Decomposing Thick Subcategories of the Stable Module Category

Let 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 mod kG. It is shown that every thick tensor-ideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C = ∐ i∈I Ci into indecomposable thick tensor-ideals. This decomposition follows from a decomposition of the corresponding idempotent kG-module EC into indecomposable modules. If C = CW 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 = ⋃n i=1Wi of W into connected components.