Cohomology Rings of Finite Groups Algebras and Applications Volume 3 Editors: F. Van Oystaeyen University ofA ntwerp, UIA, Wilrijk, Belgium A. Verschoren University ofA ntwerp, RUCA, Antwerp, Belgium Advisory Board: M. Artin Massachusetts Institute of Technology Cambridge, MA, USA A. Bondal Moscow State University, Moscow, Russia I. Reiten Norwegian University of Science and Technology Trondheim, Norway The theory of rings, algebras and their representations has evolved into a well-defined subdiscipline of general algebra, combining its proper methodology with that of other disciplines and thus leading to a wide variety of applications ranging from algebraic geometry and number theory to theoretical physics and robotics. Due to this, many recent results in these domains were dispersed in the literature, making it very hard for researchers to keep track of recent developments. In order to remedy this, Algebras and Applications aims to publish carefully refereed monographs containing up-to-date information about progress in the field of algebras and their representations, their classical impact on geometry and algebraic topology and applications in related domains, such as physics or discrete mathematics. Particular emphasis will thus be put on the state-of-the-art topics including rings of differential operators, Lie algebras and super-algebras, groups rings and algebras, C* algebras, Hopf algebras and quantum groups, as well as their applications. Cohomology Rings of Finite Groups With an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64 by Jon F. Carlson University of Georgia, Athens, Georgia, U.S.A. Lisa Townsley Benedictine University, Lisle, Illinois, U.S.A. Luis Valeri-Elizondo Instituto de Matematicas, UNAM, Morelia, Mexico and Mucheng Zhang University of Georgia, Athens, Georgia, U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6385-4 ISBN 978-94-017-0215-7 (eBook) DOI 10.1007/978-94-017-0215-7 Printed on acid-free paper All Rights Reserved © 2003 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2003 Softcover reprint of the hardcover 1st edition 2003 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. To our loved ones Contents Preface xi Acknowledgments XV 1. HOMOLOGICAL ALGEBRA 1 1. Introduction 1 2. Complexes and Sequences 3 3. Projective and Injective Modules 6 4. Resolutions 8 5. Ext 10 6. Tensor Products and Tor 17 2. GROUP ALGEBRAS 23 1. Introduction 23 2. Duality and Tensor Products 25 3. Induction and Restriction 28 4. Radicals, Socles and Projective Modules 33 5. Degree Shifting 35 6. The Stable Category 37 7. Group Cohomology and Change of Coefficients 44 3. PROJECTIVE RESOLUTIONS 47 1. Introduction 47 2. Minimal Resolutions 48 3. The Bar Resolution 50 4. Applications to Low Dimensional Cohomology 52 5. Restrictions, Inflations and Transfers 55 4. COHOMOLOGY PRODUCTS 61 1. Introduction 61 2. Yoneda Splices and Compositions of Chain Maps 61 3. Products and Group Algebras 64 4. Restriction, Inflation and Transfer 71 vii viii COHOMOLOGY RINGS OF FINITE GROUPS 5. Cohomology Ring Computations 72 6. Shifted Subgroups and Restrictions 76 7. Automorphisms and Cohomology 78 5. SPECTRAL SEQUENCES 87 1. Introductions 87 2. The Spectral Sequence of a Bicomplex 88 3. Products 92 4. The Lyndon-Hochschild-Serre Spectral Sequence 94 5. Extension Classes 99 6. Minimal Resolutions and Convergence 103 7. Exact Couples and the Bockstein Spectral Sequence 104 6. NORMS AND THE COHOMOLOGY OF WREATH PRODUCTS 111 1. Introduction 111 2. Wreath Products 112 3. The Norm Map 115 4. Examples and Applications 120 5. Finite Generation of Cohomology 123 7. STEENROD OPERATIONS 129 1. Introduction 129 2. The Steenrod Algebra and Modules 130 3. The Steenrod Operations on Cohomology 134 4. Cohomology and Modules Over the Steenrod Algebra 141 5. The Cohomology of Extraspecial 2-Groups 143 6. The Cohomology of Extraspecial p-Groups 151 7. Serre's Theorem on the Vanishing of Bocksteins 153 8. VARIETIES AND ELEMENTARY ABELIAN SUBGROUPS 159 1. Introduction 159 2. Filtrations on Modules 160 3. Vanishing Products of Cohomology Elements 169 4. Minimal Primes in Cohomology Rings. 