Student Arithmetic Geometry Seminar: Purity for the Brauer group

Seminar | April 6 | 4:10-5 p.m. | 891 Evans Hall

 Minseon Shin, UCB

 Department of Mathematics

I will discuss Kestutis Cesnavicius' recent preprint in which he proves a purity conjecture due to Grothendieck and Auslander–Goldman, which predicts that if $X$ is a regular Noetherian scheme and $Z \subseteq X$ is a closed subscheme of codimension $\ge 2$, then the restriction map on the cohomological Brauer groups $H^2_{\operatorname {\text {ét}}}(X , \mathbb G_m) \to H^2_{\text {ét}}(X \setminus Z , \mathbb G_m)$ is an isomorphism. The combination of several works by Gabber, including his proof of the absolute purity conjecture, settles the equi-characteristic case; Cesnavicius treats the mixed characteristic case using the tilting equivalence for perfectoid rings.