Remark 36.23.5 (0G7W)—The Stacks project

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

\[ \xymatrix{ R\Gamma (X, \mathcal{F})[-i] \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{E}) \ar[d] \ar[rr]_-{\xi \otimes \text{id}} & & R\Gamma (X, \mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F}) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{E}) \ar[d] \\ R\Gamma (X \times _ S Y, p^*\mathcal{F} \otimes q^*\mathcal{E})[-i] \ar[rr]^-{p^*\xi } & & R\Gamma (X \times _ S Y, p^*(\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F}) \otimes q^*\mathcal{E}) } \]

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.

The Stacks project

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