K-theoretic Tate-Poitou duality and the fiber of the cyclotomic trace

TL;DR

This paper establishes a spectral version of Tate-Poitou duality for algebraic K-theory spectra of number rings with p inverted, revealing the homotopy type of the fiber of the cyclotomic trace after a connective cover.

Contribution

It introduces a spectral Tate-Poitou duality for K-theory spectra and identifies the fiber of the cyclotomic trace for number rings and the sphere spectrum.

Findings

Homotopy type of the fiber of the cyclotomic trace for number rings is identified.

Homotopy type of the fiber of the cyclotomic trace for the sphere spectrum is described.

Spectral Tate-Poitou duality extends classical duality to algebraic K-theory spectra.

Abstract

Let $p\in \mathbb{Z}$ be an odd prime. We prove a spectral version of Tate-Poitou duality for the algebraic $K$-theory spectra of number rings with $p$ inverted. This identifies the homotopy type of the fiber of the cyclotomic trace $K(\mathcal{O}_{F})^{\scriptscriptstyle\wedge}_{p} \to TC(\mathcal{O}_{F})^{\wedge}_{p}$ after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic $K$-theory of $\mathbb{Z}$.