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K-theoretic Tate-Poitou duality at prime 2

Myungsin Cho

TL;DR

The paper extends K(1)-localized Tate–Poitou duality to the prime 2 by incorporating real-embedding phenomena through Zink’s completed étale site, thereby relating the homotopy fiber of the K(1)-local completion map to a desuspension of the $\mathbb{Z}_2$-Anderson dual of K(1)-localized K-theory. It builds a canonical duality map using Albert–Brauer–Hasse–Noether and Thomason descent, and provides a hypercohomological description of the fiber as a global section on the completed site. Two descent spectral sequences are constructed on the completed étale site, and their perfect $E_2$-page pairing, governed by Artin–Verdier duality, is shown to converge and be compatible with a pairing arising from the Moore spectrum structure. The main result establishes a 2-primary K-theoretic Tate–Poitou duality, identifying Fib(κ) with the desuspension of the Brown–Comenetz dual of the sphere in the 2-completed setting, and yields a 2-primary description of the cyclotomic-trace fiber for the sphere, connecting arithmetic duality to stable homotopy-theoretic invariants.

Abstract

We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the $K(1)$-localized algebraic K-theory of the ring of $p$-integers in a number field and its completion using the $\mathbb{Z}_p$-Anderson duality. This paper completes the picture by addressing the prime 2, where the real embeddings of number fields introduce extra complexities. As an application, we identify the homotopy type at prime 2 of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of the integers.

K-theoretic Tate-Poitou duality at prime 2

TL;DR

The paper extends K(1)-localized Tate–Poitou duality to the prime 2 by incorporating real-embedding phenomena through Zink’s completed étale site, thereby relating the homotopy fiber of the K(1)-local completion map to a desuspension of the -Anderson dual of K(1)-localized K-theory. It builds a canonical duality map using Albert–Brauer–Hasse–Noether and Thomason descent, and provides a hypercohomological description of the fiber as a global section on the completed site. Two descent spectral sequences are constructed on the completed étale site, and their perfect -page pairing, governed by Artin–Verdier duality, is shown to converge and be compatible with a pairing arising from the Moore spectrum structure. The main result establishes a 2-primary K-theoretic Tate–Poitou duality, identifying Fib(κ) with the desuspension of the Brown–Comenetz dual of the sphere in the 2-completed setting, and yields a 2-primary description of the cyclotomic-trace fiber for the sphere, connecting arithmetic duality to stable homotopy-theoretic invariants.

Abstract

We extend the result of Blumberg and Mandell on K-theoretic Tate-Poitou duality at odd primes which serves as a spectral refinement of the classical arithmetic Tate-Poitou duality. The duality is formulated for the -localized algebraic K-theory of the ring of -integers in a number field and its completion using the -Anderson duality. This paper completes the picture by addressing the prime 2, where the real embeddings of number fields introduce extra complexities. As an application, we identify the homotopy type at prime 2 of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of the integers.
Paper Structure (11 sections, 13 theorems, 102 equations)

This paper contains 11 sections, 13 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Local duality and construction of the duality class
  3. K-theoretic Tate local duality
  4. Construction of the duality class map
  5. Completed small étale site of Zink
  6. Sheaves of abelian groups on Zink's completed étale site
  7. Modified étale sheaf and its derived functor
  8. Hypersheaf of spectra on the completed étale site
  9. Construction of spectral sequences and pairing
  10. Construction of spectral sequences and pairing
  11. Proof of the main theorem

Key Result

Theorem 1

If $p$ is odd, there is a canonical weak equivalence between the homotopy fiber $\operatorname{Fib}(\kappa)$ and $\Sigma^{-1} I_{\mathbb{Z}_p} L_{K(1)}K({\mathcal{O}}_F[\tfrac{1}{p}])$, the desuspension of the $\mathbb{Z}_p$-Anderson dual spectrum of the $K(1)$-local algebraic K-theory of ${\mathcal induced by the $L_{K(1)}K({\mathcal{O}}_F[\tfrac{1}{2}])$-action on $\operatorname{Fib}(\kappa)$ fo

Theorems & Definitions (23)

  • Theorem 1: MR4121155
  • Theorem 2: Two-primary K-theoretic Tate-Poitou duality
  • Theorem 3: MR0175892
  • Corollary 1
  • Corollary 2
  • Theorem 4: MR4121155
  • Theorem 5
  • Definition 3.6
  • Definition 3.8
  • Lemma 3
  • ...and 13 more