Lemma 22.20.3. Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded $A$-module. There exists a homomorphism $P \to M$ of differential graded $A$-modules with the following properties

$P \to M$ is surjective,

$\mathop{\mathrm{Ker}}(\text{d}_ P) \to \mathop{\mathrm{Ker}}(\text{d}_ M)$ is surjective, and

$P$ sits in an admissible short exact sequence $0 \to P' \to P \to P'' \to 0$ where $P'$, $P''$ are direct sums of shifts of $A$.

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