## 24.23 Flat resolutions

This section is the analogue of Differential Graded Algebra, Section 22.20.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let us call a right differential graded $\mathcal{A}$-module $\mathcal{P}$ good if

1. the functor $\mathcal{N} \mapsto \mathcal{P} \otimes _\mathcal {A} \mathcal{N}$ is exact on the category of graded left $\mathcal{A}$-modules,

2. if $\mathcal{N}$ is an acyclic differential graded left $\mathcal{A}$-module, then $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}$ is acyclic,

3. for any morphism $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of ringed topoi and any differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$ and any map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}_\mathcal {D}$-algebras we have properties (1) and (2) for the pullback $f^*\mathcal{P}$ (Section 24.18) viewed as a differential graded $\mathcal{A}'$-module.

The first condition means that $\mathcal{P}$ is flat as a right graded $\mathcal{A}$-module, the second condition means that $\mathcal{P}$ is K-flat in the sense of Spaltenstein (see Cohomology on Sites, Section 21.17), and the third condition is that this holds after arbitrary base change.

Perhaps surprisingly, there are many good modules.

Lemma 24.23.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Then $j_!\mathcal{A}_ U$ is a good differential graded $\mathcal{A}$-module.

Proof. Let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. By Lemma 24.10.2 we have

$j_!\mathcal{A}_ U \otimes _\mathcal {A} \mathcal{N} = j_!(\mathcal{A}_ U \otimes _{\mathcal{A}_ U} \mathcal{N}|_ U) = j_!(\mathcal{N}_ U)$

as graded modules. Since both restriction to $U$ and $j_!$ are exact this proves condition (1). The same argument works for (2) using Lemma 24.19.2.

Consider a morphism $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of ringed topoi, a differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$, and a map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}$-algebras. We have to show that

$f^*j_!\mathcal{A}_ U = f^{-1}j_!\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}} \mathcal{A}'$

satisfies (1) and (2) for the ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ endowed with the sheaf of differential graded $\mathcal{O}'$-algebras $\mathcal{A}'$. To prove this we may replace $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ by equivalent ringed topoi. Thus by Modules on Sites, Lemma 18.7.2 we may assume that $f$ comes from a morphism of sites $f : \mathcal{C} \to \mathcal{C}'$ given by the continuous functor $u : \mathcal{C} \to \mathcal{C}'$. In this case, set $U' = u(U)$ and denote $j' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U') \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ the corresponding localization morphism. We obtain a commutative square of morphisms of ringed topoi

$\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U'), \mathcal{O}'_{U'}) \ar[rr]_{(j', (j')^\sharp )} \ar[d]_{(f', (f')^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \ar[d]^{(f, f^\sharp )} \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U), \mathcal{O}_ U) \ar[rr]^{(j, j^\sharp )} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}). }$

and we have $f'_*(j')^{-1} = j^{-1}f_*$. See Modules on Sites, Lemma 18.20.1. By uniqueness of adjoints we obtain $f^{-1}j_! = j'_!(f')^{-1}$. Thus we obtain

\begin{align*} f^*j_!\mathcal{A}_ U & = f^{-1}j_!\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}} \mathcal{A}' \\ & = j'_!(f')^{-1}\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}} \mathcal{A}' \\ & = j'_!\left( (f')^{-1}\mathcal{A}_ U \otimes _{f^{-1}\mathcal{A}|_{U'}} \mathcal{A}'|_{U'}\right) \\ & = j'_!\mathcal{A}'_{U'} \end{align*}

The first equation is the definition of the pullback of $j_!\mathcal{A}_ U$ to a differential graded module over $\mathcal{A}'$. The second equation because $f^{-1}j_! = j'_!(f')^{-1}$. The third equation by Lemma 24.19.2 applied to the ringed site $(\mathcal{C}', f^{-1}\mathcal{O})$ with sheaf of differential graded algebras $f^{-1}\mathcal{A}$ and with differential graded modules $(f')^{-1}\mathcal{A}_ U$ on $\mathcal{C}'/U'$ and $\mathcal{A}'$ on $\mathcal{C}'$. The fourth equation holds because of course we have $(f')^{-1}\mathcal{A}_ U = f^{-1}\mathcal{A}|_{U'}$. Hence we see that the pullback is another module of the same kind and we've proven conditions (1) and (2) for it above. $\square$

