Local control of group cohomology
type: Item
instance of: scholarly article
title: Local control of group cohomology
zbMATH Open document ID: 0839.20066
DOI: 10.1016/0022-4049(95)00095-E
published in: Journal of Pure and Applied Algebra
publication date: 1996-06-02T00:00:00Z
review text: A theorem of P. Webb states: If \(G\) is a finite group acting simplicially on a finite simplicial complex, \(\Delta\), in such a way that any element of \(G\) which stabilizes a cell fixes it pointwise, then the following conditions are equivalent for a fixed prime \(p\): (i) For all subgroups \(H\) of \(G\) with \(p\) dividing \(|H|\) and \(H/O_p (H)\) cyclic, \(\chi (\Delta^H)=1\); (ii) For all finitely generated \(\mathbb{Z} G\)-modules \(M\) and integers \(n\), \[ \widehat {H}^n (G, M)_p= \sum_{\sigma\in \Delta/G} (-1)^{\dim \sigma} \widehat {H}^n (G_\sigma, M)_p. \] [This is an equality of elements of the Grothendieck group of finite \(p\)-groups with respect to direct sum decompositions.] In the present paper, the author proves that under certain conditions (ii) may hold for the special case \(M= \mathbb{Z}\) even when (i) fails to hold. He applies these results to the Mathieu group, \(M_{12}\), and the Hall-Janko-Wales group, \(J_2\), acting on appropriate complexes. The method is to prove first, under appropriate hypotheses, an isomorphism of Tate cohomology and equivariant cohomology with trivial integer coefficients. The spectral sequence for equivariant cohomology is then analysed to obtain the required results.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 55U10
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 55N91
zbMATH DE Number: 810248
zbMATH Keywords: finite group acting simplicially on finite simplicial complex
zbMATH Keywords: finitely generated \(\mathbb{Z} G\)-modules
zbMATH Keywords: Grothendieck group
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: Mathieu group
zbMATH Keywords: Hall-Janko-Wales group
zbMATH Keywords: Tate cohomology
zbMATH Keywords: equivariant cohomology
zbMATH Keywords: spectral sequence
author: John S. Maginnis
reviewed by: Kevin Hutchinson
MaRDI profile type: Publication
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Wikidata QID: Q127517438
full work available at URL: 0022-4049(95)00095-e
OpenAlex ID: W2078299495
description: scientific article; zbMATH DE number 810248