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.
Proof. Let P_ k be the free A-module with generators x, y in degrees k and k + 1. Define the structure of a differential graded A-module on P_ k by setting \text{d}(x) = y and \text{d}(y) = 0. For every element m \in M^ k there is a homomorphism P_ k \to M sending x to m and y to \text{d}(m). Thus we see that there is a surjection from a direct sum of copies of P_ k to M. This clearly produces P \to M having properties (1) and (3). To obtain property (2) note that if m \in \mathop{\mathrm{Ker}}(\text{d}_ M) has degree k, then there is a map A[k] \to M mapping 1 to m. Hence we can achieve (2) by adding a direct sum of copies of shifts of A. \square
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