Lemma 69.13.1 (07UJ)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 69: Cohomology of Algebraic Spaces
Section 69.13: Coherent sheaves on Noetherian spaces
Lemma 69.13.1 (cite)

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Lemma 69.13.1. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The ascending chain condition holds for quasi-coherent submodules of $\mathcal{F}$. In other words, given any sequence

\[ \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F} \]

of quasi-coherent submodules, then $\mathcal{F}_ n = \mathcal{F}_{n + 1} = \ldots $ for some $n \geq 0$.

Proof. Choose an affine scheme $U$ and a surjective étale morphism $U \to X$ (see Properties of Spaces, Lemma 66.6.3). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma 67.23.5). If $\mathcal{F}_ n|_ U = \mathcal{F}_{n + 1}|_ U = \ldots $ then $\mathcal{F}_ n = \mathcal{F}_{n + 1} = \ldots $. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma 30.10.1. $\square$

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