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. $\square$

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