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.

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