Derived/spectral algebraic geometry is a relatively new area which features homotopy theory in algebraic geometry. I’ll introduce some of its features and the notions of smooth morphisms in spectral algebraic geometry with a view toward Brauer groups, a subject of my next talk in a week. There are no prerequisites required other than ordinary algebraic geometry, so everyone is welcome to attend.
Grothendieck posed a question of whether the natural map from the Brauer group of a scheme to its cohomological one is an isomorphism of abelian groups. It’s not true in general, but we have some positive results from Grothendieck and Gabber (and de Jong), among many others. After a brief review of Brauer groups in algebraic geometry, I’ll talk about some recent progress in the setting of derived and spectral algebraic geometry, where we can provide an affirmative answer for quasi-compact and quasi-separated (derived/spectral) schemes, and my work which extends the previous results to spectral algebraic stacks.