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.
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