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. $\square$

## Comments (0)

There are also:

• 2 comment(s) on Section 66.16: Étale morphisms of algebraic spaces

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03FU. Beware of the difference between the letter 'O' and the digit '0'.