Title: Decomposing thick subcategories of the stable module category
Author: Henning Krause
Status: Math. Ann. 313 (1999), 95-108
Address: University of Bielefeld, Germany
E-mail: henning@mathematik.uni-bielefeld.de
Abstract: Let stmod kG be the stable category of finitely generated modular
representations of a finite group G over a field k.
We prove a Krull-Remak-Schmidt theorem for thick subcategories of stmod kG.
It is shown that every thick tensor-ideal C of stmod kG (i.e. a thick
subcategory which is a tensor ideal) has a (usually infinite) unique
decomposition C=\coprod_{i\in I}C_i into indecomposable thick tensor-ideals.
This decomposition follows from a decomposition of the corresponding
idempotent kG-module E_C into indecomposable modules. If C=C_W is the thick
tensor-ideal corresponding to a closed homogeneous subvariety W of the
maximal ideal spectrum of the cohomology ring H^*(G,k), then the
decomposition of C reflects the decomposition W=\bigcup_{i=1}^nW_i of W
into connected components.