Lemma 10.139.2 (06A8)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.139: Smoothness and differentials
Lemma 10.139.2 (cite)

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Lemma 10.139.2. Let $A \to B \to C$ be ring maps with $A \to C$ smooth and $B \to C$ surjective with kernel $J \subset B$ . Then the exact sequence

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

of Lemma 10.131.9 is split exact.

Proof. This follows from the more general Lemma 10.138.10 because a smooth ring map is formally smooth, see Proposition 10.138.13. $\square$

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