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The étale topos reconstructs varieties over sub-p-adic fields

Magnus Carlson, Jakob Stix

Abstract

Let $K$ be a sub-$p$-adic field. We show that the functor sending a finite type $K$-scheme to its étale topos is fully faithful after localizing at the class of universal homeomorphisms. This generalizes a result of Voevodsky, who proved the analogous theorem for fields finitely generated over $\mathbb{Q}$. Our proof relies on Mochizuki's Hom-theorem in anabelian geometry, and a study of point-theoretic morphisms of fundamental groups of curves.

The étale topos reconstructs varieties over sub-p-adic fields

Abstract

Let be a sub--adic field. We show that the functor sending a finite type -scheme to its étale topos is fully faithful after localizing at the class of universal homeomorphisms. This generalizes a result of Voevodsky, who proved the analogous theorem for fields finitely generated over . Our proof relies on Mochizuki's Hom-theorem in anabelian geometry, and a study of point-theoretic morphisms of fundamental groups of curves.

Paper Structure

This paper contains 10 sections, 18 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on topoi
  3. Topoi and pinned morphisms
  4. Base points of topoi
  5. Morphisms between étale topoi and fundamental groups
  6. Quasi-sections and point-theoretic maps
  7. Galois sections of the generalized Jacobian
  8. Sections over Kummer-faithful fields
  9. Point-theoretic maps and the Abel-Jacobi map
  10. Étale reconstruction for sub-p-adic fields

Key Result

Theorem 1

Let $K$ be a sub-$p$-adic field. Then the functor sending a scheme $X \to \mathop{\mathrm{Spec}}\nolimits(K)$ of finite type to its étale topos $X_{\mathop{\mathrm{\acute et}}\nolimits} \to \mathop{\mathrm{Spec}}\nolimits(K)_{\mathop{\mathrm{\acute et}}\nolimits}$ is fully faithful.

Theorems & Definitions (34)

  • Theorem 1: see \ref{['thm:etale_reconstruction']}
  • Theorem 2
  • Proposition 3: see \ref{['prop:hyperbolicopenpi1']}
  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Lemma 3.7
  • Definition 3.8
  • ...and 24 more