On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions

TL;DR

This paper investigates how the Brauer dimension behaves over finitely-generated field extensions, showing it can become infinite and constructing specific algebras with prescribed index and exponent, thereby answering a longstanding open problem.

Contribution

It establishes that the Brauer dimension can be infinite over certain extensions and constructs algebras with specific index-exponent pairs, solving a problem posed by Auel et al.

Findings

Brauer dimension becomes infinite when absolute Brauer dimension is infinite.

Existence of central division algebras with prescribed index and exponent over certain fields.

Provides lower bounds on Brauer p-dimension in specific cases.

Abstract

Let $E$ be a field of absolute Brauer dimension abrd$(E)$, and $F/E$ a transcendental finitely-generated extension. This paper shows that the Brauer dimension Brd$(F)$ is infinite, if abrd$(E) = \infty $. When the absolute Brauer $p$-dimension abrd$_{p}(E)$ is infinite, for some prime number $p$, it proves that for each pair $(n, m)$ of integers with $n \ge m > 0$, there is a central division $F$-algebra of Schur index $p ^{n}$ and exponent $p ^{m}$. Lower bounds on the Brauer $p$-dimension Brd$_{p}(F)$ are obtained in some important special cases where abrd$_{p}(E) < \infty $. These results solve negatively a problem posed by Auel et al. (Transf. Groups {\bf 16}: 219-264, 2011).