171 5. The Stratification Theorem 174 9. COHOMOLOGY RINGS OF MODULES 179 1. Introduction 179 2. Generalized Bocksteins Over Elementary Abelian Groups 181 3. Rank Varieties and Cohomology Rings Over Elementary Abelian Groups 186 4. The Cohomological Support Variety of a Module 190 5. Equating the Rank and Cohomological Support Varieties 192 6. The Tensor Product Theorem 198 Contents IX 7. Properties of the Cohomological Support Varieties 202 10. COMPLEXITY AND MULTIPLE COMPLEXES 209 1. Introduction 209 2. Notes on Dimension and Rates of Growth 210 3. Complexity of Modules 214 4. Varieties for Modules With Other Coefficient Rings 218 5. Projective Resolutions as Multiple Complexes 225 11. DUALITY COMPLEXES 231 1. Introduction 231 2. Gaps in Cohomology 231 3. Poincare Duality Complexes 237 4. Differentials in the HSS 242 5. Cohen-Macaulay Cohomology Rings 245 6. Further Considerations 250 12. TRANSFERS, DEPTH AND DETECTION 255 1. Introduction 255 2. Notes on Depth and Associated Primes 256 3. Depth and the p-Rank of the Center 259 4. Varieties and Transfers 262 5. Detection and Depth-Essential Cohomology 267 6. Special Cases 272 7. Associated Primes in Cohomology 276 8. Unstable Modules 279 13. SUBGROUP COMPLEXES 283 1. Introduction 283 2. Posets of Subgroups and Cell Complexes 284 3. Homotopy Equivalences and Equivariance 287 4. Complexes of Posets of Finite Groups 292 5. The Bouc Complex 295 6. Applications to Cohomology 296 7. Decompositions of Moduless 300 8. Additional Remarks 308 9. Homology Decompositions 310 14. COMPUTER CALCULATIONS AND COMPLETION TESTS 313 1. Introduction 313 2. The Visual Cohomology Ring: Generators and Relations 314 3. Resolutions, Maps and Homogeneous Parameters 318 4. Tests for Completion 324 5. Two Special Cases 331 x COHOMOLOGY RINGS OF FINITE GROUPS Appendices: CALCULATIONS OF THE COHOMOLOGY RINGS OF GROUPS OF ORDER DIVIDING 64 by Jon F. Carlson, Luis Valero-Elizondo and Mucheng Zhang 337 INTRODUCTION 338 A- NOTATION AND REFERENCES 339 B- GROUPS OF ORDER 8 347 C- GROUPS OF ORDER 16 349 D- GROUPS OF ORDER 32 357 E- GROUPS OF ORDER 64 397 F- TABLES OF KRULL DIMENSION AND DEPTH 755 G- TABLES OF HILBERT /POINCARE SERIES 757 REFERENCES 761 INDEX 773 Preface Group cohomology has a rich history that goes back a century or more. Its origins are rooted in investigations of group theory and num ber theory, and it grew into an integral component of algebraic topology. In the last thirty years, group cohomology has developed a powerful con nection with finite group representations. Unlike the early applications which were primarily concerned with cohomology in low degrees, the in teractions with representation theory involve cohomology rings and the geometry of spectra over these rings. It is this connection to represen tation theory that we take as our primary motivation for this book. The book consists of two separate pieces. Chronologically, the first part was the computer calculations of the mod-2 cohomology rings of the groups whose orders divide 64. The ideas and the programs for the calculations were developed over the last 10 years. Several new features were added over the course of that time. We had originally planned to include only a brief introduction to the calculations. However, we were persuaded to produce a more substantial text that would include in greater detail the concepts that are the subject of the calculations and are the source of some of the motivating conjectures for the com putations. We have gathered together many of the results and ideas that are the focus of the calculations from throughout the mathematical literature. More than 10 years ago, two other books on the cohomology of finite groups were published by Dave Benson and Leonard Evens, both with a philosophy similar to our writing. Evens' text was shorter, intended for a one semester advanced graduate course. Benson was much more ambitious, covering much more material, but also with briefer exposi tions in some cases. Our aim has been to steer a course between the two texts and to update both. On the one hand, we delve deeper into recently developed material than Evens. On the other hand, we have xi