A note on Grothendieck groups of group rings and orders
type: Item
instance of: scholarly article
title: A note on Grothendieck groups of group rings and orders
zbMATH Open document ID: 0727.16007
DOI: 10.1007/BF01191163
author: Klaus W. Roggenkamp
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let R be a Dedekind domain with field of fractions K, and let \(\Lambda\) be an R-order in the separable K-algebra A. Furthermore, let \(_{\Lambda}{\mathfrak M}^ t\) denote the category of all finitely generated \(\Lambda\)-modules that are R-torsion, and let \(\alpha_{\Lambda}: K_ 0(_{\Lambda}{\mathfrak M}^ t)\to G_ 0(\Lambda)\) denote the homomorphism arising from the natural inclusion of \(_{\Lambda}{\mathfrak M}^ t\) into the category of all finitely generated \(\Lambda\)-modules. The main result of this work is the following: Let \(\Gamma\) be an R-order in A containing \(\Lambda\). Then the commutative square \(\alpha_{\Lambda}\circ Res^{\Gamma}_{\Lambda}=Res^{\Gamma}_{\Lambda}\circ \alpha_{\Gamma}: K_ 0(_{\Gamma}{\mathfrak M}^ t)\to G_ 0(\Lambda)\) is a pushout diagram. As an application, the author offers a proof of \textit{R. G. Swan}'s theorem \(G_ 0(RG)\cong G_ 0(KG)\) for R semilocal of characteristic 0 and G a finite group [Topology 2, 85-110 (1963; Zbl 0119.029)].
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 19A31
Mathematics Subject Classification ID: 18F30
zbMATH DE Number: 4200408
zbMATH Keywords: group ring
zbMATH Keywords: Dedekind domain
zbMATH Keywords: R-order
zbMATH Keywords: separable K-algebra
zbMATH Keywords: category
zbMATH Keywords: finitely generated \(\Lambda \) -modules
zbMATH Keywords: pushout diagram
zbMATH Keywords: finite group
reviewed by: Martin Lorenz
MaRDI profile type: Publication
cites work: Q3318943
cites work: Grothendieck groups of integral group rings
cites work: Q4061092
cites work: Lattices over orders. II
cites work: Q5587182
cites work: Induced representations and projective modules
cites work: The Grothendieck ring of a finite group
cites work: The Grothendieck group of a classical order of finite lattice type
full work available at URL: https://doi.org/10.1007/bf01191163
OpenAlex ID: W1995221017
description: scientific article; zbMATH DE number 4200408