The Stacks project

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

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