A satisfies the double centralizar property in A * G , an d C = Z (S )G. The next result is a direct consequence of a lemma by Ikehata [Ike81], which indicates that every unital left A-module which is a generator as a left B -module, is also a generator as left A-module ( where A is an //-separable extension of B .) P R O P O S IT IO N 2. If the skew group ring S*G is H-separable over S, then S is G-Galois over SG. Proof. By [Ike81], S is a generator as a left S *G-module. Thus by Morita theorem, S*G = End5 G(*S') and S is a finitely generated projective 5 G-module. Hence the action is G-Galois on S. □ Using this proposition, we can now show that for the centralizer A the Galois condition is equivalent to the //-separability condition of the corresponding skew group ring. This result is the analogue in the non-commutative case to the well known fact that if the ring S is commutative, the skew group ring is //-separable over S if and only if S is G-Galois over

T H E O R E M 2. The centralizer A is G-Galois over C if and only if A * G is H-separable over A. Proof. As shown in the proof of proposition 1, A * G is an Azumaya G-algebra; but A * G is free as a left A-module, hence A * G is //-separable over A by [Ike81]. The converse follows now from proposition 2. □

4

Alfaro and Szeto

3 T H E ff-S E P A R A B L E C O N D IT IO N We now study under which conditions the inseparability of the skew group ring S * G over S implies that the commutator A is a G-Galois extension. The fact that the commutator is G-Galois has been used to prove a Noether-Skolem type theorem for group-graded rings by Osterburg and Quinn in [OQ88]. All the notation from the previous section is assumed.

P R O P O S IT IO N 3. Let S *G be H-separable over S. If A satisfies the double centralizer property in A * G , then A is G-Galois over C. Proof. We first show that G acts faithfully on A. The ring S being a direct summand of 5 * G as left 5-modules, satisfies the double centralizer property on 5 * G by [Sug67]. Thus, if g £ G and 9d = d for all dE A, then gd = dg and so g £ C s& ( A ) = 5, forcing g = 1. Since 5*G is separable over 5, there exists a central element in S of trace one [Alf]. Since S satisfies the double centralizer property in 5 * G , then Z (A ) = Z (S ),a n d A G = Z (S )G; but also C&g (C&g (A )) = A, thus Z (A * G ) = Z (A )G. Hence there is a central element in A of trace one, so A * G is separable over A , thus also over A G(= Z (S )G = Z (A * G ).) Therefore A * G is an Azumaya A G-algebra. Furthermore A is a direct summand of A *G as (A-A)-bimodules, thus A is separable over A G and A *G is if-separable over A by [OKI87]. Proposition 2 now says that A is G-Galois over A G(= G.) □ Now we put together this last propposition with the results from the previous section, and we can prove the main result about equivalent conditions for the centralizer of the skew group ring to be a G-Galois extension. This theorem resembles the result in [Ike81] for group actions over commutative rings.

T H E O R E M 3. With all the notation as above; the following are equivalent: (1) (2) (3) (4)

A is G-Galois over C. A * G is H-separable over A. 5 * G is H-separable over 5, and A satisfies the doublecentralizerproperty onA *G . A * G is an Azumaya C-algebra.

Proof. The equivalence of the first two statements is theorem 2. The equivalence of the frist and third statement is theorem 1, corollary 1 and proposition 3. In the proof of proposition 3 we actually proved that (3) implies A * G is an Azumaya G-algebra and this implies A is G-Galois ; thus the equivalence follows immediately. □ Example. Let S be the division ring of quaternions, S ^ R - l + R - i + R - j + R-fc. The group G = { l , i , j , k} acts on S by conjugation. To differentiate between the element of G and the element of S we write i E G, and so on. It is easily seen that the fixed ring is R, and the center of 5 * G is also R. By [Ike81] the skew group ring 5 *G is inseparable over S. The centralizer of S in 5 * G is: A == R •i + Ri •i -f R j •j + Rfc •k. The group G also acts in A by conjugation, and A G = R. As the center of A * G is also R and there is an element of A with trace 1, then A * G is inseparable over A. Hence, by the theorem we know that A is G-Galois over R. In fact, we obtain a Galois basis given by

5

Centralizer on //-Separable Skew Group Rings

{a i = l/2 , o2= (1/2)m, a3 = ( l / 2 ) i 7, a4 = (l/2)kk; 61 = 1/2, b2 = -(l/ 2 )ii, -(l/2)ifejfe}.

63

= —( 1 /2 )jj, b4 =

When the double centralizer property of A cannot be checked directly, it is still possible to obtain the G-Galois condition on the commutator A if there is a particular H -system for 5 * G over S.

T H E O R E M 4. If S *G is H-separable over S', and there exists an H-system E A , u/t-=

with

® g~l , then A is G-Galois over C. geG

For the proof of the theorem we need to show first some conditions satisfied by the elements of C siC&&g {S * G ),:

L E M M A 1. If

a {g,h)9 ® h

€

C s«q®ss>g {S*G ), then:

geG

heG (1) OL(g^-\)gh~l E A for all g,h e G . (2) a ^ g -1) = 9a(hi) f or aU 9 ^ G , Proof o f lemma.. Since &EG, commutes with the elements in C&G8>sS*g (S *G ), then: J2

k a {g,h )kg ® h = Y j

geG

geG

h£G

heG

a (g,h)9 ®

Making a change of indices we obtain: Y

khg, hence ^hgfi^g)-1 ~ ^ (A )

h-1)(l)g~1 — ^g^hifig-1h~l

fig&hi^^g)-1

Therefore, hg Q gh, but the reverse inclusion is trivial, thus (f>hg = gg - ^ © A ®gg = 5^© j ^ A ^ O ^ r 1) = A #-1 = A *G. geG geG geG

Hence A*G = H om c(A , A ), and being A finitely generated as G-module, it is G-Galois .

□

R eferences [Alf]

Ricardo Alfaro. Separabilities and G-Galois actions, to appear in Proceedings of the X X I Ohio State-Denison Conference.

[Alf92]

Ricardo Alfaro. Non-commutative separability and group actions. P u b lica cion s M a te m a tiq u e s , 3 6 :3 5 9 -3 6 7 , 1992.

[Hir69]

Kazuhiko Hirata. Separable extensions and centralizers of rings. N a goya M a th em a tica l Journal,

[Ike81]

Shuichi Ikehata. Note on Azumaya algebras and H-separable extensions. M a th em a tica l J ou rn a l o f

[Ike90]

Shuichi Ikehata. On H-separable polynomials of degree 2. M a th em a tica l Jou rn a l o f O kayam a Uni­

[Ike91]

Shuichi Ikehata. On H-separable polynomials of prime degree. M a th em a tica l J ou rn a l o f O ka ya m a

3 5 :3 1 -4 5 , 1969. O kayam a U niversity, 2 3 :1 7 -1 8 , 1981. versity , 3 2 :5 3 -5 9 , 1990. U n iversity, 3 3 :2 1 -2 6 , 1991. [Kan65] Teruo Kanzaki. On Galois algebra over a commutative ring. Osaka J. o f M a th em a tics, 2 :3 0 9 -3 1 7 , 1965. [NS75]

Taichi Nakamoto and Kozo Sugano. Note on H-separable extensions. H okkaido M a th em a tica l J ou r­ nal, 4 :2 9 5 -2 9 9 , 1975.

[OKI87] Hiroaki Okamoto, Hiroaki Komatsu, and Shuichi Ikehata. On H-separable extensions in Azum aya algebras. M a th em a tica l Jou rnal o f O kayam a U niversity, 2 9:1 0 3-10 7 , 1987. [OQ88] James Osterburg and Declan Quinn. A Noether Skolem theorem for group-graded rings. J ou rn a l o f A lgeb ra , 113 :4 83 -4 9 0, 1988. [Sug67] Kozo Sugano. Note on semisimple extensions and separable extensions. Osaka J. o f M a th em a tics , 4 :2 6 5 -2 7 0 , 1967. [Sug82] Kozo Sugano. On H-separable extensions of two sided simple rings. H okkaido M a th em a tica l Journal, 1 1:2 4 6-25 2 , 1982. [Sug87] Kozo Sugano. On H-separable extensions of primitive rings. H okkaido M a th em a tica l Journal, 1 6 :2 0 7 211, 1987. [Sug90] Kozo Sugano. On H-separable extensions of primitive rings II. H okkaido M a th em a tica l Journal, 1 9 :3 5 -4 4 , 1990.

Contributions of PI Theory to Azumaya Algebras S. A. AMITSUR Hebrew University, Jerusalem, Israel

To Prof. Dan Zelinsky for long friendship.

1. The famous Artin-Procesi theorem states that “ An algebra A of rank n2 is an Azumaya algebra over its center - if and only if - it satisfies all polynomial identities of the matrix ring M „(Z ) over the integers Z, and its simple homomorphic images do not satisfy identities of M n_ i ( Z ) ” . Later proofs show that its suffices to require that A satisfies the Cappelli identity dn2+ i[x, y] = 0, and a certain central polynomial gn[xi V] of

does not vanish

on every simple image of A (e.g., [5] p. 66, [6] p. 100). The “necessity” part, known as the “easy” part, since it follows from known properties of Azumaya algebras, but which require some rather complicated methods of reduction to the noetherization case and henselization. The “sufficiency” part, known as the “difficult” part has now some simple straightforward proofs (e.g., [6] p. 102). Rowen has noticed in [5] (p. 65), that by using simple properties of localizations, and of Pl-theory, one can obtain a fairly easy proof also to the “easy” part, and in fact to get new proofs of the properties of Azumaya algebras. He advocates using Pl-theory to simplify some stages of the theory of Azumaya algebras, and, for example, he proves the existence of a splitting rings in the local case. The theme of this work, is to push further this approach, and to obtain some of the basic properties of Azumaya algebras, avoiding the reduction to noether and henselization.

2. up-central identities Our interest centers around universal properties of matrices, which can be expressed by identities and central identities of matrices M n(Z). A central polynomial z >y"] is a up-central polynomial (which we shall denote (ii) Let A\ = A^[ax] be the localization at il>[a\] o f A, then A —►IIA\ is an injection, which maps also cent (A ) —» II cent(AA). PROOF: Let a E Ker(A —►IIA\) then \j)[a\]ma = 0 and we can choose the same m for all A. Then clearly 1 •a = (]C ^[ga])* a = 0 for some large k. The rest is evident. 3 APPLICATION. Let A be an algebra which satisfies:

PI Theory and Azumaya Algebras

13

(i) The cappelli identity dn2+1 [x, y] = 0 (ii) The polynomial 8 n [x; y] be central in A, and up-central in all its simple images. We shall write 8 [x, y] and ommit the n. Condition (ii) is equivalent to (ii)’ For every maximal ideal M in A, A/M is a central simple algebra o f dimension n2. Next theorem, which is a major tool in the theory of Azumaya algebras is proved here from basic properties of the polynomial identities of M n(Z): THEOREM

3.1. Let A satisfy (i), (ii), and R = cent(A). If for some substitution p [x ,y ,z ]

in (2.3.3) is invertible in A, then a) The element z € A is integral o f degree n over the center R. b) A is a free over S = R[z] o f dimension n. c) S = R[z] is a separable extension o f R, and S is a maximal commutative subalgebra o f A. d) A is embedded in M n(S ); moreover A ® S = M n(S) (i.e. S splits A). Hence A satisfies R

all polynomial identities o f M n{Z). Proof:

(a) From the definition of (p in corollary 2.3.2, it follows 8 [zl ,XjZx, y] = 8 \ is also invertible in R as well as all the other factors of p. Hence for x\ = 1 ,u — z n we have by ( 2 .2 . 1 ): n

6 i z n = 'Y^Xj^i[z\

i= 1 and (fj[z\ are polynomials in R[z] of degree < n. Now (S^-1 € R , and so z n — ip\[z\ = n

n

Y x j tP j[z\- Taking the commutator [z, —] on both sides yields 0 = Y [z >x j](Pjlz] but j= 2 j= 2 by 2.3.3. we require that 8 [zx, [z,X j]zx, y f] 7^ 0, and so {[z ,X j]z x] are R -independent, hence all p>i[z] = 0 , and so z n — z] = [x, [y,z\] — e(g,h)[y, [x,z\] for all x € Lg) y £ Lh, and z € L. The elements of U 9€g Lg are known as the homogeneous elements of L. We will say that L is nilpotent if there exists a positive integer M such that [xm,

X\, . . . ,

xm

G L. 31

[x

m

-

i> [* *• »[x 2 >%i] *••]]] — 0 f° r all

32

Bergen and Grzeszczuk

A K - vector space V is an L-module if there is a vector space homomorphism ip : L — ► End,K{V) such that ip{[x,y]) = ip(x)ip(y) - t(g, h)ip(y)tp(x) for all x G Lg and y G L/*. Our main result will follow from

P R O P O S IT IO N 1 . Let A be a finite-dimensional subspace o f En dK( V) spanned by a set S such that for every a, 6 G S there exists a = a(a,b) G K* such that ab — aba G S. If every element o f S is a nilpotent transformation o f V , then A acts nilpotently on V . That is, there exists an integer N such that the composition o f any N transformations from A is 0. Proof. For any subset B of A, we will let sp(B) denote the subspace o f A spanned by B. Consider all subsets T of S with the properties that T acts nilpotently on V and for every a,b G T, there exists a = a(a,b) G K* such that ab — aba G sp(T). Without loss, we may assume that S contains 0, therefore such subsets certainly exist. From among all these subsets, choose one, U , such that the dimension of sp(U) is maximal. It suffices to show that sp(U) = A. If a, ai, a 2 , . . . , an G S we can inductively define F 0(a) = a, F i(a ,a i) = aa\—a(a,ai)a\a, and if t = Fn_ i(a ,a i, fl2 , •••, an- 1 )> then Fn(a ,a i, apply Proposition 1 with A = ip{L)

33

Engel-Type Theorems for Lie Color Algebras

and S = ^ (U geG Lg). Note that S satisfies the hypotheses of the proposition, since for all

x £ Lg and y £ Lh, ^ (x )^ (j/) - c(p,h)^(i/)^(® ) = ^([®,y]) € S.

□

We can now use Theorem 2 to prove the Lie color algebra analog of Engel’s theorem.

C O R O L L A R Y 3. If L is a finite-dimensional Lie color algebra such that adx is nilpotent for every homogeneous x € L, then L is nilpotent. Proof. Consider the map ip : L — ►EndK(L) defined as ip(l) = ad/, for all I £ L. The identity = [x,[y,z]] - e(p, h)[y, [x,z]] for all x £ Lg, y £ L/*, and z £ L implies that ip([x,y]) = ip(x)xp(y) — e(g,h)xp(y)ip(x). Thus L is an L-module and, by Theorem 2, there is an integer N such that the composition of the action of any N elements of L is 0 on L. Therefore [a?jv, [x n - i , [••• , [xi, x] •••]]] = 0 for all x, x \, . . . , xn € L, thus L is nilpotent. □

ACKNOW LEDGM ENTS The first author was supported by the University Research Council at DePaul University. Both authors were supported by KBN Grant 2 2012 91 02. Much of this work was done when the first author was a visitor at University of Warsaw, Bialystok Division and he would like to thank the University for its hospitality.

REFERENCES 1. J. Bergen and S. Montgomery, Smash products and outer derivations, Israel J. of Math.

53 (1986), 321-345. 2. M. Cohen and S. Montgomery, Group graded rings, smash products, and group actions, Trans. A.M.S. 282 (1984), 237-258. 3. D. Fischman and S. Montgomery, Biproducts of braided Hopf algebras and a double centralizer theorem for Lie color algebras, to appear. 4. I. Kaplansky, Lie Algebras and Locally Compact Groups, Chicago Lectures in Mathe­ matics, University of Chicago Press, Chicago, 1971. 5. V. K. Kharchenko, Automorphisms and Derivations of Associative Rings, Mathematics and its Applications, Soviet Series, vol. 69, Kluwer Academic Publishers, 1991. 6. S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Mathematics, vol. 818, Springer-Verlag, 1980. 7. S. Montgomery, Hopf Algebras and Their Actions on Rings, Conference Board of the Mathematical Sciences, American Mathematical Society, Providence, Rhode Island, 1993.

