Lemma 17.17.6. Let $(X, \mathcal{O}_ X)$ be a ringed space.

1. Any sheaf of $\mathcal{O}_ X$-modules is a quotient of a direct sum $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$.

2. Any $\mathcal{O}_ X$-module is a quotient of a flat $\mathcal{O}_ X$-module.

Proof. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. For every open $U \subset X$ and every $s \in \mathcal{F}(U)$ we get a morphism $j_{U!}\mathcal{O}_ U \to \mathcal{F}$, namely the adjoint to the morphism $\mathcal{O}_ U \to \mathcal{F}|_ U$, $1 \mapsto s$. Clearly the map

$\bigoplus \nolimits _{(U, s)} j_{U!}\mathcal{O}_ U \longrightarrow \mathcal{F}$

is surjective, and the source is flat by combining Lemmas 17.17.4 and 17.17.5. $\square$

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