## 32.19 Base change in top degree

For a proper morphism and a finite type quasi-coherent module the base change map is an isomorphism in top degree.

Lemma 32.19.1. Let $f : X \to Y$ be a morphism of schemes. Let $d \geq 0$. Assume

1. $X$ and $Y$ are quasi-compact and quasi-separated, and

2. $R^ if_*\mathcal{F} = 0$ for $i > d$ and every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Then we have

1. for any base change diagram

$\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

we have $R^ if'_*\mathcal{F}' = 0$ for $i > d$ and any quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$,

2. $R^ df'_*(\mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\mathcal{G}') = R^ df'_*\mathcal{F}' \otimes _{\mathcal{O}_{Y'}} \mathcal{G}'$ for any quasi-coherent $\mathcal{O}_{Y'}$-module $\mathcal{G}'$,

3. formation of $R^ df'_*\mathcal{F}'$ commutes with arbitrary further base change (see proof for explanation).

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

$\xymatrix{ X'' \ar[d]_{f''} \ar[r]_{h'} & X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y'' \ar[r]^ h & Y' \ar[r]^ g & Y }$

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

$Lh^*Rf'_*\mathcal{F}' \longrightarrow Rf''_*L(h')^*\mathcal{F}' \longrightarrow Rf''_*(h')^*\mathcal{F}'$

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

$g_*(R^ if'_*\mathcal{F}') = H^ i(Rg_*Rf'_*\mathcal{F}') = H^ i(Rf_*Rg'_*\mathcal{F}') = H^ i(Rf_*g'_*\mathcal{F}') = Rf^ i_*g'_*\mathcal{F}'$

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

$F(M) = H^ d(X', \mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\widetilde{M})$

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

$\mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\widetilde{M}_1 \to \mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\widetilde{M}_2 \to \mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\widetilde{M}_3 \to 0$

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

$F(M_1) \to F(M_2) \to F(M_3) \to 0$

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

$h'_*(h')^*\mathcal{F}' = \mathcal{F}' \otimes _{\mathcal{O}_{X'}} (f')^*\widetilde{R''}$

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. $\square$

Lemma 32.19.2. Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume $f$ is proper and $\dim (X_ y) = d$. Then

1. for $\mathcal{F} \in \mathit{QCoh}(\mathcal{O}_ X)$ we have $(R^ if_*\mathcal{F})_ y = 0$ for all $i > d$,

2. there is an affine open neighbourhood $V \subset Y$ of $y$ such that $f^{-1}(V) \to V$ and $d$ satisfy the assumptions and conclusions of Lemma 32.19.1.

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. $\square$

Lemma 32.19.3. 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 32.18.1. 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

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

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. $\square$

Lemma 32.19.4. 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 of finite presentation all of whose fibres have dimension $\leq d$,

2. $\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

Then $R^ df_*\mathcal{F}$ is an $\mathcal{O}_ X$-module of finite presentation.

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. $\square$

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