Lemma 37.11.12 (06B7)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.11: Formally smooth morphisms
Lemma 37.11.12 (cite)

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Lemma 37.11.12. Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$. Assume that $Z$ is formally smooth over $S$. Then the canonical exact sequence

\[ 0 \to \mathcal{C}_{Z/X} \to h^*\Omega _{X/S} \to \Omega _{Z/S} \to 0 \]

of Lemma 37.7.10 is short exact.

Proof. Let $Z \to Z'$ be the universal first order thickening of $Z$ over $X$. From the proof of Lemma 37.7.10 we see that our sequence is identified with the sequence

\[ \mathcal{C}_{Z/Z'} \to \Omega _{Z'/S} \otimes \mathcal{O}_ Z \to \Omega _{Z/S} \to 0. \]

Since $Z \to S$ is formally smooth we can locally on $Z'$ find a left inverse $Z' \to Z$ over $S$ to the inclusion map $Z \to Z'$. Thus the sequence is locally split, see Morphisms, Lemma 29.32.16. $\square$

Comments (2)

Comment #2654 by Ko Aoki on July 23, 2017 at 04:45

Typo in the first exact sequence: " $i^*\Omega_{X/S}$ " should be replaced by " $h^*\Omega_{X/S}$ ".

Comment #2670 by Johan on July 28, 2017 at 17:40

Thanks. Fixed here.

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