On geometric Brauer groups and Tate-Shafarevich groups

TL;DR

This paper establishes that the finiteness of the $ ext{ell}$-primary part of the geometric Brauer group implies the finiteness of the prime-to-$p$ part, extending classical theorems to varieties over finitely generated fields of positive characteristic.

Contribution

It generalizes Tate and Lichtenbaum's theorem to finitely generated fields of characteristic $p>0$, linking the finiteness of different parts of the Brauer group and Tate-Shafarevich groups.

Findings

Finiteness of $ ext{ell}$-primary Brauer group implies finiteness of prime-to-$p$ part.

Generalization of Tate and Lichtenbaum's theorem to positive characteristic fields.

Extension of Schneider's results to finitely generated fields.

Abstract

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.