Lemma 70.11.10. In Situation 70.11.1. The following conditions are equivalent

1. $\mathcal{A}_0$ is a finite type $\mathcal{O}_ X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra,

2. $\mathcal{A}_0$ is a finite type $\mathcal{O}_ X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{O}_ X$-algebra.

If these conditions hold, then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is proper.

Proof. By Morphisms of Spaces, Lemma 66.40.2 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.3. $\square$

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