Exercise 111.42.5. Let $f : X \to Y$ be an affine morphism of schemes. Prove that $H^ i(X, \mathcal{F}) = H^ i(Y, f_*\mathcal{F})$ for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$. Feel free to impose some further conditions on $X$ and $Y$ and use the agreement of Čech cohomology with cohomology for quasi-coherent sheaves and affine open coverings of separated schemes.
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