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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 Segals 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 t
In this paper we develop methods for classifying Baker-Richter-Szymiks Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing a
We develop a rigidity criterion to show that in simplicial model categories with a compatible symmetric monoidal structure, operad structures can be automatically lifted along certain maps. This is applied to obtain an unpublished result of M. J. Hop
We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we asso
We describe a special instance of the Goerss-Hopkins obstruction theory, due to Senger, for calculating the moduli of $E_infty$ ring spectra with given mod-$p$ homology. In particular, for the $2$-primary Brown-Peterson spectrum we give a chain compl
Ramification for commutative ring spectra can be detected by relative topological Hochschild homology and by topological Andre-Quillen homology. In the classical algebraic context it is important to distinguish between tame and wild ramification. Noe