The Stacks project

Lemma 76.39.4. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$. Let $U \subset B$ be an open subspace. Assume

  1. $B$ quasi-compact and quasi-separated,

  2. $U$ is quasi-compact,

  3. $f$ is of finite type

  4. $f^{-1}(U) \to U$ is an isomorphism.

Then there exists a $U$-admissible blowup $B' \to B$ such that $U$ is scheme theoretically dense in $B'$ and such that the strict transform $X'$ of $X$ maps isomorphically to an open subspace of $B'$.

Proof. This lemma is a generalization of Lemma 76.39.3. As the composition of $U$-admissible blowups is $U$-admissible (Divisors on Spaces, Lemma 71.19.2) we can proceed in stages. Pick a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ B$ with $|B| \setminus |U| = |V(\mathcal{I})|$. Replace $B$ by the blowup of $B$ in $\mathcal{I}$ and $X$ by the strict transform of $X$. After this replacement $B \setminus U$ is the support of an effective Cartier divisor $D$ (Divisors on Spaces, Lemma 71.17.4). In particular $U$ is scheme theoretically dense in $B$ (Divisors on Spaces, Lemma 71.6.4). Next, we do another $U$-admissible blowup to get to the situation where $X \to B$ is flat and of finite presentation, see Lemma 76.39.1. Note that $U$ is still scheme theoretically dense in $B$. Hence $X \to B$ is an open immersion by Lemma 76.37.5. $\square$


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Your email address will not be published.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 087F. Beware of the difference between the letter 'O' and the digit '0'.