Decomposing thick subcategories of the stable module category

Abstract

Let $\underline{{\rm mod}} kG$ be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of $\underline{{\rm mod}} kG$ . It is shown that every thick tensor-ideal $\mathcal{C}$ of $\underline{{\rm mod}} kG$ (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 $W=\bigcup_{i=1}^nW_i$ of W into connected components.