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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==
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| 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>.
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| ==Second definition==
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| 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).
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| ==See also==
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| *[[Étale fundamental group]]
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| *[[Fundamental group]]
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| ==Notes==
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| <references/>
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| [[Category:Scheme theory]]
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| [[Category:Topological methods of algebraic geometry]]
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Latest revision as of 23:34, 26 April 2014
I'm Roxana and was born on 19 October 1974. My hobbies are Association football and Computer programming.
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