Lemma 18.14.2. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. All limits and colimits exist in $\textit{Mod}(\mathcal{O})$ and the forgetful functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$ commutes with them. Moreover, filtered colimits are exact.

Proof. The final statement follows from the first as filtered colimits are exact in $\textit{Ab}(\mathcal{C})$ by Lemma 18.3.2. Let $\mathcal{I} \to \textit{Mod}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Let $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ be the limit of the diagram in $\textit{Ab}(\mathcal{C})$. By the description of this limit in Lemma 18.3.2 we see immediately that there exists a multiplication

$\mathcal{O} \times \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i \longrightarrow \mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$

which turns $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ into a sheaf of $\mathcal{O}$-modules. It is easy to see that this is the limit of the diagram in $\textit{Mod}(\mathcal{C})$. Let $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ be the colimit of the diagram in $\textit{PAb}(\mathcal{C})$. By the description of this colimit in the proof of Lemma 18.2.1 we see immediately that there exists a multiplication

$\mathcal{O} \times \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i \longrightarrow \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$

which turns $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ into a presheaf of $\mathcal{O}$-modules. Applying sheafification we get a sheaf of $\mathcal{O}$-modules $(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)^\#$, see Lemma 18.11.1. It is easy to see that $(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i)^\#$ is the colimit of the diagram in $\textit{Mod}(\mathcal{O})$, and by Lemma 18.3.2 forgetting the $\mathcal{O}$-module structure is the colimit in $\textit{Ab}(\mathcal{C})$. $\square$

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