The Stacks project

Lemma 70.11.9. In Situation 70.11.1. If $\mathcal{O}_ X \to \mathcal{A}_0$ is an integral algebra map1 and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is universally closed.

Proof. By Morphisms of Spaces, Lemma 66.9.5 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.3. $\square$

[1] In other words, the integral closure of $\mathcal{O}_ X$ in $\mathcal{A}_0$, see Morphisms of Spaces, Definition 66.48.2, equals $\mathcal{A}_0$.

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