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 k be a perfect field and \( {\displaystyle X\to {\text{Spec}}(k)} \) a faithfully flat and proper morphism of schemes with X a reduced and connected scheme. Assume the existence of a section \( {\displaystyle x:{\text{Spec}}(k)\to X} \), 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 \( {\displaystyle X\to S} \) 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 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
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)
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
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.
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