Lemma 24.23.2. et $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $0 \to \mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to 0$ be an admissible short exact sequence of differential graded $\mathcal{A}$-modules. If two-out-of-three of these modules are good, so is the third.

Proof. For condition (1) this is immediate as the sequence is a direct sum at the graded level. For condition (2) note that for any left differential graded $\mathcal{A}$-module, the sequence

$0 \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P}'' \otimes _\mathcal {A} \mathcal{N} \to 0$

is an admissible short exact sequence of differential graded $\mathcal{O}$-modules (since forgetting the differential the tensor product is just taken in the category of graded modules). Hence if two out of three are exact as complexes of $\mathcal{O}$-modules, so is the third. Finally, the same argument shows that given a morphism $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of ringed topoi, a differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$, and a map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}$-algebras we have that

$0 \to f^*\mathcal{P} \to f^*\mathcal{P}' \to f^*\mathcal{P}'' \to 0$

is an admissible short exact sequence of differential graded $\mathcal{A}'$-modules and the same argument as above applies here. $\square$

Lemma 24.23.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. An arbitrary direct sum of good differential graded $\mathcal{A}$-modules is good. A filtered colimit of good differential graded $\mathcal{A}$-modules is good.

Proof. Omitted. Hint: direct sums and filtered colimits commute with tensor products and with pullbacks. $\square$

Lemma 24.23.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. There exists a homomorphism $\mathcal{P} \to \mathcal{M}$ of differential graded $\mathcal{A}$-modules with the following properties

1. $\mathcal{P} \to \mathcal{M}$ is surjective,

2. $\mathop{\mathrm{Ker}}(\text{d}_\mathcal {P}) \to \mathop{\mathrm{Ker}}(\text{d}_\mathcal {M})$ is surjective, and

3. $\mathcal{P}$ is good.

Proof. Consider triples $(U, k, x)$ where $U$ is an object of $\mathcal{C}$, $k \in \mathbf{Z}$, and $x$ is a section of $\mathcal{M}^ k$ over $U$ with $\text{d}_\mathcal {M}(x) = 0$. Then we obtain a unique morphism of differential graded $\mathcal{A}_ U$-modules $\varphi _ x : \mathcal{A}_ U[-k] \to \mathcal{M}|_ U$ mapping $1$ to $x$. This is adjoint to a morphism $\psi _ x : j_{U!}\mathcal{A}_ U[-k] \to \mathcal{M}$. Observe that $1 \in \mathcal{A}_ U(U)$ corresponds to a section $1 \in j_{U!}\mathcal{A}_ U[-k](U)$ of degree $k$ whose differential is zero and which is mapped to $x$ by $\psi _ x$. Thus if we consider the map

$\bigoplus \nolimits _{(U, k, x)} j_{U!}\mathcal{A}_ U[-k] \longrightarrow \mathcal{M}$

then we will have conditions (2) and (3). Namely, the objects $j_{U!}\mathcal{A}_ U[-k]$ are good (Lemma 24.23.1) and any direct sum of good objects is good (Lemma 24.23.3).

