Fundamental group scheme

In mathematics, the fundamental group scheme is a group scheme canonically attached 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 Marco Antei, Michel Emsalem and Carlo Gasbarri.
First definition.
Let formula_1 be a perfect field and formula_2 a faithfully flat and proper morphism of schemes with formula_3 a reduced and connected scheme. Assume the existence of a section formula_4, then the fundamental group scheme formula_5 of formula_3 in formula_7 is defined as the affine group scheme naturally associated to the neutral tannakian category (over formula_1) of essentially finite vector bundles over formula_3.
Second definition.
Let formula_10 be a Dedekind scheme, formula_3 any connected scheme reduced and formula_12 a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section formula_13. Once we prove that the category of isomorphism classes of torsors over formula_3 (pointed over formula_7) under the action of finite and flat formula_10-group schemes is cofiltered then we define the universal torsor (pointed over formula_7) as the projective limit of all the torsors of that category. The formula_10-group scheme acting on it is called the fundamental group scheme and denoted by formula_5 (when formula_10 is the spectrum of a perfect field the two definitions coincide so that no confusion can arise).
The definition has been further generalized to some non reduced schemes.