Remark 59.97.1. Consider a cartesian diagram in the category of schemes:
\xymatrix{ X \times _ S Y \ar[d]_ p \ar[r]_ q \ar[rd]_ c & Y \ar[d]^ g \\ X \ar[r]^ f & S }
Let \Lambda be a ring and let E \in D(X_{\acute{e}tale}, \Lambda ) and K \in D(Y_{\acute{e}tale}, \Lambda ). Then there is a canonical map
Rf_*E \otimes _\Lambda ^\mathbf {L} Rg_*K \longrightarrow Rc_*(p^{-1}E \otimes _\Lambda ^\mathbf {L} q^{-1}K)
For example we can define this using the canonical maps Rf_*E \to Rc_*p^{-1}E and Rg_*K \to Rc_*q^{-1}K and the relative cup product defined in Cohomology on Sites, Remark 21.19.7. Or you can use the adjoint to the map
c^{-1}(Rf_*E \otimes _\Lambda ^\mathbf {L} Rg_*K) = p^{-1}f^{-1}Rf_*E \otimes _\Lambda ^\mathbf {L} q^{-1} g^{-1}Rg_*K \to p^{-1}E \otimes _\Lambda ^\mathbf {L} q^{-1}K
which uses the adjunction maps f^{-1}Rf_*E \to E and g^{-1}Rg_*K \to K.
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