Lemma 71.16.2. Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic space over S. Let \mathcal{A} be a quasi-coherent graded \mathcal{O}_ X-algebra. Let \pi : P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X be the relative Proj of \mathcal{A}. Let i : Z \to P be a closed subscheme. If \pi is quasi-compact and i of finite presentation, then there exists a d > 0 and a quasi-coherent finite type \mathcal{O}_ X-submodule \mathcal{F} \subset \mathcal{A}_ d such that Z = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A}).
Proof. The reader can redo the arguments used in the case of schemes. However, we will show the lemma follows from the case of schemes by a trick. Let \mathcal{I} \subset \mathcal{A} be the quasi-coherent graded ideal cutting out Z of Lemma 71.16.1. Choose an affine scheme U and a surjective étale morphism U \to X, see Properties of Spaces, Lemma 66.6.3. By the case of schemes (Divisors, Lemma 31.31.4) there exists a d > 0 and a quasi-coherent finite type \mathcal{O}_ U-submodule \mathcal{F}' \subset \mathcal{I}_ d|_ U \subset \mathcal{A}_ d|_ U such that Z \times _ X U is equal to \underline{\text{Proj}}_ U(\mathcal{A}|_ U/\mathcal{F}'\mathcal{A}|_ U). By Limits of Spaces, Lemma 70.9.2 we can find a finite type quasi-coherent submodule \mathcal{F} \subset \mathcal{I}_ d such that \mathcal{F}' \subset \mathcal{F}|_ U. Let Z' = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A}). Then Z' \to P is a closed immersion (Lemma 71.12.5) and Z \subset Z' as \mathcal{F}\mathcal{A} \subset \mathcal{I}. On the other hand, Z' \times _ X U \subset Z \times _ X U by our choice of \mathcal{F}. Thus Z = Z' as desired. \square
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