On Congruence Theorem for valued division algebras

Abstract

Let $K$ be a field equipped with a Henselian valuation, and let $D$ be a tame central division algebra over the field $K$. Denote by ${\mathrm{TK}}_1(D)$ the torsion subgroup of the Whitehead group ${\mathrm{K}}_1(D)=D^{\ast }/D^{\prime }$, where $D^{\ast }$ is the multiplicative group of $D$ and $D^{\prime }$ is its derived subgroup. Let $\mathbf{G}$ be the subgroup of $D^{\ast }$ such that ${\mathrm{TK}}_1(D)=\mathbf{G}/D^{\prime }$. In this note, we prove that either $(1+M_D)\cap \mathbf{G}\subseteq D^{\prime }$, or the residue field $\overline{K}$ has characteristic $p>0$ and the group $\mathbf{H}:=((1+M_D)\cap \mathbf{G})D^{\prime }/D^{\prime }$ is a $p$-group. Additionally, we provide examples of valued division algebras with non-trivial $\mathbf{H}$. This illustrates that, in contrast to the reduced Whitehead group \({\rm SK}_1(D)\), a complete analogue of the Congruence Theorem does not hold for \({\rm TK}_1(D)\).

arXiv commentAccepted for publication in Archiv der Mathematik