C onstructing M axim al Com m utative Subalgebras o f M atrix R ings WILLIAM C. BROWN Michigan State University, East Lansing, Michigan

1

Notation and History

Throughout this paper, k will denote an arbitrary field. By a ^-algebra, we will mean an associative fc-algebra with 1 ^ 0 . Algebra homomorphisms are assumed to take 1 to 1 and subalgebras are assumed to have the same 1 as the containing algebra. We will let MpXq(k) denote the set of all p x q matrices with entries from k. When p = q = n, the ^-algebra MnXn(k) will be denoted by M n(k). We will assume throughout that n > 2. We will let M n(k) denote the set of all maximal, commutative &-subalgebras of M n(k). Thus, R E M n(k) if and only if R is a commutative A;-subalgebra of M n(k) with the following property: If S is any commutative fc-subalgebra of M n{k) such that R C 5, then R = S. If C(*) denotes the centralizer of * in Afn(fc), then clearly a commutative &-subalgebra R C Mn(k) is an element of A4n(k) if and only if C(R) = R. The basic problem concerning commutative subalgebras of M n(k) is to classify up to isomorphism all R E A4n(k). Suppose R E A i n{k). Then dimk(R) < n2. In particular, R is a (commutative) artinian ring. Thus, R = i?i x •••x Rp where each Ri is a local ring (i.e., a commu­ tative ring containing precisely one maximal ideal). Let V = M\xn(k). Then V is a finitely generated, faithful (right) i?-module for which Hom#( V, V) = C(R) = R via the regular representation. As i?-modules, V = V Ri X - - - x VRP and each V Ri is a finitely generated, faithful R{-module. The regular representation induces an isomorphism Ri = Horn/?t(K', K) for each i = 1, . . . ,p. Thus, Ri can be identified with a maximal, commutative fc-subalgebra of Mni(k) where rat- = dimk{Vi). Hence, each maximal, commutative &-subalgebra of Mn(k) is a finite product of local al­ gebras which are maximal, commutative fc-subalgebras of possibly smaller matrix rings. For this reason, most papers which deal with the classification of algebras 35

36

Brown

in M.n{k) assume that the algebra is local. In this note, we present two interesting constructions of local ^-algebras in M n(k) which have residue class field k. We will use the notation (*, **, k) to indicate that * is a commutative ^-algebra which is local with unique maximal ideal ** and residue class field k. If (i?, J, k) is such an algebra and R E M n(k), then we will write (i?, J, k) E M n(k). If (i?, J, k) is a local &-subalgebra of Mn(fc), then J is nilpotent. Let i( J) denote the index of nilpotency of J. Since n > 2, (i?, J, &) E -Mn(&) implies J ^ (0). Thus, i(J) > 2 whenever (i?, J, k) E A fn(&). The study of local A;-algebras in M n{k) has a long history dating back at least to 1905 (See [6]). Here is a sampling of some of the more famous results about

M n(k). (A ) Ifk = C, the complex numbers, and n < 6; then there are only finitely many conjugacy classes of (R, J, C) E A4n(C). These classes are all known and completely described in [7]. Thus, if (i£, J, C) E M n(C) and n < 6, then R is conjugate (and hence isomorphic) to only one of finitely many algebras which are listed in [7]. In particular, the problem of classi­ fying local algebras in A4n(C) is completely solved for all n < 6. If n > 7, then we have quite a different sort of result.

(B) Ifk is infinite and n > 6, then there are infinitely many pairwise nonisomor­ phic (i?, J, k) E M n(k). In fact, there are infinitely many such (i£, J, k) in (B) with i(J) = 3. Proofs of these assertions can also be found in [7]. At any rate, the problem of constructing local algebras in M n(k) for large n is an interesting one which is still an active area of research. A more specialized problem than classifying local algebras in M n(k) is the study of their dimensions. This problem also has a rich history. The best general result to date is as follows: (C) If R e M n(k), then (2n)2/ 3 - 1 < dimk(R) < [n2/ 4] + 1. In (C), [n2/A] denotes the greatest integer less than or equal to n2/ 4. The upper bound in (C) was first proven by I. Schur for k = C in [6]. The general argument for any k was given by W. Gustafson in [4]. This upper bound is sharp as Example 1 below indicates. The lower bound in (C) was proven by T. J. Laffey in [5]. Laffey has also shown that dimk{R) > [3n2/ 3 — 4] when J(R)3 = (0). Here J(R) denotes the Jacobson radical of R. This bound is known to be sharp for infinitely many n. A more detailed history of this subject can be found in [2]. We finish this short history with two examples which are relevant to the constructions in the next section.

Maximal Commutative Subalgebras of Matrix Rings

37

Example 1: (Schur Algebras) Let 2 < n = r + s with r,s natural numbers such that \r — s| < 1. Let R be the following set of matrices in Mn(k). Xlr

R =

(i)

0

x e k,

xL

ze

M rxs(k)

Throughout this paper, Ia will denote the identity matrix of size a x a. R is clearly a commutative fc-subalgebra of Mn(k) of dimension dim^(i?) = 1 + rs = [n2/ 4] + 1. R is local with Jacobson radical J consisting of those matrices in (1) with x = 0. Thus, i( J) = 2. It is easy to check that (7?, J, k) E M n(k). ■ The algebra given in Example 1 has dimension as large as possible by (C). In honor of Schur’s early contributions to the results in (C), we call the algebra constructed in Example 1 a Schur algebra. Notice that any Schur algebra has dimension greater than or equal to n. For many years, it was conjectured that (72, J, k) E M n(k) implies dim^(72) > n. For example, if i(J) = 2, then dimk(R) > n. This was proven in [3]. In 1965, R.C. Courter gave the first example of a local (72, J, k) E M n(k) with dimfc(72) < n. For our purposes, Courter’s example can be described as follows: Example 2: (Courter) Let n = 14. Set 72 = J © I^k where J is the set of all 14 x 14 matrices of the following form: 0

0

0

0

*11

0

0

*11

*12

0

0

*12

*21

0

0

*21

*22

0

0

*22

Z 11

Zl2

Z 21

Z22

0

(2)

2X10

o

10x10

2/n 2/12 ■^11 ^12 ^21

^22

0

0

0

0

Zn

Zl2

2/21

3/22

0

0

0 ^21

0

Xu

X\2

Z22

#21

£ 22

In (2), the £,j’s , ' s and ztJ’s are arbitrary elements from k. OpXq denotes the zero matrix of size px q. J is a fc-subspace of Mu(k) which is closed under multiplication

38

Brown

and consists of matrices which are nilpotent of index at most 3. Thus, (5 , J, k) is a local, fc-subalgebra of M\^{k) with dimk(R) = 13 and i(J) = 3. In [3], Courter showed (5 , J, k) £ Mu{k). ■ It follows from results in [3] and [5] that Courter’s example is the smallest example with respect to n or i(J) for which dimk(R) < n. In the next section of this paper, we will give a general construction which gives Courter’s Example and many other interesting examples with dimk(R) < n as special cases.

2

The Constructions

If R £ -Mn(&), then V = M iXn(k) is a finitely generated, faithful 5-module for which Horn^V, V) = R via the regular representation. Conversely, suppose R is a finite dimensional, commutative ^-algebra, V is a finitely generated, faithful 5-module and Hom#(V, V) = R via the regular representation. Then R is iso­ morphic to a maximal, commutative &-subalgebra of Mn(k) where n = dim^(Vr). Thus, constructing maximal, commutative A?-subalgebras of M n(k) [for various n\ is equivalent to constructing pairs (5, V) where R is a finite dimensional, commu­ tative ^-algebra, V is a finitely generated, faithful i?-module and Hom/^V, V) = R via the regular representation. In this section, we will give two different procedures for constructing such pairs. Suppose B is a commutative ring and TV is a 5-module. Let r be a positive integer. We will let N r denote the direct sum of r copies of N and B K N r the idealization of the B module N r. Suppose (6, ni , . . . , nr) denotes a typical element of B IX N r. Thus, b £ B and ni , . . . , nr £ TV. Then B t< N r is a commutative ring with addition and multiplication defined in the usual ways: (3)

( 6, ^i, . . . ,nr) + ( 6 ' , ^ , . . . , ^ ) = (6 + 6',ni + n [ , . . . , n r + n'). (6, ni , . . . , nr)(b', n'x, . . . , nfr) = (66', ni6' + n[b, . . . , nr6' + nlrb).

If B is a ^-algebra, then TV is a k-vector space via the embedding of k into B and B K N r is a ^-algebra via (6, rci,. . . , nr)x = (6x, n\x, . . . , nrx) for x £ k. Consider the B module B r © TV. We will let (6i,. . . , 6r, n), 6Z - £ 5 , n 6 TV, denote a typical element in B r © TV. It is easy to check that B r © TV is a B K TVrmodule with scalar multiplication defined by r

(4)

(£>i,. . . 6r,n ) ( 6 , n 1, . . . , n r) -

(b ib ,. . . , brb, nb + J 2 n ibi)l—l

If TV is a finitely generated (faithful) 5-module, then B r © TV is a finitely generated (faithful) 5 K TVr-module. The following theorem is proven in [2].

39

Maximal Commutative Subalgebras of Matrix Rings

(D ) Suppose B is a commutative ring and TV is a faithful B-module. For any positive integer r, V = B r © TV is a faithful R = B X N r-module, and Hom/^V, V) = R via the regular representation. If B is a finite dimensional fc-algebra and TV is a finitely generated 5-module, then (D) implies R = B x N r E M p(k) where p = dim^i?7,® TV). If B — (5 , m, k) is a local fc-algebra, then R is also local with Jacobson radical J — m tx N r and residue class field k. Notice i(J) = i(m) + 1. The construction in (D) can be used to produce local fc-algebras (5, J, k) E M p(k) with dimk(R) < p• Suppose (5,m,fc) is a finite dimensional, local kalgebra and TV is a finitely generated, faithful 5-module. Then (R = 5 D< N r, J = m X N r, k) is a local fc-algebra in M p(k) where p = dim/c(5r ® TV). Furthermore, p — dimk{R) = (r — l)[dimjk(5) — dimjk(iV)]. Hence, if r > 2 and dim^(5) > dimk{N), then dim^(5) < p. For example, suppose 5 is a Schur algebra given in Example 1. Then (5,ra,fc) E M n(k) with i(m) — 2 and dim^(5) = [n2/4] + l. Let N = M iXn(k). Then is a finitely generated, faithful 5-module and dimA;(5) > dimA:{N) whenever n > 4. Thus, (R = B X N r,J = m X N r,k) E M p(k) where p = rdimk(B) + n. If r > 2 and n > 4, then dim^i?) < p. If r = 2 and n = 4, then the reader can easily check that 5 X iV2 is isomorphic to the algebra given in Example 2. Thus, Courter’s Example can be constructed from a Schur algebra using (D). The construction (5,7V) —> (R,V) given in (D) is called a Ci-construction in [1]. There is a second construction given in [1] which produces local algebras in M n{k) which in general are not Ci-constructions. We briefly sketch this second construction. Suppose ( 5, m, k) is a finite dimensional, local fc-algebra. We assume m ^ (0). Let N be a finitely generated, faithful 5-module such that HomJ g(Ar, N) = 5 via the regular representation. We have seen that every local ^-algebra (5, m, &) E M n{k) determines such a pair (5,7V = M\xn{k)). Let Soc(5) denote the socle of 5 . Then there exists a z E Soc(5) such that dimfc(7Vz) = 1. A proof of this easy assertion can be found in [1], Fix z E soc(5) with dim^TVz) = 1. Let X be an indeterminate over 5 . Let B\ and N\ denote the following kalgebra and 5-module respectively:

= (mXBX * - zy

K =

In Equation (5), (mX, X 2 — z) denotes the ideal in B[X\ generated by m X and X 2 — z. Let x denote the image of X in B\. Then B\ — B[x\. Every element in 5 i can be written uniquely in the form b + ax for some b E 5 and a E k. B\

40

Brown

is a finite dimensional, local fc-algebra with Jacobson radical rriB1 = m + kx and residue class field k. Also, ^(m^) = max{3,z(m)}. The B-module N\ becomes a Bi-module when multiplication by x is defined as follows: (n,n'z)x = (nfz,nz). The reader can check that this operation is well defined and that N\ is a finitely generated, faithful Bi-module. The following theorem is proven in [1]. (E) Suppose (J5i, N\) is constructed from (B, N) as above. Then Homjg^TVi, JVi) = Bi via the regular representation. The construction in (E) yields another method for producing local algebras in M p(k). Suppose (B, m, A:) £ M n(k). Set N = MiXn(k). Choose z £ Soc(B) such that dimk(Nz) = 1 and construct the pair (Bi,iVi) given above. (E) implies (BijrriBnk) £ M n+i(k). The construction (B,7V) —* (Bi,iVi) is called a C2construction. Example 11 in [2] is a (^-construction, but not a Ci-construction. Courter’s Example is a Ci-construction, but not a (^-construction. Thus, the two constructions given here are independent of each other and provide new methods for producing maximal, commutative subalgebras of Mn(k).

References [1] W.C. Brown, “Two Constructions of Maximal Commutative Subalgebras of n x n Matrices” , preprint. [2] W.C. Brown and F.W. Call, “Maximal Commutative Subalgebras of n x n Matrices,” Communications in Algebra, to appear. [3] R.C. Courter, “The Dimension of Maximal Commutative Subalgebras of K n,” Duke Mathematical Journal, 32, 225-232 (1965). [4] W.H. Gustafson, “On Maximal Commutative Algebras of Linear Transforma­ tions,” Journal of Algebra 42, 557-563 (1976). [5] T.J. Laffey, “The Minimal Dimension of Maximal Commutative Subalgebras of Full Matrix Algebras,” Linear Algebra and its Applications, 71, 199-212 (1985). [6] I. Schur, “Zur Theorie der Vertauschbaren Matrizen,” J. Reine Angew. Math., 130, 66-76 (1905). [7] D.A. Suprunenko and R.I. Tyschkevich, “Commutative Matrices,” Academic Press, New York, (1968).

Galois Extensions over Local Number Rings LINDSAY N. CHILDS State University of New York at Albany, Albany, New York

To Daniel Zelinsky, with best wishes.

Thirty years ago, generalizing classical Galois theory to extensions of commutative and non-commutative rings was a problem which attracted broad attention.

Professor Zelinsky contributed

several papers to the literature in this area.

(In fact, my

original thesis problem was to generalize a paper on Galois theory of continuous transformation rings by Rosenberg and Zelinsky [RZ55].)

Thus it seems appropriate to offer a paper on

Galois extensions to a proceedings in his honor. The purpose of this paper is to show how formal groups can help give a complete description of the group of H-Galois extensions of the valuation ring R of a finite extension of (D^, when H is a finite abelian p-power rank R-Hopf algebra with connected dual.

Except for the section on embedding rank p group

schemes in dimension one formal groups, the paper is an exposition of 20-year old results which should be more widely known than they seem to be. My thanks to J. Lubin for permission to include his Theorem 3, and to Karl Zimmermann, David Moss and Alan Koch for many stimulating discussions. 41

42

Childs

1.

Basic definitions. Let R be a commutative ring, S a commutative R-algebra which

is a finitely generated projective R-module (= algebra”) . S.

Then

"S is a finite R-

Let G be a finite group of R-algebra automorphisms S is aGalois extension of R with group G if any of six

well-known equivalent conditions (Theorem 1.3 of [CHR65]) hold. Of these, we single out: c)

the R-module map j from the crossed product D(S, G)

= {][

to EndD (S)given by j(s K

O

Is^ in S> u_)(t) = s S ®R H o m ^ R G ,

given by f) is some s

When

R) s Homs (SG, S)

h(s ®t) (o ) = sor(t) is an isomorphism; for any maximal ideal m of S and any o * 1 in G, there in S so that o ( s )

- s * m.

applied to an extension S 2 R of local or global number

rings, this last condition says that for any maximal ideal m of S, the inertia group I m = { o in G| is trivial. If I

m

is trivial, then m is unramified over R.

Thus if S/R is

Galois with group G, then S is unramified over R.

This means

that if S and R are the rings of integers of a randomly chosen Galois extension L 2 K of algebraic number fields with Galois group G, it is unlikely that S will be a Galois extension of R

of

Galois Extensions over Local Number Rings

43

with group G (indeed, if S = Z, never); whereas if S 2 R are localizations at some prime of R of rings of integers of a Galois extension L 2 K of algebraic number fields, then with finitely many exceptions (= primes dividing the discriminant of S/R), S will be a Galois extension of R with group G. Now let H be a cocommutative R-Hopf algebra which is finitely generated and projective as R-module. Hom^(H, R ) , a commutative R-Hopf algebra.

Let H

*

=

Chase and Sweedler

[CS69] extended conditions c) and e) above to finite R-algebras S such that S is an H-module algebra or an H*-comodule algebra, as follows:

c)

the R-module map j: S ® H -» EndR (S) given by j(s ® h) (t)

= sh(t) for all t in S, is bijective; e)

it

if a: S -> S « H , by «(t) =

r-i

) t.,. ® t._., is the

^

li)

Iz )

(t) *

H -comodule structure map on S, then the R-module map h: S ®R S -» S ®R H given by

*

h(s ® t) = (s ® l)a(t) =

£ s*"(l) ® ^(2) (t) *

If Y = Spec(S), X = Spec(R) and G = Spec(H )

;'’s kijective.

(which makes

sense if H is cocommutative), then condition e) for a Galois extension is a translation of the statement that Y -> X is a principal homogeneous space for the finite group scheme G.

Thus

while some of us were thinking (and continue to think) about Galois extensions, algebraic geometers (e.g. [MR70],

[Ro73]) were

44

Childs

independently studying principal homogeneous spaces for group schemes, as we will observe in more detail below. Condition c) is the key to Galois descent.

By Morita

theory, there is an equivalence of categories between the category of left R-modules and the category of left EndR (S)modules, given by the functor S ®R - .

An S-module M therefore

descends, that is, is of the form S ®R M Q for some R-module M Q , iff the S-action on M extends to an action by EndR (S).

Now if

S/R is a Hopf Galois extension with Hopf R-algebra H, then getting an EndR (S)-action on M is equivalent to finding an H-module action on M which is compatible with the S-module action (i.e. so that one gets an action by the smash product S # H ) . This criterion for descent for Galois extensions is a conceptual simplification of the general condition for faithfully flat descent (c.f. [K074] or Waterhouse [Wa79], Chapter 17).

Of

course it also works for modules with structure such as algebras, Hopf algebras, etc., provided the H-module action respects those structures (e.g. is a measuring on algebras). Condition f) for Galois extensions with group G does not extend to Hopf Galois extensions in the sense of requiring S/R to be unramified.

In fact, if S is a Hopf Galois extension of R,

rings of integers of number fields, with Hopf algebra H, then the *

discriminant of S/R = the discriminant of H /R.

