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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)