The Stacks project

Proof. By Morphisms, Lemma 29.28.4 and the fact that $f$ is closed, we can find an affine open neighbourhood $V$ of $y$ such that the fibres over points of $V$ all have dimension $\leq d$. Thus we may assume $X \to Y$ is a proper morphism all of whose fibres have dimension $\leq d$ with $Y$ affine. We will show that (2) holds, which will immediately imply (1) for all $y \in Y$.

By Lemma 32.13.2 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as a cofiltered limit with $X_ i \to Y$ proper and of finite presentation and such that both $X \to X_ i$ and transition morphisms are closed immersions. For some $i$ we have that $X_ i \to Y$ has fibres of dimension $\leq d$, see Lemma 32.18.1. For a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $R^ pf_*\mathcal{F} = R^ pf_{i, *}(X \to X_ i)_*\mathcal{F}$ by Cohomology of Schemes, Lemma 30.2.3 and Leray (Cohomology, Lemma 20.13.8). Thus we may replace $X$ by $X_ i$ and reduce to the case discussed in the next paragraph.

Assume $Y$ is affine and $f : X \to Y$ is proper and of finite presentation and all fibres have dimension $\leq d$. It suffices to show that $H^ p(X, \mathcal{F}) = 0$ for $p > d$. Namely, by Cohomology of Schemes, Lemma 30.4.6 we have $H^ p(X, \mathcal{F}) = H^0(Y, R^ pf_*\mathcal{F})$. On the other hand, $R^ pf_*\mathcal{F}$ is quasi-coherent on $Y$ by Cohomology of Schemes, Lemma 30.4.5, hence vanishing of global sections implies vanishing. Write $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ as a cofiltered limit of affine schemes with $Y_ i$ the spectrum of a Noetherian ring (for example a finite type $\mathbf{Z}$-algebra). We can choose an element $0 \in I$ and a finite type morphism $X_0 \to Y_0$ such that $X \cong Y \times _{Y_0} X_0$, see Lemma 32.10.1. After increasing $0$ we may assume $X_0 \to Y_0$ is proper (Lemma 32.13.1) and that the fibres of $X_0 \to Y_0$ have dimension $\leq d$ (Lemma 32.18.1). Since $X \to X_0$ is affine, we find that $H^ p(X, \mathcal{F}) = H^ p(X_0, (X \to X_0)_*\mathcal{F})$ by Cohomology of Schemes, Lemma 30.2.4. This reduces us to the case discussed in the next paragraph.

Assume $Y$ is affine Noetherian and $f : X \to Y$ is proper and all fibres have dimension $\leq d$. In this case we can write $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ as a filtered colimit of coherent $\mathcal{O}_ X$-modules, see Properties, Lemma 28.22.7. Then $H^ p(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ p(X, \mathcal{F}_ i)$ by Cohomology, Lemma 20.19.1. Thus we may assume $\mathcal{F}$ is coherent. In this case we see that $(R^ pf_*\mathcal{F})_ y = 0$ for all $y \in Y$ by Cohomology of Schemes, Lemma 30.20.9. Thus $R^ pf_*\mathcal{F} = 0$ and therefore $H^ p(X, \mathcal{F}) = 0$ (see above) and we win. $\square$