Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions

Abstract

Let $E$ be a field, $p$ a prime number and $F/E$ a finitely-generated extension of transcendency degree $t$. This paper shows that if the absolute Galois group $\mathcal{G}_{E}$ is of nonzero cohomological $p$-dimension cd$_{p}(E)$, then the field $F$ has Brauer $p$-dimension Brd$_{p}(F) \ge t$ except, possibly, in case $p = 2$, the Sylow pro-2-subgroups of $\mathcal{G}_{E}$ are of order 2, and $F$ is a nonreal field. It announces that Brd$_{p}(F)$ is infinite whenever $t \ge 1$ and the absolute Brauer $p$-dimension abrd$_{p}(E)$ is infinite; moreover, for each pair $(m, n)$ of integers with $1 \le m \le n$, there exists a central division $F$-algebra of exponent $p ^{m}$ and Schur index $p ^{n}$.