This observation

of Greither [Gr92] and others reduces to the unramified condition when S is a Galois extension with group G, for then H s R © R e .. .© R

*

= RG

(|GI copies of R) as R-algebra, hence the

*

Galois Extensions over Local Number Rings

45

discriminant of H* = R = disc(S/R), so no prime of R ramifies in S.

Since wild ramification is possible for Hopf Galois

extensions of number rings, Hopf Galois theory has interesting potential applications in Galois module theory.

See [Ch87],

[Gr92] or [Tay92].

2.

Local Hopf algebras and formal groups. The study of Galois extensions, even with abelian (=

commutative and cocommutative, and finite) Hopf algebras, over valuation rings of local fields, is complicated by the richness of the array of Hopf algebras over such rings.

The rank p case

is understood by the work of Tate and Oort [T070], but even 2

describing Hopf algebra orders inside KC, C cyclic of order p , is difficult: see [Gr92] or [Un94].

Thus it seems to be useful

to approach the subject using formal groups.

Such is the point

of view we will adopt in the remainder of the paper. The starting point is

Oort's Embedding Theorem ([Oo67], (2.4); [Oo74], (3.1)). valuation ring R.

[MR70],

(5.1); [MZ70],

Let K be a finite extension of 0^, with

Let H be a finite abelian Hopf R-algebra which

is connected as an algebra and of p-power rank.

Let N = Spec(H).

There is a connected p-divisible group scheme G over Spec(R) so that N embeds in G.

It follows, e.g. by work of Lubin, that if N is a finite

46

Childs

subgroup of a formal group G, then there is a formal group G' and an isogeny f: G -» G', defined over R, with kernel N. are formal groups of dimension n, n-tuple of power series f = (f ker(f) is finite, where

then an f

(If G, G'

isogeny f:G ->G'is an

n ) such that

= { x e inn I f (x) = 0} m is the maximal ideal of the

the algebraic closure of K.)

valuation

ring of

That is, N fits into a short exact

sequence of formal groups, 0 -» N -> G -» G' -> 0 . In this situation, N may be represented as N = Spec(H), where H = R[[x]]/(f) with comultiplication A induced by A(h(x)) = h(G(x, y ) ). 1 =

To see that H is a Hopf algebra, ,

set

and

J = I • R[[y]] + R[[x]] ® I c R[[x]] • R[[y]] « R[[x, y]]. To show that A is well-defined, we observe that A(fi (x)) = fi (F1 (x, y),

Fn (x, y ) ) ;

since f is a homomorphism from F to G, we have, for each i, that ^(F^x,

y) , ..., Fn (x, y ) )

= G i (f1 (x) ,... ,fn (x) , f ^ y ) , from which it

is clear that A(f^(x))is in J.

to 0, and the

antipode s, induced by

s(g(x)) = g ([-1]1 (x)f

fn (y)) » The counit sends x

[-l]n (x)) ,

is also well-defined modulo I because [-l]G «f = f©[-i]F .

Thus H

is in fact an R-Hopf algebra, and the set of S-valued points of Spec(H) is {a in Sn | f^(a) = 0 for i = 1, ..., n} = N(S). = Spec(H).

So N

Galois Extensions over Local Number Rings

3.

47

Principal homogeneous spaces for subgroups of formal groups

If N is a finite abelian group scheme defined over Spec(R), then PH(N) denotes the group of principal homogeneous spaces for N over Spec(R).

This is the same as the group PH(H) of

(isomorphism classes of) H-Galois objects over R, or, *

*

equivalently, the group Gal(H ) of H -Galois extensions of R, where H

•ft

= H o m ^ H , R) is the dual Hopf algebra to H, representing

the Cartier dual of N. Over the years there has been some considerable interest in computing PH(N) in one or more of its equivalent formulations. To cite two recent examples:

globally, there is a homomorphism, *

the Picard invariant map, from PH(N) to C1(H ) given by viewing a Galois extension S of R with Hopf algebra H* as a rank one it

projective H -module.

Taylor [Tay92] has recently shown that in

certain number-theoretic situations the kernel of the Picard invariant map relates closely to the values of certain p-adic L-functions.

Greither's recent LNM [Gr92b] gives an exposition *

of work on Gal(H ) when H

*

= RG, G cyclic of prime power order,

in both global and local settings.

When R is the valuation ring of a local field and H is a connected abelian R-Hopf algebra, then Oort's embedding theorem yields a rather nice description of PH(H). short exact sequence

Namely, take the

0 -» N -» G -> G' -» 0 over X = Spec(R), and apply cohomology in the finite topology. Mazur did this in ([Mz70], Corollary 2.7), and showed that PH(N) * H°(X, G')/Im(H°(X, G ) . Here H°(X, G) is just the R-valued points of G.

More explicitly,

if G has dimension n, then H°(X, G) a mn , m the maximal ideal of R, with group operation on mn defined by the formal group G: a +G /3 = F (a , 0).

Proposition 1.

If we denote this group by i»Gn , then

PH(N) s mG ,n /f(mGn ) .

In this way, embedding N in a short exact sequence of formal groups gives an approach to understanding the Galois extensions of R with Hopf algebra H * , N = Spec(H). Utilizing this description of PH(N), Mazur and Roberts [MR70] determined the cardinality of PH(N), namely, |PH(N)| = |n (R) |-qord(det(Jac(f)(0))) where N(R) = Alg(H, R) is the group of R-points of N, q = |R/m|, and Jac(f)(0) is the Jacobian matrix of partial derivatives

In the remainder of this paper we will look at these results when G and G' have dimension one.

Galois Extensions over Local Number Rings 4.

49

Dimension one embeddings.

To utilize the description of PH(H) it is convenient to embed N = Spec(H) into a recognizable formal group.

The Oort

embedding generally embeds N in a formal group of large dimension (c.f. [OM], p. 333, Remark). dimension one embeddings.

In this section we look at

These are what one would hope to have

if the Hopf algebra H is monogenic (i.e. a quotient of a polynomial ring in one variable): e.g. the Tate-Oort Hopf algebras.

Example 1.

When R contains

the p th roots of unity,

here is a way of embedding a generically split group scheme of rank p into a dimension one formal group. Let ( be a primitive pth root of unity in R and let X = 1 - £.

Let ab = X in R where a and b are both non-units, and

consider the Tate-Oort Hopf algebra H

3.

= R [x ] / ( -

a^_1x ) .

Then

H_3. ® K is split, since x^ - a^_1x has p roots in K, namely, x = 0 and x = w 1a for u a primitive p-1 st root of unity in K, i = 0, 1, ..., p - 2 . Consider the formal group law

(x, y) = x + y + bxy.

Then

it is easy to verify by induction that the image of any natural number q under the map [ ): Z -> EndfG^) is given by (q](x) = In particular,

(1+b)q - 1 ------b

.

50

Childs

[P](x) = px + (p )bx2 + ... + bp-1xp = bp - 1 (cx + ... + xp ) for some c in R with cR = ap_1R.

Set

h(x) = [p](x)/bp_1 = cx + ... + xp Since [p](x) is an endomorphism of G^, it is easy to see that h(x) is a homomorphism from

to G ^ .

Thus we have a short

exact sequence of formal groups: (1 )

0 -» ID -» Gb -» G p -» 0

where D = ker(h(x)) = Spec(R[x]/h(x)). Notice that D is generically split, since h(x) =

(l+b)p - 1

has p roots in K since K contains

Thus D is determined by

its discriminant, which is h'(0)PR = cpR = (ap-1)pR. Spec(H ) and R[x]/h(x) a a

h

cl

Thus

D =

.

From the sequence (1) and Proposition 1 above, we have a description of PH (ID) , namely, PH (ID) a G _(R)/h(G (R)) bF where G^(R) = m, the maximal ideal of R with group structure given by the formal group G^ On the other hand, Roberts [Ro73], and subsequently Hurley [Hu87] described PH(D), namely, PH(ID) a U

(R)/Ub (R)P b

where Ub (R) = {u e U(R) [Gr92], II 2.1).

| u a 1

(mod bR)}.

(c.f. Greither

We can recover Roberts' description from the

formal group description, as follows:

Galois Extensions over Local Number Rings

Proposition 2. ip: G

bp

51

If p/bp is in m, then the map

(R)/h(Gb (R)) -* U _(R)/Ub (R)P bp

induced by sending s to 1 + bp s for s in m, is an isomorphism of groups.

Proof.

It is routine to check that the map sending s to

1 + bp s is a homomorphism from G _(R) to U fR) and induces a bp bp homomorphism on factor groups.

To show \/j is 1-1, suppose for t

in m, tff(t) = (1 + bs)p = ^f(h(s)).

We must check that s is in m.

But ^ _

t

—

(l+b)p - 1

/

bp and upon expanding the right side, it becomes clear that, assuming that a is in m, then if t is in m, so is s. To show ip is onto, let u = 1 + bps be in U _(R) . bp m then u is in the image of ip.

(mod m) and let v = 1 + bt.

If s is in

If s is a unit, then let -s s tp

Then one checks easily that uvp = 1

+ bpw with w in m provided that p/bp is in m, completing the proof.

This description of PH(D) is a formal group version of an argument of Greither [Gr92].

52

Childs

Example 2.

Let F be a formal group of dimension one and

height h defined over R - 0R , K a local number field.

Suppose

N ([pn ])/ the Newton polygon of [pn ]p(x), has a vertex at (pr , b) where r < nh and b > 0.

Then there are homomorphisms f : F -> F',

g: F' -¥ F of formal groups so that [pn ] = g«f and ker(f) is a congruence subgroup of ker[pn ] of rank pr consisting of all roots of [pn ] whose valuations are the negatives of the slopes of the Newton polygon of [pn ] to the left of (pr , b ) . may represent ker(f) by H = R[[x]]/(f(x)).

In this way we

If we wish to realize

H as a quotient of R[ x ] , we apply the Weierstrass Preparation Theorem to f(x):

f(x) - h(x)u(x) where h(x) is a Weierstrass

polynomial, i.e. a monic polynomial of degree pr which is r congruent modulo m to xp , whose Newton polygon is a vertical translate of that portion of the Newton polygon of [pn ] with abscissas s pr .

Then H * R[x]/h(x) as R-algebras.

One has considerable freedom in obtaining Hopf R-algebras in this way, by use of the standard generic dimension one formal group of height h.

This is a formal group Ffc defined over

Zp[[t^,..•th _ 1 ]] with the property that the endomorphism [p]^ in End(Ft ) has the form h-1

(*)

i [p]t (x) = pxgQ (x) + £ t^x1* g ^ x )

h + xp gh (x)

i=l where g^(x) is a unit of Z p [

, ..., t^]][[x]] for all i < h,

and 9jj(x) is a unit of

•••»

D [ [x] ].

[Lu79]. Any specialization t -» a = (a^, ..., complete normal local domain containing

See, e.g., in R^1 1 , R a

gives a dimension one

Galois Extensions over Local Number Rings

53

formal group Fa of height h defined over R. Given any finite abelian p-group r of order pr and exponent pe , if K is a finite extension of Op with valuation ring R, which is sufficiently ramified over Op, we may find for any h sufficiently large, a specialization a in R*1”1 such that the Newton polygon of [pe ]a (x) has a vertex with abscissa pr , and the congruence subgroup of ker[pe ] corresponding to the vertex at pr is isomorphic to T.

For details see [CZ94].

For example, for fixed b = ord(/3), (3 in R such that 0 < b < 1 = ord(p), choose h so large that b(p^ - 1) < p*1 - p. Let Ft be the standard generic formal group of height h. t.^ = 0, and t2 = t 3 = ... = tj1_1 = 0. [p]^ has a vertex at (p, b ) .

Set

Then the Newton polygon of

For since b(p** - 1) < p*1 - p, we

have -1 + b

-b

so the slope of the line joining (1, 1) and (p, b) is less than the slope of the line joining (p, b) and (ph , 0).

So by the

Local Factorization Principle of Lubin [Lu79l, f: Fg -» F' and g: F' -»

[pl„ = where P for some formal group F', and ker f =

1 - b {a in ker[p]- I ord(a) = ------ > u {0} p p - 1

is a congruence subgroup

of ker[p]„ consisting of all roots of [p]ft of valuation a P P Then

1 - b ----- . p — 1

= R[[x]]/(f(x)) is a rank p Hopf algebra which represents

the group scheme ker(f). By Lubin's Lemma (Lu64], Lemma 4.1.2), we can assume that [P]g* f and g all are sums of terms of degree * 1

(mod p - 1 ) .

So

54

Childs

f(x) = cx + uxP + ...

where ord(c) = ord(p//3

and ord(u) = 0.

By the Weierstrass Preparation Theorem, f(x) = h(x)u(x) in R[[x]] where h(x) » cQx + x^ with ord(cQ) * ord(c) and u(x) is an invertible power series. In this way, given any fixed discriminant a^R, setting = 0 with o r d (0) = p/a, we can find a Tate-Oort Hopf algebra H

= R[x]/ (c.x - xP) with that discriminant, as the representing c0

0

Hopf algebra of the kernel of some homomorphism of dimension one formal groups. However, it is not clear from this construction which TateOort Hopf R-algebra Hc

is (i.e what cQ is), since varying c Q by

a unit of R changes H

generically (i.e. over K ) . C0

By being more generic about the above construction, Lubin has shown us

Theorem 3.

Let R be the valuation ring of a finite

extension K of 0^.

Any Tate-Oort R-Hopf algebra which is

connected with connected dual may be realized as the representing algebra of a congruence subgroup

of [p]Ffor

some formal

group F

of dimension one defined over R.

Proof.

Let

= R[x]/(xp - bx) be aTate-Oort

algebra, where 0 < ord(b) < 1 = ord(p).

Hopf R1

Suppose ord(b) > --- for r+1

some r, and choose h so large that r(p-l) < p*1 - p.

Let

be

the standard generic formal group of height h, as above, with

Galois Extensions over Local Number Rings

[P]t(x ) as

55

above.

Specialize to 0 for i > 1, and set r r+l , , t1 = z in S = Zp[[z» P / z , P /z ]], a complete local domain with r r+1 . . . maximal ideal JJI = (z,p/z, p /z ). L u b m ' s Local Factorization (*)•

Principle applies to specializations of this generality. the image of [p]t (x) by [p]z (x).

Call

Then [p]2 (x) = px + zx^u + ...

with u a unit of S (use Lubin's Lemma, as needed), so the Newton h__ P

polygon of [p] (x) has a vertex with abscissa p if ------------ is 2p h-p+(p-i) in the maximal ideal Ml of S. p Ph - P

„h

But

pPh - p - r ( p - l )

,,

h

2P _ P + (P -1 )

,

.

ZP - p - r ( p - l )

p ^ P ’ 1) z

r(p-1)+(p-l)

is a product of elements in Jit, and so[p]_(x) has a z

vertex with

abscissa p. Then [p]_(x) = q (x)r (x) z z z

where q_(x) is a Weierstrass z

polynomial, r (x) is a power series in S[[x]], and under any z specialization z -» c of S to a discrete valuation ring, the valuation of any root of q_(x) is greater than the valuation of c all roots of **c (x) .

Invoking Lubin's

0, n p o p ord(a_n ) a nr/(r+1).

(Here, ord(p) = 1.)

Thus the Newton

56

Childs

polygon of u(z) lies above the cone with vertex (0, 0) defined by the half-lines from (0, 0) through (1, 0) and from (0, 0) through (-(r+1), r). 1 Suppose --- < ord(b) < 1. r+l

Then the Newton polygon of

pu(z) - bz has vertices at (0, 1) and at (1, ord(b)). ord(b) > l/(r+l), then

For if

ord(b)-l -r --------- > , so the edge between 1 r+l

(0, 1) and (1, ord(b)) has a less negative slope than the edge out of (0, 1) to the left; also, since ord(b) < 1, the edge out of (1, ord(b)) to the right has non-negative

slope, hence there

is a vertex at (1, ord(b)). Now we apply a result of Lazard [Lz63].

Let m - 1 - ord(b).

Then the Laurent series w(z) = pu(z) - bz converges for valuation m.

a in K of

To see this, we see easily that for n > 0, since m

> 0, ord(pan « a s n - » + «, while for n < 0, n r ord(pan ----- . r+l

00

Set w(z) ■> £ cn zI1' where c i = Pa i for * 56 1 > c i = b + n=—» Following Lazard, for any m, set ord(w, m) = inf(ord(c^) + im), n(w, m) = least i so that ord(w, m) = ord(c^) + im, N(w, m) = largest i so that ord(w, m) = ord(c^) + im. Then for m = 1 - ord(b) = the negative of the slope of the edge

Galois Extensions over Local Number Rings

57

between (0, 1) and (1, ord(b)) in the Newton polygon of w ( z ) , it is easy to check that ord(w, m) = 1, n(w, m) = 0, N(w, m) = 1. Applying Proposition 2 of [Lz63], w(z) factors in K[x] into w(z) = P(z)g(z) where P(z) is a polynomial of degree N(w, m) - n(w, m) = 1 with a root of valuation m, and g(z) is a Laurent series which converges for z = a of valuation m. Since P(z) has degree 1, there exists a root c of w(z) in K of valuation m = 1 - ord(b) > 0.

