Spherical sector

2014-02-01T17:05:46Z

198.137.20.217:

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]]

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Spherical sector