Involutions and the Brauer group in derived algebraic geometry

Classical results of Albert and Saltman (extended by Knus–Parimala–Srinivas) connect the existence of involutions on Azumaya algebras to 2-torsion in the Brauer group. We introduce involutive versions of the derived Picard and Brauer groups of Toën and Antieau—Gepner and relate them to their classical and non-involutive counterparts. Our invariant can be used to recover the involutive Brauer group of Parimala--Srinivas when the latter is defined; however, they no longer agree even for closed points due to the existence of shifted perfect pairings. We compute these invariants for the sphere spectrum and other examples. As a consequence, we deduce a derived enhancement of an exact sequence due to Parimala--Srinivas. To carry this out, we investigate how the data of an involution on A is reflected in additional structure on its (derived) category of modules and use the theory of Poincaré ∞-categories developed by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle.  This is joint work with Viktor Burghardt and Noah Riggenbach.