On geometric Brauer groups and Tate-Shafarevich groups

Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\mathrm{\ell }$-primary part of $\mathrm{B}\mathrm{r}(X_{K^s}{)}^{G_K}$ for a single prime $\mathrm{\ell }\mathrm{\ne }p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{B}\mathrm{r}(X_{K^s}{)}^{G_K}$, 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.