Abstract
Let F be a p-adic field (i.e., any finite extension of ), R the ring of integers of F, Λ any R-order in a semisimple F-algebra Σ, α: Λ → Λ an R-automorphism of Λ, T = ⟨ t ⟩, the infinite cyclic group, Λα[t], the α-twisted polynomial ring over Λ and Λα[T], the α-twisted Laurent series ring over Λ. In this article, we study higher K-theory of Λ, Λα[t], and Λα[T].
More precisely, we prove in Section 1 that for all n ≥ 1, SK
2n−1(Λ) is a finite p-group if Σ is a direct product of matrix algebra over fields, in partial answer to an open question whether is a finite p-group if G is any finite group. So, the answer is affirmative if
splits.
We also prove that NK n (Λ; α): = ker(K n (Λα[t]) → K n (Λ)) is a p-torsion group and also that for n ≥ 2 there exists isomorphisms
Finally, we prove that NK n (Λα[T]) is p-torsion. Note that if G is a finite group and Λ =RG such that α(G) = G, then Λα[T] is the group ring RV where V is a virtually infinite cyclic group of the form V = G ⋊α T, where α is an automorphism of G and the action of the infinite cyclic group T = ⟨ t ⟩ on G is given by α(g) = tgt −1 for all g ∈ G.
2000 Mathematics Subject Classification:
Notes
Communicated by C. Pedrini.