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
\[ H^ n(X \times Y, p^*\mathcal{F} \otimes _{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) = \bigoplus \nolimits _{a + b = n} H^ a(X, \mathcal{F}) \otimes _ k H^ b(Y, \mathcal{G}) \]
or just show that this holds when one takes dimensions. Extra points for “clean” solutions.
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