The Stacks project

Closed immersions of finite presentation correspond to quasi-coherent sheaves of ideals of finite type.

Lemma 29.21.7. 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. On any affine open $\mathop{\mathrm{Spec}}(R) \subset X$ we have $i^{-1}(\mathop{\mathrm{Spec}}(R)) = \mathop{\mathrm{Spec}}(R/I)$ and $\mathcal{I} = \widetilde{I}$. Moreover, $\mathcal{I}$ is of finite type if and only if $I$ is a finite $R$-module for every such affine open (see Properties, Lemma 28.16.1). And $R/I$ is of finite presentation over $R$ if and only if $I$ is a finite $R$-module. Hence we win. $\square$

Comments (1)

Comment #1681 by Michele Serra on

Suggested slogan: Closed immersions correspond to quasi-coherent sheaves of ideals of finite type.

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