## 24.25 K-injective differential graded modules

This section is the analogue of Injectives, Section 19.12 in the setting of sheaves of differential graded modules over a sheaf of differential graded algebras.

Lemma 24.25.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{N}_ t \to \mathcal{N}'_ t$ of graded $\mathcal{A}$-modules such that an object $\mathcal{I}$ of $\textit{Mod}(\mathcal{A})$ is injective if and only if for every solid diagram

\[ \xymatrix{ \mathcal{N}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{N}'_ t \ar@{..>}[ru] } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A})$ making the diagram commute.

**Proof.**
This is true in any Grothendieck abelian category, see Injectives, Lemma 19.11.6. By Lemma 24.11.1 the category $\textit{Mod}(\mathcal{A})$ is a Grothendieck abelian category.
$\square$

Definition 24.25.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A diffential graded $\mathcal{A}$-module $\mathcal{I}$ is said to be *graded injective*^{1} if $\mathcal{M}$ viewed as a graded $\mathcal{A}$-module is an injective object of the category $\textit{Mod}(\mathcal{A})$ of graded $\mathcal{A}$-modules.

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.
$\square$

Lemma 24.25.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{M}_ t \to \mathcal{M}'_ t$ of acyclic differential graded $\mathcal{A}$-modules such that for an object $\mathcal{I}$ of $\textit{Mod}(\mathcal{A}, \text{d})$ the following are equivalent

$\mathcal{I}$ is graded injective, and

for every solid diagram

\[ \xymatrix{ \mathcal{M}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ t \ar@{..>}[ru] } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A}, \text{d})$ making the diagram commute.

**Proof.**
Let $T$ and $\mathcal{N}_ t \to \mathcal{N}'_ t$ be as in Lemma 24.25.1. Denote $F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A})$ the forgetful functor. Let $G$ be the left adjoint functor to $F$ as in Lemma 24.24.1. Set

\[ \mathcal{M}_ t = G(\mathcal{N}_ t) \to G(\mathcal{N}'_ t) = \mathcal{M}'_ t \]

This is an injective map of acyclic differential graded $\mathcal{A}$-modules by Lemma 24.24.2. Since $G$ is the left adjoint to $F$ we see that there exists a dotted arrow in the diagram

\[ \xymatrix{ \mathcal{M}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ t \ar@{..>}[ru] } \]

if and only if there exists a dotted arrow in the diagram

\[ \xymatrix{ \mathcal{N}_ t \ar[r] \ar[d] & F(\mathcal{I}) \\ \mathcal{N}'_ t \ar@{..>}[ru] } \]

Hence the result follows from the choice of our collection of arrows $\mathcal{N}_ t \to \mathcal{N}_ t'$.
$\square$

Lemma 24.25.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $S$ and for each $s$ an acyclic differential graded $\mathcal{A}$-module $\mathcal{M}_ s$ such that for every nonzero acyclic differential graded $\mathcal{A}$-module $\mathcal{M}$ there is an $s \in S$ and an injective map $\mathcal{M}_ s \to \mathcal{M}$ in $\textit{Mod}(\mathcal{A}, \text{d})$.

**Proof.**
Before we start recall that our conventions guarantee the site $\mathcal{C}$ has a set of objects and morphisms and a set $\text{Cov}(\mathcal{C})$ of coverings. If $\mathcal{F}$ is a differential graded $\mathcal{A}$-module, let us define $|\mathcal{F}|$ to be the sum of the cardinality of

\[ \coprod \nolimits _{(U, n)} \mathcal{F}^ n(U) \]

as $U$ ranges over the objects of $\mathcal{C}$ and $n \in \mathbf{Z}$. Choose an infinite cardinal $\kappa $ bigger than the cardinals $|\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})|$, $|\text{Arrows}(\mathcal{C})|$, $|\text{Cov}(\mathcal{C})|$, $\sup |I|$ for $\{ U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})$, and $|\mathcal{A}|$.

