On p-torsions of geometric Brauer groups

Let X be a smooth projective integral variety over a finitely generated field k of characteristic $p>0$. We show that the finiteness of the exponent of the p-primary part of Br(Xkˢ)Gₖ is equivalent to the Tate conjecture for divisors, generalizing D'Addezio's theorem for abelian varieties to arbitrary smooth projective varieties. As a result, we show that the cokernel of Brₙᵣ(K(X)) → Br(Xkˢ)Gₖ is of finite exponent and complete the p-primary part of the generalization of Artin-Grothendieck's theorem to higher relative dimensions.