Involutive Brauer groups and Poincaré rings

TL;DR

The paper develops a derived, symmetry-aware framework for Brauer and Picard theory in the presence of a base involution, formulating Poincaré Picard and Poincaré Brauer spaces using the theory of Poincaré ∞-categories. It introduces Poincaré ring spectra and schemes with involution to produce R-linear Poincaré ∞-categories, and defines invertible Poincaré objects and units, connecting them to hermitian forms and shifted dualities. A main achievement is the construction of a Poincaré Brauer space whose π0 encodes involutive Azumaya data via genuine involutions, together with descent results and norm fiber sequences that relate the involutive Brauer group to classical Brauer data and Parimala–Srinivas invariants. The work provides explicit computations for fundamental examples (e.g. spheres and real K-theory), a derived Saltman-type theorem, and a Morita-theoretic perspective on Azumaya algebras with involution, enabling a refined, derived understanding of Brauer groups in the presence of involution.

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.

Theorems & Definitions (285)

• Theorem 1.1.1
• Remark 1.1.2
• Theorem 1.2.1
• Theorem 1.2.2: Theorem \ref{['thm: Fausk for pnpic and schemes with good quotient']}
• Theorem 1.2.3: Corollary \ref{['cor: Brauer classes represented by Azumaya algebras with genuine involution']}
• Remark 1.2.4
• Example 1.2.5: Examples \ref{["ex:br and br' for alg closed field"]} & \ref{['ex:pnbr_closed_point_ramified']}, Remark \ref{['rmk:non_classical_PnBr_classes']}
• Theorem 1.2.6: Theorem \ref{['thm: Satlman theorem in text']}
• Definition 2.0.2: Definition 1.2.1, CDHHLMNNSI
• Definition 2.0.4: Definitions 1.2.2, 1.2.8, CDHHLMNNSI