Description:

Title: Involutive Brauer groups and Poincare infinity categories

Abstract: An Azumaya algebra with anti-involution is an Azumaya algebra $A$ together with an equivalence $\sigma:A\to A^{\mathrm{op}}$ such that $\sigma^{\mathrm{op}}\circ \sigma=\mathrm{id}_A$. One (but certainly not the only) reason to care about such objects is that they are very closely connected to simply connected group schemes, originally due to work of Weil and since extended by several authors. Partially motivated by this, Parimala and Srinivas constructed involutive Brauer groups, a version of the Brauer group for Azumaya algebras with anti-involution, in the cases when the induced $C_2$ action on the center is trivial or Galois. A consequence of the existance of this invariant is a new proof of a theorem of Saltman.

In this talk, I will explain how in joint work with Burghardt and Yang, we use Poincare infinity categories and ideas of Toen and Antieau-Gepner to construct invariants, called the Poincare Picard and Poincare Brauer spaces, of rings which captures a derived version of the above story. I will explain how our invariant recovers that of Parimala and Srinivas when the latter is defined, and how the existance of our invariant leads to a proof of a spectral version of Saltman's theorem.