Lemma 30.16.1 (0B5T)—The Stacks project

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. $\square$

The Stacks project

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