Lemma 22.21.4. Let (A, \text{d}) be a differential graded algebra. Let M be a differential graded A-module. There exists a homomorphism M \to I of differential graded A-modules such that
M \to I is a quasi-isomorphism, and
I has property (I).
Proof. Set M = M_0. We inductively choose short exact sequences
0 \to M_ i \to I_ i \to M_{i + 1} \to 0
where the maps M_ i \to I_ i are chosen as in Lemma 22.21.3. This gives a “resolution”
0 \to M \to I_0 \xrightarrow {f_0} I_1 \xrightarrow {f_1} I_1 \to \ldots
Denote I the differential graded A-module with graded parts
I^ n = \prod \nolimits _{i \geq 0} I^{n - i}_ i
and differential defined by
\text{d}_ I(x) = f_ i(x) + (-1)^ i \text{d}_{I_ i}(x)
for x \in I_ i^{n - i}. With these conventions I is indeed a differential graded A-module. Recalling that each I_ i has a two step filtration 0 \to I_ i' \to I_ i \to I_ i'' \to 0 we set
F_{2i}I^ n = \prod \nolimits _{j \geq i} I^{n - j}_ j \subset \prod \nolimits _{i \geq 0} I^{n - i}_ i = I^ n
and we add a factor I'_{i + 1} to F_{2i}I to get F_{2i + 1}I. These are differential graded submodules and the successive quotients are products of shifts of A^\vee . By Lemma 22.19.1 we see that the inclusions F_{i + 1}I \to F_ iI are admissible monomorphisms. Finally, we have to show that the map M \to I (given by the augmentation M \to I_0) is a quasi-isomorphism. This follows from Homology, Lemma 12.26.3. \square
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