Lemma 30.10.1. Let $X$ be a Noetherian scheme. 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 a finite affine open covering. On each member of the covering we get stabilization by Algebra, Lemma 10.51.1. Hence the lemma follows. $\square$

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