Let $\mathcal{F} \subset \mathcal{M}$ be an inclusion of differential graded $\mathcal{A}$-modules. Suppose given a set $K$ and for each $k \in K$ a triple $(U_ k, n_ k, x_ k)$ consisting of an object $U_ k$ of $\mathcal{C}$, integer $n_ k$, and a section $x_ k \in \mathcal{M}^{n_ k}(U_ k)$. Then we can consider the smallest differential graded $\mathcal{A}$-submodule $\mathcal{F}' \subset \mathcal{M}$ containing $\mathcal{F}$ and the sections $x_ k$ for $k \in K$. We can describe

\[ (\mathcal{F}')^ n(U) \subset \mathcal{M}^ n(U) \]

as the set of elements $x \in \mathcal{M}^ n(U)$ such that there exists $\{ f_ i : U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})$ such that for each $i \in I$ there is a finite set $T_ i$ and morphisms $g_ t : U_ i \to U_{k_ t}$

\[ f_ i^*x = y_ i + \sum \nolimits _{t \in T_ i} a_{it}g_ t^*x_{k_ t} + b_{it}g_ t^*\text{d}(x_{k_ t}) \]

for some section $y_ i \in \mathcal{F}^ n(U)$ and sections $a_{it} \in \mathcal{A}^{n - n_{k_ t}}(U_ i)$ and $b_{it} \in \mathcal{A}^{n - n_{k_ t} - 1}(U_ i)$. (Details omitted; hints: these sections are certainly in $\mathcal{F}'$ and you show conversely that this rule defines a differential graded $\mathcal{A}$-submodule.) It follows from this description that $|\mathcal{F}'| \leq \max (|\mathcal{F}|, |K|, \kappa )$.

Let $\mathcal{M}$ be a nonzero acyclic differential graded $\mathcal{A}$-module. Then we can find an integer $n$ and a nonzero section $x$ of $\mathcal{M}^ n$ over some object $U$ of $\mathcal{C}$. Let

\[ \mathcal{F}_0 \subset \mathcal{M} \]

be the smallest differential graded $\mathcal{A}$-submodule containing $x$. By the previous paragraph we have $|\mathcal{F}_0| \leq \kappa $. By induction, given $\mathcal{F}_0, \ldots , \mathcal{F}_ n$ define $\mathcal{F}_{n + 1}$ as follows. Consider the set

\[ L = \{ (U, n, x)\} \{ U_ i \to U\} _{i \in I}, (x_ i)_{i \in I})\} \]

of triples where $U$ is an object of $\mathcal{C}$, $n \in \mathbf{Z}$, and $x \in \mathcal{F}_ n(U)$ with $\text{d}(x) = 0$. Since $\mathcal{M}$ is acyclic for each triple $l = (U_ l, n_ l, x_ l) \in L$ we can choose $\{ (U_{l, i} \to U_ l\} _{i \in I_ l} \in \text{Cov}(\mathcal{C})$ and $x_{l, i} \in \mathcal{M}^{n_ l - 1}(U_{l, i})$ such that $\text{d}(x_{l, i}) = x|_{U_{l, i}}$. Then we set

\[ K = \{ (U_{l, i}, n_ l - 1, x_{l, i}) \mid l \in L, i \in I_ l\} \]

and we let $\mathcal{F}_{n + 1}$ be the smallest differential graded $\mathcal{A}$-submodule of $\mathcal{M}$ containing $\mathcal{F}_ n$ and the sections $x_{l, i}$. Since $|K| \leq \max (\kappa , |\mathcal{F}_ n|)$ we conclude that $|\mathcal{F}_{n + 1}| \leq \kappa $ by induction.

By construction the inclusion $\mathcal{F}_ n \to \mathcal{F}_{n + 1}$ induces the zero map on cohomology sheaves. Hence we see that $\mathcal{F} = \bigcup \mathcal{F}_ n$ is a nonzero acyclic submodule with $|\mathcal{F}| \leq \kappa $. Since there is only a set of isomorphism classes of differential graded $\mathcal{A}$-modules $\mathcal{F}$ with $|\mathcal{F}|$ bounded, we conclude.
$\square$

Definition 24.25.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A diffential graded $\mathcal{A}$-module $\mathcal{I}$ is *K-injective* if for every acyclic differential graded $\mathcal{M}$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I}) = 0 \]

Please note the similarity with Derived Categories, Definition 13.31.1.

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.
$\square$

Lemma 24.25.9. 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 $\mathcal{I}$ be a K-injective and graded injective object of $\textit{Mod}(\mathcal{A}, \text{d})$. For every solid diagram in $\textit{Mod}(\mathcal{A}, \text{d})$

\[ \xymatrix{ \mathcal{M} \ar[r]_ a \ar[d]_ b & \mathcal{I} \\ \mathcal{M}' \ar@{..>}[ru] } \]

where $b$ is injective and $\mathcal{M}$ is acyclic a dotted arrow exists making the diagram commute.

