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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. 34 (2021), 411-473
MSC (2020): Primary 19D55
Published electronically: January 27, 2021
MathSciNet review: 4280864
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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
MR Author ID: 1237972

Akhil Mathew
Affiliation: Department of Mathematics, University of Chicago,5734 S University Ave, Chicago, IL 60637
MR Author ID: 891016

Matthew Morrow
Affiliation: CNRS & Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Paris, France
MR Author ID: 859672

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