The Stacks project

Lemma 24.28.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. If $\mathcal{N}$ is good as a left differential graded $\mathcal{A}$-module, then we have $\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N} = \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ for all differential graded $\mathcal{A}$-modules $\mathcal{M}$.

Proof. Let $\mathcal{P} \to \mathcal{M}$ be a quasi-isomorphism where $\mathcal{P}$ is a good (right) differential graded $\mathcal{A}$-module. To prove the lemma we have to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ is a quasi-isomorphism. The cone $C$ on the map $\mathcal{P} \to \mathcal{M}$ is an acyclic right differential graded $\mathcal{A}$-module. Hence $C \otimes _\mathcal {A} \mathcal{N}$ is acyclic as $\mathcal{N}$ is assumed good as a left differential graded $\mathcal{A}$-module. Since $C \otimes _\mathcal {A} \mathcal{N}$ is the cone on the maps $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{M} \otimes _\mathcal {A} \mathcal{N}$ as a complex of $\mathcal{O}$-modules we conclude. $\square$

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