The Stacks project

Lemma 22.7.5. Let $(A, \text{d})$ be a differential graded algebra. Let $L_1 \to L_2 \to \ldots \to L_ n$ be a sequence of composable homomorphisms of differential graded $A$-modules. There exists a commutative diagram

\[ \xymatrix{ L_1 \ar[r] & L_2 \ar[r] & \ldots \ar[r] & L_ n \\ M_1 \ar[r] \ar[u] & M_2 \ar[r] \ar[u] & \ldots \ar[r] & M_ n \ar[u] } \]

in $\text{Mod}_{(A, \text{d})}$ such that each $M_ i \to M_{i + 1}$ is an admissible monomorphism and each $M_ i \to L_ i$ is a homotopy equivalence.

Proof. The case $n = 1$ is without content. Lemma 22.7.4 is the case $n = 2$. Suppose we have constructed the diagram except for $M_ n$. Apply Lemma 22.7.4 to the composition $M_{n - 1} \to L_{n - 1} \to L_ n$. The result is a factorization $M_{n - 1} \to M_ n \to L_ n$ as desired. $\square$

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