The Stacks project

Lemma 65.16.5. The base change of an étale morphism of algebraic spaces by any morphism of algebraic spaces is étale.

Proof. Let $X \to Y$ be an étale morphism of algebraic spaces over $S$. Let $Z \to Y$ be a morphism of algebraic spaces. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then $U \to Y$ is étale, hence in the diagram

\[ \xymatrix{ W \times _ Y U \ar[d] \ar[r] & W \ar[d] \\ Z \times _ Y X \ar[r] & Z } \]

the top horizontal arrow is étale. Moreover, the left vertical arrow is surjective and étale (verification omitted). Hence we conclude that the lower horizontal arrow is étale by Lemma 65.16.3. $\square$

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