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Remarks on K(1)-local K-theory

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Abstract

We prove two basic structural properties of the algebraic K-theory of rings after K(1)-localization at an implicit prime p. Our first result (also recently obtained by Land–Meier–Tamme by different methods) states that \(L_{K(1)} K(R)\) is insensitive to inverting p on R; we deduce this from recent advances in prismatic cohomology and \(\mathrm {TC}\). Our second result yields a Künneth formula in K(1)-local K-theory for adding p-power roots of unity to R.

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Notes

  1. As in Remark 2.13 below, one could replace its use below with that of the étale comparison theorem of [4].

  2. \(\delta \)-rings also arise as the natural structure on the homotopy groups of K(1)-local \(E_\infty \)-ring spectra (where they are often called \(\theta \)-algebras or Frobenius algebras), cf. [16]. We will not use this fact here.

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Acknowledgements

We thank Lars Hesselholt, Jacob Lurie, and Peter Scholze for helpful discussions. We are also grateful to the anonymous referee for their helpful comments on the first version of this paper. The third author would like to thank the University of Copenhagen for its hospitality during which some of this work was done. The first author was supported by NSF Grant #1801689 as well as grants from the Packard and Simons Foundations. The second author was supported by Lars Hesselholt’s Niels Bohr professorship, the University of Bonn, and the Max Planck Institute for Mathematics. This work was done while the third author was a Clay Research Fellow.

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Authors and Affiliations

  1. University of Michigan, Ann Arbor, USA

    Bhargav Bhatt

  2. Max Planck Institute for Mathematics, Bonn, Germany

    Dustin Clausen

  3. University of Chicago, Chicago, USA

    Akhil Mathew

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  1. Bhargav Bhatt
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Bhatt, B., Clausen, D. & Mathew, A. Remarks on K(1)-local K-theory. Sel. Math. New Ser. 26, 39 (2020). https://doi.org/10.1007/s00029-020-00566-6

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