Section 24.30 (0FTY): Equivalences of derived categories

Lemma 24.30.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $\varphi : \mathcal{A} \to \mathcal{B}$ is a homomorphism of differential graded $\mathcal{O}$ -algebras which induces an isomorphism on cohomology sheaves, then

$D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d}), \quad \mathcal{M} \longmapsto \mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}$

is an equivalence of categories.

Proof. Recall that the restriction functor

$\textit{Mod}^{dg}(\mathcal{B}, \text{d}) \to \textit{Mod}^{dg}(\mathcal{A}, \text{d}),\quad \mathcal{N} \mapsto res_\varphi \mathcal{N}$

is a right adjoint to

$\textit{Mod}^{dg}(\mathcal{A}, \text{d}) \to \textit{Mod}^{dg}(\mathcal{B}, \text{d}),\quad \mathcal{M} \mapsto \mathcal{M} \otimes _\mathcal {A} \mathcal{B}$

See Section 24.17. Since restriction sends quasi-isomorphisms to quasi-isomorphisms, we see that it trivially has a left derived extension (given by restriction). This functor will be right adjoint to $- \otimes _\mathcal {A}^\mathbf {L} \mathcal{B}$ by Derived Categories, Lemma 13.30.1. The adjunction map

$\mathcal{M} \to res_\varphi (\mathcal{M} \otimes _\mathcal {A}^\mathbf {L} \mathcal{B})$

is an isomorphism in $D(\mathcal{A}, \text{d})$ by our assumption that $\mathcal{A} \to \mathcal{B}$ is a quasi-isomorphism of (left) differential graded $\mathcal{A}$ -modules. In particular, the functor of the lemma is fully faithful, see Categories, Lemma 4.24.4. It is clear that the kernel of the restriction functor $D(\mathcal{B}, \text{d}) \to D(\mathcal{A}, \text{d})$ is zero. Thus we conclude by Derived Categories, Lemma 13.7.2. $\square$

The Stacks project

24.30 Equivalences of derived categories

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