Commutative ring objects in pro-categories and generalized Moore spectra

TL;DR

This paper introduces a rigidity criterion for lifting operad structures in symmetric monoidal model categories, demonstrating that certain towers of generalized Moore spectra are E-infinity algebras in pro-spectra, with applications to Adams resolutions.

Contribution

It develops a new rigidity criterion for operad lifting and applies it to prove towers of Moore spectra are E-infinity algebras in pro-spectra, extending understanding of structured ring spectra.

Findings

Towers of generalized Moore spectra are E-infinity algebras in pro-spectra.

Operadic model categories have homotopically tractable endomorphism operads.

Adams resolutions satisfy the rigidity criterion for operad lifting.

Abstract

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 This is applied to obtain an unpublished result of M. J. Hopkins that certain towers of generalized Moore spectra, closely related to the K(n)-local sphere, are E-infinity algebras in the category of pro-spectra. In addition, we show that Adams resolutions automatically satisfy the above rigidity criterion. In order to carry this out we develop the concept of an operadic model category, whose objects have homotopically tractable endomorphism operads.