Lemma 71.17.9. 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 71.6.10).
Proof. By Lemmas 71.17.2 and 71.6.2 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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