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→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.
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 language | English |
---|---|
Journal | Journal of the American Mathematical Society |
Volume | 34 |
Issue number | 2 |
Pages (from-to) | 411-473 |
ISSN | 0894-0347 |
DOIs | |
Publication status | Published - 2021 |