To have a bit more control over our blowups we introduce the following standard terminology.

Definition 71.19.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \subset X$ be an open subspace. A morphism $X' \to X$ is called a $U$-admissible blowup if there exists a closed immersion $Z \to X$ of finite presentation with $Z$ disjoint from $U$ such that $X'$ is isomorphic to the blowup of $X$ in $Z$.

We recall that $Z \to X$ is of finite presentation if and only if the ideal sheaf $\mathcal{I}_ Z \subset \mathcal{O}_ X$ is of finite type, see Morphisms of Spaces, Lemma 67.28.12. In particular, a $U$-admissible blowup is a proper morphism, see Lemma 71.17.11. Note that there can be multiple centers which give rise to the same morphism. Hence the requirement is just the existence of some center disjoint from $U$ which produces $X'$. Finally, as the morphism $b : X' \to X$ is an isomorphism over $U$ (see Lemma 71.17.4) we will often abuse notation and think of $U$ as an open subspace of $X'$ as well.

Lemma 71.19.2. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open subspace. Let $b : X' \to X$ be a $U$-admissible blowup. Let $X'' \to X'$ be a $U$-admissible blowup. Then the composition $X'' \to X$ is a $U$-admissible blowup.

Proof. Immediate from the more precise Lemma 71.17.12. $\square$

Lemma 71.19.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space. Let $U, V \subset X$ be quasi-compact open subspaces. Let $b : V' \to V$ be a $U \cap V$-admissible blowup. Then there exists a $U$-admissible blowup $X' \to X$ whose restriction to $V$ is $V'$.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ V$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is disjoint from $U \cap V$ and such that $V'$ is isomorphic to the blowup of $V$ in $\mathcal{I}$. Let $\mathcal{I}' \subset \mathcal{O}_{U \cup V}$ be the quasi-coherent sheaf of ideals whose restriction to $U$ is $\mathcal{O}_ U$ and whose restriction to $V$ is $\mathcal{I}$. By Limits of Spaces, Lemma 70.9.8 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ whose restriction to $U \cup V$ is $\mathcal{I}'$. The lemma follows. $\square$

Lemma 71.19.4. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U \subset X$ be a quasi-compact open subspace. Let $b_ i : X_ i \to X$, $i = 1, \ldots , n$ be $U$-admissible blowups. There exists a $U$-admissible blowup $b : X' \to X$ such that (a) $b$ factors as $X' \to X_ i \to X$ for $i = 1, \ldots , n$ and (b) each of the morphisms $X' \to X_ i$ is a $U$-admissible blowup.

Proof. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}_ i)$ is disjoint from $U$ and such that $X_ i$ is isomorphic to the blowup of $X$ in $\mathcal{I}_ i$. Set $\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_ n$ and let $X'$ be the blowup of $X$ in $\mathcal{I}$. Then $X' \to X$ factors through $b_ i$ by Lemma 71.17.10. $\square$

Lemma 71.19.5. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. Let $U, V$ be quasi-compact disjoint open subspaces of $X$. Then there exist a $U \cup V$-admissible blowup $b : X' \to X$ such that $X'$ is a disjoint union of open subspaces $X' = X'_1 \amalg X'_2$ with $b^{-1}(U) \subset X'_1$ and $b^{-1}(V) \subset X'_2$.

Proof. Choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}$, resp. $\mathcal{J}$ such that $X \setminus U = V(\mathcal{I})$, resp. $X \setminus V = V(\mathcal{J})$, see Limits of Spaces, Lemma 70.14.1. Then $|V(\mathcal{I}\mathcal{J})| = |X|$. Hence $\mathcal{I}\mathcal{J}$ is a locally nilpotent sheaf of ideals. Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$ is quasi-compact there exists an $n > 0$ such that $\mathcal{I}^ n \mathcal{J}^ n = 0$. We may and do replace $\mathcal{I}$ by $\mathcal{I}^ n$ and $\mathcal{J}$ by $\mathcal{J}^ n$. Whence $\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing up in $\mathcal{I} + \mathcal{J}$. This is $U \cup V$-admissible as $|V(\mathcal{I} + \mathcal{J})| = |X| \setminus |U| \cup |V|$. We will show that $X'$ is a disjoint union of open subspaces $X' = X'_1 \amalg X'_2$ as in the statement of the lemma.

Since $|V(\mathcal{I} + \mathcal{J})|$ is the complement of $|U \cup V|$ we conclude that $V \cup U$ is scheme theoretically dense in $X'$, see Lemmas 71.17.4 and 71.6.4. Thus if such a decomposition $X' = X'_1 \amalg X'_2$ into open and closed subspaces exists, then $X'_1$ is the scheme theoretic closure of $U$ in $X'$ and similarly $X'_2$ is the scheme theoretic closure of $V$ in $X'$. Since $U \to X'$ and $V \to X'$ are quasi-compact taking scheme theoretic closures commutes with étale localization (Morphisms of Spaces, Lemma 67.16.3). Hence to verify the existence of $X'_1$ and $X'_2$ we may work étale locally on $X$. This reduces us to the case of schemes which is treated in the proof of Divisors, Lemma 31.34.5. $\square$

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