Galois Closure of Essentially Finite Morphisms

Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$ dominating $f$ by an $X$-morphism $\lambda:\hat{X}_Y\to Y$ and universal for this property. Moreover we show that $\lambda:\hat{X}_Y\to Y$ 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 $f:Y \to X$ under a finite group scheme satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$, $Y$ has a fundamental group scheme $\pi_1 (Y,y)$ fitting in a short exact sequence with $\pi_1 (X,x)$.