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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