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

Take an affine open subscheme $U=\mathrm{Spec}A\subseteq Y$. Properness and flatness survive base change, and $A$ is reduced because $Y$ is reduced. A bounded complex of finite locally free modules computes the cohomology of $\mathcal{F}$ on $X_U$.

If some $H^p(X_U,\mathcal{F})$ were nonzero, choose the largest such $p=n$. All groups above degree $n$ would vanish, while every fiber group in degree $n$ vanishes by hypothesis. Part (iii) would force $H^n(X_U,\mathcal{F})=0$, a contradiction. Hencefor every affine $U\subseteq Y$ and every $p\ge 0$. These groups compute the sections of the higher direct images over affine opens, so $R^p{f}_{\ast }\mathcal{F}=0$ for all $p$, including $f_{\ast }\mathcal{F}=0$ when $p=0$. The Leray spectral sequence now gives