Genus of division algebras over fields with infinite transcendence degree

TL;DR

This paper proves the finiteness of the genus of finite-dimensional division algebras over certain infinitely generated fields, showing that the genus is trivial for exponent 2 algebras and certain algebraic groups.

Contribution

It establishes the finiteness of the genus for division algebras over fields with infinite transcendence degree and proves triviality of the genus for specific algebraic groups.

Findings

Genus of division algebras over these fields is finite.

Genus of exponent 2 division algebras is trivial.

Genus of G2-type algebraic groups over such fields is trivial.

Abstract

We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let $K$ be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if $D$ is a central division $K$-algebra, then ${\bf gen}(D)$ consists of Brauer classes $[D']$ such that $[D]$ and $[D']$ generate the same subgroup of $Br(K)$. In particular, the genus of any division $K$-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if $char(K) \ne 2$, we prove that the genus of a simple algebraic group of type $\mathrm{G}_2$ over such a field $K$ is trivial.