Lemma 22.19.1. Let $(A, \text{d})$ be a differential graded algebra. Let $I \to M$ be an injective homomorphism of differential graded $A$-modules. If $I$ is graded injective, then $I \to M$ is an admissible monomorphism.

## 22.19 Injective modules and differential graded algebras

If $(A, \text{d})$ is a differential graded algebra and $I$ is an object of $\text{Mod}_{(A, \text{d})}$ then we say *$I$ is injective as a graded $A$-module* or sometimes *$I$ is graded injective* to mean that $I$ is a injective object of the abelian category $\text{Mod}_ A$ of graded $A$-modules.

**Proof.**
This is immediate from the definitions.
$\square$

Let $(A, \text{d})$ be a differential graded algebra. If $M$ is a left, resp. right differential graded $A$-module, then

with $A$-module structure constructed in Section 22.18 is a right, resp. left differential graded $A$-module by the discussion in Section 22.13. By Remarks 22.13.5 and 22.13.6 there evaluation map of Section 22.18

is a homomorphism of left, resp. right differential graded $A$-modules

Lemma 22.19.2. Let $(A, \text{d})$ be a differential graded algebra. If $M$ is a left differential graded $A$-module and $N$ is a right differential graded $A$-module, then

**Proof.**
The first equality is Lemma 22.13.2 and the second equality was shown in Section 22.12.
$\square$

Lemma 22.19.3. Let $(A, \text{d})$ be a differential graded algebra. Then we have

and

as functors in the differential graded $A$-module $M$.

**Proof.**
This is clear from the discussion above.
$\square$

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