Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π 1 (X, x) ab → π 1 (Alb X/S , 0 Alb X/S ) induced by the canonical morphism from X to its Albanese scheme Alb X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS τ X/S ) ∨ → π 1 (X, x) ab → π 1 (Alb X/S , 0 Alb X/S ) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber Xη of X can be extended to a finite and commutative pointed torsor over X.

a b s t r a c tLet X be a reduced connected k-scheme pointed at a rational point x ∈ X (k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphismthis Galois closure is a torsor p :X Y → X dominating f by an X -morphism λ :X Y → Y and universal for this property. Moreover, we show that λ :X Y → Y is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X . We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f : Y → X under a finite group scheme satisfying the condition H 0 (Y , O Y ) = k, Y has a fundamental group scheme π 1 (Y , y) fitting in a short exact sequence with π 1 (X, x).

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. Comment: in French. Final version (finally!)

On montre que le morphisme canonique ϕ : π 1 (X η , x η ) → π 1 (X, x) η entre le schéma en groupes fondamental de la fibre générique X η d'un schéma X sur un schéma de Dedekind connexe et la fibre générique du schéma en groupes fondamental de X est toujours fidèlement plat. On donnera ensuite des conditions nécessaires et suffisantes pour qu'un G-torseur fini, dominé et pointé au dessus de X η puisse être étendu sur X. On décrira des exemples où ϕ : π 1 (X η , x η ) → π 1 (X, x) η est un isomorphisme.

We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack X over a field k. This is done by describing the Nori fundamental gerbe of an essentially finite cover of X. A similar result is also obtained for the S-fundamental gerbe.