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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-126/2/i/solution