Azumaya Objects in Triangulated Bicategories

Abstract

We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutative ring. We also discuss tilting theory as an application of invertibility in triangulated bicategories.

23 pages; final version; to appear in Journal of Homotopy and Related Structures

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Cited by in corpus (6)

• Brauer groups and étale cohomology in derived algebraic geometry
• Invertible braided tensor categories
• Brauer groups and Galois cohomology of commutative ring spectra
• Brauer groups for commutative $S$-algebras
• Derived Azumaya algebras and generators for twisted derived categories
• Commutative ring spectra