Incoherence of integral group rings.
type: Item
instance of: scholarly article
title: Incoherence of integral group rings.
zbMATH Open document ID: 1070.20003
published in: Journal of Algebra
publication date: 2005-04-21T00:00:00Z
review text: An integral group ring \(\mathbb{Z} G\) is called coherent if finitely presented \(\mathbb{Z} G\)-modules and homomorphisms form an Abelian category. The aim of the paper is to prove that the integral group ring of a group containing a direct product of free groups is incoherent. In order to achieve this aim coherent augmented rings are characterized by means of derived modules, and it is proved that the integral group ring of the direct product of two free groups satisfies, in fact, a stronger property than incoherence, called the infinite kernel property: there is an exact sequence \( 0\to P\to\mathbb{Z} G^b\to\mathbb{Z} G^a\), where \(a\) and \(b\) are positive integers and \(P\) is a projective \(\mathbb{Z} G\)-module of infinite rank.
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16P70
zbMATH DE Number: 2159734
zbMATH Keywords: coherent rings
zbMATH Keywords: derived modules
zbMATH Keywords: infinite kernel property
zbMATH Keywords: Abelian categories
zbMATH Keywords: finitely presented modules
zbMATH Keywords: products of free groups
zbMATH Keywords: projective modules
author: F. E. A. Johnson
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.03.014
OpenAlex ID: W1967408398
cites work: Some remarks on the invariant basis property
cites work: The universal skew field of fractions ofa tensor product of free rings
cites work: The Nonfiniteness of Nil
cites work: K-theory and dynamics. I
cites work: On Some Groups which cannot be Finitely Presented
cites work: Surface groups inside mapping class groups
cites work: On normal subgroups of direct products
cites work: SURFACE FIBRATIONS AND AUTOMORPHISMS OF NONABELIAN EXTENSIONS
cites work: Q3997989
cites work: Periodic resolutions for finite groups
cites work: Free subgroups in linear groups
cites work: Algebraic K-theory of generalized free products. II
DOI: 10.1016/J.JALGEBRA.2004.03.014
description: scientific article; zbMATH DE number 2159734