Remark 36.23.5. Let $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $a : X \to S$ and $b : Y \to S$ be morphisms of schemes. Let $\mathcal{F}$, $\mathcal{G}$ be quasi-coherent $\mathcal{O}_ X$-modules and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $\xi \in H^ i(X, \mathcal{G})$ with pullback $p^*\xi \in H^ i(X \times _ S Y, p^*\mathcal{G})$. Then the following diagram is commutative

where the unadorned tensor products are over $\mathcal{O}_{X \times _ S Y}$. The horizontal arrows are from Cohomology, Remark 20.31.2 and the vertical arrows are (36.23.0.2) hence given by pulling back followed by cup product on $X \times _ S Y$. The diagram commutes because the global cup product (on $X \times _ S Y$ with the sheaves $p^*\mathcal{G}$, $p^*\mathcal{F}$, and $q^*\mathcal{E}$) is associative, see Cohomology, Lemma 20.31.5.

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