The Stacks project

Lemma 24.29.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$, $\mathcal{B}$ be differential graded $\mathcal{O}$-algebras. Let $\mathcal{N}$ be a differential graded $(\mathcal{A}, \mathcal{B})$-bimodule. Then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}(\mathcal{N}, -) : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d}) \]

is right adjoint to

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

Proof. This follows from Derived Categories, Lemma 13.30.1 and Lemma 24.17.3. $\square$

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