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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
Received: 12 December 2012
Revised: 15 August 2013
Accepted: 5 October 2013
Published: 7 April 2014
Proposed: Richard Thomas
Seconded: Ralph Cohen, Bill Dwyer