Lemma 10.70.10. Let $R$ be a domain, $I \subset R$ an ideal, and $a \in I$ a nonzero element. Then the affine blowup algebra $R[\frac{I}{a}]$ is a domain.

Proof. Suppose $x/a^ n$, $y/a^ m$ with $x \in I^ n$, $y \in I^ m$ are elements of $R[\frac{I}{a}]$ whose product is zero. Then $a^ N x y = 0$ in $R$. Since $R$ is a domain we conclude that either $x = 0$ or $y = 0$. $\square$

There are also:

• 1 comment(s) on Section 10.70: Blow up algebras

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).