Complete cohomological functors on groups
type: Item
instance of: scholarly article
title: Complete cohomological functors on groups
zbMATH Open document ID: 0656.16013
DOI: 10.1016/0166-8641(87)90015-0
author: T. V. Gedrich
author: K. W. Gruenberg
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: If \(\Lambda\) is a ring and \(A\) is a \(\Lambda\)-module, then a terminal completion of \(Ext^*_{\Lambda}(A,-)\) is shown to exist if, and only if, \(Ext^ j_{\Lambda}(A,P)=0\) for all projective \(\Lambda\)-modules \(P\) and all sufficiently large \(j\). Such a terminal completion exists for every \(A\) if, and only if, the supremum of the injective lengths of all projective \(\Lambda\)-modules, silp \(\Lambda\), is finite. Analogous results hold for \(Ext^*_{\Lambda}(-,A)\) and involve spli \(\Lambda\), the supremum of the projective lengths of the injective \(\Lambda\)-modules. When \(\Lambda\) is an integral group ring \({\mathbb{Z}}G\), spli \({\mathbb{Z}}G\) is finite implies silp \({\mathbb{Z}}G\) is finite. Also the finiteness of spli is preserved under group extensions. If \(G\) is a countable soluble group, then spli \({\mathbb{Z}}G\) is finite if, and only if, the Hirsch number of \(G\) is finite.
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16D50
zbMATH DE Number: 4071132
zbMATH Keywords: terminal completions
zbMATH Keywords: projective modules
zbMATH Keywords: injective lengths
zbMATH Keywords: projective lengths
zbMATH Keywords: injective modules
zbMATH Keywords: integral group rings
zbMATH Keywords: soluble groups
zbMATH Keywords: Hirsch numbers
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90015-0
OpenAlex ID: W1981045209
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description: scientific article; zbMATH DE number 4071132