Lemma 10.70.6. Let $R$ be a ring. Let $I = (a_1, \ldots , a_ n)$ be an ideal of $R$. Let $a = a_1$. Then there is a surjection

$R[x_2, \ldots , x_ n]/(a x_2 - a_2, \ldots , a x_ n - a_ n) \longrightarrow \textstyle {R[\frac{I}{a}]}$

whose kernel is the $a$-power torsion in the source.

Proof. Consider the ring map $P = \mathbf{Z}[t_1, \ldots , t_ n] \to R$ sending $t_ i$ to $a_ i$. Set $J = (t_1, \ldots , t_ n)$. By Example 10.70.5 we have $P[\frac{J}{t_1}] = P[x_2, \ldots , x_ n]/(t_1 x_2 - t_2, \ldots , t_1 x_ n - t_ n)$. Apply Lemma 10.70.3 to the map $P \to A$ to conclude. $\square$

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