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The Stacks project

Lemma 31.30.5. Let S be a scheme. Let \mathcal{A} be a quasi-coherent graded \mathcal{O}_ S-algebra. Let p : X = \underline{\text{Proj}}_ S(\mathcal{A}) \to S be the relative Proj of \mathcal{A}. If \mathcal{A} is generated by \mathcal{A}_1 over \mathcal{A}_0 and \mathcal{A}_1 is a finite type \mathcal{O}_ S-module, then p is projective.

Proof. Namely, the morphism associated to the graded \mathcal{O}_ S-algebra map

\text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A}

is a closed immersion X \to \mathbf{P}(\mathcal{A}_1), see Constructions, Lemma 27.18.5. \square


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