Next, consider triples $(U, k, x)$ where $U$ is an object of $\mathcal{C}$, $k \in \mathbf{Z}$, and $x$ is a section of $\mathcal{M}^ k$ (not necessarily annihilated by the differential). Then we can consider the cone $C_ U$ on the identity map $\mathcal{A}_ U \to \mathcal{A}_ U$ as in Remark 24.22.5. The element $x$ will determine a map $\varphi _ x : C_ U[-k - 1] \to \mathcal{A}_ U$, see Remark 24.22.5. Now, since we have an admissible short exact sequence

$0 \to \mathcal{A}_ U \to C_ U \to \mathcal{A}_ U[1] \to 0$

we conclude that $j_{U!}C_ U$ is a good module by Lemma 24.23.2 and the already used Lemma 24.23.1. As above we conclude that the direct sum of the maps $\psi _ x : j_{U!}C_ U \to \mathcal{M}$ adjoint to the $\varphi _ x$

$\bigoplus \nolimits _{(U, k, x)} j_{U!}C_ U \longrightarrow \mathcal{M}$

is surjective. Taking the direct sum with the map produced in the first paragraph we conclude. $\square$

Remark 24.23.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A sheaf of graded sets on $\mathcal{C}$ is a sheaf of sets $\mathcal{S}$ endowed with a map $\deg : \mathcal{S} \to \underline{\mathbf{Z}}$ of sheaves of sets. Let us denote $\mathcal{O}[\mathcal{S}]$ the graded $\mathcal{O}$-module which is the free $\mathcal{O}$-module on the graded sheaf of sets $\mathcal{S}$. More precisely, the $n$th graded part of $\mathcal{O}[\mathcal{S}]$ is the sheafification of the rule

$U \longmapsto \bigoplus \nolimits _{s \in \mathcal{S}(U),\ \deg (s) = n} s \cdot \mathcal{O}(U)$

With zero differential we also may consider this as a differential graded $\mathcal{O}$-module. Let $\mathcal{A}$ be a sheaf of graded $\mathcal{O}$-algebras Then we similarly define $\mathcal{A}[\mathcal{S}]$ to be the graded $\mathcal{A}$-module whose $n$th graded part is the sheafification of the rule

$U \longmapsto \bigoplus \nolimits _{s \in \mathcal{S}(U)} s \cdot \mathcal{A}^{n - \deg (s)}(U)$

If $\mathcal{A}$ is a differential graded $\mathcal{O}$-algebra, the we turn this into a differential graded $\mathcal{O}$-module by setting $\text{d}(s) = 0$ for all $s \in \mathcal{S}(U)$ and sheafifying.

Lemma 24.23.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a differential graded $\mathcal{A}$-algebra. Let $\mathcal{S}$ be a sheaf of graded sets on $\mathcal{C}$. Then the free graded module $\mathcal{A}[\mathcal{S}]$ on $\mathcal{S}$ endowed with differential as in Remark 24.23.5 is a good differential graded $\mathcal{A}$-module.

Proof. Let $\mathcal{N}$ be a left graded $\mathcal{A}$-module. Then we have

$\mathcal{A}[\mathcal{S}] \otimes _\mathcal {A} \mathcal{N} = \mathcal{O}[\mathcal{S}] \otimes _\mathcal {O} \mathcal{N} = \mathcal{N}[\mathcal{S}]$

where $\mathcal{N}[\mathcal{S}$ is the graded $\mathcal{O}$-module whose degree $n$ part is the sheaf associated to the presheaf

$U \longmapsto \bigoplus \nolimits _{s \in \mathcal{S}(U)} s \cdot \mathcal{N}^{n - \deg (s)}(U)$

It is clear that $\mathcal{N} \to \mathcal{N}[\mathcal{S}]$ is an exact functor, hence $\mathcal{A}[\mathcal{S}$ is flat as a graded $\mathcal{A}$-module. Next, suppose that $\mathcal{N}$ is a differential graded left $\mathcal{A}$-module. Then we have

$H^*(\mathcal{A}[\mathcal{S}] \otimes _\mathcal {A} \mathcal{N}) = H^*(\mathcal{O}[\mathcal{S}] \otimes _\mathcal {O} \mathcal{N})$

as graded sheaves of $\mathcal{O}$-modules, which by the flatness (over $\mathcal{O})$ is equal to

$H^*(\mathcal{N})[\mathcal{S}]$

as a graded $\mathcal{O}$-module. Hence if $\mathcal{N}$ is acyclic, then $\mathcal{A}[\mathcal{S}] \otimes _\mathcal {A} \mathcal{N}$ is acyclic.

