Cohomology of permutation representation in negative dimensions
type: Item
instance of: scholarly article
title: Cohomology of permutation representation in negative dimensions
zbMATH Open document ID: 0849.20035
author: Ixin Wen
published in: Journal of Algebra
publication date: 1996-06-03T00:00:00Z
review text: Let the finite group \(G\) act on the finite set \(X\). The author splices together a standard chain complex for \(X\) and a standard cochain complex for \(X\) to get an acyclic \(G\)-complex \(C_*(X; G)\), infinite in both directions. Then if \(A\) is a \(G\)-module he defines \(H^n(X;G,A)\) to be the cohomology in dimension \(n\) of \(\text{Hom}_G(C_*(X;G),A)\), but he does not give any qualitative properties that might motivate the construction. Note that one gets Tate cohomology by taking \(X = G\) with action by left multiplication. The author makes boundary maps explicit in terms of stabilizer groups and orbit representatives.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 869806
zbMATH Keywords: standard chain complex
zbMATH Keywords: standard cochain complex
zbMATH Keywords: acyclic \(G\)-complex
zbMATH Keywords: Tate cohomology
zbMATH Keywords: action
zbMATH Keywords: boundary maps
zbMATH Keywords: stabilizer groups
zbMATH Keywords: orbit representatives
reviewed by: Wilberd van der Kallen
MaRDI profile type: Publication
full work available at URL: jabr.1996.0058
OpenAlex ID: W1978247190
DOI: 10.1006/JABR.1996.0058
description: scientific article; zbMATH DE number 869806