The Stacks project

Lemma 24.25.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $T$ be a set and for each $t \in T$ let $\mathcal{I}_ t$ be a graded injective diffential graded $\mathcal{A}$-module. Then $\prod \mathcal{I}_ t$ is a graded injective differential graded $\mathcal{A}$-module.

Proof. This is true because products of injectives are injectives, see Homology, Lemma 12.27.3, and because products in $\textit{Mod}(\mathcal{A}, \text{d})$ are compatible with products in $\textit{Mod}(\mathcal{A})$ via the forgetful functor. $\square$

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