Lemma 22.20.4. 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 such that
P \to M is a quasi-isomorphism, and
P has property (P).
Proof. Set M = M_0. We inductively choose short exact sequences
0 \to M_{i + 1} \to P_ i \to M_ i \to 0
where the maps P_ i \to M_ i are chosen as in Lemma 22.20.3. This gives a “resolution”
\ldots \to P_2 \xrightarrow {f_2} P_1 \xrightarrow {f_1} P_0 \to M \to 0
Then we set
P = \bigoplus \nolimits _{i \geq 0} P_ i
as an A-module with grading given by P^ n = \bigoplus _{a + b = n} P_{-a}^ b and differential (as in the construction of the total complex associated to a double complex) by
\text{d}_ P(x) = f_{-a}(x) + (-1)^ a \text{d}_{P_{-a}}(x)
for x \in P_{-a}^ b. With these conventions P is indeed a differential graded A-module. Recalling that each P_ i has a two step filtration 0 \to P_ i' \to P_ i \to P_ i'' \to 0 we set
F_{2i}P = \bigoplus \nolimits _{i \geq j \geq 0} P_ j \subset \bigoplus \nolimits _{i \geq 0} P_ i = P
and we add P'_{i + 1} to F_{2i}P to get F_{2i + 1}. These are differential graded submodules and the successive quotients are direct sums of shifts of A. By Lemma 22.16.1 we see that the inclusions F_ iP \to F_{i + 1}P are admissible monomorphisms. Finally, we have to show that the map P \to M (given by the augmentation P_0 \to M) is a quasi-isomorphism. This follows from Homology, Lemma 12.26.2. \square
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