Lemma 24.28.7. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{A}, \mathcal{A}', \mathcal{A}'' be differential graded \mathcal{O}-algebras. Let \mathcal{N} and \mathcal{N}' be a differential graded (\mathcal{A}, \mathcal{A}')-bimodule and (\mathcal{A}', \mathcal{A}'')-bimodule. Assume that the canonical map
\mathcal{N} \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \longrightarrow \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'
in D(\mathcal{A}'', \text{d}) is a quasi-isomorphism. Then we have
(\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' = \mathcal{M} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}')
as functors D(\mathcal{A}, \text{d}) \to D(\mathcal{A}'', \text{d}).
Proof.
Choose a good differential graded \mathcal{A}-module \mathcal{P} and a quasi-isomorphism \mathcal{P} \to \mathcal{M}, see Lemma 24.23.7. Then
\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}') = \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'
and we have
(\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' = (\mathcal{P} \otimes _\mathcal {A} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}'
Thus we have to show the canonical map
(\mathcal{P} \otimes _\mathcal {A} \mathcal{N}) \otimes _{\mathcal{A}'}^\mathbf {L} \mathcal{N}' \longrightarrow \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'
is a quasi-isomorphism. Choose a quasi-isomorphism \mathcal{Q} \to \mathcal{N}' where \mathcal{Q} is a good left differential graded \mathcal{A}'-module (Lemma 24.23.7). By Lemma 24.28.6 the map above as a map in the derived category of \mathcal{O}-modules is the map
\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{Q} \longrightarrow \mathcal{P} \otimes _\mathcal {A} \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}'
Since \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{Q} \to \mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}' is a quasi-isomorphism by assumption and \mathcal{P} is a good differential graded \mathcal{A}-module this map is an quasi-isomorphism by Lemma 24.28.5 (the left and right hand side compute \mathcal{P} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{Q}) and \mathcal{P} \otimes _\mathcal {A}^\mathbf {L} (\mathcal{N} \otimes _{\mathcal{A}'} \mathcal{N}') or you can just repeat the argument in the proof of the lemma).
\square
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