Sur l'existence du sch\'ema en groupes fondamental

TL;DR

This paper proves the existence of the fundamental group scheme for certain schemes over Dedekind schemes, including cases with reduced fibers or normal schemes, and introduces a quasi-finite fundamental group scheme classifying specific torsors.

Contribution

It establishes the existence of the fundamental group scheme under new conditions and introduces the quasi-finite fundamental group scheme and Galois torsors in this context.

Findings

Existence of fundamental group scheme for schemes with reduced fibers or normal schemes.

Introduction of the quasi-finite fundamental group scheme classifying quasi-finite torsors.

Definition of Galois torsors analogous to Galois covers in étale fundamental group theory.

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 \'etale fundamental group.