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Group Cohomology and Algebraic Cycles


Part of Cambridge Tracts in Mathematics

  • Author: Burt Totaro, University of California, Los Angeles
  • Date Published: June 2014
  • availability: Available
  • format: Hardback
  • isbn: 9781107015777
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About the Authors
  • 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.

    • Contains many concrete examples and computations
    • Early chapters synthesize background material from topology, algebraic geometry and commutative algebra
    • Includes a list of open problems at the end
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    Reviews & endorsements

    'The author is one of the world experts on the Chow ring of algebraic cycles on the classifying space of an algebraic group and its interplay with the classical mod p cohomology ring. With a focus on finite groups, this text develops in parallel the theory of these two important families of rings. Very recent results about deep structural properties are presented here for the first time in book form, including, notably, Symonds's calculation of the Castelnuovo–Mumford regularity of group cohomology and its consequences. Some results about group cohomology are improvements on the literature, and many of the parallel results about Chow rings are new. The book is recommended for advanced students and researchers interested in seeing some of the lovely ways in which representation theory, algebraic topology, algebraic geometry, and commutative algebra fruitfully interact.' Nicholas Kuhn, University of Virginia

    'This attractively written book provides a very readable and up-to-date account of the cohomology of groups. The emphasis is on the geometric point of view provided by the Chow ring of the classifying space. A particularly nice feature is that Symonds's recent proof of the regularity conjecture and several of its generalizations are discussed in detail.' David J. Benson, University of Aberdeen

    'Cohomology of groups is usually developed algebraically via resolutions, and topologically via classifying spaces. This unique and attractively written book develops the subject from the point of view of algebraic geometry … The book is full of computational examples that make accessible what could have been a very abstract subject. It is written at a level that could be used for a graduate course in cohomology of groups.' Mathematical Reviews

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    Customer reviews

    13th Sep 2015 by CUPEditorial

    MAA Reviews, Felipe Zaldivar [stars not assigned by MAA]: "This important monograph collects in a systematic way, and in some cases improves on, most of the recent developments in [group cohomology]. ... The whole book emphasizes the advantages provided by the algebro-geometric approach to group cohomology, and at the same time gives new perspectives to important open questions on algebraic geometry. Even taking into account the high level of the subject, the exposition is nicely systematic, making it accessible to graduate students and researchers in adjacent areas, and even including a chapter on open problems."

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    Product details

    • Date Published: June 2014
    • format: Hardback
    • isbn: 9781107015777
    • length: 246 pages
    • dimensions: 229 x 152 x 18 mm
    • weight: 0.53kg
    • contains: 2 b/w illus.
    • availability: Available
  • Table of Contents

    1. Group cohomology
    2. The Chow ring of a classifying space
    3. Depth and regularity
    4. Regularity of group cohomology
    5. Generators for the Chow ring
    6. Regularity of the Chow ring
    7. Bounds for p-groups
    8. The structure of group cohomology and the Chow ring
    9. Cohomology mod transfers is Cohen–Macaulay
    10. Bounds for group cohomology and the Chow ring modulo transfers
    11. Transferred Euler classes
    12. Detection theorems for cohomology and Chow rings
    13. Calculations
    14. Groups of order p⁴
    15. Geometric and topological filtrations
    16. The Eilenberg–Moore spectral sequence in motivic cohomology
    17. The Chow–Künneth conjecture
    18. Open problems.

  • Author

    Burt Totaro, University of California, Los Angeles
    Burt Totaro is a Professor of Mathematics at the University of California, Los Angeles. He was elected a Fellow of the Royal Society in 2009.

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