The Stacks project

Lemma 69.17.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties:

  1. $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism,

  2. the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$,

  3. there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective ├ętale morphism. As blowing up commutes with flat base change (Lemma 69.17.3) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma 31.32.4. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 69.17: Blowing up

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Your email address will not be published.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 085T. Beware of the difference between the letter 'O' and the digit '0'.