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

For a smooth and proper variety $X$ over an algebraically closed field $k$ of characteristic $p>0$, the group $Br(X)[p^{\mathrm{\infty }}]$ is a direct sum of finitely many copies of ${\mathbb{Q}}_p\mathrm{/}{\mathbb{Z}}_p$ 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 $NS(X)[p^{\mathrm{\infty }}]$; (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 $2$.