Lemma 10.134.15 (00S5)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.134: The naive cotangent complex
Lemma 10.134.15 (cite)

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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 $J = \mathop{\mathrm{Ker}}(\beta )$ .

Proof. See Lemmas 10.134.2 and 10.134.14. $\square$

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