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