K-theory and topological cyclic homology of henselian pairs

TL;DR

This paper proves a deep connection between algebraic K-theory and topological cyclic homology for henselian pairs, extending classical rigidity theorems and establishing new equivalences for p-adic rings.

Contribution

It generalizes classical rigidity results and shows that the cyclotomic trace induces an equivalence between K-theory and TC in large degrees for p-adic rings.

Findings

Cyclotomic trace is an equivalence in large degrees for p-adic K-theory and TC.

K-theory with finite coefficients is continuous for certain complete noetherian rings.

A new finiteness property of TC with finite coefficients is established.

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.