The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

TL;DR

This paper demonstrates an eigensplitting of the fiber sequence for the cyclotomic trace of the sphere spectrum, revealing new duality and summand structures in algebraic K-theory and topological cyclic homology.

Contribution

It introduces a novel eigensplitting of the fiber sequence for the cyclotomic trace, generalizing known splittings and identifying summands via Anderson duality.

Findings

Identification of summands as covers of $ ext{Z}_p$-Anderson duals

Self-duality of the $K(1)$-localized fiber sequence for $ ext{Z}$

Intrinsic characterization of the summand $Z$ in the splitting

Abstract

Let $p\in \mathbb Z$ be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum $\mathbb S$ admits an "eigensplitting" that generalizes known splittings on $K$-theory and $TC$. We identify the summands in the fiber as the covers of $\mathbb Z_{p}$-Anderson duals of summands in the $K(1)$-localized algebraic $K$-theory of $\mathbb Z$. Analogous results hold for the ring $\mathbb Z$ where we prove that the $K(1)$-localized fiber sequence is self-dual for $\mathbb Z_{p}$-Anderson duality, with the duality permuting the summands by $i\mapsto p-i$ (indexed mod $p-1$). We explain an intrinsic characterization of the summand we call $Z$ in the splitting $TC(\mathbb Z)^{\wedge}_{p}\simeq j \vee \Sigma j'\vee Z$ in terms of units in the $p$-cyclotomic tower of $\mathbb Q_{p}$.