Moody's induction theorem
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.
finitely generated abelian by finite group
scientific article; zbMATH DE number 4057714
scientific article