A purity theorem for vector bundles and Azumaya algebras in dimension 3

Time: 10:00-11:45, 05/03/2026 (Thursday)
Venue: Room 612, A6, Institute of Mathematics-VAST

Online (Join Zoom Meeting) link: https://zoom.us/j/99636681387?pwd=0WscBnehOJig68SqctGluVuA3RwraE.1

Abstract: Azumaya algebras provide a geometric incarnation of the Brauer group and play a central role in purity and descent problems in algebraic geometry. In this talk, we discuss several fundamental results of Gabber concerning the structure and behavior of Azumaya algebras over schemes. Topics include the extension and uniqueness of Azumaya algebras across closed subsets of codimension at least two, finite flat descent for Azumaya algebras, and the relationship between Azumaya algebras and cohomological Brauer groups.

These results form key technical inputs in proofs of purity theorems for the Brauer group, including Gabber’s proof of local purity in low dimensions.

References:

1. [Ga81] O.Gabber, “Some theorems on Azumaya algebras” in The Brauer group (Les Plans-sur-Bex, Switzerland, 1980), Lecture Notes in Math. 844, Springer, Berlin, 1981, 129–209.
2. [Gro68] Grothendieck, “Le groupe de Brauer, I-III: Exemples et compléments” in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968, 88–188.