Lemma 22.20.1. Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module. If $F_\bullet $ is a filtration as in property (P), then we obtain an admissible short exact sequence

\[ 0 \to \bigoplus \nolimits F_ iP \to \bigoplus \nolimits F_ iP \to P \to 0 \]

of differential graded $A$-modules.

**Proof.**
The second map is the direct sum of the inclusion maps. The first map on the summand $F_ iP$ of the source is the sum of the identity $F_ iP \to F_ iP$ and the negative of the inclusion map $F_ iP \to F_{i + 1}P$. Choose homomorphisms $s_ i : F_{i + 1}P \to F_ iP$ of graded $A$-modules which are left inverse to the inclusion maps. Composing gives maps $s_{j, i} : F_ jP \to F_ iP$ for all $j > i$. Then a left inverse of the first arrow maps $x \in F_ jP$ to $(s_{j, 0}(x), s_{j, 1}(x), \ldots , s_{j, j - 1}(x), 0, \ldots )$ in $\bigoplus F_ iP$.
$\square$

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