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Siddharth Mathur: Searching for the impossible Azumaya algebra

Tid: Ti 2021-09-21 kl 13.15 - 14.15

Plats: Institut Mittag-Leffler, Seminar Hall Kuskvillan (alt. Zoom, meeting ID: 921 756 1880)

Föreläsare: Siddharth Mathur (Institut Mittag-Leffler)

Abstract: In two 1968 seminars, Grothendieck used the framework of étale 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.