Brauer groups and étale cohomology in derived algebraic geometry

Abstract

In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the étale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories.

71 pages; submitted; comments welcomed

References in corpus (2)

• Higher Topos Theory
• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map

Cited by in corpus (22)

• Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map
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• The higher Morita category of $E_n$-algebras
• Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
• Duality and Tilting for Commutative DG Rings
• Brauer groups and Galois cohomology of commutative ring spectra
• Twisted differential cohomology
• The Morel-Voevodsky localization theorem in spectral algebraic geometry
• Kaledin's degeneration theorem and topological Hochschild homology
• Twisted iterated algebraic K-theory and topological T-duality for sphere bundles
• A local-global principle for the telescope conjecture
• Counterexamples to Hochschild--Kostant--Rosenberg in characteristic $p$
• A^1-homotopy invariance in spectral algebraic geometry
• K-theory of endomorphisms via noncommutative motives
• Weibel's conjecture for twisted K-theory
• Adjoining roots in homotopy theory
• Commutative ring spectra
• A categorification of the quantum Lefschetz principle
• Sur la catégorie dérivée des faisceaux tordus
• The relative Brauer group of $K(1)$-local spectra
• Twisted Spectra Revisited
• Algebraic K-theory of finitely projective modules on ${\mathbb{E}}_{\infty }$-rings