Talk
Searching for the impossible Azumaya algebra
Talk by Dr. Siddharth Mathur
Speaker invited by: Prof. Dr. Andrew Kresch
Date: 25.04.22 Time: 13.15 - 14.45 Room: Y27H25
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. To attend remotely: https://seminarlive.mnf.uzh.ch/semlive/index.php?id=module&module=AGOS&semester=fs22