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 18.29.2. Let $\mathcal{C}$ be a site. Let $W$ be a quasi-compact object of $\mathcal{C}$.

  1. The functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \mapsto \mathcal{F}(W)$ commutes with coproducts.

  2. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}(W)$ commutes with direct sums.

Proof. Proof of (1). Taking sections over $W$ commutes with filtered colimits with injective transition maps by Sites, Lemma 7.17.5. If $\mathcal{F}_ i$ is a family of sheaves of sets indexed by a set $I$. Then $\coprod \mathcal{F}_ i$ is the filtered colimit over the partially ordered set of finite subsets $E \subset I$ of the coproducts $\mathcal{F}_ E = \coprod _{i \in E} \mathcal{F}_ i$. Since the transition maps are injective we conclude.

Proof of (2). Let $\mathcal{F}_ i$ be a family of sheaves of $\mathcal{O}$-modules indexed by a set $I$. Then $\bigoplus \mathcal{F}_ i$ is the filtered colimit over the partially ordered set of finite subsets $E \subset I$ of the direct sums $\mathcal{F}_ E = \bigoplus _{i \in E} \mathcal{F}_ i$. A filtered colimit of abelian sheaves can be computed in the category of sheaves of sets. Moreover, for $E \subset E'$ the transition map $\mathcal{F}_ E \to \mathcal{F}_{E'}$ is injective (as sheafification is exact and the injectivity is clear on underlying presheaves). Hence it suffices to show the result for a finite index set by Sites, Lemma 7.17.5. The finite case is dealt with in Lemma 18.3.2 (it holds over any object of $\mathcal{C}$). $\square$


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