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.

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

    \[ \bigoplus \nolimits _{j = 1, \ldots , r} \mathcal{O}_ X(d_ j) \longrightarrow \mathcal{F}. \]
  2. For any $p$ the cohomology group $H^ p(X, \mathcal{F})$ is a finite $A_0$-module.

  3. If $p > 0$, then $H^ p(X, \mathcal{F}(d)) = 0$ for all $d$ large enough.

  4. 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AG6. Beware of the difference between the letter 'O' and the digit '0'.