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 $\mathrm{Br}\left(X\right)\left[p^{\infty }\right]$ is a direct sum of finitely many copies of $Q_p/Z_p$ and $H^3(X,Z_p(1\left)\right)\left[p^{\infty }\right]$, 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 $\mathrm{NS}\left(X\right)\left[p^{\infty }\right]$; (3) if X is an abelian variety, then $J=0$. Using Crew's formula and Ekedahl's inequality, we compute the dimension of U for surfaces and for abelian 3-folds. We show that, if X is ordinary, then the unipotent subgroup of $\mathrm{Br}(X\times Y)$ is isomorphic to the unipotent subgroup of $\mathrm{Br}\left(Y\right)$. Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree 2.