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Searching for the impossible Azumaya Algebra

Siddharth Mathur, Arizona/Düsseldorf
Tuesday, May 19, 2020 - 11:00am

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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