Lemma 32.18.3. Let $f : X \to Y$ be a morphism of schemes. Let $d \geq 0$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Assume

1. $f$ is a proper morphism all of whose fibres have dimension $\leq d$,

2. $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module of finite type.

Then $R^ df_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module of finite type.

Proof. The module $R^ df_*\mathcal{F}$ is quasi-coherent by Cohomology of Schemes, Lemma 30.4.5. The question is local on $Y$ hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. Then it suffices to prove that $H^ d(X, \mathcal{F})$ is a finite $R$-module.

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 32.17.1. 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 $\leq d$. We can write $\mathcal{F}$ as a quotient of a finitely presented $\mathcal{O}_ X$-module $\mathcal{F}'$, see Properties, Lemma 28.22.8. The map $H^ d(X, \mathcal{F}') \to H^ d(X, \mathcal{F})$ is surjective, as we have $H^{d + 1}(X, \mathop{\mathrm{Ker}}(\mathcal{F}' \to \mathcal{F})) = 0$ by the vanishing of higher cohomology seen in Lemma 32.18.2 (or its proof). Thus we reduce to the case discussed in the next paragraph.

Assume $Y = \mathop{\mathrm{Spec}}(R)$ is affine and $f : X \to Y$ is proper and of finite presentation and all fibres have dimension $\leq d$ and $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation. Write $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ as a cofiltered limit of affine schemes with $Y_ i = \mathop{\mathrm{Spec}}(R_ i)$ 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 32.17.1). After increasing $0$ we can assume there is a coherent $\mathcal{O}_{X_0}$-module $\mathcal{F}_0$ which pulls back to $\mathcal{F}$, see Lemma 32.10.2. By Lemma 32.18.1 we have

$H^ d(X, \mathcal{F}) = H^ d(X_0, \mathcal{F}_0) \otimes _{R_0} R$

This finishes the proof because the cohomology module $H^ d(X_0, \mathcal{F}_0)$ is finite by Cohomology of Schemes, Lemma 30.19.2. $\square$

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