Lemma 10.139.3 (06A9)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.139: Smoothness and differentials
Lemma 10.139.3 (cite)

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Lemma 10.139.3. Let $A \to B \to C$ be ring maps. Assume $A \to C$ is surjective (so also $B \to C$ is) and $A \to B$ smooth. Denote $I = \mathop{\mathrm{Ker}}(A \to C)$ and $J = \mathop{\mathrm{Ker}}(B \to C)$ . Then the sequence

$0 \to I/I^2 \to J/J^2 \to \Omega _{B/A} \otimes _ B B/J \to 0$

of Lemma 10.134.7 is exact.

Proof. This follows from the more general Lemma 10.138.11 because a smooth ring map is formally smooth, see Proposition 10.138.13. $\square$

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