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Group Cohomology and Algebraic Cycles

Book Description

Group cohomology reveals a deep relationship between algebra and topology, and its recent applications have provided important insights into the Hodge conjecture and algebraic geometry more broadly. This book presents a coherent suite of computational tools for the study of group cohomology and algebraic cycles. Early chapters synthesize background material from topology, algebraic geometry, and commutative algebra so readers do not have to form connections between the literatures on their own. Later chapters demonstrate Peter Symonds's influential proof of David Benson's regularity conjecture, offering several new variants and improvements. Complete with concrete examples and computations throughout, and a list of open problems for further study, this book will be valuable to graduate students and researchers in algebraic geometry and related fields.

Table of Contents

  1. Cover
  2. Series
  3. Title
  4. Copyright
  5. Dedication
  6. Table of Contents
  7. Preface
  8. 1 Group Cohomology
    1. 1.1 Definition of group cohomology
    2. 1.2 Equivariant cohomology and basic calculations
    3. 1.3 Algebraic definition of group cohomology
  9. 2 The Chow Ring of a Classifying Space
    1. 2.1 The Chow group of algebraic cycles
    2. 2.2 The Chow ring of a classifying space
    3. 2.3 The equivariant Chow ring
    4. 2.4 Basic computations
    5. 2.5 Transfer
    6. 2.6 Becker-Gottlieb transfer for Chow groups
    7. 2.7 Groups in characteristic p
    8. 2.8 Wreath products and the symmetric groups
    9. 2.9 General linear groups over finite fields
    10. 2.10 Questions about the Chow ring of a finite group
  10. 3 Depth and Regularity
    1. 3.1 Depth and regularity in terms of local cohomology
    2. 3.2 Depth and regularity in terms of generators and relations
    3. 3.3 Duflot’s lower bound for depth
  11. 4 Regularity of Group Cohomology
    1. 4.1 Regularity of group cohomology and applications
    2. 4.2 Proof of Symonds’s theorem
  12. 5 Generators for the Chow Ring
    1. 5.1 Bounding the generators of the Chow ring
    2. 5.2 Optimality of the bounds
  13. 6 Regularity of the Chow Ring
    1. 6.1 Bounding the regularity of the Chow ring
    2. 6.2 Motivic cohomology
    3. 6.3 Steenrod operations on motivic cohomology
    4. 6.4 Regularity of motivic cohomology
  14. 7 Bounds for p-Groups
    1. 7.1 Invariant theory of the group Z/p
    2. 7.2 Wreath products
    3. 7.3 Bounds for the Chow ring and cohomology of a p-group
  15. 8 The Structure of Group Cohomology and the Chow Ring
    1. 8.1 The norm map
    2. 8.2 Quillen’s theorem and Yagita’s theorem
    3. 8.3 Yagita’s theorem over any field
    4. 8.4 Carlson’s theorem on transfer
  16. 9 Cohomology mod Transfers Is Cohen-Macaulay
    1. 9.1 The Cohen-Macaulay property
    2. 9.2 The ring of invariants modulo traces
  17. 10 Bounds for Group Cohomology and the Chow Ring Modulo Transfers
  18. 11 Transferred Euler Classes
    1. 11.1 Basic properties of transferred Euler classes
    2. 11.2 Generating the Chow ring
  19. 12 Detection Theorems for Cohomology and Chow Rings
    1. 12.1 Nilpotence in group cohomology
    2. 12.2 The detection theorem for Chow rings
  20. 13 Calculations
    1. 13.1 The Chow rings of the groups of order 16
    2. 13.2 The modular p-group
    3. 13.3 Central extensions by G[sub(m)]
    4. 13.4 The extraspecial group E[sub(p[sup(3)])]
    5. 13.5 Calculations of the topological nilpotence degree
  21. 14 Groups of Order p[sup(4)]
    1. 14.1 The wreath product Z/3 ∿ Z/3
    2. 14.2 Geometric and topological filtrations
    3. 14.3 Groups of p[sup(4)] order for p ≥ 5
    4. 14.4 Groups of order 81
    5. 14.5 A 1-dimensional group
  22. 15 Geometric and Topological Filtrations
    1. 15.1 Summary
    2. 15.2 Positive results
    3. 15.3 Examples at odd primes
    4. 15.4 Examples for p = 2
  23. 16 The Eilenberg-Moore Spectral Sequence in Motivic Cohomology
    1. 16.1 Motivic cohomology of flag bundles
    2. 16.2 Leray spectral sequence for a divisor with normal crossings
    3. 16.3 Eilenberg-Moore spectral sequence in motivic cohomology
  24. 17 The Chow Künneth Conjecture
  25. 18 Open Problems
  26. Appendix: Tables
  27. References
  28. Index