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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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