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Vector bundles trivialized by proper morphisms and the fundamental group scheme

Part of: Algebraic groups (Co)homology theory

Published online by Cambridge University Press:  24 February 2010

Indranil Biswas  and
João Pedro P. Dos Santos
Indranil Biswas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, ([email protected])
João Pedro P. Dos Santos
Affiliation:
Université de Paris 6, Institut de Mathématiques de Jussieu, 175, Rue du Chevaleret, 75013 Paris, France, ([email protected])
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Abstract

Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

Keywords

essentially finite vector bundlefundamental group schemetrivialization

MSC classification

Secondary: 14L15: Group schemes 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 225 - 234
Copyright
Copyright © Cambridge University Press 2010

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