The Stacks project

Lemma 18.28.4. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}$-modules. Assume that every object $U$ of $\mathcal{C}$ has a covering $\{ U_ i \to U\} _{i \in I}$ such that $\mathcal{F}(U_ i)$ is a flat $\mathcal{O}(U_ i)$-module. Then $\mathcal{F}^\# $ is a flat $\mathcal{O}^\# $-module.

Proof. Let $\mathcal{G} \subset \mathcal{G}'$ be an inclusion of $\mathcal{O}^\# $-modules. We have to show that

\[ \mathcal{G} \otimes _{\mathcal{O}^\# } \mathcal{F}^\# \longrightarrow \mathcal{G}' \otimes _{\mathcal{O}^\# } \mathcal{F}^\# \]

is injective. By Lemma 18.26.1 the source of this arrow is the sheafification of the presheaf $\mathcal{G} \otimes _{p, \mathcal{O}} \mathcal{F}$ and similarly for the target. If $U$ is an object of $\mathcal{C}$ such that $\mathcal{F}(U)$ is a flat $\mathcal{O}(U)$-module, then

\[ (\mathcal{G} \otimes _{p, \mathcal{O}} \mathcal{F})(U) = \mathcal{G}(U) \otimes _{\mathcal{O}(U)} \mathcal{F}(U) \longrightarrow \mathcal{G}'(U) \otimes _{\mathcal{O}(U)} \mathcal{F}(U) = (\mathcal{G}' \otimes _{p, \mathcal{O}} \mathcal{F})(U) \]

is injective. Hence we reduce to showing: given a map of presheaves $f : \mathcal{H} \to \mathcal{H}'$ on $\mathcal{C}$ such that every $U$ in $\mathcal{C}$ has a covering $\{ U_ i \to U\} _{i \in I}$ with $\mathcal{H}(U_ i) \to \mathcal{H}'(U_ i)$ injective, then $f^\# $ is injective. This we leave to the reader as an exercise. $\square$

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