Involutive Brauer groups and Poincaré rings

Abstract

We use the formalism of Poincaré $\infty$-categories, as developed by Calmès-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle, to define and study moduli stacks of line bundles with $λ$-hermitian pairings and of Morita equivalence classes of Azumaya algebras equipped with an involution. Our moduli spaces give rise to enhancements of the ordinary Picard and Brauer groups which incorporate the data of an involution on the base; we will refer to these new invariants as the Poincaré Picard group and the Poincaré Brauer group. We show that we can recover the involutive Brauer group of Parimala-Srinivas from the Poincaré Brauer group when the former is defined; however, they no longer agree even for closed points due to the existence of shifted perfect pairings. A natural context for Poincaré Picard and Poincaré Brauer groups is a category of highly structured ring spectra which we refer to as Poincaré rings. We also compute these invariants for the sphere spectrum and other examples. As a consequence, we deduce a derived enhancement of a classical theorem of Saltman.