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

Second definition[]

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

Notes[]

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

[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)

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Fundamental group scheme

Contents

First definition[]

Second definition[]

See also[]

Notes[]