Lemma 22.16.1. Let $(A, \text{d})$ be a differential graded algebra. Let $M \to P$ be a surjective homomorphism of differential graded $A$-modules. If $P$ is projective as a graded $A$-module, then $M \to P$ is an admissible epimorphism.

## 22.16 Projective modules and differential graded algebras

If $(A, \text{d})$ is a differential graded algebra and $P$ is an object of $\text{Mod}_{(A, \text{d})}$ then we say *$P$ is projective as a graded $A$-module* or sometimes *$P$ is graded projective* to mean that $P$ is a projective object of the abelian category $\text{Mod}_ A$ of graded $A$-modules as in Section 22.15.

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

Lemma 22.16.2. Let $(A, d)$ be a differential graded algebra. Then we have

and

for any differential graded $A$-module $M$.

**Proof.**
Immediate from the definitions.
$\square$

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