Lemma 24.28.1. In the situation above, the functor (24.28.0.1) composed with the localization functor K(\textit{Mod}(\mathcal{A}', \text{d})) \to D(\mathcal{A}', \text{d}) has a left derived extension D(\mathcal{B}, \text{d}) \to D(\mathcal{A}', \text{d}) whose value on a good right differential graded \mathcal{B}-module \mathcal{P} is f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N}.
Proof. Recall that for any (right) differential graded \mathcal{B}-module \mathcal{M} there exists a quasi-isomorphism \mathcal{P} \to \mathcal{M} with \mathcal{P} a good differential graded \mathcal{B}-module. See Lemma 24.23.7. Hence by Derived Categories, Lemma 13.14.15 it suffices to show that given a quasi-isomorphism \mathcal{P} \to \mathcal{P}' of good differential graded \mathcal{B}-modules the induced map
f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \longrightarrow f^*\mathcal{P}' \otimes _\mathcal {A} \mathcal{N}
is a quasi-isomorphism. The cone \mathcal{P}'' on \mathcal{P} \to \mathcal{P}' is a good differential graded \mathcal{A}-module by Lemma 24.23.2. Since we have a distinguished triangle
\mathcal{P} \to \mathcal{P}' \to \mathcal{P}'' \to \mathcal{P}[1]
in K(\textit{Mod}(\mathcal{B}, \text{d})) we obtain a distinguished triangle
f^*\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}' \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}'' \otimes _\mathcal {A} \mathcal{N} \to f^*\mathcal{P}[1] \otimes _\mathcal {A} \mathcal{N}
in K(\textit{Mod}(\mathcal{A}', \text{d})). By Lemma 24.23.8 the differential graded module f^*\mathcal{P}'' \otimes _\mathcal {A} \mathcal{N} is acyclic and the proof is complete. \square
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