The Stacks project

Lemma 76.19.13. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $h : Z \to X$ be a formally unramified morphism of algebraic spaces over $B$. Assume that $Z$ is formally smooth over $B$. Then the canonical exact sequence

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

of Lemma 76.15.13 is short exact.

Proof. