Commutative ring spectra

TL;DR

This survey explores the fundamental aspects, constructions, and homological theories of commutative ring spectra, highlighting their examples, structures, and obstructions in algebraic topology.

Contribution

It provides a comprehensive overview of commutative ring spectra, including new constructions, homology theories, and the analysis of their algebraic and geometric properties.

Findings

Presented two constructions for commutative ring spectra: Thom spectra and Segal's construction.

Defined and discussed topological Hochschild and Andre-Quillen homology for ring spectra.

Explored obstruction theory, etale extensions, and computed Picard and Brauer groups for commutative ring spectra.

Abstract

In this survey paper on commutative ring spectra we present some basic features of commutative ring spectra and discuss model category structures. As a first interesting class of examples of such ring spectra we focus on (commutative) algebra spectra over commutative Eilenberg-MacLane ring spectra. We present two constructions that yield commutative ring spectra: Thom spectra associated to infinite loop maps and Segal's construction starting with bipermutative categories. We define topological Hochschild homology, some of its variants, and topological Andre-Quillen homology. Obstruction theory for commutative structures on ring spectra is described in two versions. The notion of etale extensions in the spectral world is tricky and we explain why. We define Picard groups and Brauer groups of commutative ring spectra and present examples.