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

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, soThe Leray spectral sequence therefore has only its zeroth row, and its edge maps give

For $X=U\cup V$, the Mayer-Vietoris sequence for sheaf cohomology is the long exact sequence

We prove the required vanishing by induction on the number $m$ of open sets. The case $m=1$ is an assumption. Put $U=U_1\cup \cdots \cup U_{m-1}$ and $V=U_m$. The induction hypothesis gives $H^p(U,\mathcal{F})=0$ for every $p$. The intersections $U_i\cap U_m$ cover $U\cap V$, and every nonempty finite intersection among them is one of the intersections in the hypothesis, so the same induction gives $H^p(U\cap V,\mathcal{F})=0$. We also have $H^p(V,\mathcal{F})=0$. Exactness of the Mayer-Vietoris sequence now yields

The finite complex computing cohomology in a proper flat family gives a bounded complex $K^{\bullet }$ of finite locally free $A$-modules such that, for every $A$-module $M$,In particular, $M=A$ computes $H^p(X,\mathcal{F})$ and $M=\kappa (s)$ computes $H^p(X_s,\mathcal{F}(s))$.

Since $H^p(K^{\bullet })=0$ for $p>n$, the finite exact tail above degree $n$ can be split successively: its last differential is surjective onto a projective module, hence splits, and induction moves left. Removing the resulting contractible summands leaves a finite locally free complex ending in degree $n$. Thereforeand after tensoring with $\kappa (s)$ the same formula computes the fiber cohomology.

If $H^n(X,\mathcal{F})=0$, the last differential is surjective, and remains so after every base change; hence every $H^n(X_s,\mathcal{F}(s))$ vanishes. Conversely, if all fiber groups vanish, the finitely generated cokernel $C$ satisfies $C{\otimes }_A\kappa (s)=0$ for every $s\in \mathrm{Spec}A$. Localizing and applying Nakayama lemma gives $C_s=0$ for every $s$, so $C=0$. Thus

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