On a Localisation Sequence for the K-Theory of Skew Power Series Rings

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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=Zₚ[[G]] happens to be the Iwasawa algebra of a p-adic Lie group G H Zₚ, this set S is Venjakob's canonical Ore set. In particular, our result implies that 0--> Kₙ₊₁(Zₚ[[G]])--> Kₙ₊₁(Zₚ[[G]]ₛ)--> Kₙ(Zₚ[[G]],Zₚ[[G]]ₛ)--> 0 is split exact for each n≥ 0. We also prove the corresponding result for the localisation of Zₚ[[G]][1/p] with respect to the Ore set S*. Both sequences play a major role in non-commutative Iwasawa theory.