66.16 Étale morphisms of algebraic spaces
This section really belongs in the chapter on morphisms of algebraic spaces, but we need the notion of an algebraic space étale over another in order to define the small étale site of an algebraic space. Thus we need to do some preliminary work on étale morphisms from schemes to algebraic spaces, and étale morphisms between algebraic spaces. For more about étale morphisms of algebraic spaces, see Morphisms of Spaces, Section 67.39.
Lemma 66.16.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$, $U'$ be schemes over $S$.
If $U \to U'$ is an étale morphism of schemes, and if $U' \to X$ is an étale morphism from $U'$ to $X$, then the composition $U \to X$ is an étale morphism from $U$ to $X$.
If $\varphi : U \to X$ and $\varphi ' : U' \to X$ are étale morphisms towards $X$, and if $\chi : U \to U'$ is a morphism of schemes such that $\varphi = \varphi ' \circ \chi $, then $\chi $ is an étale morphism of schemes.
If $\chi : U \to U'$ is a surjective étale morphism of schemes and $\varphi ' : U' \to X$ is a morphism such that $\varphi = \varphi ' \circ \chi $ is étale, then $\varphi '$ is étale.
Proof.
Recall that our definition of an étale morphism from a scheme into an algebraic space comes from Spaces, Definition 65.5.1 via the fact that any morphism from a scheme into an algebraic space is representable.
Part (1) of the lemma follows from this, the fact that étale morphisms are preserved under composition (Morphisms, Lemma 29.36.3) and Spaces, Lemmas 65.5.4 and 65.5.3 (which are formal).
To prove part (2) choose a scheme $W$ over $S$ and a surjective étale morphism $W \to X$. Consider the base change $\chi _ W : W \times _ X U \to W \times _ X U'$ of $\chi $. As $W \times _ X U$ and $W \times _ X U'$ are étale over $W$, we conclude that $\chi _ W$ is étale, by Morphisms, Lemma 29.36.18. On the other hand, in the commutative diagram
\[ \xymatrix{ W \times _ X U \ar[r] \ar[d] & W \times _ X U' \ar[d] \\ U \ar[r] & U' } \]
the two vertical arrows are étale and surjective. Hence by Descent, Lemma 35.14.4 we conclude that $U \to U'$ is étale.
To prove part (3) choose a scheme $W$ over $S$ and a morphism $W \to X$. As above we consider the diagram
\[ \xymatrix{ W \times _ X U \ar[r] \ar[d] & W \times _ X U' \ar[d] \ar[r] & W \ar[d] \\ U \ar[r] & U' \ar[r] & X } \]
Now we know that $W \times _ X U \to W \times _ X U'$ is surjective étale (as a base change of $U \to U'$) and that $W \times _ X U \to W$ is étale. Thus $W \times _ X U' \to W$ is étale by Descent, Lemma 35.14.4. By definition this means that $\varphi '$ is étale.
$\square$
Definition 66.16.2. Let $S$ be a scheme. A morphism $f : X \to Y$ between algebraic spaces over $S$ is called étale if and only if for every étale morphism $\varphi : U \to X$ where $U$ is a scheme, the composition $f \circ \varphi $ is étale also.
If $X$ and $Y$ are schemes, then this agree with the usual notion of an étale morphism of schemes. In fact, whenever $X \to Y$ is a representable morphism of algebraic spaces, then this agrees with the notion defined via Spaces, Definition 65.5.1. This follows by combining Lemma 66.16.3 below and Spaces, Lemma 65.11.4.
Lemma 66.16.3. 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,
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),
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),
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 66.16.1 (1). Since also the projection $W \times _ X U \to W$ is surjective and étale, we conclude from Lemma 66.16.1 (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 65.11.6. By assumption the morphism $U \to Y$ is étale, and hence $U \to V$ is étale by Lemma 66.16.1 (2).
We omit the proof that (2) and (3) are also equivalent to (1).
$\square$
Lemma 66.16.4. The composition of two étale morphisms of algebraic spaces is étale.
Proof.
This is immediate from the definition.
$\square$
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.
$\square$
Lemma 66.16.6. Let $S$ be a scheme. Let $X, Y, Z$ be algebraic spaces. Let $g : X \to Z$, $h : Y \to Z$ be étale morphisms and let $f : X \to Y$ be a morphism such that $h \circ f = g$. Then $f$ is étale.
Proof.
Choose a commutative diagram
\[ \xymatrix{ U \ar[d] \ar[r]_\chi & V \ar[d] \\ X \ar[r] & Y } \]
where $U \to X$ and $V \to Y$ are surjective and étale, see Spaces, Lemma 65.11.6. By assumption the morphisms $\varphi : U \to X \to Z$ and $\psi : V \to Y \to Z$ are étale. Moreover, $\psi \circ \chi = \varphi $ by our assumption on $f, g, h$. Hence $U \to V$ is étale by Lemma 66.16.1 part (2).
$\square$
Lemma 66.16.7. Let $S$ be a scheme. If $X \to Y$ is an étale morphism of algebraic spaces over $S$, then the associated map $|X| \to |Y|$ of topological spaces is open.
Proof.
This is clear from the diagram in Lemma 66.16.3 and Lemma 66.4.6.
$\square$
Finally, here is a fun lemma. It is not true that an algebraic space with an étale morphism towards a scheme is a scheme, see Spaces, Example 65.14.2. But it is true if the target is the spectrum of a field.
Lemma 66.16.8. Let $S$ be a scheme. Let $X \to \mathop{\mathrm{Spec}}(k)$ be étale morphism over $S$, where $k$ is a field. Then $X$ is a scheme.
Proof.
Let $U$ be an affine scheme, and let $U \to X$ be an étale morphism. By Definition 66.16.2 we see that $U \to \mathop{\mathrm{Spec}}(k)$ is an étale morphism. Hence $U = \coprod _{i = 1, \ldots , n} \mathop{\mathrm{Spec}}(k_ i)$ is a finite disjoint union of spectra of finite separable extensions $k_ i$ of $k$, see Morphisms, Lemma 29.36.7. The $R = U \times _ X U \to U \times _{\mathop{\mathrm{Spec}}(k)} U$ is a monomorphism and $U \times _{\mathop{\mathrm{Spec}}(k)} U$ is also a finite disjoint union of spectra of finite separable extensions of $k$. Hence by Schemes, Lemma 26.23.11 we see that $R$ is similarly a finite disjoint union of spectra of finite separable extensions of $k$. This $U$ and $R$ are affine and both projections $R \to U$ are finite locally free. Hence $U/R$ is a scheme by Groupoids, Proposition 39.23.9. By Spaces, Lemma 65.10.2 it is also an open subspace of $X$. By Lemma 66.13.1 we conclude that $X$ is a scheme.
$\square$
Comments (1)
Comment #8589 by Max L. on