The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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