Lemma 31.32.11 (0809)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.32: Blowing up
Lemma 31.32.11 (cite)

numberstags

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

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).

Comments (0)

Post a comment