Applications of the circle product with a right Com-module to the theory of commutative ring spectra.

If Com is the reduced commutative operad, the category of Com-algebras in spectra is the category of non-unital commutative ring spectra. The theme of this survey is that many important constructions on Com-algebras are given by taking the derived circle product with well-chosen right Com-modules. We show that examples of constructions arising this way include K⊗I, the tensor product of a based space K with such an algebra I, and TQ(I), the Topological André-Quillen homology spectrum of I. We then show how filtrations of right Com-modules can be used to filter such constructions. A natural decreasing filtration on right Com-modules is described. When specialized to the Com-bimodule Com, this defines the augmentation ideal tower of I, built out of the extended powers of TQ(I). A less studied natural increasing filtration is described. When specialized to the right Com-module used to define TQ(I), one gets a filtration on TQ(I) built out of I and the spaces in the Lie cooperad. There are two versions of this in the literature, and our setting here makes it easy to prove that these agree. Much of this applies with Com replaced by a more general reduced operad, and we make a few remarks about this. This article is part of the theme issue 'Derived Lie algebras in geometry and topology'.