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 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 (not necessarily reduced)[4] 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).

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.