The Stacks project

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). $\square$


Comments (2)

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 is étale since we conclude this from the fact that the composition is étale and is étale and surjective.


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