The Stacks project

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

  1. $P \to M$ is surjective,

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

  3. $P$ 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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