The Stacks project

Lemma 66.28.12. A closed immersion $i : Z \to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to i_*\mathcal{O}_ Z)$ is of finite type (as an $\mathcal{O}_ X$-module).

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective ├ętale morphism. By Lemma 66.28.4 we see that $i' : Z \times _ X U \to U$ is of finite presentation if and only if $i$ is. By Properties of Spaces, Section 65.30 we see that $\mathcal{I}$ is of finite type if and only if $\mathcal{I}|_ U = \mathop{\mathrm{Ker}}(\mathcal{O}_ U \to i'_*\mathcal{O}_{Z \times _ X U})$ is. Hence the result follows from the case of schemes, see Morphisms, Lemma 29.21.7. $\square$

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