Higher Algebraic K-Theory of p-Adic Orders and Twisted Polynomial and Laurent Series Rings Over p-Adic Orders

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, Λ_α[t], the α-twisted polynomial ring over Λ and Λ_α[T], the α-twisted Laurent series ring over Λ. In this article, we study higher K-theory of Λ, Λ_α[t], and Λ_α[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 (Λ_α[t]) → K _n (Λ)) is a p-torsion group and also that for n ≥ 2 there exists isomorphisms

Finally, we prove that NK _n (Λ_α[T]) is p-torsion. Note that if G is a finite group and Λ =RG such that α(G) = G, then Λ_α[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.

Key Words:

• Farrell–Jones conjecture
• Higher algebraic K-theory
• Twisted polynomials and Laurent series rings
• p-Adic orders in algebras
• Virtually infinite cyclic groups

2000 Mathematics Subject Classification:

• Primary 19D34, 19D35
• Secondary 16S34, 16H05

Notes

Communicated by C. Pedrini.