Finally, consider a morphism $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of ringed topoi, a differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$, and a map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}$-algebras. Then it is straightforward to see that

$f^*\mathcal{A}[\mathcal{S}] = \mathcal{A}'[f^{-1}\mathcal{S}]$

which finishes the proof that our module is good. $\square$

Lemma 24.23.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. There exists a homomorphism $\mathcal{P} \to \mathcal{M}$ of differential graded $\mathcal{A}$-modules with the following properties

1. $\mathcal{P} \to \mathcal{M}$ is a quasi-isomorphism, and

2. $\mathcal{P}$ is good.

First proof. Let $\mathcal{S}_0$ be the sheaf of graded sets (Remark 24.23.5) whose degree $n$ part is $\mathop{\mathrm{Ker}}(\text{d}_\mathcal {M}^ n)$. Consider the homomorphism of differential graded modules

$\mathcal{P}_0 = \mathcal{A}[\mathcal{S}_0] \longrightarrow \mathcal{M}$

where the left hand side is as in Remark 24.23.5 and the map sends a local section $s$ of $\mathcal{S}_0$ to the corresponding local section of $\mathcal{M}^{\deg (s)}$ (which is in the kernel of the differential, so our map is a map of differential graded modules indeed). By construction the induced maps on cohomology sheaves $H^ n(\mathcal{P}_0) \to H^ n(\mathcal{M})$ are surjective. We are going to inductively construct maps

$\mathcal{P}_0 \to \mathcal{P}_1 \to \mathcal{P}_2 \to \ldots \to \mathcal{M}$

Observe that of course $H^*(\mathcal{P}_ i) \to H^*(\mathcal{M})$ will be surjective for all $i$. Given $\mathcal{P}_ i \to \mathcal{M}$ denote $\mathcal{S}_{i + 1}$ the sheaf of graded sets whose degree $n$ part is

$\mathop{\mathrm{Ker}}(\text{d}_{\mathcal{P}_ i}^{n + 1}) \times _{\mathcal{M}^{n + 1}, \text{d}} \mathcal{M}^ n$

Then we set

$\mathcal{P}_{i + 1} = \mathcal{P}_ i \oplus \mathcal{A}[\mathcal{S}_{i + 1}]$

as graded $\mathcal{A}$-module with differential and map to $\mathcal{M}$ defined as follows

1. for local sections of $\mathcal{P}_ i$ use the differential on $\mathcal{P}_ i$ and the given map to $\mathcal{M}$,

2. for a local section $s = (p, m)$ of $\mathcal{S}_{i + 1}$ we set $\text{d}(s)$ equal to $p$ viewed as a section of $\mathcal{P}_ i$ of degree $\deg (s) + 1$ and we map $s$ to $m$ in $\mathcal{M}$, and

3. extend the differential uniquely so that the Leibniz rule holds.

This makes sense because $\text{d}(m)$ is the image of $p$ and $\text{d}(p) = 0$. Finally, we set $\mathcal{P} = \mathop{\mathrm{colim}}\nolimits \mathcal{P}_ i$ with the induced map to $\mathcal{M}$.

The map $\mathcal{P} \to \mathcal{M}$ is a quasi-isomorphism: we have $H^ n(\mathcal{P}) = \mathop{\mathrm{colim}}\nolimits H^ n(\mathcal{P}_ i)$ and for each $i$ the map $H^ n(\mathcal{P}_ i) \to H^ n(\mathcal{M})$ is surjective with kernel annihilated by the map $H^ n(\mathcal{P}_ i) \to H^ n(\mathcal{P}_{i + 1})$ by construction. Each $\mathcal{P}_ i$ is good because $\mathcal{P}_0$ is good by Lemma 24.23.6 and each $\mathcal{P}_{i + 1}$ is in the middle of the admissible short exact sequence $0 \to \mathcal{P}_ i \to \mathcal{P}_{i + 1} \to \mathcal{A}[\mathcal{S}_{i + 1}] \to 0$ whose outer terms are good by induction. Hence $\mathcal{P}_{i + 1}$ is good by Lemma 24.23.2. Finally, we conclude that $\mathcal{P}$ is good by Lemma 24.23.3. $\square$

Second proof. We urge the reader to read the proof of Differential Graded Algebra, Lemma 22.20.4 before reading this proof. Set $\mathcal{M} = \mathcal{M}_0$. We inductively choose short exact sequences

$0 \to \mathcal{M}_{i + 1} \to \mathcal{P}_ i \to \mathcal{M}_ i \to 0$

where the maps $\mathcal{P}_ i \to \mathcal{M}_ i$ are chosen as in Lemma 24.23.4. This gives a “resolution”

$\ldots \to \mathcal{P}_2 \xrightarrow {f_2} \mathcal{P}_1 \xrightarrow {f_1} \mathcal{P}_0 \to \mathcal{M} \to 0$

Then we let $\mathcal{P}$ be the differential graded $\mathcal{A}$-module defined as follows

1. as a graded $\mathcal{A}$-module we set $\mathcal{P} = \bigoplus _{a \leq 0} \mathcal{P}_{-a}[-a]$, i.e., the degree $n$ part is given by $\mathcal{P}^ n = \bigoplus \nolimits _{a + b = n} \mathcal{P}_{-a}^ b$,

2. the differential on $\mathcal{P}$ is as in the construction of the total complex associated to a double complex given by

$\text{d}_\mathcal {P}(x) = f_{-a}(x) + (-1)^ a \text{d}_{\mathcal{P}_{-a}}(x)$

for $x$ a local section of $\mathcal{P}_{-a}^ b$.

With these conventions $\mathcal{P}$ is indeed a differential graded $\mathcal{A}$-module; we omit the details. There is a map $\mathcal{P} \to \mathcal{M}$ of differential graded $\mathcal{A}$-modules which is zero on the summands $\mathcal{P}_{-a}[-a]$ for $a < 0$ and the given map $\mathcal{P}_0 \to \mathcal{M}$ for $a = 0$. Observe that we have

$\mathcal{P} = \mathop{\mathrm{colim}}\nolimits _ i F_ i\mathcal{P}$

where $F_ i\mathcal{P} \subset \mathcal{P}$ is the differential graded $\mathcal{A}$-submodule whose underlying graded $\mathcal{A}$-module is

$F_ i\mathcal{P} = \bigoplus \nolimits _{i \geq -a \geq 0} \mathcal{P}_{-a}[-a]$

It is immediate that the maps

$0 \to F_1\mathcal{P} \to F_2\mathcal{P} \to F_3\mathcal{P} \to \ldots \to \mathcal{P}$

$0 \to F_ i\mathcal{P} \to F_{i + 1}\mathcal{P} \to \mathcal{P}_{i + 1}[i + 1] \to 0$

By induction and Lemma 24.23.2 we find that $F_ i\mathcal{P}$ is a good differential graded $\mathcal{A}$-module. Since $\mathcal{P} = \mathop{\mathrm{colim}}\nolimits F_ i\mathcal{P}$ we find that $\mathcal{P}$ is good by Lemma 24.23.3.

