The Stacks project

Lemma 22.37.2. Let $R$ be a ring. Let $(A, \text{d})$ and $(B, \text{d})$ be differential graded algebras over $R$. Let $N$ be a differential graded $(A, B)$-bimodule. Assume that

  1. $N$ defines a compact object of $D(B, \text{d})$,

  2. if $N' \in D(B, \text{d})$ and $\mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(N, N'[n]) = 0$ for $n \in \mathbf{Z}$, then $N' = 0$, and

  3. the map $H^ k(A) \to \mathop{\mathrm{Hom}}\nolimits _{D(B, \text{d})}(N, N[k])$ is an isomorphism for all $k \in \mathbf{Z}$.

Then

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

gives an $R$-linear equivalence of triangulated categories.

Proof. By Lemma 22.33.7 the functor $M \longmapsto M \otimes _ A^\mathbf {L} N$ is fully faithful. By Lemma 22.33.5 the functor $N' \longmapsto R\mathop{\mathrm{Hom}}\nolimits (N, N')$ is a right adjoint. By assumption (3) the kernel of $R\mathop{\mathrm{Hom}}\nolimits (N, -)$ is zero. Hence the result follows from Derived Categories, Lemma 13.7.2. $\square$

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