Generalized Burnside rings and group cohomology
type: Item
instance of: scholarly article
title: Generalized Burnside rings and group cohomology
zbMATH Open document ID: 1114.19001
author: Robert Hartmann
author: Ergün Yalçın
published in: Journal of Algebra
publication date: 2007-04-16T00:00:00Z
full work available at URL: http://hdl.handle.net/11693/11588
review text: Let \(G\) be a finite group, \(X\) a finite \(G\)-set, \(M\) a \({\mathbb Z}G\)-module and \(H_X^*(G,M)\) the cohomology of \(G\) associated to \(X\) with coefficients in \(M\). A positioned \(G\)-set is a pair of the form \((X,u)\), where \(X\) is a \(G\)-set and \(u\) is a class in \(H_X^n(G,M)\). The set of isomorphism classes of positioned \(G\)-sets form a semi-ring and the cohomological Burnside ring \(B^n(G,M)\) is defined as the Grothendieck ring of this semi-ring. This is a generalization of so called crossed Burnside ring and the monomial Burnside ring. Let \(Cl(G)\) be the set of conjugacy classes of subgroups of \(G\) and let \(W_G(H)=N_G(H)/H\) be the Weyl group of \(H\) in \(G\). The ghost ring of the cohomological Burnside ring is \(\beta^n(G,M)=\oplus_{[H]\in Cl(G)} ({\mathbb Z}H^n(H,M))^{W_G(H)}.\) Then there exists an exact sequence of abelian groups \[ 0\rightarrow B^n(G,M)\rightarrow\beta^n(G,M)\rightarrow Obs^n(G,M)\rightarrow 0, \] where \(Obs^n(G,M)=\oplus_{[H]\in Cl(G)} E_H,\) \(E_H={\hat H}^0(W_G(H),{\mathbb Z}H^n(H,M))\) being the Tate group (this is the fundamental theorem for cohomological Burnside rings).
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 13D15
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 18F30
Mathematics Subject Classification ID: 20C99
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 5143570
zbMATH Keywords: cohomology of groups
zbMATH Keywords: monomial G-sets
zbMATH Keywords: generalized Burnside rings
MaRDI profile type: Publication
OpenAlex ID: W2113264902
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DOI: 10.1016/J.JALGEBRA.2006.10.037
description: scientific article; zbMATH DE number 5143570