Lemma 29.34.16 (02K4)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.34: Smooth morphisms
Lemma 29.34.16 (cite)

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Lemma 29.34.16. Let $f : X \to Y$ , $g : Y \to S$ be morphisms of schemes. Assume $f$ is smooth. Then

$0 \to f^*\Omega _{Y/S} \to \Omega _{X/S} \to \Omega _{X/Y} \to 0$

(see Lemma 29.32.9) is short exact.

Proof. The algebraic version of this lemma is the following: Given ring maps $A \to B \to C$ with $B \to C$ smooth, then the sequence

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

of Algebra, Lemma 10.131.7 is exact. This is Algebra, Lemma 10.139.1. $\square$

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