Brauer groups and étale cohomology in derived algebraic geometry
Copyright policy )Brauer groups and étale cohomology in derived algebraic geometry
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.
Comment: 71 pages; submitted; comments welcomed
- University of Washington United States
- University of Chicago United States
- Purdue University West Lafayette United States
- Purdue University System United States
- University of Mary United States
14F22, commutative ring spectra, Azumaya algebras, 18E30, Nonabelian homotopical algebra, Mathematics - Rings and Algebras, 18G55, 14D20, Derived categories, triangulated categories, Mathematics - Algebraic Geometry, Algebraic moduli problems, moduli of vector bundles, derived algebraic geometry, moduli spaces, Mathematics - Algebraic Topology, 14F22, 18G55, Brauer groups of schemes, Brauer groups
14F22, commutative ring spectra, Azumaya algebras, 18E30, Nonabelian homotopical algebra, Mathematics - Rings and Algebras, 18G55, 14D20, Derived categories, triangulated categories, Mathematics - Algebraic Geometry, Algebraic moduli problems, moduli of vector bundles, derived algebraic geometry, moduli spaces, Mathematics - Algebraic Topology, 14F22, 18G55, Brauer groups of schemes, Brauer groups
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