𝐾-theory and topological cyclic homology of henselian pairs

Abstract

Given a henselian pair ( R , I ) (R, I) of commutative rings, we show that the relative K K -theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace K → T C K \to \mathrm {TC} . This yields a generalization of the classical Gabber–Gillet–Thomason–Suslin rigidity theorem (for mod n n coefficients, with n n invertible in R R ) and McCarthy’s theorem on relative K K -theory (when I I is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between p p -adic K K -theory and topological cyclic homology for a large class of p p -adic rings. In addition, we show that K K -theory with finite coefficients satisfies continuity for complete noetherian rings which are F F -finite modulo p p . Our main new ingredient is a basic finiteness property of T C \mathrm {TC} with finite coefficients.