The map sending z to c is a

local homomorphism from S to R, sending in to m, because ord(c) > 0, ord(p/c) = ord(b) > 0 and °rd(pr /cr+1) > r - (r+1) ( ^-) = 0 r+1 Specializing z to c yields u(c)p/c = -b and [p]c (x) = (-bx + x^))vc (x)/ where vc (x) is in R[[x]]. Thus the roots r of -bx + x^ in K form a congruence subgroup of ker[p]c , and so there is a homomorphism fc : Fq -> G for some formal group G such that ker(f ) = T, and H = R[[x]]/f (x) represents T = c c ker(fc ) .

By the Weierstrass preparation theorem, fc (x) = (-bx + xp )h(x),

where h(x) is an invertible power series in R[[x]]. H s R[x]/(-bx + x^) as R-algebras.

Thus

Since a Tate-Oort R-Hopf

algebra is uniquely determined by its structure as R-algebra, the proof is complete.

The above is not exactly Lubin's argument: one could choose the height h of

he showed that

to be 2; however the

58

Childs

Tate-Oort Hopf algebra need not represent a congruence subgroup of [p] if the valuation of b is too small.

5.

Counting principal homogeneous spaces. Let R be the valuation ring of K, a finite extension of 0^.

Let H = R[[x]]/(f) be a finite Hopf algebra which represents the kernel of an isogeny f: F -> G of formal groups of dimension one. We've seen that then PHS(H) s G(R)/f(F(R)) = In that setting, the local Euler characteristic theorem of Mazur and Roberts ([MR70], Prop. 8.1) asserts that (*)

|PHS(H)| = |# AlgR (H, R)| q°rd f'(0),

where q = |R/m| and ord is the valuation on K normalized so that ord(p) = e, the absolute ramification index of K over 0^. When F, G are formal groups of dimension one, then the proof of Rasala ([MR70], pp. 225-6) may be obtained, using Proposition 1, as an application of ideas in Frohlich [Fr68].

Here is how

the proof goes.

Proposition 4.

Suppose H = R[ [x] ]/ (f (x)) , where f:F -> G is

a homomorphism of formal groups of dimension one.

Suppose

[P]F (x) = f(x)u(x) where f(x) is a Weierstrass polynomial of degree p*1 and u(x) is a constant multiple of an invertible power series.

Suppose also the logarithm and exponential maps of F and

G converge in some neighborhood of zero.

Then

|PHS(H)| = |# AlgR (H, R)|-q°rd f'(0).

59

Galois Extensions over Local Number Rings

Proof.

Following Frohlich, given the formal group F of

dimension one, let P(F) denote the group of points of F, namely, P(F) = in, the maximal ideal of the valuation ring of the algebraic closure K of K, with group operation given by +F . P(F, K) = P(F) n K.

Set A(F) = jj {ker[pn ]p : P(F) -> P(F)}, the

torsion subgroup of P(F), and let HomfF.., G J\

A(F).

a ,

Let

A(F, K) = A(F) n K.

v ) be the logarithm map.

Let i F in

Then ker{£„: P(F) -» in,} = r

+

If f is in Hom(F, G) then the diagram P(F) ->F m+ f

1

| f ' ( 0) lG

P (G)

-

m+

commutes ([Fr68], p. 112).

So we can build from ^F and f the two

pairs of exact sequences of K-valued points:

0 -> A(F, K) •I f

0 -> A(G, K)

P(F, K)

P(F, K)/A(F, K)

-If

0

i

P(G, K) -> P(G, K)/A(G, K)

0

and 0 -> P(F, K)/A(F, K) -»F m+ I

f

m+ / l F ( P(F, K ) ) -> 0

I f '( o )

0 -> P(G, K)/A(G, K) -> m+

->

I

m +/eG (P(G, K) ) -» 0

The snake lemma applied to the first pair of sequences gives

60

Childs

0 -» ker f -» ker f -» 0 -» A(G, K)/f(A(F, K ) ) P(G, K)/f(P(F, K))

P(G, K)/A(G, K ) f ( P ( F , K ) )

0.

The snake lemma applied to the second pair of sequences gives 0 -» 0 -» T -> P(G, K)/A(G, K)-f(P(F, K ) ) -> m +/f' (0)m+ -» U

0

where T, U are the kernel and cokernel of the map from m + /fF (P(F,K)) to m + / m + / is isomorphic to a sub-[/(b)-m odule algebra of the linear dual U(b)* of the enveloping algebra U(b) [Levasseur, 1986, Theorem 2.2]. This [/(b)-m odule algebra is a polynomial algebra in dime b variables and is a subalgebra of the algebra of representative functions R ( b) of U(b) (the maximal locally finite submodule of ?7(b)*). Injective hulls for the other simple b-modules are obtained by applying automorphisms of 17(b)* to E^(C) and all can be explicitly realized as submodules of -R(b) [Levasseur, 1986, Lemma 3.3]. In fact, for a finite dimensional simple b-module S, Eb(S) is isomorphic to the tensor product (over the base field C) of S with Eb(C) [Dahlberg, 1984, Theorem 11]. Hence in the case of a solvable Lie algebra, the injective hulls of simple modules can be explicitly constructed relative to the choice of a Poincare-Birkhoff-Witt basis of the enveloping algebra [Dahlberg, 1984, Levasseur, 1986]. In the case of a semisimple Lie algebra g, little is known about the structure of injective hulls of simple g-modules. It is an easy consequence of Weyl’s Theorem on complete reducibility that essential extensions of simple g-modules cannot be locally finite [Dahlberg, 1989, Remark 1]. However, injective hulls of artinian sl(2,C )modules are locally artinian [Dahlberg, 1989]. Also, if I) is a Cartan subalgebra of 0 , the f) locally finite dual of a Verma module M (A) for A integral and dominant is an injective hull of L (A) in the category O , but in general, this object is not injective in the category of all g-modules [Jantzen, 1983, 4.10]. In this paper we apply the results on injective hulls of simple modules over solv­ able Lie algebras to the standard Borel subalgebra b+ of a simple Lie algebra g. We show that these modules are g-submodules of U ( q)* under the left translation action 67

68

Dahlberg

and are essential extensions o f a submodule isomorphic to a simple g-module (The­ orems 5.1 and 5.2). In particular, if b is a Borel subalgebra of g then the injective hull o f the 1 -dimensional trivial b-module, Ef,(C), may be realized as a subalgebra of {7(g)* which is stable under the left translation action of g, and thereby becomes a sub-f/(g)-m odule algebra o f {7(g)*. This sub-{7(g)-module algebra is an essential extension o f the 1 -dimensional trivial g-module C (Theorem 6 ). Other essential ex­ tensions can be obtained from this one by twisting the g-action via automorphisms of 17(g)* induced by the action of the adjoint group of g on 17(g). Since the resulting extensions are all submodules o f 17(g)*, we may sum these submodules to obtain an essential extension o f the trivial module containing all of these submodules. In the case of g = sl( 2,C ), we can obtain more precise information about our extensions. Indeed, we construct an essential extension of each simple highest weight module L(X) which properly contains eM (A ), the Infinite dual of the Verma module Af(A), and is locally artinian with simple factors L( A) and L(sa.A) in the finite dimensional case and L( A) otherwise (Theorems 7.3 and 7.4). Finally, we compute the adjoint action of SL( 2,C ) on E ^ ( C). I would like to thank Andy Magid for his continued support and encouragement, and Marc D. Montalvo for his work on the Mathematica programs which allowed me to compute the adjoint action of S L (2,C ) on {7(sZ(2,C))*. 2. T he L inear D ual of 17(g) Let g be an n-dimensional Lie algebra over the field o f complex numbers C. Then 17(g), the universal enveloping algebra o f g, is a cocommutative Hopf algebra with comultiplication A defined by A (x ) = x 0 l + l 0 x f o r x G g , counit e where e is the augmentation map o f 17(g) with kernel gZ7(g) = I7(g)g = 7(g), and antipode v where v is the principal antiautomorphism of 17(g). The linear dual of 17(g), C7(g)* = hom e (17(g),C), is therefore a commutative and associative algebra over C with < / i / 2 ,^ > = < f \ ® h , A (u ) > for / i , / 2 G {7(g)*, u G 17(g) where we identify C 0 C with C and < / , u > = f(u) for / G f7(g)*,u G 17(g). The multiplicative identity is e (for details, consult [Sweedler, 1969] or [Dixmier, 1977, Chapter 2]). 17(g)* has a {7(g)-bimodule structure defined by < a / 6,u > = < / , 6ua > for a, 6, u G 17(g) and / G U(g)*. Define 7 ,p : g — > End 17(g)* by 7 (x)f = x f , x G g , / G 17(g)* (left translation) and p{x)f = f x (principal antiautomorphism followed by right translation). Then 7 and p are representations of g by derivations on {7(g)* [Dixmier, 1977, 2.7.7]. These representations extended to {7(g) make {7(g)* into a (left) {7(g)-module algebra [Sweedler, 1969, Example (c), p.154]. 17(g)* has an involution / / defined by < / , u > = < / , u > . The relationship between 7 and p is ( 7 {u)f)y = p(u)f. It is easy to see that 17(g)* is an injective cogenerator of the category of all left 17(g)-modules and hence contains an isomorphic copy of the injective hull, E 0(5 ), of each simple left {7(g)-module. Fix an ordered basis X i . . . X n o f g and define for I = ( » i ,. . . , i n), J = ( ji , . . . , j n) € l f x ' = . x j r . ..* * * ,

k= 1

|i| = £ > , k=l su z= Y [ sik,n *:=i

69

Simple Modules over Semisimple Lie Algebras

where Sij denotes the Kronecker delta. Then { X 1//!}/e N n is a Poincare-BirkhoffW itt (PB W ) basis for t7(fl), and with respect to this basis, we may identify the algebra 17(b)* with the algebra C [[x i , . . . , x n]] where x 1 = x\l . . . is defined by

< x I , X J/Jl The algebra isomorphism is given by W

—

/ •— >^2 < f,x*ll\ >

[Dixmier, 1977, 2.7.5].

X1

I We may now interpret 7 and p as representations of b in the Lie algebra of deriva­ tions of C [ [ * ! , . . . , x n]] although we caution that these maps depend upon the choice of a PB W basis of 17(g). Via either of these representations, C [[x i, . . . , x n]] becomes a left g-module. If we let (fl) denote the linear span of the ordered monomials X 1 for |J| > q then C [# i, . . . , x n\ identifies with all / E !7(fl)* such that / = 0 on Z/M (fl) for Q 0. Another subalgebra of interest is the algebra of representative functions on U(g): R (s) = { f £ U ( g)* |dime / T O < 00} = { / G 17(g)* |dime U (fl)/ < 00} [Hochschild, 1959, p.99].

P R O P O S IT IO N . Let fl = j ® t where j and t are subalgebras of g and let n : U(g) = jl/(fl) ® U(t) — > U(t) be the right l-module projection. Then tt*

: 17(f)* — > U(o)*

where < 7r* (/),u > = < / , 7r(u) > for f e U ( l ) * , u e U ( g ) is an algebra monomorphism. Furthermore, if

1t m - t — >Derc U(ty and 7 0 : fl — > Derc U(g)* denote the left translation representations of I and fl respectively, then 7 0(* )(* -* (/)) =

tt*

(7 ((x ) ( /) )

f o r x e t and f G U(i)*. Proof:. It is clear that jI7(fl) is a coideal of I7(fl). Hence tt is a coalgebra epimorphism and therefore 7r* is an algebra monomorphism [Sweedler, 1969, 1.4.7, 1.4.1]. The rest is an easy calculation. □ Suppose now that we choose an ordered basis X \, . . . , X m for I where m = dim I. Then with respect to the corresponding PB W basis of U (t) we may identify U(l)* with C [[t/i,. . . , 2/m]] where < y^u > = 1 if u = X* and is 0 otherwise. Extend this basis to an ordered basis X i , . . . , X n of fl and identify 17(g)* with C [ [ z i ,. . . ,x n]] via the corresponding P B W basis of {7(g) as above. Then ir*(U(i)*) identifies with C [ [ « i , . . . , x m]] where n*{yi) = x {.

70

Dahlberg

C O R O L L A R Y . In the above notation, if for m

1

< i < m

q

7t(Xi) = Y^ j= 1

o r-> ° Vj

e C [[2/ 1 , . . . , ym]]

then m

p.

= S 7r* ( ^ ) ^ T j= 1 J

By the Proposition and its Corollary we may now identify U(t)* with 7r*(l7 (t)*) = C [[x i,. .. ,a;m]] as a sub-Z7(t)-module algebra of 17(g)* = C [[x i,... ,x n]] under the left translation representation 7 = 7 g restricted to the subalgebra {. However, since clearly tt* ( u m = { f e u ( gy |< f,}U(o) > = 0 } it follows that J7(t)* is also a sub-l7(g)-module algebra of {7(g)* under 1977, 2.7.16]. 3.

I njective hulls

o f simple

7

[Dixmier,

{7(b+)-MODULES

Let b be a solvable Lie algebra and set t = [b,b]. Then b has a nilpotent subalgebra n such that b = n + t [Hochschild, I960]. There is a basis [Levasseur, 1986] -X i,. . . , X m of t such that for X E b [X,

c Xj] C

[Xi,Xj] E

CX j

CJXp,

and 1 < i , j < m.

p 17(0)* be the injective algebra homomorphism obtained from the projection U(g) = xTU(g)

0

U(b+)

U(b+)

of right U(b+)-modules. Then n*(Eb+(C)) is a sub-U(g)-module algebra o/17(g)* where g acts by left translation. Furthermore, 7r*(Eb+(C)) is an essential extension of the 1-dimensional trivial g-module C. Proof. First, we note that 7r*(2 ?b+( dm where d is the index of nilpotency of n+ , we have J(n+ )IAI C by Proposition 4. Hence our term lies in /TBJ(n+ )lAl C H BU ^ ( n ^ ) C t/[lBl+m](b+ ), and thus / will vanish on this term for \A\, \B\ » 0. Assume next that [Xi,Ya] = H G f). Then H Bx g . . . [ x g , y

j . .. x g

= a,HBx g . . . x g l ' H x g - ' x g g ... x g

- aj(oj - l) H BX g . . . X g g X g - ' X g g . . . X g - « * Bx x g ■■■x % : x g - ‘ x z : : ■■■

-a [H )a ,H B £ X g . .. X g ^ X g - ' X g g ... X g j=i+1 - Oiiat - 1)H b X % . ~ X % £ X % - lX%-_\ ••• The last expression belongs to {7[IBI+Ii4l” 1l(b+ ) and again / will vanish on these terms for |A|, |J9| ^ 0. Hence we’ve shown that 7 (Ya)7r*f = 0 on n~ 17(g) and (b+ ) for q » 0 so 7 {Ya)n*f G C[a:i, . . . , x r+i\ and thus r+ i

8

7 0 ra)|ir*(B,+ (Q) = X ) PiWT. dxi *=1 where the pi G C [# i, . . . , xr+j] and a is any simple root. Now if (3 G \ B then /? = a\ H hajk where a* G B and each partial sum [ . . . [ r a. , [ y a„ y a i] ] ...] .

Since 7 is a representation of g 7(1^) = [7(T a J, [•••[lO 'a J ,

By the above, each derivation ^f(Yai)\n*(Eb+(Q) has coefficients in C [ x i,... ,x r+/]. Hence so does 7 (F/3)|,r*(Eb+(Q)- This implies that n*(E^+(C)) is stable under the derivations 7 (Yp) for /? G J?+ , and hence has the structure of a sub- I7(g)-module algebra of C[[xx, . . . , x2r+/]] = U(g)*. Finally, Tr*(Eb+(C)) is an essential extension of the trivial g-module C. For if 0 ^ 7r*/ G 7r^(£7b-h(C)), we can find u G U(b+) such that 0 / c = w / G C, But 7(u)7r*/ = 7T*(uf) = 7r*(c) = c is nonzero in C □ C O R O L L A R Y . In the notation of the Theorem, 7r*(i?(b+ )) is a sub-U(g)-module algebra ofU(g)*. Proof. From the proof of the Theorem,

7r*(R(b+)) = C [ x i , . .. , x r+i;ex p (zx r+i);x G C, 1 < i < I] is stable under r+/ tW

I ci* .

Q

*r+i) = E P ^ ’ *=i

*

P i € C [ x i , . . . , x r+J]

□

Simple Modules over Semisimple Lie Algebras

5.2. In order to construct essential extensions of the simple g-modules, we need more precise information about the form of the derivations 7 (1^, ), 1 < i < /• Recall our PB W basis from §5.1 Y c H b X a /A\B\C\ for A, C £ FT and B £ NJ, resulting from our ordered basis for g

The corresponding dual basis X\, . . . , X2 r+l £ U (g) = C [[# i,. .. , 2?2r+z]] is such that

1 < i < r

xi corresponds to X i

1 < i = 0, we may assume C = 0 = ( 0 ,... ,0) £ Nr . Now (*)

H BX AYai = YaiH BX A + [H B,Y ai] X A + H B [X A, Yai)

so Xj will vanish on the first term on the right-hand side and also on the second term since [H B,Y ai] G YaiU(l)). If \B\ > 0,Xj will vanish on the last term as well, so we only get a non-zero value when I = (A, 0 , 0 ).

76

Dahlberg Case 1 : j = r + fc, 1 < k < 1. Then < x r+fc,u > = 1 for u = Hk and 0 otherwise. We can write

[X A,Y ai] = [X ,X S r v- i' X " iYai\ = [X\ Yai\X $Ti,- i' X " + X '[ X « r = < x r+fc, X ,ar_ (l_ i)H iX Z ~ {l- i)~ l X " - j r ,o r_ (l_ X1.

