## Marco Antei ; Michel Emsalem ; Carlo Gasbarri - Sur l'existence du schéma en groupes fondametal

epiga:5436 - Épijournal de Géométrie Algébrique, June 8, 2020, Volume 4 - https://doi.org/10.46298/epiga.2020.volume4.5436
Sur l'existence du schéma en groupes fondametal

Authors: Marco Antei ; Michel Emsalem ; Carlo Gasbarri

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.

Volume: Volume 4
Published on: June 8, 2020
Accepted on: June 8, 2020
Submitted on: May 7, 2019
Keywords: Mathematics - Algebraic Geometry,Mathematics - Number Theory,14G99, 14L15, 14L30, 11G99