Involutions and the Brauer group in derived algebraic geometry

Classical results of Albert and Saltman (extended by Knus–Parimala–Srinivas) have established a connection between the existence of (anti-)involutions on the central simple algebras A used to define the Brauer group and 2-torsion Brauer classes. Moreover, the presence of more general forms of involutions is related to the behavior of Brauer classes under corestriction along quadratic extensions. In this work, we introduce and study a generalization of these ideas to derived algebraic geometry. We investigate how the data of an involution on A  is reflected in additional structure on its category of modules. Using the theory of Poincaré \infty-categories developed by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, we introduce involutive versions of the Picard and Brauer group and relate them to their non-involutive counterparts.