The Stacks project

Lemma 66.4.13. Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces over $S$.

  1. The morphism $f$ is locally separated.

  2. The morphism $f$ is (quasi-)separated in the sense of Definition 66.4.2 above if and only if $f$ is (quasi-)separated in the sense of Section 66.3.

In particular, if $f : X \to Y$ is a morphism of schemes over $S$, then $f$ is (quasi-)separated in the sense of Definition 66.4.2 if and only if $f$ is (quasi-)separated as a morphism of schemes.

Proof. This is the equivalence of (1) and (2) of Lemma 66.4.12 combined with the fact that any morphism of schemes is locally separated, see Schemes, Lemma 26.21.2. $\square$

Comments (0)

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Your email address will not be published.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 03KY. Beware of the difference between the letter 'O' and the digit '0'.