The Stacks project

Lemma 28.22.8. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Then we can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ with $\mathcal{F}_ i$ of finite presentation and all transition maps $\mathcal{F}_ i \to \mathcal{F}_{i'}$ surjective.

Proof. Write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ as a filtered colimit of finitely presented $\mathcal{O}_ X$-modules (Lemma 28.22.7). We claim that $\mathcal{G}_ i \to \mathcal{F}$ is surjective for some $i$. Namely, choose a finite affine open covering $X = U_1 \cup \ldots \cup U_ m$. Choose sections $s_{jl} \in \mathcal{F}(U_ j)$ generating $\mathcal{F}|_{U_ j}$, see Lemma 28.16.1. By Sheaves, Lemma 6.29.1 we see that $s_{jl}$ is in the image of $\mathcal{G}_ i \to \mathcal{F}$ for $i$ large enough. Hence $\mathcal{G}_ i \to \mathcal{F}$ is surjective for $i$ large enough. Choose such an $i$ and let $\mathcal{K} \subset \mathcal{G}_ i$ be the kernel of the map $\mathcal{G}_ i \to \mathcal{F}$. Write $\mathcal{K} = \mathop{\mathrm{colim}}\nolimits \mathcal{K}_ a$ as the filtered colimit of its finite type quasi-coherent submodules (Lemma 28.22.3). Then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i/\mathcal{K}_ a$ is a solution to the problem posed by the lemma. $\square$

Comments (2)

Comment #2653 by BB on

The limit in the statement of the lemma should probably be a colimit.

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