The Stacks project

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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