Lemma 27.22.11. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A}$ is the directed colimit of its finite type quasi-coherent $\mathcal{O}_ X$-subalgebras.

**Proof.**
Omitted. Hint: Compare with the proof of Lemma 27.22.7.
$\square$

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