The Stacks project

Lemma 30.10.4. Let $X$ be a Noetherian scheme. Every quasi-coherent $\mathcal{O}_ X$-module is the filtered colimit of its coherent submodules.

Proof. This is a reformulation of Properties, Lemma 28.22.3 in view of the fact that a finite type quasi-coherent $\mathcal{O}_ X$-module is coherent by Lemma 30.9.1. $\square$

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