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.
Original languageEnglish
JournalJournal of the American Mathematical Society
Issue number2
Pages (from-to)411-473
Publication statusPublished - 2021

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