Lemma 31.34.4. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open subscheme. 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 31.32.12.
$\square$

## 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Your email address will not be published.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).

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

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)