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Homotopy pro-nilpotent structured ring spectra and topological Quillen localization

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Abstract

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.

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Acknowledgements

The author would like to thank John E. Harper and Niko Schonsheck for inspiring discussions and helpful suggestions. The author would like to thank Michael Ching, Martin Frankland, Mark W. Johnson and Jérôme Scherer for helpful conversations. The author is grateful to Oscar Randal-Williams for detailed and helpful critical comments on an early draft of this paper. The author would like to thank the anonymous referee for detailed suggestions. The author was supported in part by the Simons Foundation: Collaboration Grants for Mathematicians #638247, and by the National Natural Science Foundation of China No. 11871284; 12001474; 12261091; 12271183.

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  1. Department of Mathematics, Nankai University, 94 Weijin Road, Nankai District, Tianjin, 300071, People’s Republic of China

    Yu Zhang

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Zhang, Y. Homotopy pro-nilpotent structured ring spectra and topological Quillen localization. J. Homotopy Relat. Struct. 17, 511–523 (2022). https://doi.org/10.1007/s40062-022-00316-9

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