\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 18.3.2. Let $\mathcal{C}$ be a site.

1. All limits and colimits exist in $\textit{Ab}(\mathcal{C})$.

2. Limits are the same as the corresponding limits of abelian presheaves over $\mathcal{C}$ (i.e., commute with taking sections over objects of $\mathcal{C}$).

3. Finite direct sums are the same as the corresponding finite direct sums in the category of abelian pre-sheaves over $\mathcal{C}$.

4. A colimit is the sheafification of the corresponding colimit in the category of abelian presheaves.

5. Filtered colimits are exact.

Proof. By Lemma 18.2.1 limits and colimits of abelian presheaves exist, and are described by taking limits and colimits on the level of sections over objects.

Let $\mathcal{I} \to \textit{Ab}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Let $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ be the limit of the diagram as an abelian presheaf. By Sites, Lemma 7.10.1 this is an abelian sheaf. Then it is quite easy to see that $\mathop{\mathrm{lim}}\nolimits _ i \mathcal{F}_ i$ is the limit of the diagram in $\textit{Ab}(\mathcal{C})$. This proves limits exist and (2) holds.

By Categories, Lemma 4.24.5, and because sheafification is left adjoint to the inclusion functor we see that $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}$ exists and is the sheafification of the colimit in $\textit{PAb}(\mathcal{C})$. This proves colimits exist and (4) holds.

Finite direct sums are the same thing as finite products in any abelian category. Hence (3) follows from (2).

Proof of (5). The statement means that given a system $0 \to \mathcal{F}_ i \to \mathcal{G}_ i \to \mathcal{H}_ i \to 0$ of exact sequences of abelian sheaves over a directed set $I$ the sequence $0 \to \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact as well. A formal argument using Homology, Lemma 12.5.8 and the definition of colimits shows that the sequence $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i \to 0$ is exact. Note that $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ i$ is the sheafification of the map of presheaf colimits which is injective as each of the maps $\mathcal{F}_ i \to \mathcal{G}_ i$ is injective. Since sheafification is exact we conclude. $\square$

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