This section is the analogue of Derived Categories, Section 13.29.
Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module. We say $P$ has property (P) if it there exists a filtration
by differential graded submodules such that
$P = \bigcup F_ pP$,
the inclusions $F_ iP \to F_{i + 1}P$ are admissible monomorphisms,
the quotients $F_{i + 1}P/F_ iP$ are isomorphic as differential graded $A$-modules to a direct sum of $A[k]$.
In fact, condition (2) is a consequence of condition (3), see Lemma 22.16.1. Moreover, the reader can verify that as a graded $A$-module $P$ will be isomorphic to a direct sum of shifts of $A$.
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 of differential graded $A$-modules.
\[ 0 \to \bigoplus \nolimits F_ iP \to \bigoplus \nolimits F_ iP \to P \to 0 \]
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 $\bigoplus F_ iP$. $\square$
The following lemma shows that differential graded modules with property (P) are the dual notion to K-injective modules (i.e., they are K-projective in some sense). See Derived Categories, Definition 13.31.1.
Lemma 22.20.2. Let $(A, \text{d})$ be a differential graded algebra. Let $P$ be a differential graded $A$-module with property (P). Then for all acyclic differential graded $A$-modules $N$.
\[ \mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(P, N) = 0 \]
Proof. We will use that $K(\text{Mod}_{(A, \text{d})})$ is a triangulated category (Proposition 22.10.3). Let $F_\bullet $ be a filtration on $P$ as in property (P). The short exact sequence of Lemma 22.20.1 produces a distinguished triangle. Hence by Derived Categories, Lemma 13.4.2 it suffices to show that
for all acyclic differential graded $A$-modules $N$ and all $i$. Each of the differential graded modules $F_ iP$ has a finite filtration by admissible monomorphisms, whose graded pieces are direct sums of shifts $A[k]$. Thus it suffices to prove that
for all acyclic differential graded $A$-modules $N$ and all $k$. This follows from Lemma 22.16.2. $\square$
Lemma 22.20.3. 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 with the following properties
$P \to M$ is surjective,
$\mathop{\mathrm{Ker}}(\text{d}_ P) \to \mathop{\mathrm{Ker}}(\text{d}_ M)$ is surjective, and
$P$ sits in an admissible short exact sequence $0 \to P' \to P \to P'' \to 0$ where $P'$, $P''$ are direct sums of shifts of $A$.
Proof. Let $P_ k$ be the free $A$-module with generators $x, y$ in degrees $k$ and $k + 1$. Define the structure of a differential graded $A$-module on $P_ k$ by setting $\text{d}(x) = y$ and $\text{d}(y) = 0$. For every element $m \in M^ k$ there is a homomorphism $P_ k \to M$ sending $x$ to $m$ and $y$ to $\text{d}(m)$. Thus we see that there is a surjection from a direct sum of copies of $P_ k$ to $M$. This clearly produces $P \to M$ having properties (1) and (3). To obtain property (2) note that if $m \in \mathop{\mathrm{Ker}}(\text{d}_ M)$ has degree $k$, then there is a map $A[k] \to M$ mapping $1$ to $m$. Hence we can achieve (2) by adding a direct sum of copies of shifts of $A$. $\square$
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
where the maps $P_ i \to M_ i$ are chosen as in Lemma 22.20.3. This gives a “resolution”
Then we set
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
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
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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