Lemma 17.9.7. Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\mathcal{F}_ i, f_{ii'})$ be a system over $I$ consisting of sheaves of $\mathcal{O}_ X$-modules (see Categories, Section 4.21). Let $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ be the colimit. Assume (a) $I$ is directed, (b) $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module, and (c) $X$ is quasi-compact. Then there exists an $i$ such that $\mathcal{F}_ i \to \mathcal{F}$ is surjective. If the transition maps $f_{ii'}$ are injective then we conclude that $\mathcal{F} = \mathcal{F}_ i$ for some $i \in I$.

Proof. Let $x \in X$. There exists an open neighbourhood $U \subset X$ of $x$ and finitely many sections $s_ j \in \mathcal{F}(U)$, $j = 1, \ldots , m$ such that $s_1, \ldots , s_ m$ generate $\mathcal{F}$ as $\mathcal{O}_ U$-module. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ we may assume that each $s_ j$ comes from a section of $\mathcal{F}_ i$ for some $i \in I$. Hence, since $X$ is quasi-compact we can find a finite open covering $X = \bigcup _{j = 1, \ldots , m} U_ j$, and for each $j$ an index $i_ j$ and finitely many sections $s_{jl} \in \mathcal{F}_{i_ j}(U_ j)$ whose images generate the restriction of $\mathcal{F}$ to $U_ j$. Clearly, the lemma holds for any index $i \in I$ which is $\geq$ all $i_ j$. $\square$

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