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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