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- W1979156701 abstract "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 $f:Yto X$ satisfying the condition $H^0(Y,mathcal{O}_Y)=k$; this Galois closure is a torsor $p:hat{X}_Yto X$ dominating $f$ by an $X$-morphism $lambda:hat{X}_Yto Y$ and universal for this property. Moreover we show that $lambda:hat{X}_Yto Y$ 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 $pi_1 (X,x)$." @default.
- W1979156701 created "2016-06-24" @default.
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- W1979156701 date "2009-01-12" @default.
- W1979156701 modified "2023-09-27" @default.
- W1979156701 title "Galois Closure of Essentially Finite Morphisms" @default.
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