The Stacks project

Lemma 70.16.3. 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 $i : Z \to X$ be a closed subspace. Let $U \subset X$ be an open. Assume that

  1. $\pi $ is quasi-compact,

  2. $i$ of finite presentation,

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

  4. $U$ is quasi-compact,

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

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

Proof. We use the same trick as in the proof of Lemma 70.16.2 to reduce to the case of schemes. Let $\mathcal{I} \subset \mathcal{A}$ be the quasi-coherent graded ideal cutting out $Z$ of Lemma 70.16.1. Choose an affine scheme $W$ and a surjective ├ętale morphism $W \to X$, see Properties of Spaces, Lemma 65.6.3. By the case of schemes (Divisors, Lemma 31.31.5) there exists a $d > 0$ and a quasi-coherent finite type $\mathcal{O}_ W$-submodule $\mathcal{F}' \subset \mathcal{I}_ d|_ W \subset \mathcal{A}_ d|_ W$ such that (a) $Z \times _ X W$ is equal to $\underline{\text{Proj}}_ W(\mathcal{A}|_ W/\mathcal{F}'\mathcal{A}|_ W)$ and (b) the support of $\mathcal{A}_ d|_ W/\mathcal{F}'$ is disjoint from $U \times _ X W$. By Limits of Spaces, Lemma 69.9.2 we can find a finite type quasi-coherent submodule $\mathcal{F} \subset \mathcal{I}_ d$ such that $\mathcal{F}' \subset \mathcal{F}|_ W$. Let $Z' = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A})$. Then $Z' \to P$ is a closed immersion (Lemma 70.12.5) and $Z \subset Z'$ as $\mathcal{F}\mathcal{A} \subset \mathcal{I}$. On the other hand, $Z' \times _ X W \subset Z \times _ X W$ by our choice of $\mathcal{F}$. Thus $Z = Z'$. Finally, we see that $\mathcal{A}_ d/\mathcal{F}$ is supported on $X \setminus U$ as $\mathcal{A}_ d|_ W/\mathcal{F}|_ W$ is a quotient of $\mathcal{A}_ d|_ W/\mathcal{F}'$ which is supported on $W \setminus U \times _ X W$. Thus the lemma follows. $\square$


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