Lemma 18.11.2. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$ The sheafification functor

is exact.

Lemma 18.11.2. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$ The sheafification functor

\[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}^\# ), \quad \mathcal{F} \longmapsto \mathcal{F}^\# \]

is exact.

**Proof.**
This is true because it holds for sheafification $\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})$. See the discussion in Section 18.3.
$\square$

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