Lemma 20.31.5. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The relative cup product of Remark 20.28.7 is commutative in the sense that the diagram

$\xymatrix{ Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \ar[r] \ar[d]_\psi & Rf_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \ar[d]^{Rf_*\psi } \\ Rf_*L \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*K \ar[r] & Rf_*(L \otimes _{\mathcal{O}_ X}^\mathbf {L} K) }$

is commutative in $D(\mathcal{O}_ Y)$ for all $K, L$ in $D(\mathcal{O}_ X)$. Here $\psi$ is the commutativity constraint on the derived category (Lemma 20.46.5).

Proof. Omitted. $\square$

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