K-theory and topological cyclic homology of Henselian pairs

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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→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 TC with finite coefficients.
TidsskriftJournal of the American Mathematical Society
Udgave nummer2
Sider (fra-til)411-473
StatusUdgivet - 2021

Bibliografisk note

Funding Information:
We are grateful to Benjamin Antieau, Bhargav Bhatt, Lars Hesselholt, Thomas Nikolaus, and Peter Scholze for helpful discussions. We thank the referees for many helpful comments on an earlier version of the paper. The second author would like to thank the Universit? Paris 13, the Institut de Math?matiques de Jussieu-Paris Rive Gauche, and the University of Copenhagen for hospitality during which parts of this work were done.

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