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 $R$.