Henning Krause

Decomposing thick subcategories of the stable module category

Abstract.
Let $\underline{\text{\rm mod}}kG$ be the stable category of finitely 
generated modular representations of a finite group $G$ over a field $k$.
We prove a Krull--Schmidt theorem for thick subcategories of 
$\underline{\text{\rm mod}}kG$. It is shown that every thick tensor-ideal
$\mathcal C$ of $\underline{\text{\rm mod}}kG$ (i.e. a thick subcategory
which is a tensor ideal) has a (usually infinite) unique decomposition
$\mathcal C = \coprod_{i\in I}\mathcal C_i$ into indecomposable thick
tensor-ideals. This decomposition follows from a decomposition of the
corresponding idempotent $kG$-module $E_{\mathcal C}$ into indecomposable
modules. If $\mathcal C = \mathcal C_W$ 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 $\mathcal C$ reflects the decomposition $W = \bigcup_{i=1}^n W_i$
of $W$ into connected components.