Scuola Normale Superiore, Aula Volterra.<\/p>\n
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<\/span><\/span> 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<\/span><\/span> algebras!<\/p>\n In this talk, I will outline a strategy to construct a Brauer class that cannot be represented by an Azumaya<\/span><\/span> algebra<\/span><\/span>. 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<\/span><\/span> algebra<\/span><\/span>. At the end, I will reveal the unexpected conclusion of the experiment.<\/span><\/p>\n Further information is available on the event page<\/a> on the Indico platform.<\/p>\n","protected":false},"excerpt":{"rendered":" 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…<\/p>\n