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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