Moody's induction theorem
type: Item
instance of: scholarly article
title: Moody's induction theorem
zbMATH Open document ID: 0648.16009
author: A. Weiss
author: Gerald Cliff
published in: Illinois Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(\Gamma\) be a finitely generated abelian by finite group and let S*\(\Gamma\) be a crossed product of a left noetherian ring S with \(\Gamma\). Let \(G_ 0(S*\Gamma)\) denote the Grothendieck group of the category of all finitely generated (S*\(\Gamma)\)-modules. For any subgroup F of \(\Gamma\) there is a map \(G_ 0(S*F)\to G_ 0(S*\Gamma)\) given by sending the class [M] of an (S*F)-module M to the class \([(S*\Gamma)\otimes_{S*F}M]\) of the induced module. The purpose of this paper is to give a proof of the recent induction theorem of \textit{J. Moody} [Induction theorems for infinite groups (Ph. Thesis, Columbia Univ. 1986); see also Bull. Am. Math. Soc., New Ser. 17, 113-116 (1987; Zbl 0644.16014)]. Let \(\alpha\) be the sum of the maps from \(\sum G_ 0(S*F)\) to \(G_ 0(S*\Gamma)\), where F varies over all finite subgroups of \(\Gamma\). Then \(\alpha\) is surjective (Moody's Theorem). Let H be a polycyclic by finite group and let k be a noetherian ring. The map from \(\sum G_ 0(kF)\) to \(G_ 0(kH)\), given by the sum of inductions from finite subgroups F of H, is surjective (Moody's Theorem for polycyclic by finite groups). The authors use Moody's theorem to prove that the Goldie rank of the group ring kH of a polycyclic by finite group H over an arbitrary division ring k is equal to the least common multiple of the orders of the finite subgroups of H.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 4057714
zbMATH Keywords: finitely generated abelian by finite group
zbMATH Keywords: crossed product
zbMATH Keywords: left noetherian ring
zbMATH Keywords: Grothendieck group
zbMATH Keywords: induction theorem
zbMATH Keywords: polycyclic by finite group
zbMATH Keywords: Moody's theorem
zbMATH Keywords: Goldie rank
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
Recommended article: Brauer induction for \(G_ 0\) of certain infinite groups
Recommended article: Induced resolutions and Grothendieck groups of polycyclic-by-finite groups
Recommended article: Q3032421
Recommended article: Q3349049
Recommended article: Induction theorems for infinite groups
Recommended article: On the cohomology of polycyclic-by-finite groups
Recommended article: Q4305842
Recommended article: Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups.
Recommended article: Q3035459
Recommended article: Rings Graded by Polycyclic-by-Finite Groups
description: scientific article; zbMATH DE number 4057714