The Stacks project

Remark 68.4.1. Before we give the proof of the next lemma let us recall some facts about étale morphisms of schemes:

  1. An étale morphism is flat and hence generalizations lift along an étale morphism (Morphisms, Lemmas 29.36.12 and 29.25.9).

  2. An étale morphism is unramified, an unramified morphism is locally quasi-finite, hence fibres are discrete (Morphisms, Lemmas 29.36.16, 29.35.10, and 29.20.6).

  3. A quasi-compact étale morphism is quasi-finite and in particular has finite fibres (Morphisms, Lemmas 29.20.9 and 29.20.10).

  4. An étale scheme over a field $k$ is a disjoint union of spectra of finite separable field extension of $k$ (Morphisms, Lemma 29.36.7).

For a general discussion of étale morphisms, please see Étale Morphisms, Section 41.11.

Comments (0)

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Your email address will not be published.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 03II. Beware of the difference between the letter 'O' and the digit '0'.