On p-torsions of geometric Brauer groups

TL;DR

This paper explores the relationship between the finiteness of p-primary parts of geometric Brauer groups and the Tate conjecture, extending classical theorems to higher-dimensional varieties over fields of positive characteristic.

Contribution

It generalizes D'Addezio's theorem from abelian varieties to all smooth projective varieties and links the finiteness of Brauer group parts to the Tate conjecture in positive characteristic.

Findings

Finiteness of p-primary Brauer group parts is equivalent to the Tate conjecture for divisors.

The cokernel of the Brauer group map has finite exponent.

Completes the p-primary generalization of Artin-Grothendieck's theorem.

Abstract

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate conjecture for divisors, generalizing D'Addezio's theorem for abelian varieties to arbitrary smooth projective varieties. In combination with the Leray spectral sequence for rigid cohomology derived from the Berthelot conjecture recently proved by Ertl-Vezzani, we show that the cokernel of $\mathrm{Br}_{\mathrm{nr}}(K(X)) \rightarrow \mathrm{Br}(X_{k^s})^{G_k}$ is of finite exponent. This completes the $p$-primary part of the generalization of Artin-Grothendieck's theorem on relations between Brauer groups and Tate-Shafarevich groups to higher relative dimensions.