The Stacks project

Lemma 32.9.4. Let $X \to Y$ be a closed immersion of schemes. Assume $Y$ quasi-compact and quasi-separated. Then $X$ can be written as a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ of schemes over $Y$ where $X_ i \to Y$ is a closed immersion of finite presentation.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ Y$ be the quasi-coherent sheaf of ideals defining $X$ as a closed subscheme of $Y$. By Properties, Lemma 28.22.3 we can write $\mathcal{I}$ as a directed colimit $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{I}_ i$ of its quasi-coherent sheaves of ideals of finite type. Let $X_ i \subset Y$ be the closed subscheme defined by $\mathcal{I}_ i$. These form an inverse system of schemes indexed by $I$. The transition morphisms $X_ i \to X_{i'}$ are affine because they are closed immersions. Each $X_ i$ is quasi-compact and quasi-separated since it is a closed subscheme of $Y$ and $Y$ is quasi-compact and quasi-separated by our assumptions. We have $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ as follows directly from the fact that $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{I}_ a$. Each of the morphisms $X_ i \to Y$ is of finite presentation, see Morphisms, Lemma 29.21.7. $\square$

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