The Stacks project

Universally injective étale maps are open immersions.

Lemma 66.51.2. Let $S$ be a scheme. Let $f : X \to Y$ be an étale and universally injective morphism of algebraic spaces over $S$. Then $f$ is an open immersion.

Proof. Let $T \to Y$ be a morphism from a scheme into $Y$. If we can show that $X \times _ Y T \to T$ is an open immersion, then we are done. Since being étale and being universally injective are properties of morphisms stable under base change (see Lemmas 66.39.4 and 66.19.5) we may assume that $Y$ is a scheme. Note that the diagonal $\Delta _{X/Y} : X \to X \times _ Y X$ is étale, a monomorphism, and surjective by Lemma 66.19.2. Hence we see that $\Delta _{X/Y}$ is an isomorphism (see Spaces, Lemma 64.5.9), in particular we see that $X$ is separated over $Y$. It follows that $X$ is a scheme too, by Proposition 66.50.2. Finally, $X \to Y$ is an open immersion by the fundamental theorem for étale morphisms of schemes, see Étale Morphisms, Theorem 41.14.1. $\square$

Comments (1)

Comment #1287 by on

Suggested slogan: Universally injective \'etale maps are open immersions.

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Your email address will not be published.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 05W5. Beware of the difference between the letter 'O' and the digit '0'.