The Stacks project

Lemma 29.36.19. Let

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[dl]^ q \\ & S } \]

be a commutative diagram of morphisms of schemes. Assume that

  1. $f$ is surjective, and étale,

  2. $p$ is étale, and

  3. $q$ is locally of finite presentation1.

Then $q$ is étale.

Proof. By Lemma 29.34.19 we see that $q$ is smooth. Thus we only need to see that $q$ has relative dimension $0$. This follows from Lemma 29.28.2 and the fact that $f$ and $p$ have relative dimension $0$. $\square$

[1] In fact this is implied by (1) and (2), see Descent, Lemma 35.14.3. Moreover, it suffices to assume that $f$ is surjective, flat and locally of finite presentation, see Descent, Lemma 35.14.5.

Comments (0)

There are also:

  • 3 comment(s) on Section 29.36: Étale morphisms

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Your email address will not be published.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 02K6. Beware of the difference between the letter 'O' and the digit '0'.