Lemma 24.29.8. 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 diagram
\[ \xymatrix{ D(\mathcal{A}, \text{d}) \ar[d]_{Rf_*} \ar[rr]_{forget} & & D(\mathcal{O}_\mathcal {C}) \ar[d]^{Rf_*} \\ D(\mathcal{B}, \text{d}) \ar[rr]^{forget} & & D(\mathcal{O}_\mathcal {D}) } \]
commutes.
Proof.
Besides identifying some categories, this lemma follows immediately from Lemma 24.29.6.
We may view $(\mathcal{O}_\mathcal {C}, 0)$ as a differential graded $\mathcal{O}_\mathcal {C}$-algebra by placing $\mathcal{O}_\mathcal {C}$ in degree $0$ and endowing it with the zero differential. It is clear that we have
\[ \textit{Mod}(\mathcal{O}_\mathcal {C}, 0) = \text{Comp}(\mathcal{O}_\mathcal {C}) \quad \text{and}\quad D(\mathcal{O}_\mathcal {C}, 0) = D(\mathcal{O}_\mathcal {C}) \]
Via this identification the forgetful functor $\textit{Mod}(\mathcal{A}, \text{d}) \to \text{Comp}(\mathcal{O}_\mathcal {C})$ is the “pushforward” $\text{id}_{\mathcal{C}, *}$ defined in Section 24.18 corresponding to the identity morphism $\text{id}_\mathcal {C} : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{C}, \mathcal{O}_\mathcal {C})$ of ringed topoi and the map $(\mathcal{O}_\mathcal {C}, 0) \to (\mathcal{A}, \text{d})$ of differential graded $\mathcal{O}_\mathcal {C}$-algebras. Since $\text{id}_{\mathcal{C}, *}$ is exact, we immediately see that
\[ R\text{id}_{\mathcal{C}, *} = forget : D(\mathcal{A}, \text{d}) \longrightarrow D(\mathcal{O}_\mathcal {C}, 0) = D(\mathcal{O}_\mathcal {C}) \]
The exact same reasoning shows that
\[ R\text{id}_{\mathcal{D}, *} = forget : D(\mathcal{B}, \text{d}) \longrightarrow D(\mathcal{O}_\mathcal {D}, 0) = D(\mathcal{O}_\mathcal {D}) \]
Moreover, the construction of $Rf_* : D(\mathcal{O}_\mathcal {C}) \to D(\mathcal{O}_\mathcal {D})$ of Cohomology on Sites, Section 21.19 agrees with the construction of $Rf_* : D(\mathcal{O}_\mathcal {C}, 0) \to D(\mathcal{O}_\mathcal {D}, 0)$ in Definition 24.29.2 as both functors are defined as the right derived extension of pushforward on underlying complexes of modules. By Lemma 24.29.6 we see that both $Rf_* \circ R\text{id}_{\mathcal{C}, *}$ and $R\text{id}_{\mathcal{D}, *} \circ Rf_*$ are the derived functors of $f_* \circ forget = forget \circ f_*$ and hence equal by uniqueness of adjoints.
$\square$
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