Lemma 31.31.5. 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. Let $U \subset X$ be an open. Assume that

1. $p$ is quasi-compact,

2. $i$ of finite presentation,

3. $U \cap p(i(Z)) = \emptyset$,

4. $U$ is quasi-compact,

5. $\mathcal{A}_ n$ is a finite type $\mathcal{O}_ S$-module for all $n$.

Then there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_ S$-submodule $\mathcal{F} \subset \mathcal{A}_ d$ with (a) $Z = \underline{\text{Proj}}_ S(\mathcal{A}/\mathcal{F}\mathcal{A})$ and (b) the support of $\mathcal{A}_ d/\mathcal{F}$ is disjoint from $U$.

Proof. Let $\mathcal{I} \subset \mathcal{A}$ be the sheaf of quasi-coherent graded ideals constructed in Lemma 31.31.1. Let $U_ i$, $R_ i$, $A_ i$, $I_ i$, $f_{i, j}$, $h_{i, j, k}$, and $d$ be as constructed in the proof of Lemma 31.31.4. Since $U \cap p(i(Z)) = \emptyset$ we see that $\mathcal{I}_ d|_ U = \mathcal{A}_ d|_ U$ (by our construction of $\mathcal{I}$ as a kernel). Since $U$ is quasi-compact we can choose a finite affine open covering $U = W_1 \cup \ldots \cup W_ m$. Since $\mathcal{A}_ d$ is of finite type we can find finitely many sections $g_{t, s} \in \mathcal{A}_ d(W_ t)$ which generate $\mathcal{A}_ d|_{W_ t} = \mathcal{I}_ d|_{W_ t}$ as an $\mathcal{O}_{W_ t}$-module. To finish the proof, note that by Properties, Lemma 28.22.3 there exists a finite type $\mathcal{F} \subset \mathcal{I}_ d$ such that all the local sections

$h_{i, j, k}f_{i, j}^{(d - \deg (h_{i, j, k}))/\deg (f_{i, j})} \quad \text{and}\quad g_{t, s}$

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

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