Lemma 71.11.10. In Situation 71.11.1. The following conditions are equivalent
\mathcal{A}_0 is a finite type \mathcal{O}_ X-module and \mathcal{A} is of finite type as an \mathcal{A}_0-algebra,
\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.
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