Fundamental group scheme


In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme. 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.

First definition

Let be a perfect field and a faithfully flat and proper morphism of schemes with a reduced and connected scheme. Assume the existence of a section, then the fundamental group scheme of in is defined as the affine group scheme naturally associated to the neutral tannakian category of essentially finite vector bundles over.

Second definition

Let be a Dedekind scheme, any connected scheme and a faithfully flat morphism of finite type. Assume the existence of a section. Once we prove that the category of isomorphism classes of torsors over under the action of finite and flat -group schemes is cofiltered then we define the universal torsor as the projective limit of all the torsors of that category. The -group scheme acting on it is called the fundamental group scheme and denoted by .