Brauer spaces for commutative rings and structured ring spectra

TL;DR

This paper develops a homotopy-theoretic framework for Brauer groups and spectra in algebra and topology, connecting algebraic K-theory with structured ring spectra to deepen understanding of their algebraic and topological properties.

Contribution

It introduces a novel definition of Brauer spectra for commutative rings using algebraic K-theory and extends this to structured ring spectra, providing new insights into their homotopy-theoretic structure.

Findings

Defined Brauer spectra with homotopy groups matching classical invariants

Established two-fold non-connected deloopings of spectra of units

Linked algebraic and topological concepts through homotopy-theoretic methods

Abstract

Using an analogy between the Brauer groups in algebra and the Whitehead groups in topology, we first use methods of algebraic K-theory to give a natural definition of Brauer spectra for commutative rings, such that their homotopy groups are given by the Brauer group, the Picard group and the group of units. Then, in the context of structured ring spectra, the same idea leads to two-fold non-connected deloopings of the spectra of units.