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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