The Stacks project

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.

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.

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

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

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_{(A, \text{d})}}(A[k], M) = \mathop{\mathrm{Ker}}(\text{d} : M^{-k} \to M^{-k + 1}) \]


\[ \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(A[k], M) = H^{-k}(M) \]

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

Proof. Immediate from the definitions. $\square$

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