Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

The aim of this paper is to show that homotopy pro-nilpotent structured ring spectra are \({ \mathsf {TQ} }\)-local, where structured ring spectra are described as algebras over a spectral operad \({ \mathcal {O} }\). Here, \({ \mathsf {TQ} }\) is short for topological Quillen homology, which is weakly equivalent to \({ \mathcal {O} }\)-algebra stabilization. An \({ \mathcal {O} }\)-algebra is called homotopy pro-nilpotent if it is equivalent to a limit of nilpotent \({ \mathcal {O} }\)-algebras. Our result provides new positive evidence to a conjecture by Francis–Gaisgory on Koszul duality for general operads. As an application, we simultaneously extend the previously known 0-connected and nilpotent \({ \mathsf {TQ} }\)-Whitehead theorems to a homotopy pro-nilpotent \({ \mathsf {TQ} }\)-Whitehead theorem.