Operations on ring structures preserved by normalized automorphisms of group rings
type: Item
instance of: scholarly article
title: Operations on ring structures preserved by normalized automorphisms of group rings
zbMATH Open document ID: 0930.20005
author: Frauke Bleher
author: Ted Chinburg
published in: Journal of Algebra
publication date: 2000-02-09T00:00:00Z
review text: Let \(G\) be a finite group and \(R\) a commutative ring. The authors study the action of an automorphism \(\alpha\) of \(RG\) preserving the augmentation. Suppose that \(R\) is an integral domain of characteristic \(0\) such that no prime divisor of the order of \(G\) is invertible in \(R\). Then, the authors prove that \(\alpha\) commutes with the \(\lambda\)-ring structure of the Grothendieck group \(G_0(kG)\) for \(k\) being any field. The method is to elaborate the relation between the \(\lambda\)-ring structure and the Adams operations. Let now \(R\) be the ring of integers in a number field. Then, the authors use this result to prove that \(\alpha(X\cdot M)=\alpha(X)\cdot\alpha(M)\) where \(X\) is an element of the Grothendieck group \(G^R_0(RG)\) of \(RG\)-lattices and \(M\) is an object in one of the following \(G^R_0(RG)\)-modules: the Grothendieck group of finitely generated projective \(RG\)-modules, the locally free class group, the kernel subgroup, the torsion subgroup of the Grothendieck group of finitely generated \(R\)-modules. The method is to exhibit Fröhlich's Hom-description of class groups as well as Queyrut's Hom-description for the torsion part of the Grothendieck group of finitely generated \(RG\)-modules.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 1309127
zbMATH Keywords: integral representations
zbMATH Keywords: integral group rings
zbMATH Keywords: lattices
zbMATH Keywords: finite groups
zbMATH Keywords: actions
zbMATH Keywords: \(\lambda\)-rings
zbMATH Keywords: Grothendieck groups
zbMATH Keywords: Adams operations
zbMATH Keywords: finitely generated projective modules
zbMATH Keywords: locally free class groups
reviewed by: Alexander Zimmermann
MaRDI profile type: Publication
full work available at URL: jabr.1998.7749
OpenAlex ID: W2011720249
cites work: Q3999421
cites work: Tensor products and a conjecture of Zassenhaus
cites work: Autoequivalences of blocks and a conjecture of Zassenhaus
cites work: Q3922829
cites work: Q3966247
cites work: Q3701536
cites work: S-groupes des classes d'un ordre arithmétique
cites work: Q4011123
cites work: Q4003498
cites work: Q4271282
cites work: Q4126536
cites work: Induced representations and projective modules
cites work: Q3658084
DOI: 10.1006/JABR.1998.7749
description: scientific article; zbMATH DE number 1309127