The Stacks project

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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