In mathematics, the fundamental group scheme is a group scheme canonically associated to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental group. Although its existence was conjectured by Alexander Grothendieck, the first construction is due to Madhav Nori, who only worked on schemes over fields. A generalisation to schemes over Dedekind schemes is due to Carlo Gasbarri.
Let k be a perfect field and X → Spec ( k ) a faithfully flat and proper morphism of schemes with X a reduced and connected scheme. Assume the existence of a section x : Spec ( k ) → X , then the fundamental group scheme π 1 ( X , x ) of X in x is defined as the affine group scheme naturally associated to the neutral tannakian category (over k ) of essentially finite vector bundles over X .
Let S be a Dedekind scheme, X any connected scheme (not necessarily reduced) and X → S a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section x : S → X . Once we prove that the category of isomorphism classes of torsors over X (pointed over x ) under the action of finite and flat S -group schemes is cofiltered then we define the universal torsor (pointed over x ) as the projective limit of all the torsors of that category. The S -group scheme acting on it is called the fundamental group scheme and denoted by π 1 ( X , x ) (when S is the spectrum of a perfect field the two definitions coincide so that no confusion can arise).
