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The Stacks project

Lemma 71.11.10. In Situation 71.11.1. 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 67.40.2 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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