The Stacks project

Lemma 65.16.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$, $U'$ be schemes over $S$.

  1. 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 $X$.

  2. 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.

  3. 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 64.5.1 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 64.5.4 and 64.5.3 (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. $\square$

Comments (2)

Comment #7430 by Taro konno on

In the proof of part (2), is etale by Lemma 29.36.18 instead of Lemma 29.36.19, I think. And there is typo: "To prove part (2)" should be "To prove part (3)"

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