Chromatic convergence for the algebraic K-theory of the sphere spectrum

TL;DR

This paper proves a form of Waldhausen's Chromatic Convergence Conjecture for the algebraic K-theory of the sphere spectrum, showing a key map is a connective cover and identifying its fiber in K-theoretic terms.

Contribution

It establishes the chromatic convergence for the algebraic K-theory of the sphere spectrum and identifies the fiber of the map in K-theoretic terms, advancing understanding of chromatic phenomena.

Findings

The map from K(S) to its chromatic completion is a connective cover.

The fiber of this map is explicitly identified in K-theoretic terms.

The chromatic convergence map is shown to be an inclusion of a wedge summand.

Abstract

We show that the map from $K({\mathbb S})$ to its chromatic completion is a connective cover and identify the fiber in $K$-theoretic terms. We combine this with recent work of Land-Mathew-Meier-Tamme to prove a form of "Waldhausen's Chromatic Convergence Conjecture": we show that the map $K({\mathbb S}_{(p)})_{(p)}\to \mathop{\rm holim} K(L^{f}_{n}{\mathbb S})_{(p)}$ is the inclusion of a wedge summand.