Brauer groups and \'etale cohomology in derived algebraic geometry

TL;DR

This paper explores Brauer groups and Azumaya algebras within derived algebraic geometry, providing new computational tools, fundamental results, and applications such as the vanishing of the Brauer group of the sphere spectrum and uniqueness theorems in stable homotopy theory.

Contribution

It introduces new foundational results on Brauer groups in derived settings, including local geometricity, étale local triviality, and a local-global principle, along with computational methods and applications.

Findings

Brauer group of the sphere spectrum vanishes

Established local geometricity of moduli spaces of perfect modules

Proved étale local triviality of Azumaya algebras

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 \'etale local triviality of Azumaya algebras over connective derived schemes, and a local to global principle for the algebraicity of stacks of stable categories.