For an <affine morphism> $f:X\to Y$ and a <quasi-coherent sheaf> $\mathcal F$, every inverse image of an affine open $V\subseteq Y$ is affine. Higher cohomology of a quasi-coherent sheaf on an affine scheme vanishes, so
$$
(R^qf_*\mathcal F)|_V=H^q(f^{-1}V,\mathcal F)=0\qquad(q>0).
$$
The <Leray spectral sequence> therefore has only its zeroth row, and its edge maps give
$$
\boxed{H^p(X,\mathcal F)\simeq H^p(Y,f_*\mathcal F)\quad(p\ge0).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 126 / 2 / i / Solution

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A morphism $f:X\to Y$ is affine when the inverse image of every affine open subscheme of $Y$ is affine. For a <quasi-coherent sheaf> $\mathcal F$ on $X$, one has $R^qf_*\mathcal F=0$ for $q>0$.

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