/= (A ,0 ,0 )€ N 2r+ ‘

By (**) and the above discussion we have [X A, Yai) = X '[ X “ r (' _i), Yai) X " + X ' x £ - (,- ° [X ", YQi). We claim that [X " , YaJ cannot involve X a|., and hence x r_(/_i) will vanish on the last term in the sum on the right-hand side. But this is clear since [X^, Yai ] G and (3 — a a fi oti for any (3 G J?+ . Hence < x r_ (I_ 0 , [ ^ Y Qj] > = < x r_ (l_ ih X '[ X % - {' - ‘ \ Y ai} X " >

- X /ar_(,_i)(ar_(,_r_(i_i) = - x 2 r _ (i_ iy

□

-2

for i4 = ( 0 , . . . , 0 ,

0

otherwise.

> , 0, . . . , 0)

Simple Modules over Semisimple Lie Algebras T H E O R E M . Let ( m i , . . . , m j) G C*. Then

(1) I l( U (g ) )e x p ( ^ 2 m jXr+ j) S* L ( A)

j= 1 I for A = ^ m j uj where u>i,. . . ,u>/ are the fundamental weights. j= i ( 2 ) C [ # i ,... ,£ r+ /]ex p ( ^ * =1

is an essential extension of i

l (U ( g ) ) e x p ( ^ 2 mi x r+j) ^ X(A). i= i

Proa/. Let v = exp ( ^ =1 raj-av+j). By Theorem 3 (1), l(X i)v = 0

for

1 < i < r

and 7 (H i)v = miV = A(H i)v

for

1 < i = < f , g ~ l .u > = < f,A d (g )~ l {u) > where g G G , f G 17(g)*, and u G 17(g) where Ad denotes the adjoint representation. This action has the following prop­ erties which are easily verified: (1)

9-(u -f) = (g .u )(g .f)

(2)

g.u = (g.u)

for g G G , f G 17(g)*, and u G 17(g) L E M M A . Identify E b+ (C) with the g-submodule 7r*(Eb+(C)) ofU (g)* as in The­ orem 5.1. Then fo r g G G, g .E b+(C) and E gb+(C) are isomorphic as U (g.b^ )modules. P roof U(g)* is an injective l7(y.b+ )-module for each g G G [Dahlberg, 1984, Propo­ sition 4]. Hence U(g)* contains a submodule isomorphic to E gb+ (C), and we shall identify these two modules. Now p.i?b+(C) is a t/(^.b+ )-module by ( 1 ). Since G fixes C ,g .E b+(C ) is an essential extension of the trivial p.b+ -module C. Hence g.E b+(C) C E 9'b+(C) and so E b+ (C) C g ^ l .Egtb+(C). But p_ 1 .E^.b+(C) is a J7(b+ )-module via u .(g~ l .f ) = (^“ V u X f l " 1. / ) = 9~ l -{{g -u ).f) G g ~ l .Eg%h+ (C) where u G U (b+) and / G E g^ ( C). Let 0 ^ fl” 1. / G g ~ l .Eg^ ( C). There exists g.u G ?7(^.b+ ) such that 0 ^ (g -u ).f = c G C. Hence u.(g~l .f ) = g ~ l .((g .u ).f) = # _1.c = c, and so p " 1 .Eg b+(C) is an essential extension of C as a C/(b+ )-module. Therefore, £ b+(C) = g ~ l .Eg^b+(C) and so p.E b+(C) = Egtb+(C). □ T H E O R E M . Let g be a semisimple Lie algebra and let G denote the adjoint group ° f 9* (1) For any Borel subalgebra b of g, E b(C) has a g-module structure. ( 2 ) Identify E b+(C) with the g-submodule 7r*(Eb+(C)) ofU (g)* as in Theorem 5.1. Then E = Y , 9 - E b+ ( Q = g£G

£

E b(C)

Borel subalgebras f>C0

is a g-submodule o fU (g )* which is an essential extension o f the trival gmodule C. P roof (1) Any Borel subalgebra b of g can be written as g.b+ for some g G G [Dixmier, 1977, 1 . 10 .20]. By the Lemma , E b(C) = E g.b+(C) = g .E b+(C). Let u G U(g) and / G E b+ (C). Then u .(g .f) = {gg~l .u ).{g .f) = g .{{g ~ l .u ).f) by ( 1 ). By Theorem 5.1, E b+ (C) may be considered as a g-submodule of U (g)* and so ( 1

and has a basis given by vn?m = XiX™eCX2 + W m- i for m > 0. We have 'y(Y )(vnyTn) = (c 7l)^n+lfm9 7(^0(^n,m) = (c 2 n )vn#3]] under 7 we have the following: ( 1 ) For n, fc E Z + we have

7 ( y fc/fc '-)(4 ) = £ ( - i ) fc+i( * I (2) 7 (L7(g))(a:J) = M (0 ) /o r n G Z+. P roof (1) This follows by induction on k. (2) Since j ( X ) ( x ^ ) = 0 = ^f(H){x^)^x^ is a maximal vector of highest weight 0. Hence there is a surjective 0-homomorphism : M (0 ) — > 7f(U (Q ))(xz) defined by (1 ® 1) = £3 [Dixmier, 1977, 7.1.8 (i)]. Now (1) implies that j ( u ) restricted to 7 (U(g))(x%) is injective for all non-zero u E C/(n“” )) = C [y ], so (f>is injective and the result follows [Dixmier, 1977, 7.1.8(vi)]. □ Finally, let G = 5 L (2 ,C ) act on Z7(0)* = C [[£ i,£ 2 ,£ 3]] as in §6 . Since G acts via automorphisms (i.e. by substitution of variables), it suffices to calculate the translates of £ 1 , £ 2 , £3 by an element g G G. We shall see that it will be sufficient to calculate the translates of £ i ,£ 2,#3 by elements of the form etx and ety. We have X = e~tx.£ = £ H = e~ tx.h = h + 2tx Y = e~ tx.y = y — th — t2x. Suppose that Y n = (y —th —t2x ) n, n = 1 , 2 , 3 , . . . is expressed as a linear combi­ nation of ordered monomials yt3ht2x %1. The following results are easily proved by induction: (1 )

The sum of those terms of the form y k is

84

Dahlberg (2)

The sum of those terms of the form x k is

£ k= (3)

(*-!)! W *

k tn+kx k.

]

The term involving only h is —(n — l)\tnh.

Therefore, by using the above formulas, we obtain (i) Y n = n\

— (n — 1)! tnh —n! tn+ 1 x,

n > 1 (modulo higher degree terms)

and using (i) we can show (ii)

Y xH n~x = —(—2)n” ti! tt+1 x,

i > 0 (modulo higher degree terms).

If / i G U(q)* corresponds to Xi € C [[x i,X 2 ,xs]] under the isomorphism U(q)* ^ C [[x i,X 2 ,X 3]] then etx.xi =

^

< f i ,e ~ tx.yXsh%2x Xl >

( t i,t 2 jtsjGN3

=

< f u Y iaH i2X il > / i 1 !i2 !i 3 !xi, x ^ x j 3

Y , (»1 **2 **3 ) GN3

so it suffices to find the linear term of Y X3H t2X %1 when expressed as a linear com­ bination of the ordered monomials y U3hU2x Ul since /» vanishes on all other ordered monomials. Since X = x, it is clear that we only need to consider the expressions Y %3H %2. By routinecalculations using (i) and (ii) above we obtain f p—2x2 tar i i e .®i = ®i + 1 - -— -— ; 1 — tx 3 e tac.®2 = ®2 + log (1 — tx 3);

t*

x3

e -®3 = -— — . 1 —1®3 Similarly, for the action o f e-ty we have X = e ~ ty.x = —t2y + th + x H = e~ ty.h = h - 2 t y

Y = e~ ty.y = y. Suppose that X n = ( —t2y + th + x )n, n = 1 ,2 ,3 ,... is expressed as a linear combination o f ordered monomials y*s/i*2x*1. Again we have similar results as above: (1) The sum o f those terms o f the form x k is

(2) The sum o f those terms of the form y k is

(3) The term involving only h is (—l ) n -1 (n — 1 )! tnh.

Simple Modules over Semisimple Lie Algebras

85

Similarly, by using the above formulas, we obtain (iii)

X n = (—l ) n(n\tn+1y - (n - l)\tnh - n!tn- xx),

n > 1

modulo higher degree terms, and using (iii) we can show (iv)

H n- ' X ' = ( —1)*(—2)n - ‘ z!t’ +xy,

i > 0 (modulo higher degree terms).

Using (iii) and (iv) one can calculate the action of ety on x i , x 2,xj: exp (ty ) •x i =

Xl

1 + tx i exp (ty) •x 2 = x 2 + log (1 + tx i) ^g“ 2l 2 exp (ty) •x 3 = x 3 - 1 + ,■ 1 + tX\

The maximal torus T of diagonal matrices g = diag(u,u~1) in G acts via

g •x i = u~2x i g -x 2 = x 2

g - x 3 = u2x 3. Now let e S L ( 2,C ). There are two cases: (i) a ^ O . Then

_ /

1

9 ~ Va“ xc

0\ f l

ab\ ( a

0 \

1 / V°

1 / V°

a V

= exp (a~l cy) exp (abx) diag(a, a " 1). Thus we have (b + dx\)(d — 6x 3 ) — bde~2x2 ^

Xl (a + cxi)(d — 6 x3 ) —bee"

2*2

x 2 + log [(a + cxi)(d — 6 x3 ) - bce~2x2]

g •x 2 =

_ (a + c x i) ( —c + ax 3 ) + ace~2x2 ^ X3

(a + c x i)(d — 6x 3 ) — bce~2x*

(ii) a = 0. Then 9

-(-!-■:) 0

l\fl

- \ - i

o )\ o

_/

—b~1d\

1

A

( 6-x 0

0\ V

= exp (x) exp (—y) exp (x) exp (—b~ldx) diag( 6_1, b). We have g - x 1 = -b d +

b ( d - b x 3) X1 X3 — b~l dx\ + e - 2 *2

g - x 2 = x 2 + log (xxx 3 - 6-1 dxi + e-2 *2) 9-x3 =

—6 -2 xi * 1*3 ~ 6_1dxi + e-2 **'

Dahlberg

86 R eferences

1. R.P.Dahlberg, Injective hulls of Lie modules, J.Algebra 87 (1984), 458-471. 2. _______, Injective hulls of simple sl(2,C) modules are locally artinian, Proc. Amer.Math.Soc. 107 (1989), 35-37. 3. J.Dixmier, Enveloping Algebras, North-Holland, New York, 1977. 4. G.P.Hochschild, Algebraic Lie algebras and representative functions, Illinois J. Math. 3 (1959), 499-523. 5. ______ , Algebraic Lie algebras and representativefunctions,supplements, Illinois J.Math. 4 (1960), 609-618. 6. J.Humphreys, Introduction to Lie algebras and representation theory, Springer Verlag, New York, 1972. 7. J.C.Jantzen, Einhullende Algebren halbeinfacher Lie-Algebren, Springer-Verlag, Berlin, 1983. 8. T.Levasseur, Cohomologie des algebres de Lie nilpotentes et enveloppes injectives, Bull. Soc. Math. FVance 100 (1976), 377-383. 9. ______ , L fenveloppe injective du module trivial sur une algebre de Lie resoluble, Bull.Soc.Math. FVance 110 (1986), 49-61. 10. ______ , Private Communication 6/30/87. 11. M.E.Sweedler, Hopf algebras, Benjamin, New York, 1969. 12. M.Vergne, Cohomologie des algebres de Lie nilpotentes, Bull.Soc.Math.FVance 98 (1970), 81116.

Smoothing Coherent Torsion-Free Sheaves

Amassa Fauntleroy, Department of Mathematics, North Carolina State University, Raleigh, NC 27695

Introduction Let F be a coherent torsion free sheaf on the noetherian normal scheme X . If F has a pseudo-determinant in P ic (X ) (see below for definitions) then there exists a sheaf of ideals I in Ox such that if a : Z —►X is the blow-up of X along / , then the strict transform of F is locally free on Z.

1

Algebraic Version

Let R be a noetherian local integrally closed domain with quotient field k. Let M be a finitely generated torsion-free i?-module. We shall say that M has a pseudo-determinant if there is an element L £ Pic(R) and a mapping ip : AdM —►L where d = r k ( M ) such that ipp is an isomorphism at all height one primes p of R. Since rank one projectives are reflexive it follows d d that L ~ (A M)**. Thus M has a pseudo-determinant precisely when [A M]** is in Pic(R) and hence L is unique up to isomorphism. When M has a pseudo-determinant we define the determinantal ideal of M , I-det(M ), of M as follows: If c : L (& L* —►R is the natural isomorphism, then I-det(M ) is the image in R of the composition A M ® L* L ® L* R. For any prime ideal p of height one in i?, both maps in this composition become isomorphisms upon localization at p. It follows that I-det(M) is not contained in any height one prime so that ht(I-det(M )) > 2. The following lemma is well-known. We give a proof here for the convenience of the reader. 87

88

Fauntleroy

Lem m a

1.1

Let Q be a prime ideal of R. Then the following are equivalent:

(i) M q is a free J?g-module. d (ii) A (M q) is a free Rq-module. (iii) I R q is a principal ideal where I = I-det(M ).

P roof. The implications (i) =£► (i) => (iii) being immediate we show(iii) => (i). Fix an isomorphism A of I R q with R q and let e denote the composition A M q 0 L* —> IRq —►Rq There exists / £ L* and m i ,. . . rrid in M q such that e ( f 0 (mi A ... A rrid)) = 1. Let be the (d — l)-form obtained from mi A ... A rrid by deleting m,- for 1 < i < d. Then for m £ M q we must have a relation of the form d bm = ^ 2 6*m i> i=l

biinRg

since m i ,. . . , m^ is a basis of M q 0 Rq K over K . Then e ( f ® (hm A w,))

=

b •e ( f 0 (m A to,-))

=

E ty •e ( f 0 rrij A Wi)

=

± 6 te ( / ® (mi A ... A md)) = ± 6 t-

Let ai = e ( / 0 m A Wi). Then b{ = ±ba{ and b(m — E )ia tm t) = 0, thus m = ± a ,m t-. Since the set { m i , . . . ,m d} spans M q and rank M q = d, this set is actually a basis and M q is free. Thus (iii) => (i) and the proof of the lemma is complete.

Corollary. If q is a prime ideal of R at which M q is not Rq-iree, then q D I-det(M ).

Note that the above proof works under the more general setting L £ Pic(R), A M —►L given and the image of A M 0 L~l —> L 0 L~l —> R is principle. We do not actually d need L = (A M)**. When R is not necessarily local we say that the R-module M has a pseudo-determinant if Mp has one over Rp for every prime ideal p of R.

Smoothing Coherent Torsion-Free Sheaves 2

89

. G eom etric Version

Let X be a normal variety over the algebraically closed field k and F a coherent torsion free O^-module of rank d. We say F has a pseudo-determinant if T(V, F ) has a pseudo­ determinant as a r(V ,0*)-m odule for every affine open V of X . If (L,ip) is a pseudod determinant we denote by detF the image under ip of A F and by I-detF the determinantal ideal defined locally as in Section 1. Assume now that such an F is given with pseudo­ determinant L € P ic (X ) and that F is actually reflexive and generated by its global sections. Let a : Z —> X be the blow-up of X along I-detF. Recall (c.f. Raynaud, Section 4.1) that for any quasi-coherent sheaf G on X , the strict transform &(G) of G is the sheaf cr*(G)/N (G ) where N (G ) is the subsheaf of cr*G generated by sections whose support lies along the exceptional divisor of a.

Lemma

2 . 1 . If G is torsion free, then N(G) is the torsion subsheaf of cr*G and hence a(G) = 0 yields the sequence 0 ^ N(G) ® 0z K -+ { cj*G) z ®o2 K -> a{G )z 0 o 2 K -> 0 But the map on the right is an isomorphism since cr*G and crG agree at the generic point of Z. Thus N (G) 0 oz I< = 0 and N(G) C T. d . . d—1 Consider now the map a*L. If x € cr*F is a torsion element and w E A F is any (d — l)-form, then p*(x A w) is torsion hence zero. It follows that (p* factors through A &{F).

P roposition

2 .2

d

. The image of A d'(F) in cr*L is the invertible sheaf IO z 0 o*L.

P roof. By definition we have an exact sequence A F ® L 1 —►IO x —►0.

90

Fauntleroy

d Applying the right exact functor a* and using the fact that (p* factors through A crF yields the exact sequence A &F ® a 'L - 1 - » IO z - » 0. To obtain the desired conclusion we simply tensor this last sequence with the invertible sheaf cr*L.

C orollary 2.5. The strict transform v ( F ) of F is locally free on Z. d

P roof. The question being local, we may use the fact that A &(F) surjects onto a locally free sheaf and the proof of Lemma 1.1 to obtain the corollary. Note that if X is locally factorial, every coherent torsion free sheaf has a pseudo­ determinant.

