Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Étale Reconstruction for $\mathbb{F}_p(t)$-Schemes

Zachary Berens

TL;DR

The paper extends étale-site reconstruction from characteristic 0 to positive characteristic by leveraging perfections and absolute weak normalizations. It adapts Voevodsky’s strategy to char p via Kummer theory and the tame fundamental group to isolate prime-to‑$p$ data, proving that perfections of finite type schemes over infinite AFG fields in characteristic $p$ are reconstructible from étale sites; in particular, Mor$_{K^{p^{-∞}}}(X,Y)$ bijects with Mor$_{K_{ ext{ét}}}^ullet(X_{ ext{ét}},Y_{ ext{ét}})$ for $X=X_0^{ ext{perf}}$ and $Y$ finite type over $K^{p^{-∞}}$. Moreover, perfections of finite type schemes over $K$ are étale reconstructible, yielding $X ot o Y$ is determined by their étale topoi. The work aligns with parallel developments (e.g., CHW) and clarifies how to handle Artin–Schreier phenomena by passing to the tame part of $ ext{π}_1^{ ext{ét}}( ext{G}_m)$. Overall, it extends the toposic reconstruction paradigm to positive characteristic and to non-finite type contexts, highlighting the role of perfection and weak normalizations in capturing scheme structure through étale data.

Abstract

Voevodsky proved that normal schemes of finite type over finitely generated fields of characteristic $0$ can be reconstructed from their étale sites. Let $K$ be a field that is finitely generated over $\mathbb{F}_p(t)$. Grothendieck conjectured that perfections of finite type $K$-schemes can be reconstructed from their étale sites. Adapting Voevodsky's methods, we prove this.

Étale Reconstruction for $\mathbb{F}_p(t)$-Schemes

TL;DR

The paper extends étale-site reconstruction from characteristic 0 to positive characteristic by leveraging perfections and absolute weak normalizations. It adapts Voevodsky’s strategy to char p via Kummer theory and the tame fundamental group to isolate prime-to‑ data, proving that perfections of finite type schemes over infinite AFG fields in characteristic are reconstructible from étale sites; in particular, Mor bijects with Mor for and finite type over . Moreover, perfections of finite type schemes over are étale reconstructible, yielding is determined by their étale topoi. The work aligns with parallel developments (e.g., CHW) and clarifies how to handle Artin–Schreier phenomena by passing to the tame part of . Overall, it extends the toposic reconstruction paradigm to positive characteristic and to non-finite type contexts, highlighting the role of perfection and weak normalizations in capturing scheme structure through étale data.

Abstract

Voevodsky proved that normal schemes of finite type over finitely generated fields of characteristic can be reconstructed from their étale sites. Let be a field that is finitely generated over . Grothendieck conjectured that perfections of finite type -schemes can be reconstructed from their étale sites. Adapting Voevodsky's methods, we prove this.

Paper Structure

This paper contains 9 sections, 28 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Characteristic $0$
  3. Positive Characteristic
  4. Preparations
  5. Generalities on Étale Sites
  6. Étalified Rational Points
  7. The Meat
  8. Kummer Theory Preliminaries
  9. The Main Theorem

Key Result

Theorem 1.1

If $X$ and $Y$ are $k$-schemes of finite type and $X$ is normal, there is a bijection where the right-hand side is the set of ($2$-)isomorphism classes of "admissible" $\operatorname{Spec}(k)_{\operatorname{\acute et}}$-morphisms from $X_{\operatorname{\acute et}}$ to $Y_{\operatorname{\acute et}}$.

Theorems & Definitions (64)

  • Theorem 1.1: Voe, https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Etale_topologies_published.pdf#page=10
  • Conjecture 1.2: The Toposic Hom Conjecture
  • Remark 1.3
  • Definition 1.4: Stacks, https://stacks.math.columbia.edu/tag/0EUK
  • Lemma 1.7: Rush, https://www.sciencedirect.com/science/article/pii/0022404982900329
  • Lemma 1.8
  • proof
  • Lemma 1.9
  • proof
  • Theorem 1.10
  • ...and 54 more