The Brauer Group of a Surface over a Finite Field

Abstract: This is an English translation of the author's 1989 note in Russian, published in a collection "Arithmetic and Geometry of Varieties" (V.E. Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67. Let $X$ be be an absolutely irreducible smooth projective surface over a finite field $k$ of odd characteristic, let $Br(X)$ be the (commutative periodic) Brauer group of $X$ and $DIV Br(X)$ the subgroup of its divisible elements. We write $Br(X){DIV}$ for the quotient $Br(X)/DIV Br(X)$ and $Br(X){DIV}(2)$ for its (finite) $2$-primary component. We prove that the order of $Br(X)_{DIV}(2)$ is a full square under the following additional assumptions on $\bar{X}=X\times \bar{k}$ where $ \bar{k}$ is an algebraic closure of $k$. There is no 2-torsion in the N\'eron-Severi group of $\bar{X}$. The surface $\bar{X}$ admits a lifting to characteristic 0. The proof is based on constructions of author's paper (Math. USSR Izv. 20 (1983), 203-234) and Wu's Theorem that relates Stiefel-Whitney classes and Steenrod squares.