Vector bundles trivialized by proper morphisms and the fundamental group scheme

Publisher： Cambridge University Press

E-ISSN： 1475-3030|10|2|225-234

ISSN： 1474-7480

Source： Journal of the Institute of Mathematics of Jussieu, Vol.10, Iss.2, 2010-02, pp. : 225-234

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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.

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