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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 (3)

Comment #1681 by Michele Serra on

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

Comment #8113 by Elías Guisado on

The additional equivalence with “ is a finitely presented -module” follows easily from the statement (and maybe it is worth adding it?). On the one hand, if is of finite presentation, it follows that is of finite type by 17.11.3. For the converse, consider the exact sequence If is of finite type, then given , we can find a surjection in some open neighborhood of , and, hence, we can replace by in the last sequence (after restricting) while maintaining exactness. Thus, is of finite presentation.

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