Lemma 70.9.1 (07V9)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 70: Limits of Algebraic Spaces
Section 70.9: Applications
Lemma 70.9.1 (cite)

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Lemma 70.9.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$ . Every quasi-coherent $\mathcal{O}_ X$ -module is a filtered colimit of finitely presented $\mathcal{O}_ X$ -modules.

Proof. We may view $X$ as an algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ , see Spaces, Definition 65.16.2 and Properties of Spaces, Definition 66.3.1. Thus we may apply Proposition 70.8.1 and write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i$ of finite presentation over $\mathbf{Z}$ . Thus $X_ i$ is a Noetherian algebraic space, see Morphisms of Spaces, Lemma 67.28.6. The morphism $X \to X_ i$ is affine, see Lemma 70.4.1. Conclusion by Cohomology of Spaces, Lemma 69.15.2. $\square$

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