Decomposing thick subcategories of the stable module category
Let \(\underline{\text{mod}} kG\) be the stable category of finitely generated modular representations of a finite group \(G\) over a field \(k\). The main result of the paper is a Krull-Schmidt theorem for thick subcategories of \(\underline{\text{mod}} kG\). It is shown that every thick tensor-ideal \(\mathcal C\) of \(\underline{\text{mod}} kG\) (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 \(W=\bigcup^n_{i=1}W_i\) of \(W\) into connected components.
finitely generated modular representations
scientific article; zbMATH DE number 1336974