Lemma 24.33.4. Let g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) and \varphi : g^*\mathcal{A} \to \mathcal{A}' be as above. Then the functor Lg^* : D(\mathcal{A}, \text{d}) \to D(\mathcal{A}', \text{d}) maps \mathit{QC}(\mathcal{A}, \text{d}) into \mathit{QC}(\mathcal{A}', \text{d}).
Proof. Let U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}') with image U = u(U') in \mathcal{C}. Let pt denote the category with a single object and a single morphism. Denote (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}'(U')) and (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U)) the ringed topoi as indicated endowed with the differential graded algebras \mathcal{A}'(U) and \mathcal{A}(U). Of course we identify the derived category of differential graded modules on these with D(\mathcal{A}'(U'), \text{d}) and D(\mathcal{A}(U), \text{d}). Then we have a commutative diagram of ringed topoi
\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}'(U')) \ar[rr]_{U'} \ar[d] & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \ar[d]^ g \\ (\mathop{\mathit{Sh}}\nolimits (pt), \mathcal{O}(U)) \ar[rr]^ U & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) }
each endowed with corresponding differential graded algebras. Pullback along the lower horizontal morphism sends M in D(\mathcal{A}, \text{d}) to R\Gamma (U, K) viewed as an object in D(\mathcal{A}(U), \text{d}). Pullback by the left vertical arrow sends M to M \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U'). Going around the diagram either direction produces the same result (Lemma 24.28.4) and hence we conclude
R\Gamma (U', Lg^*K) = R\Gamma (U, K) \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U')
Finally, let f' : U' \to V' be a morphism in \mathcal{C}' and denote f = u(f') : U = u(U') \to V = u(V') the image in \mathcal{C}. If K is in \mathit{QC}(\mathcal{A}, \text{d}) then we have
\begin{align*} R\Gamma (V', Lg^*K) \otimes _{\mathcal{A}'(V')}^\mathbf {L} \mathcal{A}'(U') & = R\Gamma (V, K) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}'(V') \otimes _{\mathcal{A}'(V')}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (V, K) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (V, K) \otimes _{\mathcal{A}(V)}^\mathbf {L} \mathcal{A}(U) \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (U, K) \otimes _{\mathcal{A}(U)}^\mathbf {L} \mathcal{A}'(U') \\ & = R\Gamma (U', Lg^*K) \end{align*}
as desired. Here we have used the observation above both for U' and V'. \square
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