The Stacks project

Lemma 10.70.3. Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal and $a \in I$. Set $J = IS$ and let $b \in J$ be the image of $a$. Then $S[\frac{J}{b}]$ is the quotient of $S \otimes _ R R[\frac{I}{a}]$ by the ideal of elements annihilated by some power of $b$.

Proof. Let $S'$ be the quotient of $S \otimes _ R R[\frac{I}{a}]$ by its $b$-power torsion elements. The ring map

\[ S \otimes _ R R[\textstyle {\frac{I}{a}}] \longrightarrow S[\textstyle {\frac{J}{b}}] \]

is surjective and annihilates $a$-power torsion as $b$ is a nonzerodivisor in $S[\frac{J}{b}]$. Hence we obtain a surjective map $S' \to S[\frac{J}{b}]$. To see that the kernel is trivial, we construct an inverse map. Namely, let $z = y/b^ n$ be an element of $S[\frac{J}{b}]$, i.e., $y \in J^ n$. Write $y = \sum x_ is_ i$ with $x_ i \in I^ n$ and $s_ i \in S$. We map $z$ to the class of $\sum s_ i \otimes x_ i/a^ n$ in $S'$. This is well defined because an element of the kernel of the map $S \otimes _ R I^ n \to J^ n$ is annihilated by $a^ n$, hence maps to zero in $S'$. $\square$

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