Complexity and multiple complexes
type: Item
instance of: scholarly article
title: Complexity and multiple complexes
zbMATH Open document ID: 0593.20062
DOI: 10.1007/BF01166459
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183688
review text: Let G be a finite group and let R be a commutative ring with unit. One of the most basic problems in group cohomology is that of finding suitable projective resolutions for RG-modules. The main result of this paper shows that if \(R=k\) is a field of characteristic \(p>0\), and if M is any finitely generated kG-module then there exists a projective resolution of M that is a tensor product of periodic complexes. The constructed resolutions are usually not minimal but do have the same rate of growth (complexity) as the corresponding minimal resolutions. An integral version of the main theorem for RG-lattices is also proved. As an application of the results we present a new proof of G. Carlsson's theorem that a finite group acting freely on a product of n spheres of the same dimension must have p-rank at most n for all primes p.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18G10
zbMATH DE Number: 3953096
zbMATH Keywords: group cohomology
zbMATH Keywords: projective resolutions for RG-modules
zbMATH Keywords: finitely generated kG-module
zbMATH Keywords: tensor product of periodic complexes
zbMATH Keywords: p-rank
author: David Benson
author: Jon F. Carlson
MaRDI profile type: Publication
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OpenAlex ID: W4254311771
Wikidata QID: Q105419669
description: scientific article; zbMATH DE number 3953096