Lemma 20.31.4. 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 associative in the sense that the diagram

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

is commutative in $D(\mathcal{O}_ Y)$ for all $K, L, M$ in $D(\mathcal{O}_ X)$.

**Proof.**
Going around either side we obtain the map adjoint to the obvious map

\begin{align*} Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*M) & = Lf^*(Rf_*K) \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*(Rf_*L) \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*(Rf_*M) \\ & \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L \otimes _{\mathcal{O}_ X}^\mathbf {L} M \end{align*}

in $D(\mathcal{O}_ X)$.
$\square$

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