The Stacks project

Lemma 31.30.2. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ be the relative Proj of $\mathcal{A}$. If $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{O}_ S$-algebras, then $p$ is of finite type and $\mathcal{O}_ X(d)$ is a finite type $\mathcal{O}_ X$-module.

Proof. The assumption implies that $p$ is quasi-compact, see Lemma 31.30.1. Hence it suffices to show that $p$ is locally of finite type. Thus the question is local on the base and target, see Morphisms, Lemma 29.15.2. Say $S = \mathop{\mathrm{Spec}}(R)$ and $\mathcal{A}$ corresponds to the graded $R$-algebra $A$. After further localizing on $S$ we may assume that $A$ is a finite type $R$-algebra. The scheme $X$ is constructed out of glueing the spectra of the rings $A_{(f)}$ for $f \in A_{+}$ homogeneous. Each of these is of finite type over $R$ by Algebra, Lemma 10.57.9 part (1). Thus $\text{Proj}(A)$ is of finite type over $R$. To see the statement on $\mathcal{O}_ X(d)$ use part (2) of Algebra, Lemma 10.57.9. $\square$

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Your email address will not be published.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 07ZY. Beware of the difference between the letter 'O' and the digit '0'.