Definition 67.39.1. 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.
67.39 Étale morphisms
The notion of an étale morphism of algebraic spaces was defined in Properties of Spaces, Definition 66.16.2. Here is what it means for a morphism to be étale at a point.
Lemma 67.39.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:
$f$ is étale,
for every $x \in |X|$ the morphism $f$ is étale at $x$,
for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is étale,
for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is étale,
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,
there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi $ is étale,
for every commutative diagram
where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is étale,
there exists a commutative diagram
where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ surjective such that the top horizontal arrow is étale, and
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.
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
Proof. Combine Properties of Spaces, Lemmas 66.16.3, 66.16.5 and 66.16.4. Some details omitted. $\square$
Lemma 67.39.3. The composition of two étale morphisms of algebraic spaces is étale.
Proof. This is a copy of Properties of Spaces, Lemma 66.16.4. $\square$
Lemma 67.39.4. 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 66.16.5. $\square$
Lemma 67.39.5. 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 66.16.2. By Properties of Spaces, Lemma 66.16.3 we see this means there exists a diagram as in Lemma 67.22.1 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. $\square$
Lemma 67.39.6. An étale morphism of algebraic spaces is smooth.
Proof. The proof is identical to the proof of Lemma 67.39.5. It uses the fact that an étale morphism of schemes is smooth (by definition of an étale morphism of schemes). $\square$
Lemma 67.39.7. An étale morphism of algebraic spaces is flat.
Proof. The proof is identical to the proof of Lemma 67.39.5. It uses Morphisms, Lemma 29.36.12. $\square$
Lemma 67.39.8. An étale morphism of algebraic spaces is locally of finite presentation.
Proof. The proof is identical to the proof of Lemma 67.39.5. It uses Morphisms, Lemma 29.36.11. $\square$
Lemma 67.39.9. 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 67.39.8 and 67.28.5. $\square$
Lemma 67.39.10. An étale morphism of algebraic spaces is unramified.
Proof. The proof is identical to the proof of Lemma 67.39.5. It uses Morphisms, Lemma 29.36.5. $\square$
Lemma 67.39.11. 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 66.16.6. $\square$
Lemma 67.39.12. 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 66.16.3 we see this means there exists a diagram as in Lemma 67.22.1 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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