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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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$K$-theory and topological cyclic homology of henselian pairs
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by Dustin Clausen, Akhil Mathew and Matthew Morrow
J. Amer. Math. Soc. 34 (2021), 411-473
Published electronically: January 27, 2021


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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Bibliographic Information
  • Dustin Clausen
  • Affiliation: Matematiske Fag, Københavns Universitet, Universitetsparken 5, 2100 København
  • MR Author ID: 1237972
  • Email:
  • Akhil Mathew
  • Affiliation: Department of Mathematics, University of Chicago,5734 S University Ave, Chicago, IL 60637
  • MR Author ID: 891016
  • Email:
  • Matthew Morrow
  • Affiliation: CNRS & Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Paris, France
  • MR Author ID: 859672
  • Email:
  • 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.
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 34 (2021), 411-473
  • MSC (2020): Primary 19D55
  • DOI:
  • MathSciNet review: 4280864