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In mathematics, the '''fundamental group scheme''' is a [[group scheme]] canonically associated to a [[Scheme (mathematics)|scheme]] over a Dedekind scheme (e.g. the spectrum of a [[field (mathematics)|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,<ref>M. V. Nori ''On the Representations of the Fundamental Group'', Compositio Mathematica, Vol. 33, Fasc. 1, (1976), p. 29-42</ref><ref>T. Szamuely ''Galois Groups and Fundamental Groups.'' Cambridge Studies in Advanced Mathematics, Vol. 117 (2009)</ref> who only worked on schemes over fields. A generalisation to schemes over Dedekind schemes is due to Carlo Gasbarri.<ref>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</ref>


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==First definition==
Let <math>k</math> be a perfect field and <math>X\to \text{Spec}(k)</math> a faithfully flat and proper morphism of schemes with <math>X</math> a reduced and connected scheme. Assume the existence of a section <math>x:\text{Spec}(k)\to X</math>, then the fundamental group scheme <math>\pi_1(X,x)</math> of <math>X</math>  in <math>x</math> is defined as the affine group scheme naturally associated to the neutral [[tannakian category]] (over <math>k</math>) of [[essentially finite vector bundle]]s over <math>X</math>.
 
==Second definition==
Let <math>S</math> be a Dedekind scheme, <math>X</math> any connected scheme (not necessarily reduced)<ref>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.</ref> and <math>X\to S</math> a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section <math>x:S\to X</math>. Once we prove that the [[Category (mathematics)|category]] of isomorphism classes of [[torsor]]s  over <math>X</math> (pointed over <math>x</math>) under the action of finite and flat <math>S</math>-[[group scheme]]s is cofiltered then we define the universal torsor (pointed over <math>x</math>) as the projective limit of all the torsors of that category. The <math>S</math>-group scheme acting on it is called the fundamental group scheme and denoted by <math>\pi_1(X,x)</math> (when <math>S</math> 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==
<references/>
 
[[Category:Scheme theory]]
[[Category:Topological methods of algebraic geometry]]

Revision as of 10:35, 8 November 2013

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→Spec(k) a faithfully flat and proper morphism of schemes with X a reduced and connected scheme. Assume the existence of a section x:Spec(k)→X, then the fundamental group scheme π1(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→S a faithfully flat morphism of finite type (not necessarily proper). Assume the existence of a section x:S→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 π1(X,x) (when S is the spectrum of a perfect field the two definitions coincide so that no confusion can arise).

See also

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.
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