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 19.13.4. Let $\mathcal{A}$ be a Grothendieck abelian category. Then

  1. $D(\mathcal{A})$ has both direct sums and products,

  2. direct sums are obtained by taking termwise direct sums of any complexes,

  3. products are obtained by taking termwise products of K-injective complexes.

Proof. Let $K^\bullet _ i$, $i \in I$ be a family of objects of $D(\mathcal{A})$ indexed by a set $I$. We claim that the termwise direct sum $\bigoplus _{i \in I} K^\bullet _ i$ is a direct sum in $D(\mathcal{A})$. Namely, let $I^\bullet $ be a K-injective complex. Then we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(\bigoplus \nolimits _{i \in I} K^\bullet _ i, I^\bullet ) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(\bigoplus \nolimits _{i \in I} K^\bullet _ i, I^\bullet ) \\ & = \prod \nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(K^\bullet _ i, I^\bullet ) \\ & = \prod \nolimits _{i \in I} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(K^\bullet _ i, I^\bullet ) \end{align*}

as desired. This is sufficient since any complex can be represented by a K-injective complex by Theorem 19.12.6. To construct the product, choose a K-injective resolution $K_ i^\bullet \to I_ i^\bullet $ for each $i$. Then we claim that $\prod _{i \in I} I_ i^\bullet $ is a product in $D(\mathcal{A})$. This follows from Derived Categories, Lemma 13.29.5. $\square$


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