Theorem 24.25.13. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. For every differential graded $\mathcal{A}$-module $\mathcal{M}$ there exists a quasi-isomorphism $\mathcal{M} \to \mathcal{I}$ where $\mathcal{I}$ is a graded injective and K-injective differential graded $\mathcal{A}$-module. Moreover, the construction is functorial in $\mathcal{M}$.

Proof. Let $R$ and $\mathcal{M}_ r \to \mathcal{M}'_ r$ be a set of morphisms of $\textit{Mod}(\mathcal{A}, \text{d})$ found in Lemma 24.25.11. Let $M$ with transformation $\text{id} \to M$ be as constructed in Lemma 24.25.12 using $R$ and $\mathcal{M}_ r \to \mathcal{M}'_ r$. Using transfinite recursion we define a sequence of functors $M_\alpha$ and natural transformations $M_\beta \to M_\alpha$ for $\alpha < \beta$ by setting

1. $M_0 = \text{id}$,

2. $M_{\alpha + 1} = M \circ M_\alpha$ with natural transformation $M_\beta \to M_{\alpha + 1}$ for $\beta < \alpha + 1$ coming from the already constructed $M_\beta \to M_\alpha$ and the maps $M_\alpha \to M \circ M_\alpha$ coming from $\text{id} \to M$, and

3. $M_\alpha = \mathop{\mathrm{colim}}\nolimits _{\beta < \alpha } M_\beta$ if $\alpha$ is a limit ordinal with the coprojections as transformations $M_\beta \to M_\alpha$ for $\alpha < \beta$.

Observe that for every differential graded $\mathcal{A}$-module the maps $\mathcal{M} \to M_\beta (\mathcal{M}) \to M_\alpha (\mathcal{M})$ are injective quasi-isomorphisms (as filtered colimits are exact).

Recall that $\textit{Mod}(\mathcal{A}, \text{d})$ is a Grothendieck abelian category. Thus by Injectives, Proposition 19.11.5 (applied to the direct sum of $\mathcal{M}_ r$ for all $r \in R$) there is a limit ordinal $\alpha$ such that $\mathcal{M}_ r$ is $\alpha$-small with respect to injections for every $r \in R$. We claim that $\mathcal{M} \to M_\alpha (\mathcal{M})$ is the desired functorial embedding of $\mathcal{M}$ into a graded injective K-injective module.

Namely, any map $\mathcal{M}_ r \to M_\alpha (\mathcal{M})$ factors through $M_\beta (\mathcal{M})$ for some $\beta < \alpha$. However, by the construction of $M$ we see that this means that $\mathcal{M}_ r \to M_{\beta + 1}(\mathcal{M}) = M(M_\beta (\mathcal{M}))$ factors through $\mathcal{M}'_ r$. Since $M_\beta (\mathcal{M}) \subset M_{\beta + 1}(\mathcal{M}) \subset M_\alpha (\mathcal{M})$ we get the desired factorizaton into $M_\alpha (\mathcal{M})$. We conclude by our choice of $R$ and $\mathcal{M}_ r \to \mathcal{M}'_ r$ in Lemma 24.25.11. $\square$

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