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Mathematische Annalen

, Volume 313, Issue 1, pp 95–108 | Cite as

Decomposing thick subcategories of the stable module category

  • Henning Krause

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.

Mathematics Subject Classification (1991):20C05, 20J05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Henning Krause
    • 1
  1. 1. Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany (e-mail: henning@mathematik.uni-bielefeld.de) DE

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