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Let $X$ and $Y$ be varieties over an algebraically closed field $K$, $\phi:Y \longrightarrow X$ be a finite etale cover , and let $K(X),K(Y)$ be the function fields of $X$ and $Y$ respectively. Then what is the relationship between $Gal(K(Y) / K(X))$ and $Aut(Y / X)$ where $Aut$ here means scheme automorphisms of $Y$ preserving $\phi$?

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If $X$ and $Y$ are smooth, then the two groups are naturally isomorphic because of the correspondence betweeen dominant morphisms and field homomorphisms. –  Zhen Lin Dec 30 '12 at 0:31

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