The Stacks project

[Expose I, Corollary 9.11, SGA1]

Lemma 37.36.2. Let $f : X \to Y$ be a morphism of schemes. Assume

  1. $Y$ is integral and geometrically unibranch,

  2. at least one irreducible component of $X$ dominates $Y$,

  3. $f$ is unramified, and

  4. $X$ is connected.

Then $f$ is étale and $X$ is irreducible.

Proof. Let $X' \subset X$ be the irreducible component which dominates $Y$. This means that the generic point of $X'$ maps to the generic point of $Y$ (see for example Morphisms, Lemma 29.8.5). By Lemma 37.36.1 we see that $f$ is étale at every point of $X'$. In particular, the open subscheme $U \subset X$ where $f$ is étale contains $X'$. Note that every quasi-compact open of $U$ has finitely many irreducible components, see Descent, Lemma 35.15.3. On the other hand since $Y$ is geometrically unibranch and $U$ is étale over $Y$, the scheme $U$ is geometrically unibranch. In particular, through every point of $U$ there passes at most one irreducible component. A simple topological argument now shows that $X' \subset U$ is both open and closed. Then of course $X'$ is open and closed in $X$ and by connectedness we find $X = U = X'$ as desired. $\square$


Comments (2)

Comment #7216 by Alex Ivanov on

It seems that assumption (2) is not necessary, as it follows from assumption (4).


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Your email address will not be published.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 0GS9. Beware of the difference between the letter 'O' and the digit '0'.