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. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.