Lemma 18.28.9. Let $\mathcal{C}$ be a category. Let $\mathcal{O}$ be a presheaf of rings. Let

$0 \to \mathcal{F}'' \to \mathcal{F}' \to \mathcal{F} \to 0$

be a short exact sequence of presheaves of $\mathcal{O}$-modules. Let $\mathcal{G}$ be a presheaf of $\mathcal{O}$-modules.

1. If $\mathcal{F}$ is a flat presheaf of modules, then the sequence

$0 \to \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G} \to \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G} \to 0$

is exact.

2. If $\mathcal{C}$ is a site, $\mathcal{O}$, $\mathcal{F}$, $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{G}$ are sheaves, and $\mathcal{F}$ is flat as a sheaf of modules, then the sequence

$0 \to \mathcal{F}'' \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F}' \otimes _\mathcal {O} \mathcal{G} \to \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \to 0$

is exact.

Proof. Choose a flat presheaf of $\mathcal{O}$-modules $\mathcal{G}'$ which surjects onto $\mathcal{G}$. This is possible by Lemma 18.28.8. Let $\mathcal{G}'' = \mathop{\mathrm{Ker}}(\mathcal{G}' \to \mathcal{G})$. The lemma follows by applying the snake lemma to the following diagram

$\begin{matrix} & & 0 & & 0 & & 0 & & \\ & & \uparrow & & \uparrow & & \uparrow & & \\ & & \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G} & \to & \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G} & \to & \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G} & \to & 0 \\ & & \uparrow & & \uparrow & & \uparrow & & \\ 0 & \to & \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G}' & \to & \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G}' & \to & \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}' & \to & 0 \\ & & \uparrow & & \uparrow & & \uparrow & & \\ & & \mathcal{F}'' \otimes _{p, \mathcal{O}} \mathcal{G}'' & \to & \mathcal{F}' \otimes _{p, \mathcal{O}} \mathcal{G}'' & \to & \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}'' & \to & 0 \\ & & & & & & \uparrow & & \\ & & & & & & 0 & & \end{matrix}$

with exact rows and columns. The middle row is exact because tensoring with the flat module $\mathcal{G}'$ is exact. The proof in the case of sheaves is exactly the same. $\square$

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