Title: Searching for the impossible Azumaya algebra, Speaker(s): Siddharth Mathur, Orsay, Date and time: 25 May 2022, 14:30 (Europe/Rome), Lecture series: Algebraic and Arithmetic Geometry Seminar, Venue: Scuola Normale Superiore (Aula Volterra).
You can access the full event here: https://events.dm.unipi.it/e/72
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.
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