Minimal free resolutions of lattices over finite groups
type: Item
instance of: scholarly article
title: Minimal free resolutions of lattices over finite groups
zbMATH Open document ID: 0682.20006
author: K. W. Gruenberg
author: Peter A. Linnell
published in: Illinois Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let G be a finite group and let M be a \({\mathbb{Z}}G\)-lattice. The first author [Proc. Lond. Math. Soc., III. Ser. 48, 91-107 (1984; Zbl 0507.20004)] has proved that two minimal projective resolutions of M are in the same genus (as complexes). The main object of this article is the analogous question with respect to minimal free resolutions of M. It is shown that in general two minimal free resolutions have different rank sequences. However, the following result holds. Theorem. If \({\mathbb{Z}}G\) allows cancellation (this is e.g. the case, if the Eichler condition is present), then all minimal free resolutions of a \({\mathbb{Z}}G\)- lattice M belong to one genus class. The final part of the paper investigates the difference between minimal projective and minimal free resolutions. A short description (in the authors' own words) is that this difference is concentrated in a finite interval. Clearly the results are based on cancellation properties, generation properties (Swan modules) and the usual techniques of homological algebra. However, the authors present some interesting refinements of these methods, which are also certainly of interest with respect to other questions of integral representations of finite groups.
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 18G10
zbMATH DE Number: 4116786
zbMATH Keywords: finite group
zbMATH Keywords: \({\mathbb{Z}}G\)-lattice
zbMATH Keywords: minimal projective resolutions
zbMATH Keywords: minimal free resolutions
zbMATH Keywords: cancellation
zbMATH Keywords: Swan modules
zbMATH Keywords: integral representations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4116786