Lemma 66.16.5 (03FU)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.16: Étale morphisms of algebraic spaces
Lemma 66.16.5 (cite)

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Lemma 66.16.5. 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 66.16.3. $\square$

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