Lemma 74.6.1 (060U)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 74: Descent and Algebraic Spaces
Section 74.6: Descent of finiteness properties of modules
Lemma 74.6.1 (cite)

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Lemma 74.6.1. Let $X$ be an algebraic space over a scheme $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_ i}$ -module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_ X$ -module.

Proof. This follows from the case of schemes, see Descent, Lemma 35.7.1, by étale localization. $\square$

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