The Stacks project

Lemma 31.32.11. Let $X$ be a scheme. Let $b : X' \to X$ be a blowup of $X$ in a closed subscheme. The pullback $b^{-1}D$ is defined for all effective Cartier divisors $D \subset X$ and pullbacks of meromorphic functions are defined for $b$ (Definitions 31.13.12 and 31.23.4).

Proof. By Lemmas 31.32.2 and 31.13.2 this reduces to the following algebra fact: Let $A$ be a ring, $I \subset A$ an ideal, $a \in I$, and $x \in A$ a nonzerodivisor. Then the image of $x$ in $A[\frac{I}{a}]$ is a nonzerodivisor. Namely, suppose that $x (y/a^ n) = 0$ in $A[\frac{I}{a}]$. Then $a^ mxy = 0$ in $A$ for some $m$. Hence $a^ my = 0$ as $x$ is a nonzerodivisor. Whence $y/a^ n$ is zero in $A[\frac{I}{a}]$ as desired. $\square$


Comments (0)

There are also:

  • 7 comment(s) on Section 31.32: 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Your email address will not be published.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 0809. Beware of the difference between the letter 'O' and the digit '0'.