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 étale 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 085N. Beware of the difference between the letter 'O' and the digit '0'.