English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

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