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.

本文的目的是证明同伦亲幂零结构环谱是\({ \mathsf {TQ} }\) -局部的，其中结构环谱被描述为谱运算\({ \mathcal {O } }\)。这里，\({ \mathsf {TQ} }\)是拓扑 Quillen 同调的缩写，它弱等价于\({ \mathcal {O} }\) -代数稳定。一个\({ \mathcal {O} }\) -代数称为同伦 pro-nilpotent 如果它等价于幂零极限\({ \mathcal {O} }\)-代数。我们的结果为弗朗西斯-盖斯戈里关于一般歌剧的科祖尔对偶性的猜想提供了新的积极证据。作为一个应用程序，我们同时将先前已知的 0 连通和幂零\({ \mathsf {TQ} }\) -Whitehead 定理扩展为同伦 pro-nilpotent \({ \mathsf {TQ} }\) -Whitehead 定理。