Gröbner bases and the computation of group cohomology.
type: Item
instance of: scholarly article
title: Gröbner bases and the computation of group cohomology.
zbMATH Open document ID: 1050.20036
author: David J. Green
published in: Lecture Notes in Mathematics
publication date: 2003-12-03T00:00:00Z
review text: The computation of the cohomology ring of finite groups is in general a difficult task; two major obstacles are found right away: how to find appropriate resolutions and how to determine whether we have found all the existing relations. In this book, the author explains how to overcome the above hurdles and the implementation of computer programs that perform the necessary computations to obtain complete descriptions of the cohomology ring of finite \(p\)-groups. Here are the steps carried out by the author: first he describes a method to construct a minimal resolution for the \(kG\)-module \(k\) (\(k\) is a field of characteristic \(p\)), a major ingredient in this step is to use Gröbner bases to determine a minimal set of generators for the kernel of a \(kG\)-module map. The author describes both the theoretical foundations of the algorithms and the implementations of them in the computer. The second step is to determine the product in the cohomology and a full set of relations, the key ingredient in this step is the implementation of a criterion found by J. Carlson, again the author describes in great detail both the foundation of the algorithm and the implementation in the computer. The author closes with several examples of cohomology rings providing, among other things, a presentation, restriction to subgroups and essential classes. Finally, the author refers to his www-page for more details, programs and full packages used in this book.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 16E05
Mathematics Subject Classification ID: 16Z05
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 2012371
zbMATH Keywords: group cohomology
zbMATH Keywords: Gröbner bases
zbMATH Keywords: cohomology rings
zbMATH Keywords: resolutions
zbMATH Keywords: algorithms
reviewed by: Daniel Juan Pineda
describes a project that uses: MeatAxe
MaRDI profile type: Publication
full work available at URL: b93836
OpenAlex ID: W4214647664
DOI: 10.1007/B93836
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description: scientific article; zbMATH DE number 2012371