K-theory and topological cyclic homology of henselian pairs

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.

arXiv comment59 pages, revised and final version

References in corpus (8)

• Higher Topos Theory
• On the K-theory of local fields
• Localization theorems in topological Hochschild homology and topological cyclic homology
• Nilpotence and descent in equivariant stable homotopy theory
• On the topological cyclic homology of the algebraic closure of a local field
• On the Blumberg-Mandell Künneth theorem for TP
• Infinitesimal K-theory
• The geometry of the cyclotomic trace

Cited by in corpus (8)

• Topological Hochschild homology and integral $p$-adic Hodge theory
• Some recent advances in topological Hochschild homology
• $K$-theory of valuation rings
• On The Algebraic $K$-Theory of Double Points
• On $K(1)$-local $\mathrm{TR}$
• Rigidity in equivariant algebraic $K$-theory
• Spectral Waldhausen categories, the $S_{\bullet }$-construction, and the Dennis trace
• On the Dundas-Goodwillie-McCarthy theorem