2

An Example

Let R = k[x,y,z] be a 3-dimensional polynomial algebra over the field k and let M be the cokernel of the map : R 2 —> R 5 where x 0 0 x y

o

*

y

0

z

Let m be the maximal ideal generated by x ,y and z. Then since 0 —> R2 m —> R?m —> M m —> 0 is exact, depthm(Mm) — 2 and Mm is reflexive. Since Mp is free for any prime ideal p ^ m it follows that M is reflexive. Let {e, : 1 < i < 5} be the standard basis for R 5. Then M is defined by the relations txx =

x e 1 + ye 3 + ze 4 = 0

u2

x e 2 + ye 4 + ze 5 = 0

=

Smoothing Coherent Torsion-Free Sheaves Let L C

R

91 2

be the submodule generated by ux and u2 and define maps /

3

5

:AL

AM —

5

R and g :A R —> R as follows: #(e! A ... A e5) = 1 /( u i A u 2 0 mi A ra2 A m 3) = Ux A u2 A Ux A u2 A u3 where ut- is any preimage of rat- in i?5. If mx has preimages v and v' then v with a, b in i?. Hence Ux

A

— v'

=

u2 A v A v2 A v3 =

= u\ A u2 A (u' + aux + 6u2) A v 2 A v 3 = u\ A u2 A vr A v2 A v3 It follows from similar calculations using m2, m 3 that / is well defined. Let ^ = g of. Then ^ induces an isomorphism A L P® ( A M ) p -> Rp for all height one primes

2

p of R. Thus A L ~ R is identified with the dualof

Computing(uxA u 2 ® A ej A e^) for i < j < k we find I-det(M ) = I = just the ideal generated by the 2 x 2 minors of (j).

(x,

A M

y , z )2. This is

Let a : Z X = Speci? be the blow-up of X along the sheaf of ideals 1 0 x- Since I = m2, Z is isomorphic to the blow-up of X at the point (0, 0 ,0). The scheme Z is covered by three open affines Zx, Zy and Zx where, for example, r ( Z x , 0z ) =

R[ot, /? ],

xa =

y,

x/3 =

z.

A similar description can be given for the other two open affines Zy, Z z. Note that M x is generated by elements of the form t / x where £ is a linear form in e 1 ? . . . , e 5 . On the affine open set a (M ) = E 0 is generated by e3, e± and e 5. Similarly, cr(M) is free of rank 3 over Zy and Zz.

References [1]

M. Raynaud, Flat modules in algebraic geometry, Com positio Math. 24 (1972), pp. 11-31.

au

Projective Covers and Quasi-Isomorphisms

MARK A. GODDARD Department of Mathematical Sciences, The University of Akron, Akron, Ohio 44325-4002

1

. Introduction

In this paper, the concept of the projective cover of a module is generalized to the case of complexes of finitely generated modules over a Noetherian local ring. The projective cover of a complex possesses many but not all of the key properties of the projective cover in the module case. Using the fact that the projective cover is a quasi-isomorphism, we see that the projective cover is closely related to the free resolution and minimal free resolution of a complex (see Roberts, 1980). We begin with some introductary terminology. In this paper, R will be a commutative ring with identity. A complex C of R-modules is a sequence of R-module homomorphisms . . . —►C n —^ C n- i —> . . . —►C i -A C o —►• ••

satisfying 6;_i o < $,• = 0 for all i 6 Z. The maps Si are called the boundary maps of the complex C. If ker will be said to be surjective if each map iis surjective. A quasi-isomorphism is a map of complexes C

D which induces an isomorphism

on homology, i.e. each induced map H i ( C ) -A R ;(D ) is an isomorphism. 93

Goddard

94

Generalizing the definition of a projective 12-module, we define a projective complex to be a complex Q satisfying the condition that for any complexes C and D , any map of complexes Q D C the diagram

C and any surjective map of complexes

\ \ \ Q can be completed by a map of complexes. It is worth noting that every complex of projective 12-modules does not form a projective complex. (See Goddard (1993) for an example.) Nonetheless, the collection of all projective complexes can be easily described as seen in the following proposition: PROPOSITION 1 .1 A bounded above complex P is a projective complex if and only if P is an exact sequence o f projective modules. Proof: See Roberts (1986), proposition 2.1. □ By a generalization of the categorical definition of the projective cover of a module, we now define the projective cover of a complex: DEFINITION: The projective cover of a complex C is a complex P of projective modules and a map of complexes P C satisfying the following two conditions: 1.

For any complex Q of projective modules, and any map of complexes Q —> C , the diagram

\ \ \ Q can be completed by a map of complexes.

Projective Covers and Quasi-Isomorphisms

95

2. The diagram

P

C

\ \ \ P can only be completed by maps of complexes which are automorphisms of Pi in each degree. Note that a complex of projective modules satisfying only condition definition above is called a projective precover of C. 2

1

of the

. Properties o f the Projective Cover

In this section, we shall consider some of the important properties of the projec­ tive cover of a module and compare and contrast these with the properties of the projective cover of a complex. The projective cover P M of a module, M , is a unique surjective map from a projective module P onto M . The existence of the projective cover may be shown in the case where M is a finitely generated module over a Noetherian local ring. We will see that the existence and unique­ ness of projective covers is preserved in the case of complexes as is the surjective property. The projective cover of a complex need not be a projective complex however. For an example of a projective cover which is not itself projective, see Goddard (1993). We begin with the proof of the existence of the projective cover of a complex. Although the theorem below is stated for finitely generated modules over a Noetherian local ring, we may replace this with modules over a perfect ring or any other condition guaranteeing the existence of projective covers in the module case. THEOREM

2 .1

I f C is a bounded above complex o f finitely generated modules

over a Noetherian local ring then C has a projective cover. Proof: Let P 0 ^ Co be the projective cover of Co whose existence is guaranteed by the fact that Co is a finitely generated module over a Noetherian local ring and let Pi = 0 for all i < 0. From this point we proceed inductively to construct the projective cover of the remainder of the complex. Assuming that we have completed the ith step

96

Goddard

Pi

/nr

fr+i

^»+l

*

P i-l

r t ______t /nr

* W -l

we make use of the standard pullback construction given by the set

s = {(p>c) € ker(b{) © Ci+ 1 |£+i(c) = Ci(p)}. Since 5 is a submodule of

Pi © Ci+1 and hence is itself finitely generated, we

can construct the projective cover Pi+ 1 A 5 of 5 to obtain the commutative diagram

Pi+i —

—

S

Pi

Pt

L>i-1

If we define 6l+1 = tti o e and et+1 = 7r2 o e then this construction clearly yields a complex P of projective modules and a map of complexes P A C . All that remains to be shown is that this map satisfies properties 1 and 2 stated in the definition of the projective cover. First we let Q C be a map of complexes from a complex of projective modules Q into C. To satisfy condition 1, we must find a map of complexes Q A P so that e o h = . For i < 0 we let hi equal the zero map and we define ho to be the map completing the diagram

a—

P„ \

Co t

\

j for all j < i. To construct h,-+x we consider the diagram

Projective Covers and Quasi-Isomorphisms

Pi+l— —

97

S ------- ^

Pi

C7i+1— ^

%+1

—

Define /

: Q i+1 ->

Qx—

—

Q i -1

S by /( g ) = (hi o gi+1(q),i+i(q)). Since Pi+i is the

projective cover of 5 , there exists a map Qi+i h^ Pi+x such that e o h,-+ 1 = / . It follows easily that e, +1 o /ii+1 = (f>i+1 and that the map h defined in this manner still commutes with the boundary maps. Next, let P

P be a map of complexes satisfying e o h = e. We must show

that each map Pi ^ Pi is an automorphism of P*. For * < 0 this is clear since each map hi is the zero map and since P Q Co is the projective cover of Co> we know that ho is an automorphism as well. Proceeding inductively, let us assume that hj is an automorphism for all j < i and show that h,-+1 must be an automorphism as well.

We can define an automorphism seen that / makes the diagram

S

S by f(p, c) = (A;(p), c). It can easily be

98

Goddard

Pi+r J*i+1

P i+ r

commute and since P,+i A 5 is the projective cover of 5 , it follows easily that hi+i is an automorphism of Pi+1 and our proof is complete. □ In the same way that the definition of a projective cover can be generalized to yield an injective or flat cover in the case of modules (Enochs, 1981), we can also define the injective or flat cover of a complex. Proceeding exactly as in the proof of the last theorem, we can prove the existence of the injective cover of any bounded above complex of modules over a Noetherian ring. Next we consider the question of uniqueness. By generalizing the proof used in the module case, we can easily prove that the projective cover of a complex is unique up to isomorphism. The result below holds for injective covers of complexes as well. PROPOSITION morphism.

2 .1

The projective cover o f a complex C is unique up to iso­

Proof: Let P A C and Q

C be projective covers of C . Since P is a projective

cover, there exists a map of complexes Q P such that e o h = . Similarly • • • • k since Q is a projective cover, there exists a map P —►Q such that 0 —►C n —* C n_ i —► . . . —►C\ —> C o —* o —► . . .

where Co, C \ ,..., C„_j are all free modules then the projective cover of C is equal to the minimal free resolution of C if and only if £{(Cj) C m C,_i for all i. We conclude this paper with a theorem which specifies the most straightforward condition which can be placed upon the minimal free resolution to guarantee that it is identical to the projective cover. We first need the following pair of lemmas.

and G —» C are two minimal free resolutions of C and F is a map of complexes making the diagram

LEMMA 3.2 / / F —> C G

Hi{F)--------- * Hi(C)

commute for all values ofi, then each map hi is an isomorphism. Proof: See Roberts (1980). □ LEMMA 3.3

If F

C

is the minimal free resolution of C then the diagram

\ \ \ can only be completed by maps of complexes which are automorphisms of Fi in each degree.

105

Projective Covers and Quasi-Isomorphisms

Proof: If h is a map of complexes completing the diagram above, then Hi() o Hi{h) = Hi() as well and by lemma 3.2, h is an isomorphism from F onto itself. □

The minimal free resolution of a complex C is identical to the projective cover of C if and only if the minimal free resolution is a surjective map of complexes. THEOREM 3.2

Proof: Since the projective cover is a surjective map of complexes, the first implication is obvious. To proceed in the other direction, let us assume that the minimal free resolution F C is surjective. By lemma 3.3, the minimal free resolution always satisfies the second defining condition of the projective cover. Thus, all we need to show is that the minimal free resolution is a projec­ tive precover. This follows immediately from lemma 3.1 since the minimal free resolution is a quasi-isomorphism and is surjective by assumption. □

REFERENCES

1. Enochs, E. (1981). Injective and Flat Covers, Envelopes, Israel Journal of Mathematics, (3)39, pp. 189-209. 2.

and Resolvents,

Goddard, M. (1993). Projective Covers of Complexes, Proceedings of the 1992 Ohio State - Denison Mathematics Conference.

3. Roberts, P. (1980). Homological Invarients of Modules Rings, Les Presses de l’Universite de Montreal.

over Commutative

4. Roberts, P. (1986) Some Remarks on the Homological Algebra of Multiple Complexes, Journal of Pure and Applied Algebra, 43, pp. 99-110. 5. Rotman, J. (1979) An Press Inc., New York.

Introduction to Homological Algebra, Academic

On Dihedral Algebras and Conjugate Splittings

DARRELL E. HAILE Department of Mathematics, Indiana University, Bloomington, Indiana, [email protected]

Let E be a field and let D be an E-central division algebra. A well known theorem of Wedderburn (1921) says that if an irreducible polynomial f ( x ) E F[x] has a root 8 in D then f ( x ) decomposes into a product of linear factors in D [x ]. More precisely there axe elements 8 \ — 6 , 6 2 ,- . in D, such that each 8 i is conjugate to 8 and f ( x ) = {x — 6 k)(x — 8 k - 1 ) •••(x — 8 2 ){x — 6 \). Note: The elements 8 { need not commute! In the same paper Wedderburn proved that if the dimension o f D over F is nine, then one can do better. In that case if 8 is in D — F then f ( x ) (which will be of degree three) decomposes over D in a very special way: there is an element y in D such that y z E F x and f ( x ) = (x —y ~ 2 8 y2)(x —y ~ 1 8 y)(x —8 ). Wedderburn used this fact to prove that every division algebra o f dimension nine is cyclic, that is has a maximal subfield (necessarily of degree three over F ) that is a cyclic extension of F. From this point on the field F will be assumed infinite. In Haile (1989, 1991) I investigated the possibility of extending this result on polynomials to division, algebras of arbitrary dimension. This inquiry was spurred in part by the discovery that such a decomposition is related to other questions of interest and in particular to the theory of Clifford algebras, which will now be described. If nx n)m ~ g(a\ , ., an)|ai,. . . , an E E }. Of course if g has degree two, then this is the classical Clifford algebra. For more information in the general case the reader is referred to the bibliography, particularly [Childs (1978), Tesser (1988), Haile & Tesser (1988), Hodges & Tesser (1989)] and their references. The connection with decompositions of the type described above is as follows: Suppose A is an E-algebra and f ( x ) E F[x]. We will say f ( x ) has a conjugate splitting in A if there is an invertible element y in A and an element 8 in A such that y n E E and f ( x ) — ( x —y~^n~ 1 ^8 y n~ 1 ) ( x —y~^n~ 2 ^8 y n~2) •••(x —y ~ l 8 y)(x —8 ) in A[x]. We call the scalar y n the associated constant for the splitting. Let i,. . . , & n-i) € F denote the determinant o f x I —A (that is, this is just another notation for f ( x ) ) . Expanding along the first row we obtain hn(a0, a i> •••? an - i 5 •••>& n-i) — £ ^ n - i ( 0>a 2 >a 3 >•••> a „ _ i ; 6i, 62, . . . , &n_ 2 , 0) + « i 5 '+ 6 n_ i T for appropriate cofactors 5 and T . Expanding these cofactors along their first columns gives fc„(ao, a i , . . . , an_ i ; 6o> •••>&n-i) — x ^ n - i ( 0 , a 2 > «3> ••• &1 ? &2 j •••, &n-2 , 0) — ai &o^n-2 ( 0, n- i /i „ _ 2 (0, a2, a3, . . . , a n_ 2; ^ , &2, . . . , &n_ 3 , 0 ) + ( - l ) na0ai •••a n_ ! -fi ( - l ) n 60&i •••6n- l - The result follows im m ediately by induction. □ P R O P O S IT IO N 2. Let c, d, e, / E L 0- If ca + d a - 1 and e/3 -fi / / 3 ”*1 are invertible, then r]n E Lo? where 77 = ( c a -fi d a _ 1 )(e/3 + /jS ” 1) ” 1. P r o o f . W e pass to A = A F , where jF denotes the algebraic closure o f jF , which we may assume contains L 0. Let r E F such that r n = &(— /3n). Then P — Pr where P

= 1 . The algebra A is generated as an jF-algebra by a and p. M oreover A acts on the

n-dimensional vector space F[a] via ^

j S i j a ip J •a* =

In particular if

we use the powers o f a as a basis o f F [a ], the element e/3 + / / 3 _1 acts diagonally, in fact (e/3 -fi / / 3 _1 ) •a k = ( ( ker -fi ( ~ kf r ~ l ) a k. It follows that the m atrix o f 77 is as follows:

0

ai

0

bo

0

a2

0 0

br

0

0 62

0

0 0

0 0

\ a _ 1 a0

0 0

0 0

0 0

a4

0

0 0

0

•' •• • •

0 0 0 0

• 0 •■• bn- 2

^ n —1 \

0 0 0 a n—1

0

/

where a t- = c(£ ler -fi ( - l / r - 1 )- 1 and 6t- = d(£*er -fi C_ t/ r _ 1 ) _ 1 - By Lem m a 1 the charac­ teristic polynom ial f ( x ) = Ckxk (say) °f *1 property that c* = 0 for all even positive fc. If we reverse the roles o f a and /3 then the argument just given shows that, except for the constant term, the even coefficients o f the characteristic polynom ial o f 77_ 1 are zero and so we conclude that Ck = 0 for 0 < fc < n, that is r}n E L q. □

110

Haile

We can now proceed to our main result. Let 6 = (a + a - 1 )/(an~ 1 + a 1-n ) G i f . Note that the conjugates ^‘ (tf), 0 < i < n —1 , are distinct: If crl(6) = 0 then a computation gives a 2n = 1 , which contradicts the fact that L q[q\ = L. Hence if d(x) denotes the minimal polynomial of 9 over F , then the degree of d(x) is n and i f = F(9). THEOREM 3. Let A be any central simple F-algebra of degree n. If the polynomial d(x) (defined above) has a root in A, then d(x) has a conjugate splitting in A. P r o o f. If d(x) has a root in A then A contains a (maximal) subfield isomorphic to i f , which we will identify with K . Let j3 G A ® F L q be as defined above. We claim there is an element s G F x such that s/3 + s ~ l f3~x is invertible in A ® F L q: The extension L q[/3]/Lq is a cyclic Galois extension of L q (but not necessarily a field). We comput the norm in this extension of the element s/3 + s - 1 /?- 1 : N l /l S sP + s ~ l P~l ) = N i /l 0( s ~1/3~1)N l /l 0(s2^2 + 1) = s~2nb~1( ( —l ) n — s2nb2). Clearly there is an element s G F x such that this last term is nonzero. For such an s the element s/3 + s ~ l f3~l is invertible. Now let U = (a + a - 1 )(s/3 + s- 1 /3- 1 )- 1 and V = —( a n_1 + a 1~n)(sf.3 + s - 1 /3- 1 ) - 1 . These are elements of A Lq fixed by r, hence in A. If c ,d G F then cU + dV = ((c — a~1d)a + (c — ad)a~1){sfi + 2. Since V g(B) = Z, Vg(Vg(B)) = Vg(Z). But B is an B-direct summand o f S and S is H-separable over B, so B = Vg(Vg(B)) ([S], (5), P. 296). Hence, B = V S(Z). m -1

2 * * 4 . Noting that Vg(Z) = Y Anng(Ij)x* by Proposition 3.1 (3), we have i-O that V g ( Z ) = B if and only if A n n g ( I j ) = { 0 } for each i = 1,..., m - 1.

