Lemma 66.16.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

$f$ is étale,

there exists a surjective étale morphism $\varphi : U \to X$, where $U$ is a scheme, such that the composition $f \circ \varphi $ is étale (as a morphism of algebraic spaces),

there exists a surjective étale morphism $\psi : V \to Y$, where $V$ is a scheme, such that the base change $V \times _ Y X \to V$ is étale (as a morphism of algebraic spaces),

there exists a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]where $U$, $V$ are schemes, the vertical arrows are étale, and the left vertical arrow is surjective such that the horizontal arrow is étale.

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Comment #2316 by Nithi Rungtanapirom on

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