Representations of Finite Groups: Local Cohomology and Support: (43 Oberwolfach Seminars)

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.

Table of Contents:
Preface.- 1 Monday.- 1.1 Overview.- 1.2 Modules over group algebras.- 1.3 Triangulated categories.- 1.4 Exercises.- 2 Tuesday.- 2.1 Perfect complexes over commutative rings.- 2.2 Brown representability and localization.- 2.3 The stable module category of a finite group.- 2.4 Exercises.- 3 Wednesday.- 3.1.- 3.2 Koszul objects and support.- 3.3 The homotopy category of injectives.- 3.4 Exercises.- 4 Thursday.- 4.1 Stratifying triangulated categories.- 4.2 Consequences of stratification.- 4.3 The Klein four group.- 4.4 Exercises.- 5 Friday.- 5.1 Localising subcategories of D(A).- 5.2 Elementary abelian 2-groups.- 5.3 Stratification for arbitrary finite groups.- 5.4 Exercises.- A Support for modules over commutative rings.- Bibliography.- Index.

Review :
From the reviews: “The book is aimed at a readership with a solid background in algebra, in particular representation theory, commutative algebra and homological algebra. The volume comprises five chapters and an appendix, and each chapter is divided into four sections. Each chapter consists of the lecture material and the exercises handled during one day at the Oberwolfach Seminar (in 2010) with the same title. … The book ends with an appendix … and there is a comprehensive bibliography.” (Nadia P. Mazza, Mathematical Reviews, March, 2013) “The manuscript under review provides a quite nice introduction to the tools used in these classification theorems and offers an excellent starting point for someone new to the area. The manuscript is based on a week-long series of lectures given by the authors to introduce people to the ideas involved in the proof of the classification of localising subcategories of Mod(kG).” (Christopher P. Bendel, Zentralblatt MATH, Vol. 1246, 2012)