The Stacks project

Lemma 10.134.15. Let $R \to S$ be a ring map of finite type. For any presentations $\alpha : R[x_1, \ldots , x_ n] \to S$, and $\beta : R[y_1, \ldots , y_ m] \to S$ we have

\[ I/I^2 \oplus S^{\oplus m} \cong J/J^2 \oplus S^{\oplus n} \]

as $S$-modules where $I = \mathop{\mathrm{Ker}}(\alpha )$ and $J = \mathop{\mathrm{Ker}}(\beta )$.

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