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.