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

Abstract

Let $B=A[[t;\sigma,\delta]]$ be a skew power series ring such that $\sigma$ 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 $B=Z_p[[G]]$ happens to be the Iwasawa algebra of a $p$-adic Lie group $G\isomorph H\rtimes Z_p$, this set $S$ is Venjakob's canonical Ore set. In particular, our result implies that $$ 0--> K_{n+1}(Z_p[[G]])--> K_{n+1}(Z_p[[G]]_S)--> K_n(Z_p[[G]],Z_p[[G]]_S)--> 0 $$ is split exact for each $n\geq 0$. We also prove the corresponding result for the localisation of $Z_p[[G]][\frac{1}{p}]$ with respect to the Ore set $S^*$. Both sequences play a major role in non-commutative Iwasawa theory.