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Sur l'existence du schéma en groupes fondamental

Marco Antei, Michel Emsalem, Carlo Gasbarri

Abstract

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of étale fundamental group.

Sur l'existence du schéma en groupes fondamental

Abstract

Let be a Dedekind scheme, a connected -scheme locally of finite type and a section. The aim of the present paper is to establish the existence of the fundamental group scheme of , when has reduced fibers or when is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of at , which classifies all the quasi-finite torsors over , pointed over . We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of étale fundamental group.

Paper Structure

This paper contains 14 sections, 6 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Objectifs et plan du travail
  3. Préliminaires
  4. Notations
  5. Torseurs et sections
  6. Un lemme technique
  7. Réduction de schéma en groupes structural
  8. Le cas où $X \to S$ est à fibres réduites
  9. Le cas normal
  10. Le cas $G \to S$ fini
  11. Le cas $G \to S$ quasi-fini
  12. Le schéma en groupes fondamental
  13. Torseurs galoisiens
  14. Appendice : Le contre-exemple de Jilong Tong

Key Result

Proposition 3.1

On suppose que pour tout point $s\in S$, $X_s$ est réduit et que $G$ un $S$-schéma en groupes fini et plat. Soient $Y\to X$ un $G$-torseur et $H'$ un sous-schéma fermé en groupes de $G_{\eta}$. On suppose que $Y_{\eta}\to X_{\eta}$ admet une réduction $Z\subset Y_\eta$ de schéma en groupes structura

Theorems & Definitions (9)

  • proof
  • Proposition 3.1
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.6
  • proof
  • Proposition 5.5
  • Proposition 5.6