Lemma 26.23.6. Let $j : X \to Y$ be a morphism of schemes. If $j$ is injective on points, then $j$ is separated.

Proof. Let $z$ be a point of $X \times _ Y X$. Then $x = \text{pr}_1(z)$ and $\text{pr}_2(z)$ are the same because $j$ maps these points to the same point $y$ of $Y$. Then we can choose an affine open neighbourhood $V \subset Y$ of $y$ and an affine open neighbourhood $U \subset X$ of $x$ with $j(U) \subset V$. Then $z \in U \times _ V U \subset X \times _ Y X$. Hence $X \times _ Y X$ is the union of the affine opens $U \times _ V U$. Since $\Delta _{X/Y}^{-1}(U \times _ V U) = U$ and since $U \to U \times _ V U$ is a closed immersion, we conclude that $\Delta _{X/Y}$ is a closed immersion (see argument in the proof of Lemma 26.21.2). $\square$

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