Algebraic and Arithmetic Geometry Seminar
# Categorified Beauville-Laszlo theorem (and related problems)

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Europe/Rome

Aula Magna (Department of Mathematics)
### Aula Magna

#### Department of Mathematics

Description

Sheaves of Azumaya algebras were introduced by Grothendieck to represent classes in the cohomological Brauer group of schemes, i.e. $Br(X) := H^2_{ét}(X;G_m)$, along the same lines every class in $H^1_{ét}(X;G_m)$ is representable by a line bundle on X. However, it turns out that not every class in $Br(X)$ can be represented by a sheaf of Azumaya algebras, as shown in the case of Mumford's normal surface. In much more recent times, Toën introduced the notion of sheaf of derived Azumaya algebra, and proved that these objects represent even nontorsion classes in $Br(X)$.

In collaboration with Federico Binda we studied two problems related to derived Azumaya algebras: the Grothendieck existence and the Beauville-Laszlo theorems. In this talk, I will survey both questions and explain how our categorified approach allows to go beyond a classical injectivity result of Grothendieck. I will finish with a brief discussion of the consequences of categorified Beauville-Laszlo that will be the object of a future work.