## Geometry & Topology

### 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, which solves a conjecture of Baker and Richter, 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.

#### Article information

Source
Geom. Topol., Volume 18, Number 2 (2014), 1149-1244.

Dates
Revised: 15 August 2013
Accepted: 5 October 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732767

Digital Object Identifier
doi:10.2140/gt.2014.18.1149

Mathematical Reviews number (MathSciNet)
MR3190610

Zentralblatt MATH identifier
1308.14021

#### Citation

Antieau, Benjamin; Gepner, David. Brauer groups and étale cohomology in derived algebraic geometry. Geom. Topol. 18 (2014), no. 2, 1149--1244. doi:10.2140/gt.2014.18.1149. https://projecteuclid.org/euclid.gt/1513732767

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