Finally, we have to show that $\mathcal{P} \to \mathcal{M}$ is a quasi-isomorphism. If $\mathcal{C}$ has enough points, then this follows from the elementary Homology, Lemma 12.26.2 by checking on stalks. In general, we can argue as follows (this proof is far too long — there is an alternative argument by working with local sections as in the elementary proof but it is also rather long). Since filtered colimits are exact on the category of abelian sheaves, we have

$H^ d(\mathcal{P}) = \mathop{\mathrm{colim}}\nolimits H^ d(F_ i\mathcal{P})$

We claim that for each $i \geq 0$ and $d \in \mathbf{Z}$ we have (a) a short exact sequence

$0 \to H^ d(\mathcal{M}_{i + 1}[i]) \to H^ d(F_ i\mathcal{P}) \to H^ d(\mathcal{M}) \to 0$

where the second arrow comes from $F_ i\mathcal{P} \to \mathcal{P} \to \mathcal{M}$ and (b) the composition

$H^ d(\mathcal{M}_{i + 1}[i]) \to H^ d(F_ i\mathcal{P}) \to H^ d(F_{i + 1}\mathcal{P})$

is zero. It is clear that the claim suffices to finish the proof.

Proof of the claim. For any $i \geq 0$ there is a map $\mathcal{M}_{i + 1}[i] \to F_ i\mathcal{P}$ coming from the inclusion of $\mathcal{M}_{i + 1}$ into $\mathcal{P}_ i$ as the kernel of $f_ i$. Consider the short exact sequence

$0 \to \mathcal{M}_{i + 1}[i] \to F_ i\mathcal{P} \to C_ i \to 0$

of complexes of $\mathcal{O}$-modules defining $C_ i$. Observe that $C_0 = \mathcal{M}_0 = \mathcal{M}$. Also, observe that $C_ i$ is the total complex associated to the double complex $C_ i^{\bullet , \bullet }$ with columns

$\mathcal{M}_ i = \mathcal{P}_ i/\mathcal{M}_{i + 1}, \mathcal{P}_{i - 1}, \ldots , \mathcal{P}_0$

in degree $-i, -i + 1, \ldots , 0$. There is a map of double complexes $C_ i^{\bullet , \bullet } \to C_{i - 1}^{\bullet , \bullet }$ which is $0$ on the column in degree $-i$, is the surjection $\mathcal{P}_{i - 1} \to \mathcal{M}_{i - 1}$ in degree $-i + 1$, and is the identity on the other columns. Hence there are maps of complexes

$C_ i \longrightarrow C_{i - 1}$

These maps are surjective quasi-isomorphisms because the kernel is the total complex on the double complex with columns $\mathcal{M}_ i, \mathcal{M}_ i$ in degrees $-i, -i + 1$ and the identity map between these two columns. Using the resulting identifications $H^ d(C_ i) = H^ d(C_{i - 1} = \ldots = H^ d(\mathcal{M})$ this already shows we get a long exact sequence

$H^ d(\mathcal{M}_{i + 1}[i]) \to H^ d(F_ i\mathcal{P}) \to H^ d(\mathcal{M}) \to H^{d + 1}(\mathcal{M}_{i + 1}[i])$

from the short exact sequence of complexes above. However, we also have the commutative diagram

$\xymatrix{ \mathcal{M}_{i + 2}[i + 1] \ar[r]_ a & T_{i + 1} \ar[r] & F_{i + 1}\mathcal{P} \ar[r] & C_{i + 1} \ar[d] \\ & \mathcal{M}_{i + 1}[i] \ar[r] \ar[u]^ b & F_ i\mathcal{P} \ar[u] \ar[r] & C_ i }$

where $T_{i + 1}$ is the total complex on the double complex with columns $\mathcal{P}_{i + 1}, \mathcal{M}_{i + 1}$ placed in degrees $-i - 1$ and $-i$. In other words, $T_{i + 1}$ is a shift of the cone on the map $\mathcal{P}_{i + 1} \to \mathcal{M}_{i + 1}$ and we find that $a$ is a quasi-isomorphism and the map $a^{-1} \circ b$ is a shift of the third map of the distinguished triangle in $D(\mathcal{O})$ associated to the short exact sequence

$0 \to \mathcal{M}_{i + 2} \to \mathcal{P}_{i + 1} \to \mathcal{M}_{i + 1} \to 0$

