The Stacks project

Lemma 89.5.3. Let $X, x_ i, U_ i \to X, u_ i$ be as in (89.3.0.1) and assume $f : Y \to X$ corresponds to $g_ i : Y_ i \to U_ i$ under $F$. Assume $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 67.49.1. Then there exists a factorization

\[ Y = Z_ m \to Z_{m - 1} \to \ldots \to Z_1 \to Z_0 = X \]

of $f$ where $Z_{j + 1} \to Z_ j$ is the normalized blowing up of $Z_ j$ at a closed point $z_ j$ lying over $\{ x_1, \ldots , x_ n\} $ if and only if for each $i$ there exists a factorization

\[ Y_ i = Z_{i, m_ i} \to Z_{i, m_ i - 1} \to \ldots \to Z_{i, 1} \to Z_{i, 0} = U_ i \]

of $g_ i$ where $Z_{i, j + 1} \to Z_{i, j}$ is the normalized blowing up of $Z_{i, j}$ at a closed point $z_{i, j}$ lying over $u_ i$.

Proof. This follows by the exact same argument as used to prove Lemma 89.4.2. $\square$


Comments (0)


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Your email address will not be published.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.




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 0BHL. Beware of the difference between the letter 'O' and the digit '0'.