Galois closure of essentially finite morphisms

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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 H 0 ( Y , O Y ) = k ; this Galois closure is a torsor p : X ˆ Y → X dominating f by an X -morphism λ : X ˆ Y → Y and universal for this property. Moreover, we show that λ : X ˆ Y → Y 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 H 0 ( Y , O Y ) = k , Y has a fundamental group scheme π 1 ( Y , y ) fitting in a short exact sequence with π 1 ( X , x ) .