The Stacks project

Lemma 71.11.8. In Situation 71.11.1. If $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{O}_ X$-algebras, then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is of finite type.

Proof. By Morphisms of Spaces, Lemma 67.23.4 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.2. $\square$

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