The Stacks project

Lemma 29.36.8. Let $f : X \to S$ be a morphism of schemes. If $f$ is flat, locally of finite presentation, and for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$, then $f$ is étale.

Proof. You can deduce this from Algebra, Lemma 10.143.7. Here is another proof.

By Lemma 29.36.7 a fibre $X_ s$ is étale and hence smooth over $s$. By Lemma 29.34.3 we see that $X \to S$ is smooth. By Lemma 29.35.12 we see that $f$ is unramified. We conclude by Lemma 29.36.5. $\square$

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Your email address will not be published.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