The Stacks project

Proof. Before giving the proofs, we explain the meaning of (c). Suppose we have an additional cartesian square

tacked onto our given diagram. If (a) holds, then there is a canonical map $\gamma : h^*R^ df'_*\mathcal{F}' \to R^ df''_*(h')^*\mathcal{F}'$. Namely, $\gamma $ is the map on degree $d$ cohomology sheaves induced by the composition

Here the first arrow is the base change map (Cohomology, Remark 20.28.3) and the second arrow complex from the canonical map $L(g')^*\mathcal{F} \to (g')^*\mathcal{F}$. Similarly, since $Rf'_*\mathcal{F}$ has no nonzero cohomology sheaves in degrees $> d$ by (a) we have $H^ d(Lh^*Rf_*\mathcal{F}') = h^*R^ df_*\mathcal{F}$. The content of (c) is that $\gamma $ is an isomorphism.

Having said this, we can check (a), (b), and (c) locally on $Y'$ and $Y''$. Suppose that $V \subset Y$ is a quasi-compact open subscheme. Then we claim (1) and (2) hold for $f|_{f^{-1}(V)} : f^{-1}(V) \to V$. Namely, (1) is immediate and (2) follows because any quasi-coherent module on $f^{-1}(V)$ is the restriction of a quasi-coherent module on $X$ (Properties, Lemma 28.22.1) and formation of higher direct images commutes with restriction to opens. Thus we may also work locally on $Y$. In other words, we may assume $Y''$, $Y'$, and $Y$ are affine schemes.

Proof of (a) when $Y'$ and $Y$ are affine. In this case the morphisms $g$ and $g'$ are affine. Thus $g_* = Rg_*$ and $g'_* = Rg'_*$ (Cohomology of Schemes, Lemma 30.2.3) and $g_*$ is identified with the restriction functor on modules (Schemes, Lemma 26.7.3). Then

which is zero by assumption (2). Hence (a) by our description of $g_*$.

Proof of (b) when $Y'$ is affine, say $Y' = \mathop{\mathrm{Spec}}(R')$. By part (a) we have $H^{d + 1}(X', \mathcal{F}') = 0$ for any quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$, see Cohomology of Schemes, Lemma 30.4.6. Consider the functor $F$ on $R'$-modules defined by the rule

By Cohomology, Lemma 20.19.1 this functor commutes with direct sums (this is where we use that $X$ and hence $X'$ is quasi-compact and quasi-separated). On the other hand, if $M_1 \to M_2 \to M_3 \to 0$ is an exact sequence, then

is an exact sequence of quasi-coherent modules on $X'$ and by the vanishing of higher cohomology given above we get an exact sequence

In other words, $F$ is right exact. Any right exact $R'$-linear functor $F : \text{Mod}_{R'} \to \text{Mod}_{R'}$ which commutes with direct sums is given by tensoring with an $R'$-module (omitted; left as exercise for the reader). Thus we obtain $F(M) = H^ d(X', \mathcal{F}') \otimes _{R'} M$. Since $R^ d(f')_*\mathcal{F}'$ and $R^ d(f')_*(\mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\widetilde{M})$ are quasi-coherent (Cohomology of Schemes, Lemma 30.4.5), the fact that $F(M) = H^ d(X', \mathcal{F}') \otimes _{R'} M$ translates into the statement given in (b).

Proof of (c) when $Y'' \to Y' \to Y$ are morphisms of affine schemes. Say $Y'' = \mathop{\mathrm{Spec}}(R'')$ and $Y' = \mathop{\mathrm{Spec}}(R')$. Then we see that $R^ df''_*(h')^*\mathcal{F}'$ is the quasi-coherent module on $Y'$ associated to the $R''$-module $H^ d(X'', (h')^*\mathcal{F}')$. Now $h' : X'' \to X'$ is affine hence $H^ d(X'', (h')^*\mathcal{F}') = H^ d(X, h'_*(h')^*\mathcal{F}')$ by the already used Cohomology of Schemes, Lemma 30.2.4. We have

as the reader sees by checking on an affine open covering. Thus $H^ d(X'', (h')^*\mathcal{F}') = H^ d(X', \mathcal{F}') \otimes _{R'} R''$ by part (b) applied to $f'$ and the proof is complete. $\square$

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. $\square$

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 32.18.1. 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 32.19.2 (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 32.18.1). 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 32.19.1 we have

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. $\square$

Proof. The proof is exactly the same as the proof of Lemma 32.19.3 except that the third paragraph can be skipped. We omit the details. $\square$