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The Stacks project

Lemma 10.70.2. Let $R$ be a ring, $I \subset R$ an ideal, and $a \in I$. Let $R' = R[\frac{I}{a}]$ be the affine blowup algebra. Then

  1. the image of $a$ in $R'$ is a nonzerodivisor,

  2. $IR' = aR'$, and

  3. $(R')_ a = R_ a$.

Proof. Immediate from the description of $R[\frac{I}{a}]$ above. $\square$

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