The Stacks project

Lemma 37.37.3. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. Let $x \in X$ with image $y \in Y$. Assume

  1. $Y$ is integral and geometrically unibranch at $y$,

  2. $f$ is locally of finite type,

  3. $g \circ f$ is étale at $x$,

  4. there is a specialization $x' \leadsto x$ such that $f(x')$ is the generic point of $Y$.

Then $f$ is étale at $x$ and $g$ is étale at $y$.

Proof. The morphism $f$ is unramified at $x$ by Morphisms, Lemmas 29.35.16 and 29.36.5. Hence $f$ is étale at $x$ by Lemma 37.37.1. Then by étale descent we see that $g$ is étale at $y$, see for example Descent, Lemma 35.14.4. $\square$


Comments (1)

Comment #8619 by on

Condition (2) isn't necessary in the presence of (3): in a neighbourhood of the morphism is of finite type by (3) combined with Lemma 29.15.8 (and the trivial fact that etale morphisms are of finite type).


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Your email address will not be published.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