Complexity and varieties for infinite groups. I
type: Item
instance of: scholarly article
title: Complexity and varieties for infinite groups. I
zbMATH Open document ID: 0886.20002
published in: Journal of Algebra
publication date: 1998-03-05T00:00:00Z
review text: The author develops the module theory and the theory of complexity and varieties for modules of type \(FP_\infty\) over \(LH{\mathfrak F}\)-groups, which have been introduced by P. H. Kropholler. As applications of these methods the author obtains the following results. Theorem 1.1. Let \(G\) be an \(LH{\mathfrak F}\)-group, \(k\) a commutative Noetherian ring of finite global dimension, \(M\) a \(kG\)-module of type \(FP_\infty\). Every element of \(\text{Ext}^*_{kG}(M,M)\), which restricts to a nilpotent element of each finite elementary abelian subgroup of \(G\), is nilpotent. Theorem 1.2. Let \(G\) be an \(LH{\mathfrak F}\)-group, \(k\) a field, \(M\) a \(kG\)-module of type \(FP_\infty\). Then there is an upper bound for the complexities of the restrictions of \(M\) to finite subgroups of \(G\).
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 1051521
zbMATH Keywords: complexity
zbMATH Keywords: varieties of modules
zbMATH Keywords: modules over group rings
zbMATH Keywords: modules of type \(FP_ \infty\)
zbMATH Keywords: restrictions
zbMATH Keywords: finite subgroups
author: David Benson
reviewed by: L. A. Kurdachenko
MaRDI profile type: Publication
full work available at URL: jabr.1996.6996
OpenAlex ID: W1987901958
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DOI: 10.1006/JABR.1996.6996
description: scientific article; zbMATH DE number 1051521