Lemma 70.11.4. 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.
Closed immersions of qcqs algebraic spaces can be approximated by finitely presented closed immersions.
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 70.9.2 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
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Comment #860 by Bhargav Bhatt on