Remarks on $p$-primary torsion of the Brauer group

TL;DR

This paper investigates the structure of the $p$-primary torsion part of the Brauer group for varieties over algebraically closed fields of characteristic $p$, revealing conditions under which certain unipotent components vanish or relate to other invariants.

Contribution

It characterizes the $p$-primary torsion of the Brauer group for various classes of varieties and computes the dimension of unipotent parts, extending previous results and providing new criteria for cohomological injectivity.

Findings

For ordinary varieties, the unipotent part of the Brauer group vanishes.

For surfaces, the finite part of the Brauer group relates to the Néron-Severi group.

For abelian varieties, the finite part of the Brauer group is trivial.

Abstract

For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^\infty]$ is a direct sum of finitely many copies of $\mathbb{Q}_p/\mathbb{Z}_p$ and an abelian group of finite exponent. The latter is an extension of a finite group $J$ by the group of $k$-points of a connected commutative unipotent algebraic group $U$. In this paper we show that (1) if $X$ is ordinary, then $U = 0$; (2) if $X$ is a surface, then $J$ is the Pontryagin dual of $NS(X)[p^\infty]$; (3) if $X$ is an abelian variety, then $J = 0$. Using Crew's formula, we compute the dimension of $U$ for surfaces and abelian $3$-folds. We show that, if $X$ is ordinary, then the unipotent subgroup of $Br(X\times Y)$ is isomorphic to the unipotent subgroup of $Br(Y)$. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree $2$.