The Stacks project

Lemma 70.17.9. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be a blowup of $X$ in a closed subspace. For any effective Cartier divisor $D$ on $X$ the pullback $b^{-1}D$ is defined (see Definition 70.6.10).

Proof. By Lemmas 70.17.2 and 70.6.2 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. $\square$


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