On the cohomology rings of small categories
type: Item
instance of: scholarly article
title: On the cohomology rings of small categories
zbMATH Open document ID: 1153.18013
author: Fei Xu
published in: Journal of Pure and Applied Algebra
publication date: 2008-09-04T00:00:00Z
review text: This paper gives generalizations of some theorems in group cohomology theory. Let \(\mathcal C\) denote a small category, \(R\) a commutative ring with identity. Then, the cohomology ring of \(\mathcal C\), with coefficients in \(R\) is defined as the cohomology ring of the topological realization of the nerve of \(\mathcal C\). The author first looks into the cohomology ring modulo nilpotents which he shows is not necessarily finitely generated, even in case of finite EI-category. His springboard are results by Słomińska. The author furthermore studies relationships between the cohomology ring of a category with the cohomology rings of its subcategories and extensions, respectively. The acknowledgments provide an insight into the people who participated in various ways in this project and the appendix provides the reader with standard material regarding the cohomology theory of small categories.
reviewed by: Radoslav M. Dimitrić
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 18A25
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20J99
zbMATH DE Number: 5321430
zbMATH Keywords: cohomology ring of a small category with coefficients in a commutative ring
zbMATH Keywords: EI-category
zbMATH Keywords: group cohomology theory
zbMATH Keywords: LHS spectral sequence
zbMATH Keywords: Grothendieck construction
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.04.004
OpenAlex ID: W1989496019
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DOI: 10.1016/J.JPAA.2008.04.004
description: scientific article; zbMATH DE number 5321430