Higher Algebraic K-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

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 Λα[T] (resp. Σα[T] be the α-twisted Laurent series ring over Λ (resp. Σ). In this paper we prove that (i) There exist isomorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{Q}\otimes K_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes G_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes K_{n}(\Sigma_{\alpha}[T])$\end{document}) for all n ≥ 1. (ii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)\simeq G_n(\Lambda_{\alpha}[T],\hat{Z}_l)$\end{document}is an l-complete profinite Abelian group for all n≥2. (iii)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm div} G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)=0$\end{document}for all n≥2. (iv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_n(\Lambda_{\alpha}[T]) \longrightarrow G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)$\end{document}is injective with uniquely l-divisible cokernel (for all n≥2). (v) K–1(Λ), K–1(Λα[T]) are finitely generated Abelian groups.