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Higher Algebraic <Emphasis Type="BoldItalic">K</Emphasis>-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

Authors:	Aderemi Kuku

Institution:	(1) Mathematics Department, The University of Iowa, 14 Maclane Hall, Iowa City, Iowa 52242, USA;(2) Max-Planck-Institut für Mathematik, Bonn, Germany

Abstract:	Let R be the ring of integers in a number field F, Λ any R-order in a semisimple F-algebra Σ, α an R-automorphism of Λ. Denote the extension of α to Σ also by α. Let Λ_αT] (resp. Σ_αT] be the α-twisted Laurent series ring over Λ (resp. Σ). In this paper we prove that (i) There exist isomorphisms $\mathbb{Q}\otimes K_{n}(\Lambda_{\alpha}T])\simeq \mathbb{Q}\otimes G_{n}(\Lambda_{\alpha}T])\simeq \mathbb{Q}\otimes K_{n}(\Sigma_{\alpha}T])$ ) for all n ≥ 1. (ii) $G^{\rm pr}_n(\Lambda_{\alpha}T],\hat{Z}_l)\simeq G_n(\Lambda_{\alpha}T],\hat{Z}_l)$ is an l-complete profinite Abelian group for all n≥2. (iii) ${\rm div} G^{\rm pr}_n(\Lambda_{\alpha}T],\hat{Z}_l)=0$ for all n≥2. (iv) $G_n(\Lambda_{\alpha}T]) \longrightarrow G^{\rm pr}_n(\Lambda_{\alpha}T],\hat{Z}_l)$ is injective with uniquely l-divisible cokernel (for all n≥2). (v) K _–1(Λ), K _–1(Λ_αT]) are finitely generated Abelian groups. Presented by Alain Verschoren.

Keywords:	K-theory Twisted Laurent series rings Semisimple algebras Orders Virtually infinite cyclic group
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