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Galois closure of essentially finite morphisms
Authors:Marco Antei  Michel Emsalem
Institution:
  • Laboratoire Paul Painlevé, U.F.R. de Mathématiques, Université des Sciences et des Technologies de Lille 1, 59 655 Villeneuve d’Ascq, France
  • Abstract:Let X be a reduced connected k-scheme pointed at a rational point xX(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:YX satisfying the condition H0(Y,OY)=k; this Galois closure is a torsor View the MathML source dominating f by an X-morphism View the MathML source and universal for this property. Moreover, we show that View the MathML source 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:YX 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).
    Keywords:Primary  14E20  14L15  Secondary  14G32  12F10  11G99
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