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, Lambda any R-order in a semisimple F-algebra Sigma, alpha an R-automorphism of Lambda. Denote the extension of alpha to Sigma also by alpha. Let Lambda(alpha)[T] ( resp. Sigma(alpha)[T] be the alpha-twisted Laurent series ring over Lambda (resp. Sigma). In this paper we prove that (i) There exist isomorphisms Q circle times K-n(Lambda(alpha)[T]) similar or equal to Q circle times. Gn(Lambda(alpha)[T]) similar or equal to Q circle times K-n(Sigma(alpha)[T]) for all n >= 1. (ii) G(n)(pr) (Lambda(alpha) [T], (Z)overcap>(l)) similar or equal to G(n) (Lambda(alpha)[T], (Z) over cap (l)) is an l-complete profinite Abelian group for all n >= 2. (iii) divG(n)(pr) (Lambda(alpha)[T], (Z) over cap (l)) = 0 for all n >= 2. (iv) G(n)(Lambda(alpha)[T]) -> G(n)(pr) (Lambda(alpha) [T], (Z) over cap (l)) is injective with uniquely l-divisible cokernel (for all n >= 2). (v) K-1(Lambda), K-1 (Lambda(alpha)[T]) are finitely generated Abelian groups.