**Proof.**
Since $\mathcal{M}$ is acyclic and $\mathcal{I}$ is K-injective, there exists a graded $\mathcal{A}$-module map $h : \mathcal{M} \to \mathcal{I}$ of degree $-1$ such that $a = \text{d}(h)$. Since $\mathcal{I}$ is graded injective and $b$ is injective, there exists a graded $\mathcal{A}$-module map $h' : \mathcal{M}' \to \mathcal{I}$ of degree $-1$ such that $h = h' \circ b$. Then we can take $a' = \text{d}(h')$ as the dotted arrow.
$\square$

Lemma 24.25.10. 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 $\mathcal{I}$ be a K-injective and graded injective object of $\textit{Mod}(\mathcal{A}, \text{d})$. For every solid diagram in $\textit{Mod}(\mathcal{A}, \text{d})$

\[ \xymatrix{ \mathcal{M} \ar[r]_ a \ar[d]_ b & \mathcal{I} \\ \mathcal{M}' \ar@{..>}[ru] } \]

where $b$ is a quasi-isomorphism a dotted arrow exists making the diagram commute up to homotopy.

**Proof.**
After replacing $\mathcal{M}'$ by the direct sum of $\mathcal{M}'$ and the cone on the identity on $\mathcal{M}$ (which is acyclic) we may assume $b$ is also injective. Then the cokernel $\mathcal{Q}$ of $b$ is acyclic. Thus we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{Q}, \mathcal{I}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{Q}, \mathcal{I})[1] = 0 \]

as $\mathcal{I}$ is K-injective. As $\mathcal{I}$ is graded injective by Remark 24.25.3 we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}', \mathcal{I}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I}) \]

is bijective and the proof is complete.
$\square$

Lemma 24.25.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $R$ and for each $r \in R$ an injective map $\mathcal{M}_ r \to \mathcal{M}'_ r$ of acyclic differential graded $\mathcal{A}$-modules such that for an object $\mathcal{I}$ of $\textit{Mod}(\mathcal{A}, \text{d})$ the following are equivalent

$\mathcal{I}$ is K-injective and graded injective, and

for every solid diagram

\[ \xymatrix{ \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{M}'_ r \ar@{..>}[ru] } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A}, \text{d})$ making the diagram commute.

**Proof.**
Let $T$ and $\mathcal{M}_ t \to \mathcal{M}'_ t$ be as in Lemma 24.25.5. Let $S$ and $\mathcal{M}_ s$ be as in Lemma 24.25.6. Choose an injective map $\mathcal{M}_ s \to \mathcal{M}'_ s$ of acyclic differential graded $\mathcal{A}$-modules which is homotopic to zero. This is possible because we may take $\mathcal{M}'_ s$ to be the cone on the identity; in that case it is even true that the identity on $\mathcal{M}'_ s$ is homotopic to zero, see Differential Graded Algebra, Lemma 22.27.4 which applies by the discussion in Section 24.22. We claim that $R = T \coprod S$ with the given maps works.

The implication (1) $\Rightarrow $ (2) holds by Lemma 24.25.9.

Assume (2). First, by Lemma 24.25.5 we see that $\mathcal{I}$ is graded injective. Next, let $\mathcal{M}$ be an acyclic differential graded $\mathcal{A}$-module. We have to show that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{M}, \mathcal{I}) = 0 \]

The proof will be exactly the same as the proof of Injectives, Lemma 19.12.3.

We are going to construct by induction on the ordinal $\alpha $ an acyclic differential graded submodule $\mathcal{K}_\alpha \subset \mathcal{M}$ as follows. For $\alpha = 0$ we set $\mathcal{K}_0 = 0$. For $\alpha > 0$ we proceed as follows:

If $\alpha = \beta + 1$ and $\mathcal{K}_\beta = \mathcal{M}$ then we choose $\mathcal{K}_\alpha = \mathcal{K}_\beta $.

