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The Stacks project

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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