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

TL;DR

This paper investigates the K-theory of skew power series rings, showing that certain localisation sequences split into short exact sequences, with applications to Iwasawa algebras of p-adic Lie groups.

Contribution

It establishes a splitting of localisation sequences in K-theory for skew power series rings with inner automorphisms, extending to Iwasawa algebras and their localisations.

Findings

The Waldhausen localisation sequence splits into short exact sequences.

For noetherian A, the sequence corresponds to a localisation sequence for a left denominator set.

In the case of Iwasawa algebras, the sequence is split exact for all n ≥ 0.

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.