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Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$.

Are the following two conditions equivalent?

  1. The function field extension $K(Y)\subset K(X)$ is Galois (in the field-theoretic sense)

  2. The quotient $X/G$ exists and the natural morphism $X/G\to Y$ is an isomorphism.

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1 Answer 1

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(2) implies (1) is OK when $X/G$ is correctly defined (e.g. if $X$ is a projective variety over a field, and $O_{X/G}=O_X^G$).

Conversely (and suppose $X$ is projective as above), by definition of $G$, we have a factorization $X\to Y$ through a finite morphism $g: X/G\to Y$. Condition (1) implies that $g$ is birational. If $Y$ is normal, then $X/G\to Y$ is an isomorphism. Otherwise, $X/G\to Y$ can be identified with the normalization map.

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