The Stacks project

Lemma 38.37.1. Consider an almost blow up square (38.37.0.1). Let $Y \to X$ be any morphism. Then the base change

\[ \xymatrix{ Y \times _ X E \ar[d] \ar[r] & Y \times _ X X' \ar[d] \\ Y \times _ X Z \ar[r] & Y } \]

is an almost blow up square too.

Proof. The morphism $Y \times _ X X' \to Y$ is proper and of finite presentation by Morphisms, Lemmas 29.41.5 and 29.21.4. The morphism $Y \times _ X Z \to Y$ is a closed immersion (Morphisms, Lemma 29.2.4) of finite presentation. The inverse image of $Y \times _ X Z$ in $Y \times _ X X'$ is equal to the inverse image of $E$ in $Y \times _ X X'$ and hence is locally principal (Divisors, Lemma 31.13.11). Let $X'' \subset X'$, resp. $Y'' \subset Y \times _ X X'$ be the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{X'}$, resp. $\mathcal{O}_{Y \times _ Y X'}$ supported on $E$, resp. $Y \times _ X E$. Clearly, $Y'' \subset Y \times _ X X''$ is the closed subscheme corresponding to the quasi-coherent ideal of sections of $\mathcal{O}_{Y \times _ Y X''}$ supported on $Y \times _ X (E \cap X'')$. Thus $Y''$ is the strict transform of $Y$ relative to the blowing up $X'' \to X$, see Divisors, Definition 31.33.1. Thus by Divisors, Lemma 31.33.2 we see that $Y''$ is the blow up of $Y \times _ X Z$ on $Y$. $\square$


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Your email address will not be published.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 0EV7. Beware of the difference between the letter 'O' and the digit '0'.