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.
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Received: 27 April 1998 / In revised form: 16 July 1998
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Krause, H. Decomposing thick subcategories of the stable module category. Math Ann 313, 95–108 (1999). https://doi.org/10.1007/s002080050252
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DOI: https://doi.org/10.1007/s002080050252