Brauer groups for commutative $S$-algebras

TL;DR

This paper develops a theory of Brauer groups for structured ring spectra, exploring Azumaya algebras, Galois properties, and examples related to topological Hochschild cohomology and local variants.

Contribution

It introduces a new framework for Brauer groups in the context of commutative S-algebras, including definitions, properties, and examples involving Galois descent and topological Hochschild cohomology.

Findings

Defined Brauer groups for structured ring spectra

Analyzed Galois theoretic properties of Azumaya algebras

Constructed examples related to topological Hochschild cohomology

Abstract

We investigate a notion of Azumaya algebras in the context of structured ring spectra and give a definition of Brauer groups. We investigate their Galois theoretic properties, and discuss examples of Azumaya algebras arising from Galois descent and cyclic algebras. We construct examples that are related to topological Hochschild cohomology of group ring spectra and we present a K(n)-local variant of the notion of Brauer groups.