The Stacks project

Lemma 57.11.1. In the situation above the following are equivalent

  1. $X$ is unramified in $L$,

  2. $Y \to X$ is étale, and

  3. $Y \to X$ is finite étale.

Proof. Observe that $Y \to X$ is an integral morphism. In each case the morphism $Y \to X$ is locally of finite type by definition. Hence we find that in each case $Y \to X$ is finite by Morphisms, Lemma 29.44.4. In particular we see that (2) is equivalent to (3). An étale morphism is unramified, hence (2) implies (1).

Conversely, assume $Y \to X$ is unramified. Let $x \in X$. We can choose an étale neighbourhood $(U, u) \to (X, x)$ such that

\[ Y \times _ X U = \coprod V_ j \longrightarrow U \]

is a disjoint union of closed immersions, see Étale Morphisms, Lemma 41.17.3. Shrinking we may assume $U$ is quasi-compact. Then $U$ has finitely many irreducible components (Descent, Lemma 35.13.3). Since $U$ is normal (Descent, Lemma 35.15.2) the irreducible components of $U$ are open and closed (Properties, Lemma 28.7.5) and we may assume $U$ is irreducible. Then $U$ is an integral scheme whose generic point $\xi $ maps to the generic point of $X$. On the other hand, we know that $Y \times _ X U$ is the normalization of $U$ in $\mathop{\mathrm{Spec}}(L) \times _ X U$ by More on Morphisms, Lemma 37.17.2. Every point of $\mathop{\mathrm{Spec}}(L) \times _ X U$ maps to $\xi $. Thus every $V_ j$ contains a point mapping to $\xi $ by Morphisms, Lemma 29.53.9. Thus $V_ j \to U$ is an isomorphism as $U = \overline{\{ \xi \} }$. Thus $Y \times _ X U \to U$ is étale. By Descent, Lemma 35.20.29 we conclude that $Y \to X$ is étale over the image of $U \to X$ (an open neighbourhood of $x$). $\square$


Comments (2)

Comment #6534 by Tim Holzschuh on

Possible typo: "Hence we find that in each case the lemma is finite by Morphisms ..." I think "the lemma" should read sth. along the lines of "the morphism".


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Your email address will not be published.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