The Stacks project

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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