The Stacks project

Lemma 28.22.11. 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. If $\mathcal{A}_1, \mathcal{A}_2 \subset \mathcal{A}$ are quasi-coherent $\mathcal{O}_ X$-subalgebras of finite type, then the image of $\mathcal{A}_1 \otimes _{\mathcal{O}_ X} \mathcal{A}_2 \to \mathcal{A}$ is also a quasi-coherent $\mathcal{O}_ X$-subalgebra of finite type (some details omitted) which contains both $\mathcal{A}_1$ and $\mathcal{A}_2$. In this way we see that the system is directed. To show that $\mathcal{A}$ is the colimit of this system, write $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{A}_ i$ as a directed colimit of finitely presented quasi-coherent $\mathcal{O}_ X$-algebras as in Lemma 28.22.10. Then the images $\mathcal{A}'_ i = \mathop{\mathrm{Im}}(\mathcal{A}_ i \to \mathcal{A})$ are quasi-coherent subalgebras of $\mathcal{A}$ of finite type. Since $\mathcal{A}$ is the colimit of these the result follows. $\square$

Comments (2)

Comment #7364 by Yijin Wang on

Typo in the proof of lemma 28.22.11: the first sentence should be 'A_1,A_2 ⊂A'

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