Example 10.70.5. Let $R$ be a ring. Let $P = R[t_1, \ldots , t_ n]$ be the polynomial algebra. Let $I = (t_1, \ldots , t_ n) \subset P$. Let $a = t_1$. With notation as in Definition 10.70.1 there is an isomorphism

$P[x_2, \ldots , x_ n]/(t_1x_2 - t_2, \ldots , t_1x_ n - t_ n) \longrightarrow \textstyle {P[\frac{I}{a}] = P[\frac{I}{t_1}]}$

sending $x_ i$ to $t_ i/t_1$. We leave it to the reader to show that this map is well defined. Since $I$ is generated by $t_1, \ldots , t_ n$ we see that our map is surjective. To see that our map is injective, the reader can argue that the source and target of our map are $t_1$-torsion free and that the map is an isomorphism after inverting $t_1$, see Lemma 10.70.2. Alternatively, the reader can use the description of the Rees algebra in Example 10.70.4. We omit the details.

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