Fundamental group scheme

In mathematics, the fundamental group scheme is a group scheme canonically associated 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 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 (not necessarily reduced)^[4] 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).

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. C. Gasbarri, Heights of Vector Bundles and the Fundamental Group Scheme of a Curve, Duke Mathematical Journal, Vol. 117, No. 2, (2003) p. 287-311
4. M. Antei, The fundamental group scheme of a non reduced scheme, Bulletin des Sciences Mathématiques, Volume 135, Issue 5, July–August 2011, Pages 531-539.