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

TL;DR

This paper explores how the derived circle product with right Com-modules can be used to construct and analyze commutative ring spectra, including filtrations and comparisons of different approaches.

Contribution

It introduces a unified framework for constructions on Com-algebras using right Com-modules and demonstrates how filtrations can be applied to these constructions.

Findings

Filtrations of right Com-modules lead to natural filtrations of commutative ring spectra.

The augmentation ideal tower of I can be constructed via a natural decreasing filtration.

The paper proves the equivalence of two existing filtrations of TQ(I).

Abstract

If Com is the reduced commutative operad, the category of Com-algebras in spectra is the category of nounital 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. Examples of constructions arising this way include the tensor product of a based space K with such an algebra I, and TQ(I), the Topological Andre-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, when specialized to the Com-bimodule Com, defines the augmentation ideal tower of I, built out of the extended powers of TQ(I). A natural increasing filtration on right Com-modules, when specialized to the right Com-module used to define TQ(I), defines 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 -- by the author and by Behrens and Rezk -- 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.