Higher algebraic \(K\)-theory for twisted Laurent series rings over orders and semisimple algebras
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. \(K_{-1}(\Lambda)\cong K_{-1}(\Lambda_{\alpha}[T])\).
scientific article; zbMATH DE number 5324368
scientific article