The Stacks project

Lemma 27.16.9. Let $S$ be a scheme and $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ S$-algebras. The morphism $\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S$ is separated.

Proof. To prove a morphism is separated we may work locally on the base, see Schemes, Section 26.21. By construction $\underline{\text{Proj}}_ S(\mathcal{A})$ is over any affine $U \subset S$ isomorphic to $\text{Proj}(A)$ with $A = \mathcal{A}(U)$. By Lemma 27.8.8 we see that $\text{Proj}(A)$ is separated. Hence $\text{Proj}(A) \to U$ is separated (see Schemes, Lemma 26.21.13) as desired. $\square$

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