The Stacks project

Lemma 26.21.1. The diagonal morphism of a morphism between affines is closed.

Proof. The diagonal morphism associated to the morphism $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is the morphism on spectra corresponding to the ring map $S \otimes _ R S \to S$, $a \otimes b \mapsto ab$. This map is clearly surjective, so $S \cong S \otimes _ R S/J$ for some ideal $J \subset S \otimes _ R S$. Hence $\Delta $ is a closed immersion according to Example 26.8.1. $\square$

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