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$ K$-theory and topological cyclic homology of henselian pairs


Authors: Dustin Clausen, Akhil Mathew and Matthew Morrow
Journal: J. Amer. Math. Soc.
MSC (2020): Primary 19D55
DOI: https://doi.org/10.1090/jams/961
Published electronically: January 27, 2021
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Abstract: Given a henselian pair $ (R, I)$ of commutative rings, we show that the relative $ K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $ K \to \mathrm {TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $ n$ coefficients, with $ n$ invertible in $ R$) and McCarthy's theorem on relative $ K$-theory (when $ I$ is nilpotent).

We deduce that the cyclotomic trace is an equivalence in large degrees between $ p$-adic $ K$-theory and topological cyclic homology for a large class of $ p$-adic rings. In addition, we show that $ K$-theory with finite coefficients satisfies continuity for complete noetherian rings which are $ F$-finite modulo $ p$. Our main new ingredient is a basic finiteness property of $ \mathrm {TC}$ with finite coefficients.


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Additional Information

Dustin Clausen
Affiliation: Matematiske Fag, Københavns Universitet, Universitetsparken 5, 2100 København
Email: dustin.clausen@math.ku.dk

Akhil Mathew
Affiliation: Department of Mathematics, University of Chicago,5734 S University Ave, Chicago, IL 60637
Email: amathew@math.uchicago.edu

Matthew Morrow
Affiliation: CNRS & Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Paris, France
Email: matthew.morrow@imj-prg.fr

DOI: https://doi.org/10.1090/jams/961
Received by editor(s): April 18, 2018
Received by editor(s) in revised form: April 22, 2020, and May 28, 2020
Published electronically: January 27, 2021
Additional Notes: The first author was supported by Lars Hesselholt’s Niels Bohr Professorship.
This work was done while the second author was a Clay Research Fellow.
Article copyright: © Copyright 2021 American Mathematical Society