Higher algebraic \(K\)-theory for twisted Laurent series rings over orders and semisimple algebras
type: Item
instance of: scholarly article
title: Higher algebraic \(K\)-theory for twisted Laurent series rings over orders and semisimple algebras
zbMATH Open document ID: 1147.19004
published in: Algebras and Representation Theory
publication date: 2008-09-10T00:00:00Z
review text: Let \(G\) be a group and \(\mathbb{Z}[G]\) its integral group ring. The Farrell-Jones isomorphism conjecture in \(K\)-theory proposes that the algebraic \(K\)-theory groups \(K_{*}(\mathbb{Z}[G])\) must be determined by \(K_{*}(\mathbb{Z}[V])\), where \(V\) runs over the virtually cyclic subgroups of \(G\) and homological data. Hence the importance of studying properties of the latter. A special kind of infinite virtually cyclic groups are semidirect products \(V\cong F\rtimes_{\alpha}\mathbb{Z}\) , where \(F\) is a finite group. The author proves several qualitative properties for higher \(K\)-theory of \(\mathbb{Z}[V]\) in a more general setup: Theorem 1: Let \(F\) be an algebraic number field with ring of integers \(R\), \(\Lambda\) any \(R\)-order in a semisimple \(F\)-algebra \(\Sigma\), and \(\alpha\) an \(R\)-automorphism of \(\Lambda\). Denote by \(\alpha\) also the extension of \(\alpha\) to \(\Sigma\). Let \(\Lambda_{\alpha}[T]\) (resp. \(\Sigma_{\alpha}[T]\)) be the \(\alpha\)-twisted Laurent polynomial ring over \(\Lambda\) (resp \(\Sigma\)). Then 1. \(G_{n}(\Lambda_{\alpha}[T])\) is a finitely generated abelian group for all \(n\geq 1\). 2. There are isomorphisms: \[ \mathbb{Q}\otimes K_{n}(\Lambda_{\alpha}[T])\cong \mathbb{Q}\otimes G_{n}(\Lambda_{\alpha}[T])\cong \mathbb{Q}\otimes K_{n}(\Sigma_{\alpha}[T]), \] for all \(n\geq 2\). The author also proves the following: under the hypotheses of Theorem 1, it also holds: 1. \(K_{-1}(\Lambda)\) is a finitely generated abelian group. 2. \(K_{-1}(\Lambda_{\alpha}[T])\) is a finitely generated abelian group. 3. \(K_{-1}(\Lambda)\cong K_{-1}(\Lambda_{\alpha}[T])\).
Mathematics Subject Classification ID: 19D35
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 5324368
zbMATH Keywords: twisted Laurent polynomial rings
zbMATH Keywords: semisimple algebras
zbMATH Keywords: orders
zbMATH Keywords: virtually infinite cyclic group
author: Aderemi O. Kuku
reviewed by: Daniel Juan Pineda
MaRDI profile type: Publication
full work available at URL: s10468-008-9085-6
OpenAlex ID: W2073848203
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DOI: 10.1007/S10468-008-9085-6
description: scientific article; zbMATH DE number 5324368