Geodesic
On the Brauer group of an algebraic variety over a finite field
Izvestiya. Mathematics , Tome 69 (2005) no. 2, pp. 331-343

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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 $V$. 
@article{IM2_2005_69_2_a2,
     author = {T. V. Zasorina},
     title = {On the {Brauer} group of an algebraic variety over a finite field},
     journal = {Izvestiya. Mathematics },
     pages = {331--343},
     publisher = {mathdoc},
     volume = {69},
     number = {2},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2005_69_2_a2/}
}
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T. V. Zasorina. On the Brauer group of an algebraic variety over a finite field. Izvestiya. Mathematics , Tome 69 (2005) no. 2, pp. 331-343. http://geodesic.mathdoc.fr/item/IM2_2005_69_2_a2/