The Stacks project

Lemma 37.67.4. In Situation 37.67.1. If $X$ and $Y$ are separated, then the pushout $Y \amalg _ Z X$ (Proposition 37.67.3) is separated. Same with “separated over $S$”, “quasi-separated”, and “quasi-separated over $S$”.

Proof. The morphism $Y \amalg X \to Y \amalg _ Z X$ is surjective and universall closed. Thus we may apply Morphisms, Lemma 29.41.11. $\square$

Comments (1)

Comment #9450 by Ivan on

Typo in the proof: says "universall" instead of "universally"

There are also:

  • 4 comment(s) on Section 37.67: Pushouts in the category of schemes, II

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Your email address will not be published.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 0E26. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 37.67.4 as pdf