The Stacks project

Lemma 24.26.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Then

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

  2. direct sums are obtained by taking direct sums of differential graded $\mathcal{A}$-modules,

  3. products are obtained by taking products of K-injective differential graded modules.

Proof. We will use that $\textit{Mod}(\mathcal{A}, \text{d})$ is an abelian category with arbitrary direct sums and products, and that these give rise to direct sums and products in $K(\textit{Mod}(\mathcal{A}, \text{d}))$. See Lemmas 24.13.2 and 24.21.3.

Let $\mathcal{M}_ j$ be a family of differential graded $\mathcal{A}$-modules. Consider the direct sum $\mathcal{M} = \bigoplus \mathcal{M}_ j$ as a differential graded $\mathcal{A}$-module. For a differential graded $\mathcal{A}$-module $\mathcal{N}$ choose a quasi-isomorphism $\mathcal{N} \to \mathcal{I}$ where $\mathcal{I}$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. See Theorem 24.25.13. Using Lemma 24.26.7 we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{N}) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{M}, \mathcal{I}) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{M}_ j, \mathcal{I}) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{M}_ j, \mathcal{I}) \end{align*}

whence the existence of direct sums in $D(A, \text{d})$ as given in part (2) of the lemma.

Let $\mathcal{M}_ j$ be a family of differential graded $\mathcal{A}$-modules. For each $j$ choose a quasi-isomorphism $\mathcal{M} \to \mathcal{I}_ j$ where $\mathcal{I}_ j$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. Consider the product $\mathcal{I} = \prod \mathcal{I}_ j$ of differential graded $\mathcal{A}$-modules. By Lemmas 24.25.8 and 24.25.4 we see that $\mathcal{I}$ is graded injective and K-injective as a differential graded $\mathcal{A}$-module. For a differential graded $\mathcal{A}$-module $\mathcal{N}$ using Lemma 24.26.7 we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}) & = \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{I}_ j) \\ & = \prod \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A}, \text{d})}(\mathcal{N}, \mathcal{M}_ j) \end{align*}

whence the existence of products in $D(\mathcal{A}, \text{d})$ as given in part (3) of the lemma. $\square$

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