The map $H^ d(\mathcal{P}_{i + 1}) \to H^ d(\mathcal{M}_{i + 1})$ is surjective because we chose our maps such that $\mathop{\mathrm{Ker}}(\text{d}_{\mathcal{P}_{i + 1}}) \to \mathop{\mathrm{Ker}}(\text{d}_{\mathcal{M}_{i + 1}})$ is surjective. Thus we see that $a^{-1} \circ b$ is zero on cohomology sheaves. This proves part (b) of the claim. Since $T_{i + 1}$ is the kernel of the surjective map of complexes $F_{i + 1}\mathcal{P} \to C_ i$ we find a map of long exact cohomology sequences

$\xymatrix{ H^ d(T_{i + 1}) \ar[r] & H^ d(F_{i + 1}\mathcal{P}) \ar[r] & H^ d(\mathcal{M}) \ar[r] & H^{d + 1}(T_{i + 1}) \\ H^ d(\mathcal{M}_{i + 1}[i]) \ar[r] \ar[u] & H^ d(F_ i\mathcal{P}) \ar[r] \ar[u] & H^ d(\mathcal{M}) \ar[r] \ar[u] & H^{d + 1}(\mathcal{M}_{i + 1}[i]) \ar[u] }$

Here we know, by the discussion above, that the vertical maps on the outside are zero. Hence the maps $H^ d(F_{i + 1}\mathcal{P}) \to H^ d(\mathcal{M})$ are surjective and part (a) of the claim follows. More precisely, the claim follows for $i > 0$ and we leave the claim for $i = 0$ to the reader (actually it suffices to prove the claim for all $i \gg 0$ in order to get the lemma). $\square$

Lemma 24.23.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{P}$ be a good acyclic right differential graded $\mathcal{A}$-module.

1. for any differential graded left $\mathcal{A}$-module $\mathcal{N}$ the tensor product $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}$ is acyclic,

2. for any morphism $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of ringed topoi and any differential graded $\mathcal{O}'$-algebra $\mathcal{A}'$ and any map $\varphi : f^{-1}\mathcal{A} \to \mathcal{A}'$ of differential graded $f^{-1}\mathcal{O}$-algebras the pullback $f^*\mathcal{P}$ is acyclic and good.

Proof. Proof of (1). By Lemma 24.23.7 we can choose a good left differential graded $\mathcal{Q}$ and a quasi-isomorphism $\mathcal{Q} \to \mathcal{N}$. Then $\mathcal{P} \otimes _\mathcal {A} \mathcal{Q}$ is acyclic because $\mathcal{Q}$ is good. Let $\mathcal{N}'$ be the cone on the map $\mathcal{Q} \to \mathcal{N}$. Then $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}'$ is acyclic because $\mathcal{P}$ is good and because $\mathcal{N}'$ is acyclic (as the cone on a quasi-isomorphism). We have a distinguished triangle

$\mathcal{Q} \to \mathcal{N} \to \mathcal{N}' \to \mathcal{Q}[1]$

in $K(\textit{Mod}(\mathcal{A}, \text{d}))$ by our construction of the triangulated structure. Since $\mathcal{P} \otimes _\mathcal {A} -$ sends distinguished triangles to distinguished triangles, we obtain a distinguished triangle

$\mathcal{P} \otimes _\mathcal {A} \mathcal{Q} \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N}' \to \mathcal{P} \otimes _\mathcal {A} \mathcal{Q}[1]$

in $K(\textit{Mod}(\mathcal{O}))$. Thus we conclude.

Proof of (2). Observe that $f^*\mathcal{P}$ is good by our definition of good modules. Recall that $f^*\mathcal{P} = f^{-1}\mathcal{P} \otimes _{f^{-1}\mathcal{A}} \mathcal{A}'$. Then $f^{-1}\mathcal{P}$ is a good acyclic (because $f^{-1}$ is exact) differential graded $f^{-1}\mathcal{A}$-module. Hence we see that $f^*\mathcal{P}$ is acyclic by part (1). $\square$

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