We shall show a characterization of a center-Galois extension B when u is in U(ZP). THEOREM 3.4 The following conditions are equivalent: (a) Z is Galois over Z p with Galois group o f order m and u e U (ZP). (b) (i) S is an H-separable extension o f B, (ii) Annz (Ij) = {0 } for each i = 1,..., m-l, and (iii) B = BPZ.

118

Ikehata and Szeto

Proof:

Let Z be Galois over Z p with Galois group

of order m. Then (i)

and (ii) hold by Theorem3.3 (1) ->( 4 ). For (iii), it can be checked that is a projective dual basis for B over B p where {

a pp f in Z, i = 1

, k for some

m-1

integer k} is a Galois coordinate system for Z and

tr(b) = ^p' ( b) for each b in B.

b = '^tr[bai)^i which is in BPZ; and so B = BPZ.

Thus

i

Conversely, it suffices by (3) of Theorem 3.3 to show that V 3 (B) = Z. I n fact, let ^

xiai be an element in V 3 (B); then, for any b in Bp, b\ ^ *'«,) = ( V 1

i

Thus

/

V i

)

b-

p\b)ai = bat = atb. Since B = BPZ, at is in Z for each i. Also, for any b in Z , so

p Xf y a - a p . Hence a,(fc-p‘(fc)) = 0 for each i and b inZ.

But Ann^Hi) = { 0 } for any i =

1 ,...,

m - 1, so a, = 0 for any i =

1 ,...,

m - 1.

Therefore V 3 (B) = Z (for Z c V s (B ) is clear).

4 S K E W P O L Y N O M IA L RINGS O YE R AN A Z U M A Y A A L G E B R A In this section, we shall give a sufficient and necessary condition for S being an Hseparable extension over an Azumaya algebra B. Our result generalizes Theorem 2.1

in ([I 2]) for S over a commutative ring B. Then a structure theorem for S over

B satisfying the Kanzaki hypothesis is derived. T H E O R E M 4.1

S is an H-separable extension over an Azumaya algebra B if and

only if S is an Azumaya Z p-algebra.

Proof: Let S be an Azumaya Z p algebra. Since S is a free left module over B, S is an H-separable extension over B ([I 1], Theorem 1). But then B = V 3 ( V ^ B )) ([S], (5), P. 296). Again, S is an H-separable extension over B so V 3 (B) is Galois over Z p by Theorem 3.2. Hence V 3 (B) is a separable subalgebra of S over Z p. Thus B = V s ( V 3 (B)) is also a separable subalgebra of S over Z p by the commutator theorem for Azumaya algebras ([DI], Theorem 4.3, P. 57). Therefore B is Azumaya over Z ([DI], Theorem 3.8, P. 55).

H -Skew Polynomial Rings and Galois Extensions

119

Conversely, Since S is an H-separable extension over B and B is an Azumaya Z-algebra, S is also an Azumaya algebra over its center ([O], Theorem 1). By Proposition 3.1, the center of S is Z p , so S is Azumaya over 2f

.

W e recall that a ring B together with a finite automorphism group G satisfies the Kanzaki hypothesis if (1) B is Azumaya over Z and (2) Z is Galois over Z G with Galois group G/Z induced by and isomorphic to G ([D

1 ],

[SW], [SM]). Now

we consider the Kanzaki hypothesis in the case G =

. W e note that when B satisfies the hypothesis, S is an H-separable extension of B.

C O R O L L A R Y 4.2

If (B,

) satisfies the Kanzaki hypothesis, then S, B p and

Z[x,p] are Azumaya Z p-algebras and S a B p®

Proof:

pZ[x,p].

Since B satisfies the Kanzaki hypothesis, B a B p® ^ p Z and B p is

Azumaya over Z p ([D

1 ],

Lemma 2, P. 119). Hence S = B[x,p] a B p® ^p Z [x ,p ].

By Theorem 4.1, S is an Azumaya Z p algebra, so Bp and Z[x,p] are Azumaya Z p-algebras ([DI], Theorem 4.4, P. 58). A similar structure theorem for the skew polynomial subrings B[xi,pi] of S for each i = l , ..., m-l can be proved. T H E O R E M 4.3 If (

B ,< p > ) satisfies the Kanzaki hypothesis, then (1) for each

i = 1 ,..., m - l , Bp' is Azumaya over Z p , and (2) f i f jr '. p '] e # p'® z,. z [ x \ p ‘] is Azumaya over Z p', where ® is over Z p'.

Proof: since

1. Fori = l,

B = BpZ * B p®ze Z ([D 1], Lemma 2, P. 119), Bp‘= (Bpz )P' /

as [Bp® z„ 2

Bp is Azumaya over ZP([D 1], Lemma 2, P.119). F ori>l,

.

V®p'

Zj

(

= Bp®z„ Z p ; and so Bp is an Azumaya Zp -algebra.

By ( 1 ), b [ * v ] .

^ ^ 5 p® z)), z j [ j :',p '] =

(b p®zPz )[x ‘, p ' ]* (b p® z, z p,® zP, z)|V,p']

Bp ®^, z|V,p']. Since Z is Galois over Zp with Galois

120

Ikehata and Szeto

group

, z j V . p '] is Azumaya over Z p'. ([1 1], Theorem 2.2, P. 23). Thus Bp,® zP, zJV .p'] -flfoc'.p'] is an isomorphism as Azumaya Z p' -algebras.

Proposition 3 .1 shows that the center of Vg(B) is Z if S is an H-separable extension over B and Theorem 3.2 states that Vg(B) is Galois over Z p with Galois group < p /V g (B )> of order m if and only if S is an H-separable extension of B. Thus we derive the following proposition for V$(B). PROPOSITION 4.4

If S is an H-separable extension of B, then V $(B) is an

Azumaya Z-algebra.

Proof:

By Theorem 3.2, V$(B) is Galois over Z p so V §(B ) is separable over Z p.

But the center of V$(B) is Z by Proposition 3.1, so V$(B) is Azumaya over Z.

REFERENCES [D 1] DeMeyer, F. R. (1965). Some Notes on the General Galois Theory of Rings,

Osaka J. Math., 2: 117-127.

[D 2] DeMeyer, F. R. (1966). Galois Theory in Separable Algebras over Commutative Rings, Illinois J. Math.. 10: 287-295. [DI]

DeMeyer, F. R. and Ingraham, E. (1971). Separable algebras over commutative rings, Lecture Notes in Mathematics, 181: Springer-Verlag, Berlin-Heidelberg-New York.

[H]

Hirata, K. (1968). Some Types of Separable Extensions, Nagoya Math J.. 33: 107-115.

[1 1]

Ikehata, S. (1981). Azumaya Algebras and Skew Polynomial Rings, Math J. Okayama Univ.. 23: 19-32.

[12]

Ikehata, S. (1984). Azumaya Algebras and Skew Polynomial Rings, II,

[13]

Ikehata, S. (1981). Note on Azumaya Algebras and H-Separable

Math J. Okayama Univ., 26: 49-57. Extensions, Math J. Okayama Univ., 23: 17-18. [N]

Nakajima, A. (1987). P-Polynomials and H-Galois Extensions, J. A lg ., 110: 124-133.

[O]

Okamoto, H. (1988). On Projective H-Separable Extensions of Azumaya Algebras, Results in Math., 14: 330-332.

//-Skew Polynomial Rings and Galois Extensions

[S]

121

Sugano, K. 1975). On a Special Type of Galois Extensions. Hokkaido MathJ. .4 : 123-128.

[SW] Szeto, G. and Wong, Y.F. (1983). On Separable Noncyclic Extensions of Rings, J. Austral. Math Soc. Series A. 34: 394-398. [SM] Szeto, G. and Ma, L.J. (1988). On Rings Whose Center Are Galois Extensions, Portugaliae Math., 45: 75-82.

Separability and the Jones Polynomial

LARS KADISON Department of M athematics and Physics, Roskilde University, 4000 Roskilde, Denmark

1

Introduction Jones’ index theory of type II\ von Neumann algebra subfactors was pub­

lished in 1983, and led in the spring of the following year to a new polynomial invariant of knots and links. Subsequently, the Jones polynomial was gener­ alized in different directions and several old problems of Tait’s in knot theory were solved. Certain key ingredients of Jones’ theory may be reduced to al­ gebra in different ways. For example, the Jones polynomial may be defined from certain traces of Ocneanu’s on a sequence of finite dimensional algebras named after Hecke. A second example: the semi-discrete index spectrum of III subfactors may be obtained from the classification of matrix norms of the 0 -1 matrices - accomplished long ago - the 0 -1 matrices arising as the inclusion matrices of the multimatrix e;-algebras Ap A, an A-A bimodule morphism defined by a,Q® a\ i-+ aoa\. Definition 2 . 1 A is called a finite separable extension of S if there exists an element / € A ®s A, an S-S bimodule homomorphism E : A —» S, and r £ k° such that 1.

a f = fa

(Va € A) and fi(f) =

1;

2. E( 1) = 1; 3. n(l ® s E ) f = fi(E s 1 ) / =

t

.

An element / satisfying axiom 1 is called a separating element, or a separability element, and its existence alone defines a separable extension of rings, a theory generalizing separable algebras and developed by Sugano and several others [3]. The existence of / is equivalent to /j, being a split epimorphism of Abimodules, which is in turn equivalent to the vanishing of relative Hochschild

Separability and the Jones Polynomial

cohomology groups

1

125

with arbitrary coefficients (n >

0 ),

H n{ A , S - - ) = 0.

The map E : A —> S satisfying axiom 2 is a conditional expectation as in operator theory, and its existence for a subalgebra S C A defines a split extension of rings. It is equivalent to requiring the subalgebra S be a direct summand in the bimodule 5 A 5 : since the inclusion map splits the kernel exact sequence of the S - S epimorphism E : A —> S, we note that A = S' © ker E. Conditional expectations and separating elements are not unique, but axiom 3 demands the existence of a conditional expectation E : A —> S and separating element in A ® 5 A n

f = t ^

a:,-

yi

»=i

such that r E k° and

( 1)

it,E(xi)yi = it,xiE(yi) = 1i=1

i= l

Indeed, a short computation reveals that we may choose x\ = y\ = 1 and Xi,Vi £ kerf? for i = 2 , . . . ,n. We will say that E and / are compatible in case they satisfy axiom 3. Now fix the notations S C A, / , E , x,-, ?/;, /, or G generates J . An ideal J is said to be reduced if J agrees with its own radical; or, equivalently, if f 2 G J entails / G J always. Our first result is the following: P r o p o s it io n

2

.

For any ideal J C A '^ , if Lt(J) is reduced then J isreduced.

Proof. Suppose J not reduced, and consider the polynomials h such that h (£ J, h2 G J. Among all such h ’s let / be one whose leading term it( /) is minimal. Then, (h ( / ) ) 2 = £ t(/2) G Lt( J) is clear . But, it{f) £ Lt (J). For, if lt ( /) G Lt(J), then by Lemma 1 it(f) = it{g) for some g G J. But then it(f — g) < it(f), whereas f — g & J, {f — g)2 € J. This is contrary to the minimality of / among the h ’s as above. □

2

Simple examples such as J = < x2+ y >, does not hold.

1

< y < x show that the converse of Prop.

Let J be an ideal in generated by monomials. If each of the generating monomials is square-free, then J is reduced.

P r o p o s it io n 3.

Proof. Let J = < m\,...,nnq > with square-free monomials mt, 1 < i < q. As­ sume there is an h £ J, h2 G J. Among such polynomials h, take / to be one with the lowest leading term relative to the by Lemma 1 . It follows, also by Lemma 1 , that p2 is divisible by one of the m* ’s, say m3. Since m3 is square-free, this implies p is divisible by m t , so that it( /) = a •p G J. A contradiction. □

Grobner Bases and Reduced Ideals

141

By combining Prop. 2 with Prop. 3 we now get the main result:

Let J C K ^ be an ideal, and E a Grobner basis for J, with respect to a given admissible order < on S. If for each g £ E its leading term it (g) is squarefree, then J is reduced. □ T h eorem .

Let B be a Grobner basis for an ideal J. Suppose, for some f ^ g both in B , lt(f) is divisible by lt(g). Clearly, then, the set B — { / } , too, is a Grobner basis for J. Continuing to throw out redundant members like / from B , one will reach a minimal Grobner basis, whose obvious definition we shall omit here. Observe that the theorem above gains its full force when applied to minimal Grobner bases.

R e fe r e n ce s

Abhyankar, S. S. (1989). On the Jacobian Conjecture: A new approach via Grobner Bases, J. Pure Applied Alg., 61: 211 Buchberger, B. (1985). Grobner bases: an algorithmic method in polynomial ideal theory, M U L T I D I M E N S I O N A L S Y S T E M T H E O R Y ( H . K. Bose, eel), Reidel, Dordrecht, Chapter 6 . Gianni, P., Trager, B., and Zacharias, G. (1989). Grobner bases and primary de­ composition of polynomial ideals, C O M P U T A T I O N A L A S P E C T S O F C O M M . A L G . (L. Robbiano, ed.), Academic Press, London and San Diego, p.15. Moller, H. M. and Mora, F. (1986). New constructive methods in classical ideal the­ ory, J. Algebra, 100: 138.

Bicomplexes and Galois Cohomology H. F. K r e i m e r Department of Mathematics, Florida State University, Tallahassee, Florida

In this paper, a new derivation of the well known long exact sequence of Galois cohomology for commutative rings is presented. An attempt to incor­ porate much of what is known about the terms and mappings of that exact sequence into a single theory is made. A filtration of a differential, graded module gives rise not only to an exact couple and a spectral sequence, but to a more elaborate structure herein called an exact octahedron. Long exact sequences are obtained by unwinding “strands” of an exact octahedron, and relationships between the exact sequences are recorded in the exact octahe­ dron. MacLane (1963), especially section 5 of chapter XI on exact couples, is a helpful resource; and a very readable account of the technique of faithfully flat descent, which will be used to interpret cohomology groups, is found in Knus and Ojanguren (1974). Unless otherwise specified by notation or context, all modules, and homomorphisms of modules are over a given commutative ring R.

1

. B IC O M P L E X E S

A bicomplex J is a bigraded module with homogeneous homomorphisms d' of bidegree (—1 , 0 ) and d " of bidegree (0 , —1 ) such that d'd' =

0,

&d" + 0 " & =

0,

d" d" =

0.

(1 .1 )

Thus a bicomplex consists of a family { XP}q} of modules and two families, d* : X Piq —>X p- i iq and d n : X Piq —>X PA-\ , of module homomorphisms, defined for 143

144

Kreimer

all integers p and q and for which conditions (1.1) are satisfied. The totalization of J is a complex in which (Tot X )n is the direct sum

^

X Pyq and the

p+q=n

differential operator is & + 5 ". The first filtration of subcomplexes of Tot X is defined by setting (FpTotX)n =

Xr,n-r- It is noteworthy that the family rE ' E' K f E ' i E 1—>, in which the homomorphisms 6!, e’, and f are induced by the additive relation

77/ - 1

and the homomorphisms e and f,

respectively.

Proof. 6777

Since dr] = rjer] =

777/

= 0, dd = drje = 0. Likewise 7 d = 7 / 7 =

= 0, and therefore d d = f ^ d = 0. From the hypothesis of exactness

146

Kreimer

follows the exactness o f the rows in the following commutative diagram. 0 —► Coker d

A

di

0 —>

K

-£•

—► Coim d

7 i

r) |

Im d

E

—> 0

d|

- y E - ^ K - ^ *

Ker d

—>

0

B y a diagram chase or a version o f the strong four lemma, there are the followe

/

ing exact sequence o f kernels: 0 —>E 1—>Ker 77 —>Ker 7 —>0, and the following exact sequence o f cokernels: 0 —>Coker 77 A Coker 7 -£• E ' —>0.

Now consider

the com m utative diagram: Coker 77

Coker 7

A

d -1

kernel-cokernel

'

rj J.

Ker d T he

-A

exact

=

Ker d

—>

sequence

for

0 this

diagram

is:

E 1-> E ' —>Ker d j Im 77, and the Yoneda com posite

0 —>Ker d f Im rj —>K '

o f these exact sequences is the desired long exact sequence. N ote that the additive relations 77/” 1 and e _ 1 7 are the same,

and the

lem m a could be proved also by considering the kernel-cokernel exact sequence for the follow ing com m utative diagram. 0

—> Coker d =

1

7

E'

A

i

Ker 77

Coker d 2 i

^

Ker 7

□ P R O P O S IT IO N 1.5.

I f D , E , and K are terms and e, f , g, h, i , j , and

k are m appings o f an exact octahedron as in figure 1.2; then another exact octah ed ron is derived by setting D r = Im i, E f = K er j k / l m j k and K ' = Ker jh / Im g k , letting the mappings efy f , h1, V, and k’ , be induced by the m appings e , f , h, i, and k, and letting the mappings g 1 and j 1 be induced by the additive relations g i~ l and j i ~ l . P roof

T h e exactness o f the couple ( E , yK , yj ,k,ye,yf f) follows from Lemma

1.4 by setting E = E , d = d = j k , j = g k , and 77 = j h , and noting that j ' k 1

Bicomplexes and Galois Cohomology

147

is the hom om orphism o f E f into E 1 induced by the additive relation ^ //~ 1( = j h f “ -1 = ji~ * k = e ~ l g k ). (£> ',K '\ (z ')2,# ', h') is shown to be an exact couple by noting that the dom ain o f definition o f the additive relation g i~ l is Im i = D ', the indeterm inancy o f g i~ l is g {Ker i) = g (Im k ) = Im g k , Ker g i~ x = z(Ker g) = z(Im z2) = z2( D ') , and Im g i~ x = Im g = Ker h. Also /i(Ker j/z ) = /i/i_ 1 (K er jf) = Ker j fl Im h = Im z fl Ker z2 = D ' fl Ker z2. The required conditions o f com m utativity are readily verified.

