The Stacks project

Lemma 31.31.4. Let $S$ be a quasi-compact and quasi-separated 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 $\mathcal{A}$. Let $i : Z \to X$ be a closed subscheme. If $p$ is quasi-compact and $i$ of finite presentation, then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_ S$-submodule $\mathcal{F} \subset \mathcal{A}_ d$ such that $Z = \underline{\text{Proj}}_ S(\mathcal{A}/\mathcal{F}\mathcal{A})$.

Proof. By Lemma 31.31.1 we know there exists a quasi-coherent graded sheaf of ideals $\mathcal{I} \subset \mathcal{A}$ such that $Z = \underline{\text{Proj}}(\mathcal{A}/\mathcal{I})$. Since $S$ is quasi-compact we can choose a finite affine open covering $S = U_1 \cup \ldots \cup U_ n$. Say $U_ i = \mathop{\mathrm{Spec}}(R_ i)$. Let $\mathcal{A}|_{U_ i}$ correspond to the graded $R_ i$-algebra $A_ i$ and $\mathcal{I}|_{U_ i}$ to the graded ideal $I_ i \subset A_ i$. Note that $p^{-1}(U_ i) = \text{Proj}(A_ i)$ as schemes over $R_ i$. Since $p$ is quasi-compact we can choose finitely many homogeneous elements $f_{i, j} \in A_{i, +}$ such that $p^{-1}(U_ i) = D_{+}(f_{i, j})$. The condition on $Z \to X$ means that the ideal sheaf of $Z$ in $\mathcal{O}_ X$ is of finite type, see Morphisms, Lemma 29.21.7. Hence we can find finitely many homogeneous elements $h_{i, j, k} \in I_ i \cap A_{i, +}$ such that the ideal of $Z \cap D_{+}(f_{i, j})$ is generated by the elements $h_{i, j, k}/f_{i, j}^{e_{i, j, k}}$. Choose $d > 0$ to be a common multiple of all the integers $\deg (f_{i, j})$ and $\deg (h_{i, j, k})$. By Properties, Lemma 28.22.3 there exists a finite type quasi-coherent $\mathcal{F} \subset \mathcal{I}_ d$ such that all the local sections

\[ h_{i, j, k}f_{i, j}^{(d - \deg (h_{i, j, k}))/\deg (f_{i, j})} \]

are sections of $\mathcal{F}$. By construction $\mathcal{F}$ is a solution. $\square$

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