Cohomology Rings of Finite Groups: with an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64 by Jon CarlsonCohomology Rings of Finite Groups: with an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64 by Jon Carlson

Cohomology Rings of Finite Groups: with an Appendix: Calculations of Cohomology Rings of Groups of…

byJon Carlson

Paperback | December 9, 2010

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This text offers comprehensive coverage of group cohomology, from introductory material through the most recent developments in the field. The primary motivation for this book is the interaction of group cohomology with representation theory, especially the geometry of support varieties over cohomology rings. The appendices, comprising computer calculations of the mod-2 cohomology rings of the groups whose orders divide 64, provide information useful for further developments in the field. A unique feature of this text is that it includes the concepts that are the subject of the calculations and are the source of some of the motivating conjectures for the computations. The programs for computing the cohomology rings were executed in the MAGMA computer algebra language. The text is a valuable resource for researchers in group cohomology and related disciplines. In addition, the book could be used as the text for an advanced graduate class or a graduate seminar.
Title:Cohomology Rings of Finite Groups: with an Appendix: Calculations of Cohomology Rings of Groups of…Format:PaperbackDimensions:792 pages, 9.25 × 6.1 × 0.04 inPublished:December 9, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048163854

ISBN - 13:9789048163854

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Table of Contents

Preface. Acknowledgements. 1: Homological Algebra. 1. Introduction. 2. Complexes and Sequences. 3. Projective and Injective Models. 4. Resolutions. 5. Ext. 6. Tensor Products and Tor. 2: Group Algebras. 1. Introduction. 2. Duality and Tensor Products. 3. Induction and Restriction. 4. Radicals, Socles and Projective Modules. 5. Degree Shifting. 6. The Stable Category. 7. Group Cohomology and Change of Coefficients. 3: Projective Resolutions. 1. Introduction. 2. Minimal Resolutions. 3. The Bar Resolution. 4. Applications to Low Dimensional Cohomology. 5. Restrictions, Inflations and Transfers. 4: Cohomology Products. 1. Introduction. 2. Yoneda Splices and Compositions of Chain Maps. 3. Products and Group Algebras. 4. Restriction, Inflation and Transfer. 5. Cohomology Ring Computations. 6. Shifted Subgroups and Restrictions. 7. Automorphisms and Cohomology. 5: Spectral Sequences. 1. Introduction. 2. The Spectral Sequence of a Biocomplex. 3. Products. 4. The Lyndon-Hochschild-Serre Spectral Sequence. 5. Extension Classes. 6. Minimal Resolutions and Convergence. 7. Exact Couples and the Bockstein Spectral Sequence. 6: Norms and the Cohomology of Wreath Products. 1. Introduction. 2. Wreath Products. 3. The Norm Map. 4. Examples and Applications. 5. Finite Generation of Cohomology. 7: Steenrod Operations. 1. Introduction. 2. The Steenrod Algebra and Modules. 3. The Steenrod Operations on Cohomology. 4. Cohomology and Modules Over the Steenrod Algebra. 5. The Cohomology of Extraspecial 2-Groups. 6. The Cohomology of Extraspecial p-Groups. 7. Serre's Theorem on the Vanishing of Bocksteins. 8: Varieties and Elementary Abelian Subgroups. 1. Introduction. 2. Filtrations on Modules. 3. Vanishing Products of Cohomology Elements. 4. Minimal Primes in Cohomology Rings. 5. The Stratification Theorem. 9: Cohomology Rings of Modules. 1. Introduction. 2. Generalized Bocksteins Over Elementary Abelian Groups. 3. Rank Varieties and Cohomology Rings Over Elementary Abelian Groups. 4. The Cohomological Support Variety of a Module. 5. Equating the Rank and Cohomological Support Varieties. 6. The Tensor Product Theorem. 7. Properties of the Cohomological Support Varieties. 10: Complexity and Multiple Complexes. 1. Introduction. 2. Notes on Dimension and Rates of Growth. 3. Complexity of Modules. 4. Varieties for Modules with Other Coefficient Rings. 5. Projective Resolutions as Multiple Complexes. 11: Duality Complexes. 1. Introduction. 2. Gaps in Cohomology. 3. Poincaré Duality Complexes. 4. Differentials in the HSS. 5. Cohen Macaulay Cohomology Rings. 6. Further Considerations. 12: Transfers, Depth and Detection. 1. Introduction. 2. Notes on Depth and Associated Primes. 3. Depth and the p-Rank of the Center. 4. Varieties and Transfers. 5. Detection and Depth-Essential Cohomology. 6. Special Cases. 7. Associated Primes in Cohomology. 8. Unstable Modules. 13: Subgroup Complexes. 1. Introduction. 2. Posets of Subgroups and Cell Complexes. 3. Homotopy Equivalences and Equivariance. 4. Complexes of Posets of Finite Groups. 5. The Bouc Complex. 6. Applications to Cohomology. 7. Decompositions of Molecules. 8. Additional Remarks. 9. Cohomology Compositions. 14: Computer Calculations and Completion Tests. 1. Introduction. 2. The Visual Cohomology Ring: Generators and Relations. 3. Resolutions, Maps and Homogeneous Parameters. 4. Tests for Completion. 5. Two Special Cases. Appendices: Calculations of the Cohomology Rings of Groups of Order Dividing 64; J.F. Carlson, L. Valero-Elizondo, Mucheng Zhang. Introduction. A. Notation and References. B. Groups of Order 8. C. Groups of Order 16. D. Groups of Order 32. E. Groups of Order 64. F. Tables of Krull Dimension and Depth. G. Tables of Hilbert/Poincaré Series. References. Index.

Editorial Reviews

From the reviews:"The cohomology of groups was developed to be . a powerful tool in the study of group representations. . Some other books concerning the cohomology of finite groups have appeared, but they are at least ten years old, and the computational tools that have been developed since then, make this book to be really valuable for both professionals and students. . This book is almost self-contained and it is very useful . for all of those that are using the cohomology instruments in their work." (Viorel Mihai Gontineac, Zentralblatt MATH, Vol. 1056 (7), 2005)