The Stacks project

Remark 21.20.6. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product

\[ Rf_*K \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} Rf_*L \longrightarrow Rf_*(K \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} L) \]

in $D(\mathcal{O}_\mathcal {D})$ for all $K, L$ in $D(\mathcal{O}_\mathcal {C})$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} Rf_*L) \to K \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} Rf_*L) = Lf^*Rf_*K \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} Lf^*Rf_*L$ (Lemma 21.19.4) with the map $Lf^*Rf_*K \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} Lf^*Rf_*L \to K \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} L$ coming from the counit $Lf^* \circ Rf_* \to \text{id}$.


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