Skew Laurent Series Ring Over a Dedekind Domain

TL;DR

This paper investigates the algebraic properties of skew Laurent series rings over Dedekind domains, establishing their noncommutative Dedekind domain structure and analyzing their K-theoretic and homological invariants.

Contribution

It demonstrates that skew Laurent series rings over Dedekind domains are noncommutative Dedekind domains and characterizes their K-theory and various dimensions.

Findings

$R$ is a noncommutative Dedekind domain.

If $\sigma$ acts trivially on the class group, then $K_0(R) o K_0(D)$ is an isomorphism.

Determines Krull, global, and stable ranks of $R$.

Abstract

We show that the formal skew Laurent series ring $R = D(\! ( x; \sigma )\! )$ over a commutative Dedekind domain $D$ with an automorphism $\sigma$ is a noncommutative Dedekind domain. If $\sigma$ acts trivially on the ideal class group of $D$, then $K_0(R)$, the Grothendieck group of $R$, is isomorphic to $K_0(D)$. Furthermore, we determine the Krull dimension, the global dimension, the general linear rank, and the stable rank of $R$.