Lemma 10.131.11 (02HQ)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.131: Differentials
Lemma 10.131.11 (cite)

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Lemma 10.131.11. Let $R \to S$ be a ring map. Let $I \subset S$ be an ideal. Let $n \geq 1$ be an integer. Set $S' = S/I^{n + 1}$. The map $\Omega _{S/R} \to \Omega _{S'/R}$ induces an isomorphism

\[ \Omega _{S/R} \otimes _ S S/I^ n \longrightarrow \Omega _{S'/R} \otimes _{S'} S/I^ n. \]

Proof. This follows from Lemma 10.131.9 and the fact that $\text{d}(I^{n + 1}) \subset I^ n\Omega _{S/R}$ by the Leibniz rule for $\text{d}$. $\square$

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