Lemma 65.16.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. $f$ is étale,

2. 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),

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

4. 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.

Proof. Let us prove that (4) implies (1). Assume a diagram as in (4) given. Let $W \to X$ be an étale morphism with $W$ a scheme. Then we see that $W \times _ X U \to U$ is étale. Hence $W \times _ X U \to V$ is étale as the composition of the étale morphisms of schemes $W \times _ X U \to U$ and $U \to V$. Therefore $W \times _ X U \to Y$ is étale by Lemma 65.16.1 (1). Since also the projection $W \times _ X U \to W$ is surjective and étale, we conclude from Lemma 65.16.1 (3) that $W \to Y$ is étale.

Let us prove that (1) implies (4). Assume (1). Choose a commutative diagram

$\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }$

where $U \to X$ and $V \to Y$ are surjective and étale, see Spaces, Lemma 64.11.6. By assumption the morphism $U \to Y$ is étale, and hence $U \to V$ is étale by Lemma 65.16.1 (2).

We omit the proof that (2) and (3) are also equivalent to (1). $\square$

Comment #2316 by Nithi Rungtanapirom on

In the last sentence of the proof of "(4) implies (1)", it should follow from Lemma 34.11.4 (Tag 02KM) instead of Lemma 54.15.1(2) that $W\to Y$ is étale since we conclude this from the fact that the composition $W\times_XU\to W\to Y$ is étale and $W\times_XU\to W$ is étale and surjective.

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