Azumaya Objects in Triangulated Bicategories

Details Azumaya Objects in Triangulated Bicategories Azumaya Objects in Triangulated Bicategories

2010-05-26

arxiv.org/abs/1005.4878v5

Mathematics

Algebraic Topology

18 pages; expanded main results and reorganized

Scientific paper

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.

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