On function fields of curves over higher local fields and their division LFD-algebras

TL;DR

This paper investigates the properties of function fields over higher local fields, establishing conditions under which their Brauer p-dimension is finite and analyzing division algebras over these fields.

Contribution

It demonstrates the finiteness of the Brauer p-dimension for certain function fields over higher local fields with finite Diophantine dimension, and characterizes locally finite-dimensional division algebras over these fields.

Findings

Finite Brauer p-dimension for specified function fields.

Locally finite-dimensional division algebras are normally locally finite.

Conditions linking higher local fields and division algebra properties.

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 trd$(K/K _{m}) \le 1$. This paper shows that if $K _{0}$ is a field of finite Diophantine dimension (for example, a finitely-generated extension of a finite or a pseudo-algebraically closed perfect field $E$), then the absolute Brauer $p$-dimension abrd$_{p}(K)$ of $K$ is finite, for every prime number $p$. Thus it turns out that if $R$ is an associative locally finite-dimensional (abbr., LFD) central division $K$-algebra, then it 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 $R$.