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The Stacks project

Lemma 22.33.5. Let R be a ring. Let (A, \text{d}) and (B, \text{d}) be differential graded R-algebras. Let N be a differential graded (A, B)-bimodule. Then the functor

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

of Lemma 22.33.2 is a left adjoint to the functor

R\mathop{\mathrm{Hom}}\nolimits (N, -) : D(B, \text{d}) \longrightarrow D(A, \text{d})

of Lemma 22.31.2.

Proof. This follows from Derived Categories, Lemma 13.30.1 and the fact that - \otimes _ A N and \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{dg}_{(B, \text{d})}}(N, -) are adjoint by Lemma 22.30.3. \square


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