Lemma 37.64.4. In Situation 37.64.1. If $X$ and $Y$ are separated, then the pushout $Y \amalg _ Z X$ (Proposition 37.64.3) 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. $\square$

There are also:

• 4 comment(s) on Section 37.64: Pushouts in the category of schemes, II

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).