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

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

Extension of torsors and prime to p fundamental group scheme

“…The notation cannot lead to any ambiguity as the two operations of taking the prime to p quotient and reducing to the generic fiber are interchangeable. That the morphism ψ (p ) is faithfully flat is, again, a consequence of [5,Proposition 5.5]. In order to prove that it is an isomorphism it is thus sufficient to prove that it is a closed immersion.…”

“…In few words we prove that in order for the torsor Y → X K to be extended to X it is sufficient (and uninterestingly necessary) to extend it with a finite and faithfully flat morphism T → X with some few extra assumptions which are often satisfied even when T → X is not a torsor. To show the usefulness of our criterion we will provide an interesting application of our main result to the base change behavior of the fundamental group scheme, as defined in [12], [13] and [5] (cf. also [10] for a more recent construction).…”

Finite torsors on projective schemes defined over a discrete valuation ring

“…Let A be a discrete valuation ring and X a projective flat A-scheme carrying an A-point x 0 . There has been some interest in constructing an affine and flat group scheme Π(X, x 0 ) over A with the property that morphisms to finite and flat group schemes Π(X, x 0 ) → G should canonically correspond to pointed G-torsors over X; see [Gas03,AEG20,MS13]. This is an analogue of Nori's theory [Nor76] (Nori deals with a base field ) and hence can be developed through (SS) a Tannakian category of semi-stable vector bundles [Nor76, Definition after Lemma 3.6], (F) the construction of fibre products of torsors [Nor82,Definition II.2] or (T) a Tannakian category of vector bundles trivialized by proper and surjective morphisms [BdS11].…”

“…The papers [Gas03] and [AEG20] follow (F), and [MS13] follows a variant of (SS). That is to say, [Gas03] and [AEG20] construct Π(X, x 0 ) by producing a universal Π(X, x 0 )-torsor X → X, and this is achieved by filtering the category of torsors under finite group schemes [AEG20,Theorem 4.2].…”

“…The papers [Gas03] and [AEG20] follow (F), and [MS13] follows a variant of (SS). That is to say, [Gas03] and [AEG20] construct Π(X, x 0 ) by producing a universal Π(X, x 0 )-torsor X → X, and this is achieved by filtering the category of torsors under finite group schemes [AEG20,Theorem 4.2]. On the other hand, [MS13] looks at locally free O X -modules on X which lie in the categories considered by Nori when pulled back to the fibres of X; see [MS13, beginning of Section 3, top of p. 346].…”