The Stacks project

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 ├ętale 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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