Lemma 70.11.7. In Situation 70.11.1. 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 $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is quasi-compact.

Proof. By Morphisms of Spaces, Lemma 66.8.8 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.1. $\square$

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