The Stacks project

Remark 36.22.2. With $X, Y, S, a, b, p, q, f$ as in the introduction to this section suppose we have an $\mathcal{O}_ X$-module $\mathcal{F}$ and an $\mathcal{O}_ Y$-module $\mathcal{G}$. Then we have

\begin{align*} & p^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{G} \\ & = p^{-1}(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X) \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}(\mathcal{O}_ Y \otimes _{\mathcal{O}_ Y} \mathcal{G}) \\ & = p^{-1}\mathcal{F} \otimes _{p^{-1}\mathcal{O}_ X} p^{-1}\mathcal{O}_ X \otimes _{f^{-1}\mathcal{O}_ S} q^{-1}\mathcal{O}_ Y \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F} \otimes _{q^{-1}\mathcal{O}_ X} \mathcal{O}_{X \times _ S Y} \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^{-1}\mathcal{F} \otimes _{q^{-1}\mathcal{O}_ X} \mathcal{O}_{X \times _ S Y} \otimes _{\mathcal{O}_{X \times _ S Y}} \mathcal{O}_{X \times _ S Y} \otimes _{q^{-1}\mathcal{O}_ Y} q^{-1}\mathcal{G} \\ & = p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} q^*\mathcal{G} \end{align*}

This is occasionally useful.

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