Genus of division algebras over fields with infinite transcendence degree

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.