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In mathematics, an '''essentially finite vector bundle''' is a particular type of [[vector bundle]] defined by 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> as the main tool in the construction of the [[fundamental group scheme]]. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural objects to study in [[algebraic geometry]]. So before recalling the definition we give this characterization: | |||
==Characterization== | |||
Let <math>X</math> be a reduced and connected [[Scheme (mathematics)|scheme]] over a perfect [[field (mathematics)|field]] <math>k</math> endowed with a section <math>x\in X(k)</math>. Then a vector bundle <math>V</math> over <math>X</math> is essentially finite if and only if there exists a [[Finite morphism|finite]] <math>k</math>-[[group scheme]] <math>G</math> and a <math>G</math>-[[torsor]] <math>p:P\to X</math> such that <math>V</math> becomes trivial over <math>P</math> (i.e. <math>p^*(V)\cong O_P^{\oplus r}</math>, where <math>r=rk(V)</math>). | |||
==Definition== | |||
{{Empty section|date=October 2012}} | |||
==Notes== | |||
<references/> | |||
[[Category:Scheme theory]] | |||
[[Category:Topological methods of algebraic geometry]] | |||
Revision as of 19:05, 1 February 2014
In mathematics, an essentially finite vector bundle is a particular type of vector bundle defined by Madhav Nori,[1][2] as the main tool in the construction of the fundamental group scheme. Even if the definition is not intuitive there is a nice characterization that makes essentially finite vector bundles quite natural objects to study in algebraic geometry. So before recalling the definition we give this characterization:
Characterization
Let be a reduced and connected scheme over a perfect field endowed with a section . Then a vector bundle over is essentially finite if and only if there exists a finite -group scheme and a -torsor such that becomes trivial over (i.e. , where ).