Description of the étale schemes over fields and fibres of étale morphisms.

Lemma 28.34.7. Fibres of étale morphisms.

1. Let $X$ be a scheme over a field $k$. The structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ is étale if and only if $X$ is a disjoint union of spectra of finite separable field extensions of $k$.

2. If $f : X \to S$ is an étale morphism, then for every $s \in S$ the fibre $X_ s$ is a disjoint union of spectra of finite separable field extensions of $\kappa (s)$.

Proof. You can deduce this from Lemma 28.33.11 via Lemma 28.34.5 above. Here is a direct proof.

We will use Algebra, Lemma 10.141.4. Hence it is clear that if $X$ is a disjoint union of spectra of finite separable field extensions of $k$ then $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Conversely, suppose that $X \to \mathop{\mathrm{Spec}}(k)$ is étale. Then for any affine open $U \subset X$ we see that $U$ is a finite disjoint union of spectra of finite separable field extensions of $k$. Hence all points of $X$ are closed points (see Lemma 28.19.2 for example). Thus $X$ is a discrete space and we win. $\square$

## Comments (1)

Comment #1039 by Jakob Scholbach on

Suggested slogan: Description of the étale site of a field.

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Your email address will not be published.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