Although its existence was conjectured by Alexander Grothendieck, the first proof of its existence is due, for schemes defined over fields, to Madhav Nori.
[1][2][3] A proof of its existence for schemes defined over Dedekind schemes is due to Marco Antei, Michel Emsalem and Carlo Gasbarri.
[4][5] 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.
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.,
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.
As mentioned in the introduction this result was due to Madhav Nori who in 1976 published his first construction of this new object
As for the name he decided to abandon the true fundamental group name and he called it, as we know it nowadays, the fundamental group scheme.
stands for Nori, in order to distinguish it from the previous fundamental groups and to its modern generalizations.
defined on regular schemes of dimension 1 had to wait about forty more years.
and the quasi finite fundamental group scheme
[4] The original definition and the first construction have been suggested by Nori for schemes
Then they have been adapted to a wider range of schemes.
a faithfully flat morphism, locally of finite type.
is built as the affine group scheme naturally associated to the neutral tannakian category (over
) of essentially finite vector bundles over
[1] Nori also proves a that the fundamental group scheme exists when
is any finite type, reduced and connected scheme over
[2] Since then several other existence results have been added, including some non reduced schemes.
a faithfully flat morphism locally of finite type.
Then the existence of the fundamental group scheme
has been proved by Marco Antei, Michel Emsalem and Carlo Gasbarri in the following situations:[4] Over a Dedekind scheme, however, there is no need to only consider finite group schemes: indeed quasi-finite group schemes are also a very natural generalization of finite group schemes over fields.
[7] This is why Antei, Emsalem and Gasbarri also defined the quasi-finite fundamental group scheme
a faithfully flat morphism, locally of finite type.
then it coincides with the étale fundamental group
any two smooth projective schemes over an algebraically closed field
[9] This result was conjectured by Nori[1] and proved by Vikram Mehta and Subramanian.