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 proof if its existence is due, for schemes defined over fields, to Madhav Nori. A proof of its existence for schemes defined over Dedekind schemes is due to Marco Antei, Michel Emsalem and Carlo Gasbarri. The (topological) fundamental group associated with a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. Although it is still being studied for the classification of algebraic varieties even in algebraic geometry, for many applications the fundamental group has been found to be inadequate for the classification of objects, such as schemes, that are more than just topological spaces. The same topological space may have indeed several distinct scheme structures, yet its topological fundamental group will always be the same. Therefore, it became necessary to create a new object that would take into account the existence of a structural sheaf together with a topological space. This led to the creation of the étale fundamental group, the projective limit of all finite groups acting on étale coverings of the given scheme . Nevertheless, in positive characteristic the latter has obvious limitations, since it does not take into account the existence of group schemes that are not étale (e.g., when the characteristic is ) and that act on torsors over , a natural generalization of the coverings. It was from this idea that Grothendieck hoped for the creation of a new true fundamental group (un vrai groupe fondamental, in French), the existence of which he conjectured, back in the early 1960s in his celebrated SGA 1, Chapitre X. More than a decade had to pass before a first result on the existence of the fundamental group scheme came to light.