On a localisation sequence for the k-theory of skew power series rings

摘要

Let B = A[[t;σ,δ]] be a skew power series ring such that σ is given by an inner automorphism of B. We show that a certain Waldhausen localisation sequence involving the K-theory of B splits into short split exact sequences. In the case that A is noetherian we show that this sequence is given by the localisation sequence for a left denominator set S in B. If B =p[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G ? H a? p, this set S is Venjakob's canonical Ore set. In particular, our result implies that 0 → K_(n+1) (Z_p[[G]]) → K_(n+1p) [[G]]S) → K_np[[G]] _p[[G]]_s) → 0 is split exact for each n ≥ 0. We also prove the corresponding result for the localisation of _p[[G]][ ~1 _p] with respect to the Ore set S. Both sequences play a major role in non-commutative Iwasawa theory.