Journal of Homotopy and Related Structures

, Volume 9, Issue 2, pp 465–493 | Cite as

Azumaya objects in triangulated bicategories

Article

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.

Keywords

Brauer group Ring spectrum Homotopical algebra 

Mathematics Subject Classification (2000)

55U99 18D35 16K50 14F22 

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State University NewarkNewarkUSA

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