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,[1][2] who only worked on schemes over fields. A generalisation to schemes over Dedekind schemes is due to Marco Antei, Michel Emsalem and Carlo Gasbarri.[3]

First definition

Let ${\displaystyle k}$ be a perfect field and ${\displaystyle X\to {\text{Spec}}(k)}$ a faithfully flat and proper morphism of schemes with ${\displaystyle X}$ a reduced and connected scheme. Assume the existence of a section ${\displaystyle x:{\text{Spec}}(k)\to X}$, then the fundamental group scheme ${\displaystyle \pi _{1}(X,x)}$ of ${\displaystyle X}$ in ${\displaystyle x}$ is defined as the affine group scheme naturally associated to the neutral tannakian category (over ${\displaystyle k}$) of essentially finite vector bundles over ${\displaystyle X}$.

Second definition

Let ${\displaystyle S}$ be a Dedekind scheme, ${\displaystyle X}$ any connected scheme reduced and ${\displaystyle X\to S}$ a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section ${\displaystyle x:S\to X}$. Once we prove that the category of isomorphism classes of torsors over ${\displaystyle X}$ (pointed over ${\displaystyle x}$) under the action of finite and flat ${\displaystyle S}$-group schemes is cofiltered then we define the universal torsor (pointed over ${\displaystyle x}$) as the projective limit of all the torsors of that category. The ${\displaystyle S}$-group scheme acting on it is called the fundamental group scheme and denoted by ${\displaystyle \pi _{1}(X,x)}$ (when ${\displaystyle S}$ 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.

See also

• Étale fundamental group
• Fundamental group

Notes

1. M. V. Nori On the Representations of the Fundamental Group, Compositio Mathematica, Vol. 33, Fasc. 1, (1976), p. 29-42
2. T. Szamuely Galois Groups and Fundamental Groups. Cambridge Studies in Advanced Mathematics, Vol. 117 (2009)
3. M. Antei, M. Emsalem, C. Gasbarri, Sur l'existence du schéma en groupes fondamental, Épijournal de Géométrie Algébrique, Volume 4, (2020)