The Stacks project

Lemma 24.28.5. In the situation above, if $\mathcal{N} \to \mathcal{N}'$ is an isomorphism on cohomology sheaves, then $t$ is an isomorphism of functors $(- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}) \to (- \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}')$.

Proof. It is enough to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N}'$ is an isomorphism on cohomology sheaves for any good differential graded $\mathcal{A}$-module $\mathcal{P}$. To do this, let $\mathcal{N}''$ be the cone on the map $\mathcal{N} \to \mathcal{N}'$ as a left differential graded $\mathcal{A}$-module, see Definition 24.22.2. (To be sure, $\mathcal{N}''$ is a bimodule too but we don't need this.) By functoriality of the tensor construction (it is a functor of differential graded categories) we see that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}''$ is the cone (as a complex of $\mathcal{O}$-modules) on the map $\mathcal{P} \otimes _\mathcal {A} \mathcal{N} \to \mathcal{P} \otimes _\mathcal {A} \mathcal{N}'$. Hence it suffices to show that $\mathcal{P} \otimes _\mathcal {A} \mathcal{N}''$ is acyclic. This follows from the fact that $\mathcal{P}$ is good and the fact that $\mathcal{N}''$ is acyclic as a cone on a quasi-isomorphism. $\square$

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