On the Brauer group of an algebraic variety over a finite field

For an arithmetic model $X\to C$ of a smooth regular projective variety $V$ over a global field $k$ of positive characteristic, we prove the finiteness of the $l$-primary component of the group $\operatorname{Br}'(X)$ under the conditions that $l$ does not divide the order of the torsion group $\bigl[\operatorname{NS}(V)\bigr]_{\text{tors}}$ and the Tate conjecture on divisorial cohomology classes is true for $V$.