The Stacks project

Lemma 31.30.1. 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 one of the following holds

  1. $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{A}_0$-algebras,

  2. $\mathcal{A}$ is generated by $\mathcal{A}_1$ as an $\mathcal{A}_0$-algebra and $\mathcal{A}_1$ is a finite type $\mathcal{A}_0$-module,

  3. there exists a finite type quasi-coherent $\mathcal{A}_0$-submodule $\mathcal{F} \subset \mathcal{A}_{+}$ such that $\mathcal{A}_{+}/\mathcal{F}\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\mathcal{A}/\mathcal{F}\mathcal{A}$,

then $p$ is quasi-compact.

Proof. The question is local on the base, see Schemes, Lemma 26.19.2. Thus we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$ and $\mathcal{A}$ corresponds to the graded $R$-algebra $A$. Then $X = \text{Proj}(A)$, see Constructions, Section 27.15. In case (1) we may after possibly localizing more assume that $A$ is generated by homogeneous elements $f_1, \ldots , f_ n \in A_{+}$ over $A_0$. Then $A_{+} = (f_1, \ldots , f_ n)$ by Algebra, Lemma 10.58.1. In case (3) we see that $\mathcal{F} = \widetilde{M}$ for some finite type $A_0$-module $M \subset A_{+}$. Say $M = \sum A_0f_ i$. Say $f_ i = \sum f_{i, j}$ is the decomposition into homogeneous pieces. The condition in (3) signifies that $A_{+} \subset \sqrt{(f_{i, j})}$. Thus in both cases we conclude that $\text{Proj}(A)$ is quasi-compact by Constructions, Lemma 27.8.9. Finally, (2) follows from (1). $\square$

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