Volume 18, issue 2 (2014)

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Brauer groups and étale cohomology in derived algebraic geometry

Benjamin Antieau and David Gepner

Geometry & Topology 18 (2014) 1149–1244
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.

Keywords
commutative ring spectra, derived algebraic geometry, moduli spaces, Azumaya algebras, Brauer groups
Mathematical Subject Classification 2010
Primary: 14F22, 18G55
Secondary: 14D20, 18E30
Publication
Revised: 15 August 2013
Accepted: 5 October 2013
Published: 7 April 2014
Proposed: Richard Thomas
Seconded: Ralph Cohen, Bill Dwyer
Authors
 Benjamin Antieau Department of Mathematics University of Washington Box 354350 Seattle, WA 98195 USA David Gepner Department of Mathematics Purdue University 150 N University Street West Lafayette, IN 47907 USA