The Stacks project

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

  1. $R\Gamma (V, M) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}(U) \to \Gamma (U, M)$ is an isomorphism in $D(\mathcal{A}(U), \text{d})$, and

  2. $R\Gamma (V, M \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}) \otimes _{\mathcal{B}(V)}^\mathbf {L} \mathcal{B}(U) \to \Gamma (U, M \otimes _\mathcal {A}^\mathbf {L} \mathcal{B})$ 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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