Constructing endotrivial modules.
type: Item
instance of: scholarly article
title: Constructing endotrivial modules.
zbMATH Open document ID: 1098.20004
published in: Journal of Pure and Applied Algebra
publication date: 2006-05-29T00:00:00Z
review text: Let \(G\) be a finite \(p\)-group and let \(k\) be an algebraically closed field of characteristic \(p\). A \(kG\)-module \(M\) is called endotrivial if its endomorphism ring over \(k\) is isomorphic to a direct sum of the trivial \(kG\)-module \(k\) and a projective module. Endotrivial modules form a finitely generated Abelian group \(T(G)\) whose structure was determined very recently. Puig showed that the torsion part of \(T(G)\) is finitely generated, Carlson-Thévenaz gave the precise structure, Bouc-Thévenaz and Alperin gave a subgroup of finite index in the torsion free part of \(T(G)\), and Carlson-Thévenaz showed that the subgroup described by Alperin is actually all of the torsion free part of \(T(G)\). In the paper under review the author gives explicit generators of \(T(G)\) by methods coming from the cohomology variety of the group. The modules are constructed by this well developed theory by modules corresponding to irreducible components of this variety.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 5028786
zbMATH Keywords: endotrivial modules
zbMATH Keywords: Dade groups
zbMATH Keywords: support varieties
zbMATH Keywords: cohomology of groups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: endomorphism rings
author: Jon F. Carlson
MaRDI profile type: Publication
full work available at URL: j.jpaa.2005.01.014
OpenAlex ID: W2000035518
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DOI: 10.1016/J.JPAA.2005.01.014
description: scientific article