Lemma 22.22.4 (09QI)—The Stacks project

Lemma 22.22.4. Let $(A, \text{d})$ be a differential graded algebra. Then

$D(A, \text{d})$ has both direct sums and products,
direct sums are obtained by taking direct sums of differential graded modules,
products are obtained by taking products of differential graded modules.

Proof. We will use that $\text{Mod}_{(A, \text{d})}$ is an abelian category with arbitrary direct sums and products, and that these give rise to direct sums and products in $K(\text{Mod}_{(A, \text{d})})$ . See Lemmas 22.4.2 and 22.5.4.

Let $M_ j$ be a family of differential graded $A$ -modules. Consider the graded direct sum $M = \bigoplus M_ j$ which is a differential graded $A$ -module with the obvious. For a differential graded $A$ -module $N$ choose a quasi-isomorphism $N \to I$ where $I$ is a differential graded $A$ -module with property (I). See Lemma 22.21.4. Using Lemma 22.22.3 we have

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

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

Let $M_ j$ be a family of differential graded $A$ -modules. Consider the product $M = \prod M_ j$ of differential graded $A$ -modules. For a differential graded $A$ -module $N$ choose a quasi-isomorphism $P \to N$ where $P$ is a differential graded $A$ -module with property (P). See Lemma 22.20.4. Using Lemma 22.22.3 we have

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

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

The Stacks project

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