The Stacks project

Lemma 22.37.1. Let $R$ be a ring. Let $(A, \text{d}) \to (B, \text{d})$ be a homomorphism of differential graded algebras over $R$, which induces an isomorphism on cohomology algebras. Then

\[ - \otimes _ A^\mathbf {L} B : D(A, \text{d}) \to D(B, \text{d}) \]

gives an $R$-linear equivalence of triangulated categories with quasi-inverse the restriction functor $N \mapsto N_ A$.

Proof. By Lemma 22.33.7 the functor $M \longmapsto M \otimes _ A^\mathbf {L} B$ is fully faithful. By Lemma 22.33.5 the functor $N \longmapsto R\mathop{\mathrm{Hom}}\nolimits (B, N) = N_ A$ is a right adjoint, see Example 22.33.6. It is clear that the kernel of $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ is zero. Hence the result follows from Derived Categories, Lemma 13.7.2. $\square$

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