A Tate cohomology sequence for generalized Burnside rings
type: Item
instance of: scholarly article
title: A Tate cohomology sequence for generalized Burnside rings
zbMATH Open document ID: 1169.19001
author: Olcay Coşkun
author: Ergün Yalçın
published in: Journal of Pure and Applied Algebra
publication date: 2009-04-16T00:00:00Z
full work available at URL: http://hdl.handle.net/11693/22701
review text: The fundamental theorem for the Burnside ring \(B(G)\) of a finite group \(G\) is an exact sequence of abelian groups \[ 0 \to B(G) \to C(G) \to \bigoplus_K \mathbb{Z}/|N_G(K):K|\,\mathbb{Z} \to 0; \] here \(C(G)\) denotes the ghost ring of \(G\). The authors show that this result follows from an exact sequence of Mackey functors \[ 0 \to \widehat{\text{Ext}}^{-1}_\gamma(\rho,D) \to D_+ @>{\varphi}>> D^+ \to \widehat{\text{Ext}}^{0}_\gamma(\rho,D) \to 0. \] Here \(D\) is a restriction functor for \(G\) over a noetherian commutative ring \(R\), as defined by \textit{R. Boltje} [J. Algebra 206, No. 1, 293--343 (1998; Zbl 0913.20001)], and the definitions of \(D_+, D^+\) and \(\varphi\) go also back to this paper. Moreover, \(\rho\) and \(\gamma\) are the restriction and conjugation algebra, respectively, of \(G\), defined in a way similar to the construction of the Mackey algebra. The Ext-groups appearing above can be described in terms of Tate cohomology for groups.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 5544384
zbMATH Keywords: Burnside ring
zbMATH Keywords: Mackey functor
zbMATH Keywords: Mackey algebra
zbMATH Keywords: Tate cohomology
zbMATH Keywords: mark homomorphism
MaRDI profile type: Publication
OpenAlex ID: W1989479254
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DOI: 10.1016/J.JPAA.2008.11.025
description: scientific article; zbMATH DE number 5544384