Lemma 30.14.2 (0AG6)—The Stacks project

Lemma 30.14.2. Let $A$ be a graded ring such that $A_0$ is Noetherian and $A$ is generated by finitely many elements of $A_1$ over $A_0$. Set $X = \text{Proj}(A)$. Then $X$ is a Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

There exists an $r \geq 0$ and $d_1, \ldots , d_ r \in \mathbf{Z}$ and a surjection

\[ \bigoplus \nolimits _{j = 1, \ldots , r} \mathcal{O}_ X(d_ j) \longrightarrow \mathcal{F}. \]
For any $p$ the cohomology group $H^ p(X, \mathcal{F})$ is a finite $A_0$-module.
If $p > 0$, then $H^ p(X, \mathcal{F}(d)) = 0$ for all $d$ large enough.
For any $k \in \mathbf{Z}$ the graded $A$-module

\[ \bigoplus \nolimits _{d \geq k} H^0(X, \mathcal{F}(d)) \]

is a finite $A$-module.

Proof. By assumption there exists a surjection of graded $A_0$-algebras

\[ A_0[T_0, \ldots , T_ n] \longrightarrow A \]

where $\deg (T_ j) = 1$ for $j = 0, \ldots , n$. By Constructions, Lemma 27.11.5 this defines a closed immersion $i : X \to \mathbf{P}^ n_{A_0}$ such that $i^*\mathcal{O}_{\mathbf{P}^ n_{A_0}}(1) = \mathcal{O}_ X(1)$. In particular, $X$ is Noetherian as a closed subscheme of the Noetherian scheme $\mathbf{P}^ n_{A_0}$. We claim that the results of the lemma for $\mathcal{F}$ follow from the corresponding results of Lemma 30.14.1 for the coherent sheaf $i_*\mathcal{F}$ (Lemma 30.9.8) on $\mathbf{P}^ n_{A_0}$. For example, by this lemma there exists a surjection

\[ \bigoplus \nolimits _{j = 1, \ldots , r} \mathcal{O}_{\mathbf{P}^ n_{A_0}}(d_ j) \longrightarrow i_*\mathcal{F}. \]

By adjunction this corresponds to a map $\bigoplus _{j = 1, \ldots , r} \mathcal{O}_ X(d_ j) \longrightarrow \mathcal{F}$ which is surjective as well. The statements on cohomology follow from the fact that $H^ p(X, \mathcal{F}(d)) = H^ p(\mathbf{P}^ n_{A_0}, i_*\mathcal{F}(d))$ by Lemma 30.2.4. $\square$

The Stacks project

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