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The Stacks project

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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