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 (X₊⌁) ^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 Br₍ₑ (K (X) ) Br (X₊⌁) ^Gₖ is of finite exponent and complete the p-primary part of the generalization of Artin-Grothendieck's theorem to higher relative dimensions.