The Stacks project

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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