On triviality of reduced Whitehead group over Henselian fields

TL;DR

This paper proves that under certain conditions on a Henselian field and a central division algebra over it, the reduced Whitehead group is trivial, extending understanding of algebraic K-theory in valued fields.

Contribution

It establishes the triviality of the reduced Whitehead group for division algebras over Henselian fields with specific value group dimensions.

Findings

Reduced Whitehead group is trivial when the value group dimension is between 1 and 3.

Results apply to division algebras of index a power of q over Henselian fields.

Extends known cases of triviality in algebraic K-theory for valued fields.

Abstract

Let $F$ be a Henselian field of $q$-cohomological dimension $3$, where $q$ is a prime. Let $\Gamma_F$ be the totally ordered abelian value group of $F$ and let $D$ be a central division algebra over $F$ of index a power of $q$ such that the characteristic of the residue field, $\overline{F}$ is coprime to $q$. We show that when $1\leq\dim_{\mathbb{F}_q}(\Gamma_F/q\Gamma_F)\leq 3$, the reduced Whitehead group of $D$ is trivial.