The Stacks project

Definition 24.29.2. Derived internal hom and derived pushforward.

1. 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. The right derived extension

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

   of the internal hom functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {B}^{dg}(\mathcal{N}, -)$ is called derived internal hom.

2. 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. The right derived extension

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

   of the pushforward $f_*$ is called derived pushforward.