
Lemma 18.28.7. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings.

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

2. Any presheaf of $\mathcal{O}$-modules is a quotient of a flat presheaf of $\mathcal{O}$-modules.

3. If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, then any sheaf of $\mathcal{O}$-modules is a quotient of a direct sum $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$.

4. If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, then any sheaf of $\mathcal{O}$-modules is a quotient of a flat sheaf of $\mathcal{O}$-modules.

Proof. Proof of (1). For every object $U$ of $\mathcal{C}$ 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. The source is flat by combining Lemmas 18.28.4 and 18.28.6 which proves (2). The sheaf case follows from this either by sheafifying or repeating the same argument. $\square$

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