A version of Waldhausen's chromatic convergence for $TC$

TL;DR

This paper proves a version of Waldhausen's chromatic convergence theorem for topological cyclic homology, showing that the map from the p-completion of TC of the sphere spectrum to the homotopy limit over chromatic layers is a weak equivalence.

Contribution

It establishes a chromatic convergence result for TC, extending Waldhausen's theorem to topological cyclic homology.

Findings

The map $TC(S)^{rown}_p o ext{holim } TC(L_n S)^{rown}_p$ is a weak equivalence.

Supports the chromatic approach to understanding TC.

Advances the understanding of the relationship between TC and chromatic layers.

Abstract

The map $TC(\mathbb{S})^{\wedge}_{p}\to \text{holim } TC(L_{n}\mathbb{S})^{\wedge}_{p}$ is a weak equivalence.