Azumaya Objects in Triangulated Bicategories

TL;DR

This paper generalizes the concept of Azumaya objects and Brauer groups within homotopy-theoretic and bicategorical frameworks, providing new tools for understanding derived categories and ring spectra.

Contribution

It introduces Azumaya objects in bicategories, characterizes them in triangulated bicategories, and applies this to define a homotopical Brauer group for derived categories and ring spectra.

Findings

Homotopical Brauer group of Eilenberg-Mac Lane spectrum equals that of the underlying ring

Characterization theorem for Azumaya objects in triangulated bicategories

Application to tilting theory in derived categories

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.