Decomposing thick subcategories of the stable module category
type: Item
instance of: scholarly article
title: Decomposing thick subcategories of the stable module category
zbMATH Open document ID: 0926.20004
author: Henning Krause
published in: Mathematische Annalen
publication date: 1999-12-01T00:00:00Z
review text: 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.
reviewed by: Henning Krause
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 1336974
zbMATH Keywords: stable categories
zbMATH Keywords: finitely generated modular representations
zbMATH Keywords: finite groups
zbMATH Keywords: Krull-Schmidt theorem
zbMATH Keywords: thick subcategories
zbMATH Keywords: unique decomposition
zbMATH Keywords: indecomposable thick tensor-ideals
zbMATH Keywords: cohomology rings
MaRDI profile type: Publication
full work available at URL: s002080050252
OpenAlex ID: W2316875497
DOI: 10.1007/S002080050252
description: scientific article; zbMATH DE number 1336974