The Stacks project

Lemma 24.29.4. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{A}$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. Let $\varphi : \mathcal{B} \to f_*\mathcal{A}$ be a homomorphism of differential graded $\mathcal{O}_\mathcal {D}$-algebras. Then

\[ Rf_* : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{B}, \text{d}) \]

is right adjoint to

\[ Lf^* : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{A}, \text{d}) \]

Proof. This follows from Derived Categories, Lemma 13.30.1 and Lemma 24.18.1. $\square$

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