## 66.39 Étale morphisms

The notion of an étale morphism of algebraic spaces was defined in Properties of Spaces, Definition 65.16.2. Here is what it means for a morphism to be étale at a point.

Definition 66.39.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. We say $f$ is étale at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is étale.

Lemma 66.39.2. 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. for every $x \in |X|$ the morphism $f$ is étale at $x$,

3. for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is étale,

4. for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is étale,

5. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is an étale morphism,

6. there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi$ is étale,

7. for every commutative diagram

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

where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is étale,

8. 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 $U \to X$ surjective such that the top horizontal arrow is étale, and

9. there exist Zariski coverings $Y = \bigcup Y_ i$ and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is étale.

Proof. Combine Properties of Spaces, Lemmas 65.16.3, 65.16.5 and 65.16.4. Some details omitted. $\square$

Lemma 66.39.3. The composition of two étale morphisms of algebraic spaces is étale.

Proof. This is a copy of Properties of Spaces, Lemma 65.16.4. $\square$

Lemma 66.39.4. The base change of an étale morphism of algebraic spaces by any morphism of algebraic spaces is étale.

Proof. This is a copy of Properties of Spaces, Lemma 65.16.5. $\square$

Lemma 66.39.5. An étale morphism of algebraic spaces is locally quasi-finite.

Proof. Let $X \to Y$ be an étale morphism of algebraic spaces, see Properties of Spaces, Definition 65.16.2. By Properties of Spaces, Lemma 65.16.3 we see this means there exists a diagram as in Lemma 66.22.1 with $h$ étale and surjective vertical arrow $a$. By Morphisms, Lemma 29.36.6 $h$ is locally quasi-finite. Hence $X \to Y$ is locally quasi-finite by definition. $\square$

Proof. The proof is identical to the proof of Lemma 66.39.5. It uses the fact that an étale morphism of schemes is smooth (by definition of an étale morphism of schemes). $\square$

Proof. The proof is identical to the proof of Lemma 66.39.5. It uses Morphisms, Lemma 29.36.12. $\square$

Lemma 66.39.8. An étale morphism of algebraic spaces is locally of finite presentation.

Proof. The proof is identical to the proof of Lemma 66.39.5. It uses Morphisms, Lemma 29.36.11. $\square$

Lemma 66.39.9. An étale morphism of algebraic spaces is locally of finite type.

Proof. An étale morphism is locally of finite presentation and a morphism locally of finite presentation is locally of finite type, see Lemmas 66.39.8 and 66.28.5. $\square$

Proof. The proof is identical to the proof of Lemma 66.39.5. It uses Morphisms, Lemma 29.36.5. $\square$

Lemma 66.39.11. Let $S$ be a scheme. Let $X, Y$ be algebraic spaces étale over an algebraic space $Z$. Any morphism $X \to Y$ over $Z$ is étale.

Proof. This is a copy of Properties of Spaces, Lemma 65.16.6. $\square$

Lemma 66.39.12. A locally finitely presented, flat, unramified morphism of algebraic spaces is étale.

Proof. Let $X \to Y$ be a locally finitely presented, flat, unramified morphism of algebraic spaces. By Properties of Spaces, Lemma 65.16.3 we see this means there exists a diagram as in Lemma 66.22.1 with $h$ locally finitely presented, flat, unramified and surjective vertical arrow $a$. By Morphisms, Lemma 29.36.16 $h$ is étale. Hence $X \to Y$ is étale by definition. $\square$

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