Exercise 111.44.6. Let $X$ and $Y$ be schemes over a field $k$ (feel free to assume $X$ and $Y$ are nice, for example qcqs or projective over $k$). Set $X \times Y = X \times _{\mathop{\mathrm{Spec}}(k)} Y$ with projections $p : X \times Y \to X$ and $q : X \times Y \to Y$. For a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and a quasi-coherent $\mathcal{O}_ Y$-module $\mathcal{G}$ prove that

or just show that this holds when one takes dimensions. Extra points for “clean” solutions.

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