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.
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.
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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