Lemma 26.21.10. Let $g : X \to Y$ be a morphism of schemes over $S$. The morphism $i : X \to X \times _ S Y$ is an immersion. If $Y$ is separated over $S$ it is a closed immersion. If $Y$ is quasi-separated over $S$ it is quasi-compact.

**Proof.**
This is a special case of Lemma 26.21.9 applied to the morphism $X = X \times _ Y Y \to X \times _ S Y$.
$\square$

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