K-THEORY AND TOPOLOGICAL CYCLIC HOMOLOGY OF HENSELIAN PAIRS
Résumé
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 cy-clotomic 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.
Domaines
K-théorie et homologie [math.KT]
Fichier principal
Clausen, Mathew, Morrow, K-theory and TC of Henselian pairs.pdf (590.8 Ko)
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