The Stacks project

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 $\bigoplus F_ iP$. $\square$

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. $\square$

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 $A$. $\square$

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. $\square$