30.16 Higher direct images along projective morphisms
We first state and prove a result for when the base is affine and then we deduce some results for projective morphisms.
Lemma 30.16.1. Let $R$ be a Noetherian ring. Let $X \to \mathop{\mathrm{Spec}}(R)$ be a proper morphism. Let $\mathcal{L}$ be an ample invertible sheaf on $X$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.
The graded ring $A = \bigoplus _{d \geq 0} H^0(X, \mathcal{L}^{\otimes d})$ is a finitely generated $R$-algebra.
There exists an $r \geq 0$ and $d_1, \ldots , d_ r \in \mathbf{Z}$ and a surjection
\[ \bigoplus \nolimits _{j = 1, \ldots , r} \mathcal{L}^{\otimes d_ j} \longrightarrow \mathcal{F}. \]
For any $p$ the cohomology group $H^ p(X, \mathcal{F})$ is a finite $R$-module.
If $p > 0$, then $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0$ for all $d$ large enough.
For any $k \in \mathbf{Z}$ the graded $A$-module
\[ \bigoplus \nolimits _{d \geq k} H^0(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) \]
is a finite $A$-module.
Proof.
By Morphisms, Lemma 29.39.4 there exists a $d > 0$ and an immersion $i : X \to \mathbf{P}^ n_ R$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ R}(1)$. Since $X$ is proper over $R$ the morphism $i$ is a closed immersion (Morphisms, Lemma 29.41.7). Thus we have $H^ i(X, \mathcal{G}) = H^ i(\mathbf{P}^ n_ R, i_*\mathcal{G})$ for any quasi-coherent sheaf $\mathcal{G}$ on $X$ (by Lemma 30.2.4 and the fact that closed immersions are affine, see Morphisms, Lemma 29.11.9). Moreover, if $\mathcal{G}$ is coherent, then $i_*\mathcal{G}$ is coherent as well (Lemma 30.9.8). We will use these facts without further mention.
Proof of (1). Set $S = R[T_0, \ldots , T_ n]$ so that $\mathbf{P}^ n_ R = \text{Proj}(S)$. Observe that $A$ is an $S$-algebra (but the ring map $S \to A$ is not a homomorphism of graded rings because $S_ n$ maps into $A_{dn}$). By the projection formula (Cohomology, Lemma 20.54.2) we have
\[ i_*(\mathcal{L}^{\otimes nd + q}) = i_*(\mathcal{L}^{\otimes q}) \otimes _{\mathcal{O}_{\mathbf{P}^ n_ R}} \mathcal{O}_{\mathbf{P}^ n_ R}(n) \]
for all $n \in \mathbf{Z}$. We conclude that $\bigoplus _{n \geq 0} A_{nd + q}$ is a finite graded $S$-module by Lemma 30.14.1. Since $A = \bigoplus _{q \in \{ 0, \ldots , d - 1} \bigoplus _{n \geq 0} A_{nd + q}$ we see that $A$ is finite as an $S$-algebra, hence (1) is true.
Proof of (2). This follows from Properties, Proposition 28.26.13.
Proof of (3). Apply Lemma 30.14.1 and use $H^ p(X, \mathcal{F}) = H^ p(\mathbf{P}^ n_ R, i_*\mathcal{F})$.
Proof of (4). Fix $p > 0$. By the projection formula we have
\[ i_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes nd + q}) = i_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes q}) \otimes _{\mathcal{O}_{\mathbf{P}^ n_ R}} \mathcal{O}_{\mathbf{P}^ n_ R}(n) \]
for all $n \in \mathbf{Z}$. By Lemma 30.14.1 we conclude that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{nd + q}) = 0$ for $n \gg 0$. Since there are only finitely many congruence classes of integers modulo $d$ this proves (4).
Proof of (5). Fix an integer $k$. Set $M = \bigoplus _{n \geq k} H^0(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. Arguing as above we conclude that $\bigoplus _{nd + q \geq k} M_{nd + q}$ is a finite graded $S$-module. Since $M = \bigoplus _{q \in \{ 0, \ldots , d - 1\} } \bigoplus _{nd + q \geq k} M_{nd + q}$ we see that $M$ is finite as an $S$-module. Since the $S$-module structure factors through the ring map $S \to A$, we conclude that $M$ is finite as an $A$-module.
$\square$
Lemma 30.16.2. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that
$S$ is Noetherian,
$f$ is proper,
$\mathcal{F}$ is coherent, and
$\mathcal{L}$ is relatively ample on $X/S$.
Then there exists an $n_0$ such that for all $n \geq n_0$ we have
\[ R^ pf_*\left(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}\right) = 0 \]
for all $p > 0$.
Proof.
Choose a finite affine open covering $S = \bigcup V_ j$ and set $X_ j = f^{-1}(V_ j)$. Clearly, if we solve the question for each of the finitely many systems $(X_ j \to V_ j, \mathcal{L}|_{X_ j}, \mathcal{F}|_{V_ j})$ then the result follows. Thus we may assume $S$ is affine. In this case the vanishing of $R^ pf_*(\mathcal{F} \otimes \mathcal{L}^{\otimes n})$ is equivalent to the vanishing of $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$, see Lemma 30.4.6. Thus the required vanishing follows from Lemma 30.16.1 (which applies because $\mathcal{L}$ is ample on $X$ by Morphisms, Lemma 29.39.4).
$\square$
Lemma 30.16.3. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a locally projective morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then $R^ if_*\mathcal{F}$ is a coherent $\mathcal{O}_ S$-module for all $i \geq 0$.
Proof.
We first remark that a locally projective morphism is proper (Morphisms, Lemma 29.43.5) and hence of finite type. In particular $X$ is locally Noetherian (Morphisms, Lemma 29.15.6) and hence the statement makes sense. Moreover, by Lemma 30.4.5 the sheaves $R^ pf_*\mathcal{F}$ are quasi-coherent.
Having said this the statement is local on $S$ (for example by Cohomology, Lemma 20.7.4). Hence we may assume $S = \mathop{\mathrm{Spec}}(R)$ is the spectrum of a Noetherian ring, and $X$ is a closed subscheme of $\mathbf{P}^ n_ R$ for some $n$, see Morphisms, Lemma 29.43.4. In this case, the sheaves $R^ pf_*\mathcal{F}$ are the quasi-coherent sheaves associated to the $R$-modules $H^ p(X, \mathcal{F})$, see Lemma 30.4.6. Hence it suffices to show that $R$-modules $H^ p(X, \mathcal{F})$ are finite $R$-modules (Lemma 30.9.1). This follows from Lemma 30.16.1 (because the restriction of $\mathcal{O}_{\mathbf{P}^ n_ R}(1)$ to $X$ is ample on $X$).
$\square$
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