Closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions.

Lemma 69.11.4. Let $S$ be a scheme. Let $f : X \to Y$ be a closed immersion of algebraic spaces over $S$. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ where the transition maps are closed immersions and the morphisms $X_ i \to Y$ are closed immersions of finite presentation.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ Y$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subspace of $Y$. By Lemma 69.9.2 we can write $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits \mathcal{I}_ i$ as the filtered colimit of its finite type quasi-coherent submodules. Let $X_ i$ be the closed subspace of $X$ cut out by $\mathcal{I}_ i$. Then $X_ i \to Y$ is a closed immersion of finite presentation, and $X = \mathop{\mathrm{lim}}\nolimits X_ i$. Some details omitted. $\square$

Comment #860 by Bhargav Bhatt on

Suggested slogan: Arbitrary closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions.

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