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Vector bundles trivialized by proper morphisms and the fundamental group scheme

Published online by Cambridge University Press:  24 February 2010

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])

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2010

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