Séminaire Arithmétique et géométrie algébrique

organisé par l'équipe Arithmétique et géométrie algébrique

Qin Yanshuai

oundedness of the p-primary torsion of geometric Brauer groups

19 décembre 2024 - 14:00Salle de séminaires IRMA

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 $Br(X_{k^s})^{G_k}$ 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 $Br_{nr}(K(X)) \rightarrow Br(X_{k^s})^{G_k}$ has 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. This is joint with Zhenghui Li