The Stacks project

The diagonal morphism for relative schemes is an immersion.

Lemma 25.21.2. Let $X$ be a scheme over $S$. The diagonal morphism $\Delta _{X/S}$ is an immersion.

Proof. Recall that if $V \subset X$ is affine open and maps into $U \subset S$ affine open, then $V \times _ U V$ is affine open in $X \times _ S X$, see Lemmas 25.17.2 and 25.17.3. Consider the open subscheme $W$ of $X \times _ S X$ which is the union of these affine opens $V \times _ U V$. By Lemma 25.4.2 it is enough to show that each morphism $\Delta _{X/S}^{-1}(V \times _ U V) \to V \times _ U V$ is a closed immersion. Since $V = \Delta _{X/S}^{-1}(V \times _ U V)$ we are just checking that $\Delta _{V/U}$ is a closed immersion, which is Lemma 25.21.1. $\square$

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