Lemma 65.27.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and a monomorphism, then $f$ is separated and locally quasi-finite.

Proof. A monomorphism is separated, see Lemma 65.10.3. By Lemma 65.27.6 it suffices to prove the lemma after performing a base change by $Z \to Y$ with $Z$ affine. Hence we may assume that $Y$ is an affine scheme. Choose an affine scheme $U$ and an étale morphism $U \to X$. Since $X \to Y$ is locally of finite type the morphism of affine schemes $U \to Y$ is of finite type. Since $X \to Y$ is a monomorphism we have $U \times _ X U = U \times _ Y U$. In particular the maps $U \times _ Y U \to U$ are étale. Let $y \in Y$. Then either $U_ y$ is empty, or $\mathop{\mathrm{Spec}}(\kappa (u)) \times _{\mathop{\mathrm{Spec}}(\kappa (y))} U_ y$ is isomorphic to the fibre of $U \times _ Y U \to U$ over $u$ for some $u \in U$ lying over $y$. This implies that the fibres of $U \to Y$ are finite discrete sets (as $U \times _ Y U \to U$ is an étale morphism of affine schemes, see Morphisms, Lemma 29.36.7). Hence $U \to Y$ is quasi-finite, see Morphisms, Lemma 29.20.6. As $U \to X$ was an arbitrary étale morphism with $U$ affine this implies that $X \to Y$ is locally quasi-finite. $\square$

## 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Your email address will not be published.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 0463. Beware of the difference between the letter 'O' and the digit '0'.