  Let $X$ be a smooth projective variety over a finitely generated field $K$ of
characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part
of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the
finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$,
generalizing a theorem of Tate and Lichtenbaum for varieties over finite
fields. For an abelian variety $A$ over $K$, we proved a similar result for the
Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian
varieties over global function fields.
