Lemma 32.7.3 (09YZ)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.7: Relative approximation
Lemma 32.7.3 (cite)

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Lemma 32.7.3. Let $X \to S$ be an integral morphism with $S$ quasi-compact and quasi-separated. Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i \to S$ finite and of finite presentation.

Proof. Consider the sheaf $\mathcal{A} = f_*\mathcal{O}_ X$ . This is a quasi-coherent sheaf of $\mathcal{O}_ S$ -algebras, see Schemes, Lemma 26.24.1. Combining Properties, Lemma 28.22.13 we can write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{A}_ i$ as a filtered colimit of finite and finitely presented $\mathcal{O}_ S$ -algebras. Then

$X_ i = \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}_ i) \longrightarrow S$

is a finite and finitely presented morphism of schemes. By construction $X = \mathop{\mathrm{lim}}\nolimits _ i X_ i$ which proves the lemma. $\square$

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