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

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 $K({\mathcal{O}}_F{)}_p^{\wedge }\to TC({\mathcal{O}}_F{)}_p^{\wedge }$ 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 $\mathbb{Z}$.

arXiv commentThis draft omits material in previous drafts including: review of homotopy theory of presheaves of spectra (incl. hypercover descent and models for derived functors); review of pairings in localized and mod-p^n K-theory; proofs of some transfer and pairing folk results in the Thomason SS; details on duality in the splitting of the fiber sequence of the cyclotomic trace. For these and more, see v3