Geometry Seminar

University of Tor Vergata, Department of Mathematics

11th of October 2022, 14:30-15:30, room “Dal Passo”

Speaker

Sid Mathur

(University of Orsay)

Title

Searching for the impossible Azumaya algebra

Abstract

In two 1968 seminars, Grothendieck used the framework of etale cohomology to extend the definition of the Brauer group to all schemes. Over a field, the objects admit a well-known algebro-geometric description: they are represented by \mathbb{P}^n-bundles (equivalently: Azumaya Algebras). Despite the utility and success of Grothendieck's definition, an important foundational aspect remains open: is every cohomological Brauer class over a scheme represented by a \mathbb{P}^n-bundle? It is not even known if smooth proper threefolds over the complex numbers have enough Azumaya algebras!

In this talk, I will outline a strategy to construct a Brauer class that cannot be represented by an Azumaya algebra. Although the candidate is algebraic, the method will leave the category of schemes and use formal-analytic line bundles to create Brauer classes. I will then explain a strange criterion for the existence of a corresponding Azumaya Algebra. At the end, I will reveal the unexpected conclusion of the experiment.