If $\alpha = \beta + 1$ and $\mathcal{K}_\beta \not= \mathcal{M}$ then $\mathcal{M}/\mathcal{K}_\beta $ is a nonzero acyclic differential graded $\mathcal{A}$-module. We choose a differential graded $\mathcal{A}$ submodule $\mathcal{N}_\alpha \subset \mathcal{M}/\mathcal{K}_\beta $ isomorphic to $\mathcal{M}_ s$ for some $s \in S$, see Lemma 24.25.6. Finally, we let $\mathcal{K}_\alpha \subset \mathcal{M}$ be the inverse image of $\mathcal{N}_\alpha $.

If $\alpha $ is a limit ordinal we set $\mathcal{K}_\beta = \mathop{\mathrm{colim}}\nolimits \mathcal{K}_\alpha $.

It is clear that $\mathcal{M} = \mathcal{K}_\alpha $ for a suitably large ordinal $\alpha $. We will prove that

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{K}_\alpha , \mathcal{I}) \]

is zero by transfinite induction on $\alpha $. It holds for $\alpha = 0$ since $\mathcal{K}_0$ is zero. Suppose it holds for $\beta $ and $\alpha = \beta + 1$. In case (1) of the list above the result is clear. In case (2) there is a short exact sequence

\[ 0 \to \mathcal{K}_\beta \to \mathcal{K}_\alpha \to \mathcal{N}_\alpha \to 0 \]

By Remark 24.25.3 and since we've seen that $\mathcal{I}$ is graded injective, we obtain an exact sequence

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{K}_\beta , \mathcal{I}) \to \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{K}_\alpha , \mathcal{I}) \to \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Mod}(\mathcal{A}, \text{d}))}(\mathcal{N}_\alpha , \mathcal{I}) \]

By induction the term on the left is zero. By assumption (2) the term on the right is zero: any map $\mathcal{M}_ s \to \mathcal{I}$ factors through $\mathcal{M}'_ s$ and hence is homotopic to zero. Thus the middle group is zero too. Finally, suppose that $\alpha $ is a limit ordinal. Because we also have $\mathcal{K}_\alpha = \mathop{\mathrm{colim}}\nolimits \mathcal{K}_\alpha $ as graded $\mathcal{A}$-modules we see that

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})} (\mathcal{K}_\alpha , \mathcal{I}) = \mathop{\mathrm{lim}}\nolimits _{\beta < \alpha } \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})} (\mathcal{K}_\beta , \mathcal{I}) \]

as complexes of abelian groups. The cohomology groups of these complexes compute morphisms in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ between shifts. The transition maps in the system of complexes are surjective by Remark 24.25.3 because $\mathcal{I}$ is graded injective. Moreover, for a limit ordinal $\beta \leq \alpha $ we have equality of limit and value. Thus we may apply Homology, Lemma 12.31.8 to conclude.
$\square$

Lemma 24.25.12. 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 $R$ be a set and for each $r \in R$ let an injective map $\mathcal{M}_ r \to \mathcal{M}'_ r$ of acyclic differential graded $\mathcal{A}$-modules be given. There exists a functor $M : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A}, \text{d})$ and a natural transformation $j : \text{id} \to M$ such that

$j_\mathcal {M} : \mathcal{M} \to M(\mathcal{M})$ is injective and a quasi-isomorphism,

for every solid diagram

\[ \xymatrix{ \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{M} \ar[d]^{j_\mathcal {M}} \\ \mathcal{M}'_ r \ar@{..>}[r] & M(\mathcal{M}) } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A}, \text{d})$ making the diagram commute.

**Proof.**
We define $M(\mathcal{M})$ as the pushout in the following diagram

\[ \xymatrix{ \bigoplus _{(r, \varphi )} \mathcal{M}_ r \ar[r] \ar[d] & \mathcal{M} \ar[d] \\ \bigoplus _{(r, \varphi )} \mathcal{M}'_ r \ar[r] & M(\mathcal{M}) } \]

where the direct sum is over all pairs $(r, \varphi )$ with $r \in R$ and $\varphi \in \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}, \text{d})}(\mathcal{M}_ r, \mathcal{M})$. Since the pushout of an injective map is injective, we see that $\mathcal{M} \to M(\mathcal{M})$ is injective. Since the cokernel of the left vertical arrow is acyclic, we see that the (isomorphic) cokernel of $\mathcal{M} \to M(\mathcal{M})$ is acyclic, hence $\mathcal{M} \to M(\mathcal{M})$ is a quasi-isomorphism. Property (2) holds by construction. We omit the verification that this procedure can be turned into a functor.
$\square$

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

$M_0 = \text{id}$,

$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

$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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