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

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 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 \(\mathbb{Q}\otimes K_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes G_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes K_{n}(\Sigma_{\alpha}[T])\)) 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(Λα[T]) are finitely generated Abelian groups.