Lemma 24.25.8 (0FSV)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 24: Differential Graded Sheaves
Section 24.25: K-injective differential graded modules
Lemma 24.25.8 (cite)

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Lemma 24.25.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$ . Let $T$ be a set and for each $t \in T$ let $\mathcal{I}_ t$ be a K-injective diffential graded $\mathcal{A}$ -module. Then $\prod \mathcal{I}_ t$ is a K-injective differential graded $\mathcal{A}$ -module.

Proof. Let $\mathcal{K}$ be an acyclic differential graded $\mathcal{A}$ -module. Then we have

$\mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{K}, \prod \nolimits _{t \in T} \mathcal{I}_ t) = \prod \nolimits _{t \in T} \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{K}, \mathcal{I}_ t)$

because taking products in $\textit{Mod}(\mathcal{A}, \text{d})$ commutes with the forgetful functor to graded $\mathcal{A}$ -modules. Since taking products is an exact functor on the category of abelian groups we conclude. $\square$

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