2

□

. Am itsur Cohomology

Let R be a com m utative ring, let 5 be a commutative algebra over iZ, let 5 ° = iZ, and let 5 n+1 = S n ® r S for each nonnegative integer n. There are n + 1 hom om orphism s o f the algebra S n into the algebra S n+1; and they may be defined recursively as follows: let

be the canonical homomorphism of R

into the J?-algebra 5 ; and for each positive integer rz, use again the symbol Sk to denote the natural extension o f the hom om orphism £* o f S n~ l into S n to a h om om orph ism €k® S o f S n = S n~ x ® R S into 5 n+1 = S u ® r S^ 0 < k < n — 1, and let e n be the hom om orphism o f S n onto the first factor o f S u® r S = 5 n+1. The hom om orphism e \t o f S n into 5 n+1 determines a functor from the category o f S n-m odules to the category o f 5 n+1-modules, and this functor will also be denoted by the sym bol £*. Note that 6{€j = ej+iSi for z < j . If F is a functor from the category o f commutative algebras over R to the category o f abelian groups, the Amitsur com plex C {S /R ^ F ) is a cocom ­ plex with terms F ( S n) for n > 1. The coboundary from F ( S n) into F ( S n+1) n

is the alternating sum

( ~ l ) fcF (£fc): and the Amitsur cohom ology groups k= 0

H n( S / R ,F ), 72 > 0, are the cohom ology groups o f this complex. Let U denote the functor which assigns to a ring its multiplicative group o f invertible ele­ ments, and let P denote the functor which assigns to a commutative ring its Picard group o f isom orphism classes o f rank one, projective modules. If T is a com m utative algebra over i2, there is a third quadrant bicom ­ plex X = C {S ,T / R ,U ) with terms X ™ = U (S P+ 1 ® R T*+1), for p > 0 and q > 0, and X p,q = 0, otherwise.

The convention o f using positive super­

scripts instead o f negative subscripts is being followed here. The hom om or­ phism ( —l ) 9d' o f U (S P+1

®

r

o f the A m itsur com plex C (S o f U (S P+1

®

r

T q+1) into U (S P+ 2 ® ®

r

r

T q+1) is the coboundary

T 9+1/ T 9+1, i7), and the hom om orphism d "

T q+1) into U (S P^ 1 ®

r

T 9+2) is the coboundary o f the Amitsur

148

Kreimer

com plex C ( S p* 1 ® r T / S p~*~1, U ). Then with respect to the first filtration o f the bicom plex, F pT ot X / F ^ 1 Tot is the Am itsur com plex C (S p* 1 ® r T / S p* 1, U ); and for the associated exact octahedron, E p,q = i f g( 5 p+1 ® r T / 5 p+1, U ). From now on, assume that T is a faithfully flat i?-m odule. Then S p+1 ® r T is a faithfully flat m odule over Sp+1, and E p’° = £T°(5,,+1 ® r T / 5 p+1,J7) = U (S P+1) (K nus and Ojanguren, 1974, Chapter V , Proposition 2.1).

To the

hom om orphism o f an i?-algebra S onto the first factor o f S ® r T , there corresponds a group hom om orphism o f P ( S ) into P (S ® r T ).

Letting

I \ P (S , T ) denote the kernel o f this group hom om orphism , K P ( •, T ) is a func­ tor from the category o f ii-algebras to the category o f abelian groups and E Pil = H \ S p + 1 ® r T / S p+ \ U ) = I< P {S p+ \ T ) by (Knus and Ojanguren, 1974, Chapter V , Proposition 2.1). To give a brief sketch o f the argument for this last assertion, regard 5 P+1 ® r T 2 as a m odule over the algebra S p + 1 ® r T 2 and an element u o f U {S p^rl ® r T 2) as an autom orphism of this module. The condition that u be a cocycle o f the Am itsur com plex C (S P^~1

®

r

T / 5 ^ 1, U )

com pletes the descent data necessary to obtain an 5 p+1-m odule M such that M

®

r

T and S p + 1

®

r

T are isom orphic £ p+1

®

r

T-m odules; and M must be

a rank one, projective S p+1-m odule, since T is faithfully flat over R. and M* are S p+1-m odules such that M

®

r

T and M 1

®

r

If M

T are isomorphic to

5 P+1 ® r T , then an isom orphism o f M onto M* may be lifted to an autom or­ phism o f

®

r

T and an element o f U (S P+ 1

®

r

T ). It follows readily that

coh om ologou s cocycles determine isom orphic 5 p+1-modules. For the associated exact octahedron, the hom om orphism j k o f E p,q into E p+1'q is induced by ( —l ) g H \ S / R , U) -> K * '1 -> H °(S / R , K P { - , T )) -* I H om _ 5 (M , A ) = H o m _ s(M , H o m _ s(M , 5 ) ) « H o m _ s (M 0 5 My 5 ), under which the m apping x ® y —* (x) • y) o f M ® s M into 5 corresponds to an element

o f A ® s A The comultiplication o f M determines a m ultiplica-

151

Bicomplexes and Galois Cohomology

V?('0(m (i)) • m (2))* There is a natural ho-

tion on A by the rule: ipxi/>(m) = (m )

m om orphism , which is an isomorphism when 5 is a finitely generated, p rojec­ tive m odule over i2, o f A ® r 5 = H o m _s(M , 5 ) ® r 5 into H o m _ s (M , 5 ® r 5 ), where the right 5-m odu le structure of 5 ® #

arises from the first factor

5 . Since M is a rank one, projective 5 2-module, H om s 2(M , M ) = 5 2; and H o m _ 5 ( M , 5 r 5 ) = H om _ 5 (M ,H om £ 2( M ,M ) ) , which is isom orphic to H o m _ 52 ( M ® s M , M ) by adjoint associativity. The isomorphism / induces an isom orphism o f Hom__s2( M ® 5 M , M ) onto H om _ 52 (e 1 (M ), M ), which is natu­ rally isom orphic to H o m _ s(M , M ). Through this sequence o f mappings, the el­ ^ ( ^ ( l ) ) * m ( 2) *

ement H \ S / R , P ) - » H 3 (S/R, U ) - » IC2' - » H 2 (S/R, P ) - . . .

(2.5)

This derivation o f the exact sequence in 2.5 is merely a refinement o f the m ethods used by Chase and Rosenberg (1965) to obtain the initial seven term

153

Bicomplexes and Galois Cohomology

exact sequence. Since exact sequences are preserved under direct limits, the exact sequence in 2.5 may be derived also by taking direct limits o f the terms o f the first exact octahedron to obtain an exact octahedron with terms D , E , K , E p,° = U (S P+1) and E Pyl = P ( 5 P+1), and then forming the derived exact octahedron.

This latter process is a variation o f the techniques employed

by V illam ayor and Zelinsky (1977) to obtain the infinite exact sequence in 2.5.

For each positive integer n, the groups Jn and i f n(J ), introduced in

V illam ayor and Zelinsky (1977), are isomorphic to the groups K™

’

and

K 2 1*1, respectively.

Let H q(S ,U ) = lim H q(S T / T ,U ) for a category T o f com m utative P-algebras T which are faithfully flat modules over R. This definition o f H q(S , U) does depend on the choice of the category.

2 . 6 . There is an exact octahedron with terms D , E , and K ,

THEOREM

such that E p,q = H q(S p+1j U) for nonnegative integers p and q ( E p’q = 0 otherw ise).

It has been shown that H ° (*,J7) and U are equivalent functors and f f 1( # , U ) and P are equivalent functors for suitably chosen categories o f com ­ m utative iZ-algebras T. The canonical homomorphism o f R into the i?-algebra S determines a group hom om orphism o f H 2(R ,U ) into H 2(S, 17), and it can be shown that K 2’

is the kernel o f this group homomorphism, (Villam ayor

and Zelinsky, 1977, Theorem 6.14). Thus H 2(R ,U ) is a generalization o f the Brauer group o f R.

But the derived exact octahedron contains much more

than one long exact sequence. In particular, there is the following com m uta­ tive diagram with exact rows, by which the Amitsur cohom ology groups with respect to the various functors H q( •, 17) are related.

I -

Hv~l (S/R, Hq+1(’ ,U))

1 -

II —

HP-l (S/R,H+1(',U))

II -*

HP-2(S/R,H (2.7)

i -

HpJti{S/R,

, U))

-

154

Kreimer

References Auslander, M. and Goldman, 0 . (1960a).

Maximal orders, Trans.

A m er .

M ath. Soc., 97: 1-24. Auslander, M. and Goldm an, 0 . (1960b). The Brauer group o f a com m utative ring, Trans. A m er. Math. Soc., 97: 367-409. Chase, S.U. and Rosenberg, A. (1965). Amitsur Cohom ology and the Brauer group, M em oirs A .M .S ., 52: 34-79. Knus, M .A . and Ojanguren, M. (1974).

Theorie de la descente et algebres

d ’Azum aya, Springer Lecture Notes, 389. M acLane, S. (1963). Homology, Springer Verlag, Berlin. Villam ayor, O.E. and Zelinsky, D. (1977). Brauer groups and Am itsur coh o­ m ology for general commutative ring extensions, Journal o f Pure and Applied Algebra, 1 0 : 19-55.

Adjoining Idempotents ANDY R. MAGID University of Oklahoma, Norman, Oklahoma

In t r o d u c t io n

Let R denote a commutative ring. Idempotents of R are elements e E R such that e2 = e; examples are the additive and multiplicative identitiy elements 0 and 1. The set B (R ) of idempotents of R forms a Boolean algebra with operations:

ei H e2 =

6-2 ]

ei U e2 = ei + e2 -

e2;

e = 1 — e;

0

=

0;

U = 1.

Now suppose B D B (R ) is a Boolean algebra extension. Is it the case that there is a ring R b containing R such that B (R b ) = B and so that the Boolean algebra inclusion B (R b ) 3 B (R ) coming from the ring inclusion R b 3 R is the given extension B 3 B (R )? More precisely, is there a commutative R algebra R —> R b and a Boolean algebra ismorphism B (R b ) —■ ►B such that the following commutes? B (R b ) — +B

\ /

B (R )

We will show in this paper that the above question has a positive answer, and furthermore discuss why the question is of interest. Our notation for iZ^, which seems to depend only on R and B , is not yet accurate: this will be remedied by making it universal with respect to the above property. Then we will also see that R b has a number of nice features, including being generated over R by idempotents and being a directed union of subalgebras isomorphic to products of copies of direct factors of R. 155

156

Magid T he B o o l e a n Sp e c t r u m

In this section, we will review the Boolean spectrum, or space of connected components, of a commutative ring. This notion is due to Pierce [P], with addi­ tional developments due to Villamayor and Zelinsky [VZ]. An exposition is given in Chapter II o f [M], which we follow here. Let I? be a commutative ring. Idempotents e o f P and open-closed subsets U of the spectrum Spec(P) o f P correspond one-one: to e we associate the subset e_ 1 ( l ) = { P E R|1 — e E P } ; and to U we associate its characteristic function \u where x u = 1 € R p if and only if P E U. (One has to show that there is a unique idempotent \u € R with this property.) Open-closed subsets of Spec(P) are unions o f connected components. Thus is is convenient to pass to the space C om p(Spec(P)) of connected components of Spec(P) which we denote -X’(iZ) and which is defined by the continuity o f the map Spec(P) — - — ►C om p(Spec(P)) = X (R ). The space X (R ) is compact and totally disconnected by construction. It also turns out to be Hausdorff, and hence profinite. The elements of X (R ) correspond to the maximal ideals of P (P ) and hence X (R ) is known as the Boolean Spectrum of P. The points o f X ( R ) are connected components of Spec(P). If P and Q are prime ideals which lie in the same connected component x then P fl B (R ) = Q fl P (P ); this set of idempotents depends only on the component x. We denote by I (x ) the ideal of R which it generates, and we let R x denote the quotient R / I(x). The ring R x is connected; that is, it has no non-trivial idempotents. For a E P , we let ax be the coset a + I (x ) of a in R x. Then for an idempotent e we have ex = 1 if and only if e = 1 on x. We generalize this notation to P modules: if M is an P module and x E X (R ) then M x = R x ® r M . In fact, the rings R x are the stalks o f a sheaf on X (P ). We consider the map of ringed spaces induced by the map p above: (Spec( R ) ,O r ) ^ ( X ( R ) , p .( O r )). Then the stalk P* (O r )x (r ),x can be identified with P x. The spaces X (-) are functorial: if / : P —* S is a homomorphism of commutative rings then there is an induced continuous function X ( f ) : X (S ) —> X (R ) which sends y E X (S ) to the component X ( f ) ( y ) of Spec(P) containing { / “ 1 (P)|P E y }. If X ( f ) ( y ) = x then / induces a homomorphism R x —> Sy by a + I ( x ) f(a )+ I (y ). As a consequence, the natural maps into a tensor product S

> S®r T

i

i

R ---------►

T

157

Adjoining Idempotents induce a commutative diagram of Boolean spectra

X(S) C x then induces a homeomorphism X —> X (C ( X , Z )). If p : X Y is a continous map then there is an induced ring homomorphism C (p) : C(Y, Z) —►C (X , Z) given by / i-> / o p , and we have X (C (p )) = p in the sense that the following diagram commutes: X ---------►X ( C ( X ,Z ) ) p|

Y

} x ( C (p))

> X (C (Y , Z ))

As an application of these concepts, we prove the following (which will be needed in the sequel, and which illustrates the essential techniques): L em m a . Let R be a commutative ring and let W be a profinite space. Then the natural map X ( R ® z C (W , Z )) -> X (R ) x W is a homeomorphism. Proof. We first consider the case where W is finite of cardinality w. Then C( W, Z) = Z x ••• x Z = Z^> so R ® C (W , Z) = Thus X (R ® C (W , Z )) = X ( R ) ^ = W x X (R ). Thus we have the result for finite W . In general, we have W = proj limWi where Wi is finite. Then C (W ,Z ) = dir lim C (W i,Z ) so R ® C (W , Z) = dir lim(Jl ® C (W h Z)). Thus X ( R ® C(W , Z )) = proj lim (X (i? ® C (W ,-,Z)) = proj lim (X (R ) x ^ ) = X (R ) x proj lim Wi = X (i? ) x W , where the third equality follows from the finite case already considered. Finally, we note that if R and 5 are commutative rings then B (R x S) = B (R ) x B (S ) from which it follows that X ( R x S) = X (R ) x X (S ). T h e A d j u n c t io n

T heorem

Using the notations and results recalled in the preceding section, we can now state the main result.

158

Magid

T h e o r e m 1 . Let R be a commutative ring, Y a proSnite topological space, and 7r : Y —►X ( R ) a continuous surjection. Then there is an R algebra Tr (Y ) = T (Y ) such that: ( 1 ) There is a homeomorphism Y —►X ( T (Y )) such that Y -> X ( T ( Y ) ) -> X (R ) = Y A X (J l); (2)

If 5 is any R algebra and h : X (S ) —►Y is a continuousmap such that X (S ) i y A

X (R ) = X (S ) - » X (R )

then there is a unique R algebra map T (Y ) —> 5 such that X (S ) (3)

X ( T ( Y ) ) = X (S ) - » X (T ( Y )) ;

T (Y ) = U(Re[ai' x . . . Re'n"'1) (directed union) where the c i , . . . , e n are partitions o f unity (e,ej = ^ and = 1 ).

We begin the proof with the construction of a canonical map from the ring of functions on the Boolean spectrum to the ring: P r o p o s itio n 2 . Let R be a commutative ring and define = $ : C (X (R ), Z )

R

by * ( / ) = £ * * /-> (* )■ *€ Z

Then $ is a ring homomorphism which induces a Boolean algebra isomorphism B (C (X (R ), Z )) -> B(JJ) and X ( $ ) : X (f? ) X (C ,(-X’(H), Z )) = X (H ) is the identity The image o f $ is the subring rL B {R ) = Aiso, $ is functorial in R: if a : R ^ S is a commutative ring homomorphism then the following diagram commutes: C (X (R ), Z) C (X (a))j

R ja

C (X (S ), Z ) ---------►5.

Proof. A continuous function / : X (R ) —►Z has finite image, so the sum in the defintion of $ is actually finite, as are the sums which will appear in this proof. Moreover, the various / “ 1 (h )’s are disjoint. The characteristic function of an inter­ section is the product of characteristic functions: x u n v = XU’ Xv- And if UC\V = 0, then the characteristic function xu u v of the union is the sum Xtz+Xv? with a similar result for sets of mutally disjoint sets. Let / , g belong to C (X (R ), Z). For simplicity

Adjoining Idempotents

159

in the followingcalculations, we will denote thecharacteristic function of / 1 (z) by Xi and the characteristic function of by i/>j. Then Xi — Yli= 1 so * (/) = £

fcx * = £

kXk^i

£

k i

k

and $ (s0 = £

m ^rn = £

m

£ m

raV’mXj

j

SO

$ ( / ) + $ (s 0 = £ ( p + 9)xPVv jm Since ( / + y )- 1 (fc) = Up+ g= fc(/“ 1 (p) fl y_ 1 (y)), its characteristic function is £

x Pv>,

p+q=k so that $ ( / +