Lemma 18.28.8 (03EW)—The Stacks project

Lemma 18.28.8. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings.

Any presheaf of $\mathcal{O}$ -modules is a quotient of a direct sum $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$ .
Any presheaf of $\mathcal{O}$ -modules is a quotient of a flat presheaf of $\mathcal{O}$ -modules.
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 $\bigoplus j_{U_ i!}\mathcal{O}_{U_ i}$ .
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 18.28.5 and 18.28.7 which proves (2). The sheaf case follows from this either by sheafifying or repeating the same argument. $\square$

The Stacks project

Comments (0)

Post a comment