A nilpotent Whitehead theorem for TQ-homology of structured ring spectra

TL;DR

This paper establishes a TQ-Whitehead theorem for nilpotent structured ring spectra, extending classical Whitehead results to the realm of algebraic topology and structured ring spectra using TQ-homology.

Contribution

It proves a TQ-Whitehead theorem for nilpotent structured ring spectra and introduces retract theorems for TQ-completion and homotopy completion.

Findings

Proved a TQ-Whitehead theorem for nilpotent structured ring spectra.

Established retract theorems for TQ-completion and homotopy completion.

Extended classical Whitehead results to structured ring spectra context.

Abstract

The aim of this short paper is to prove a TQ-Whitehead theorem for nilpotent structured ring spectra. We work in the framework of symmetric spectra and algebras over operads in modules over a commutative ring spectrum. Our main result can be thought of as a TQ-homology analog for structured ring spectra of Dror's generalized Whitehead theorem for topological spaces; here TQ-homology is short for topological Quillen homology. We also prove retract theorems for the TQ-completion and homotopy completion of nilpotent structured ring spectra.