Locally finite-dimensional central division algebras over function fields of curves over m-local fields, criterion for normality

Abstract

Let $K_m$ be an $m$-local field with an $m$-th residue field $K_0$, for some integer $m>0$, and let $K/K_m$ be a field extension of transcendence degree $1$. The paper under review shows that if $K_0$ is a field of finite Diophantine dimension ddim$(K_0)$, and $R$ is an associative locally finite-dimensional central division $K$-algebra, then $R$ is a normally locally finite algebra over $K$, that is, every nonempty finite subset $Y$ of $R$ is contained in a finite-dimensional central $K$-subalgebra ${\mathcal{R}}_Y$ of $R$.

10 pages, no figures: title changed, abstract and references updated, the proof of Lemma 2.1 deleted and replaced by a reference, minor improvements in the text