Lemma 29.21.7 (01TV)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.21: Morphisms of finite presentation
Lemma 29.21.7 (cite)

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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 October 22, 2015 at 09:58

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

Comment #8113 by Elías Guisado on February 18, 2023 at 11:24

The additional equivalence with “ $i_*\mathcal{O}_Z$ is a finitely presented $\mathcal{O}_X$ -module” follows easily from the statement (and maybe it is worth adding it?). On the one hand, if $i_*\mathcal{O}_Z$ is of finite presentation, it follows that $\mathcal{I}$ is of finite type by 17.11.3. For the converse, consider the exact sequence $\mathcal{I}\to\mathcal{O}_X\to i_*\mathcal{O}_Z\to 0.$ If $\mathcal{I}$ is of finite type, then given $x\in X$ , we can find a surjection $\mathcal{O}_U^{\oplus r}\to\mathcal{I}|_U$ in some open neighborhood $U\subset X$ of $x$ , and, hence, we can replace $\mathcal{I}$ by $\mathcal{O}_U^{\oplus r}$ in the last sequence (after restricting) while maintaining exactness. Thus, $i_*\mathcal{O}_Z$ is of finite presentation.

Comment #8222 by Stacks Project on March 03, 2023 at 12:54

I agree with what you wrote, but I am going to leave this as is for now.

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