Searching for the impossible Azumaya algebra

Moduli and Algebraic Cycles

21 September 13:15 - 14:15

Siddharth Mathur - Pontificia Universidad Católica de Chile

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 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 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.

Click here to watch the seminar

John Christian Ottem
University of Oslo
Dan Petersen
Stockholm University
David Rydh
KTH Royal Institute of Technology


Dan Petersen


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