Representations of Finite Groups: Local Cohomology and Support

  • David J. Benson
  • Srikanth Iyengar
  • Henning Krause

Part of the Oberwolfach Seminars book series (OWS, volume 43)

Table of contents

  1. Front Matter
    Pages i-x
  2. David J. Benson, Srikanth Iyengar, Henning Krause
    Pages 1-26
  3. David J. Benson, Srikanth Iyengar, Henning Krause
    Pages 27-46
  4. David J. Benson, Srikanth Iyengar, Henning Krause
    Pages 47-62
  5. David J. Benson, Srikanth Iyengar, Henning Krause
    Pages 63-78
  6. David J. Benson, Srikanth Iyengar, Henning Krause
    Pages 79-91
  7. Back Matter
    Pages 93-105

About this book

Introduction

The seminar focuses on a recent solution, by the authors, of a long standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on ideas from commutative algebra, cohomology of groups, and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. The prototype for this has been Daniel Quillen’s description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Jon Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings and over cocommutative Hopf algebras. One of the threads in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Mike Hopkins’ classification of thick subcategories of the perfect complexes over a commutative Noetherian ring, followed by a classification of localizing subcategories of its full derived category, due to Amnon Neeman. The authors have been developing an approach to address such classification problems, based on a construction of local cohomology functors and support for triangulated categories with ring of operators. The book serves as an introduction to this circle of ideas.

Keywords

Homological algebra and category theory Homological methods for commutative rings Homological methods for noncommutative rings

Authors and affiliations

  • David J. Benson
    • 1
  • Srikanth Iyengar
    • 2
  • Henning Krause
    • 3
  1. 1., Institute of MathematicsUniversity of AberdeenAberdeenUnited Kingdom
  2. 2., Department of MathematicsUniversity of NebraskaLincolnUSA
  3. 3., Fakultät für MathematikUniversität BielefeldBielefeldGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0260-4
  • Copyright Information Springer Basel AG 2012
  • Publisher Name Springer, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0259-8
  • Online ISBN 978-3-0348-0260-4
  • About this book