Lemma 24.34.2. Let \mathcal{C}, \mathcal{O} be as in Section 24.33. Let \varphi : \mathcal{A} \to \mathcal{B} be a homomorphism of differential graded \mathcal{O}-algebras which induces an isomorphism on cohomology sheaves, then the equivalence D(\mathcal{A}, \text{d}) \to D(\mathcal{B}, \text{d}) of Lemma 24.30.1 induces an equivalence \mathit{QC}(\mathcal{A}, \text{d}) \to \mathit{QC}(\mathcal{B}, \text{d}).
Proof. It suffices to show the following: given a morphism U \to V of \mathcal{C} and M in D(\mathcal{A}, \text{d}) the following are equivalent
R\Gamma (V, M) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}(U) \to \Gamma (U, M) is an isomorphism in D(\mathcal{A}(U), \text{d}), and
R\Gamma (V, M \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}) \otimes _{\mathcal{B}(V)}^\mathbf {L} \mathcal{B}(U) \to \Gamma (U, M \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}) is an isomorphism in D(\mathcal{B}(U), \text{d}).
Since the topology on \mathcal{C} is chaotic, this simply boils down to fact that \mathcal{A}(U) \to \mathcal{B}(U) and \mathcal{A}(V) \to \mathcal{B}(V) are quasi-isomorphisms. Details omitted. \square
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