# The Stacks Project

## Tag 03EI

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. $\square$

The code snippet corresponding to this tag is a part of the file sites-modules.tex and is located in lines 873–883 (see updates for more information).

\begin{lemma}
\label{lemma-sheafification-exact}
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.
\end{lemma}

\begin{proof}
This is true because it holds for sheafification
$\textit{PAb}(\mathcal{C}) \to \textit{Ab}(\mathcal{C})$.
See the discussion in Section \ref{section-abelian-sheaves}.
\end{proof}

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