Homotopy completion and topological Quillen homology of structured ring spectra

Abstract

Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (eg structured ring spectra). We prove a strong convergence theorem that shows that for $0$–connected algebras and modules over a $\left(-1\right)$–connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor.

By systematically exploiting strong convergence, we prove several theorems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topological Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre’s finiteness theorem for spaces and H R Miller’s boundedness result for simplicial commutative rings (but in reverse form). We also prove absolute and relative Hurewicz Theorems and a corresponding Whitehead Theorem for topological Quillen homology. Furthermore, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or $\mathsf{TQ}$–completion). The $\mathsf{TQ}$–completion construction can be thought of as a spectral algebra analog of Sullivan’s localization and completion of spaces, Bousfield and Kan’s completion of spaces with respect to homology and Carlsson’s and Arone and Kankaanrinta’s completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in unbounded chain complexes.