Multiple complexes and gaps in Farrell cohomology
type: Item
instance of: scholarly article
title: Multiple complexes and gaps in Farrell cohomology
zbMATH Open document ID: 1056.18006
author: Jang Hyun Jo
published in: Journal of Pure and Applied Algebra
publication date: 2004-11-18T00:00:00Z
review text: Let \(\Gamma\) be a group having a torsion free subgroup \(\Gamma'\) of finite index. The cohomological dimension of \(\Gamma'\) does not depend on the choice of \(\Gamma'\) and is called the virtual cohomological dimension of \( \Gamma\). A complete resolution of the trivial \(\Gamma\)-module \(\mathbb Z\) is an acyclic complex \(F\) of projective modules, which coincides with a projective resolution of \(\mathbb Z\) from some degree onwards. Farrell cohomology is the homology of the complex \(\Hom_\Gamma(F,N)\) for a module \(N\). For a group \(G\) let \(r(G)\) be the maximum of the \(p\)-ranks of finite elementary abelian \(p\)-subgroups of \(G\). The main results of the paper are the following. If the cohomological dimension of \(\Gamma'\) is an integer \(n\), and if the trivial \(\Gamma'\)-module \(\mathbb Z\) has a periodic resolution of period \(q\) after \(k\) steps, then the trivial \(\Gamma\)-module \(\mathbb Z\) has a periodic resolution of period \(q\) after \(n+k\) steps. If the virtual cohomological dimension of \(\Gamma\) is an integer \(n\), then the trivial \(\Gamma\)-module \(\mathbb Z\) has a projective resolution being the tensor product of \(r(\Gamma/\Gamma')\) complexes which are periodic resolutions of some period after \(n\) steps. The same holds if one replaces \(r(\Gamma/\Gamma')\) by \(r(\Gamma)\). This last statement uses subtle arguments on cohomology varieties. As an application the author shows, using a spectral sequence argument, that for groups with finite virtual cohomological dimension there is an integer \(r\) so that the vanishing of \(r+1\) consecutive degrees of Farrell cohomology implies vanishing of Farrell cohomology for all degrees.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 18G20
zbMATH DE Number: 2116174
zbMATH Keywords: periodic projective resolutions
zbMATH Keywords: virtual cohomological dimension
zbMATH Keywords: Farrell cohomology
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.03.006
OpenAlex ID: W2028147270
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DOI: 10.1016/J.JPAA.2004.03.006
description: scientific article; zbMATH DE number 2116174