Separable algebras and Brauer groups

Separable algebras are a framework that classically encompasses both étale algebras and Azumaya algebras. Etale algebras lay as the foundation of the étale topology (as the name suggests), while Azumaya algebras have to do with the Brauer group, which in turn is closely related to étale cohomology. In my work, I study the homotopical version of these objects, first introduced in homotopy theory by Balmer. I am also interested in generalizations of the Brauer group in homotopy theory that do not necessarily have to do with Azumaya algebras.

• With Niko Naumann and Luca Pol, we have studied the Galois and finite étale theory of the category of genuine G-spectra, where G is a finite group, in Separable commutative algebras in equivariant homotopy theory and Galois theory of G-spectra and the Burnside ring.
• In \(K(1)\)-local K-theory of Azumaya algebras and Localizing motives of Azumaya algebras, I studied how much information K-theory and other related invariants remember about an Azumaya algebra (specifically, whether, and how, they remember its Brauer class).
• Separability in homotopical algebra is my first paper on the subject, where I study the notion of separability in homotopical algebra, and I prove in various ways that it is particularly well behaved.