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 p>0 . We show that the finiteness of the exponent of the 𝑝-primary part of Br ⁢ ( X k s ) G k Br(X_{{kˢ}})^{{Gₖ}} is equivalent to the Tate conjecture for divisors, generalizing D’Addezio’s theorem for abelian varieties to arbitrary smooth projective varieties. In combination with the Leray spectral sequence for rigid cohomology derived from the Berthelot conjecture recently proved by Ertl–Vezzani, we show that the cokernel of Br nr ⁢ ( K ⁢ ( X ) ) → Br ⁢ ( X k s ) G k Brₙᵣ(K(X))(X_{{kˢ}})^{{Gₖ}} is of finite exponent. This completes the 𝑝-primary part of the generalization of Artin–Grothendieck’s theorem on relations between Brauer groups and Tate–Shafarevich groups to higher relative dimensions.