Galois closure of essentially finite morphisms |
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Authors: | Marco Antei Michel Emsalem |
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Affiliation: | Laboratoire Paul Painlevé, U.F.R. de Mathématiques, Université des Sciences et des Technologies de Lille 1, 59 655 Villeneuve d’Ascq, France |
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Abstract: | Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H0(Y,OY)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H0(Y,OY)=k, Y has a fundamental group scheme π1(Y,y) fitting in a short exact sequence with π1(X,x). |
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Keywords: | Primary, 14E20, 14L15 Secondary, 14G32, 12F10, 11G99 |
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