Brauer groups and étale cohomology in derived algebraic geometry

By Benjamin Antieau, D. Gepner. 2012

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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 etale local triviality of Azumaya algebras over connective derived schemes and a local to global principle for the algebraicity of stacks of stable categories. 14F22, 18G55; 14D20, 18E30

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0581592096d0eed42a07fd78cd433192c982af07

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Citations in the corpus

Higher traces, noncommutative motives, and the categorified Chern character (2015, background): “Derived Azumaya algebras over X are examples of dualizable categorical sheaves, and this gives rise to a multiplicative map from the algebraic Brauer space Bralg(X) of Antieau–Gepner [AG14] to K (2)(X), analogous to the multiplicative map from the Picard space of line bundles to ordinary K-theory.”
Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map (2011, background): “g which should be regarded as the Brauer group of an E∞ ring spectrum. The connections to the classical deﬁnitions of the Brauer group have been studied by the third author with various collaborators [6, 30]. We do not go into detail about any of the applications of this here, other than to brieﬂy o”
The Galois group of a stable homotopy theory (2014, background): “This would yield a type of descent for the Brauer spectrum of R (see for instance [1]), whose π0 consists of equivalence classes of invertible 2-modules that admit a compact”