Decomposing thick subcategories of the stable module category

Let 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 . It is shown that every thick tensor-ideal of (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module into indecomposable modules. If is the thick tensor-ideal corresponding to a closed homogeneous subvariety W of the maximal ideal spectrum of the cohomology ring , then the decomposition of reflects the decomposition of W into connected components. Received: 27 April 1998 / In revised form: 16 July 1998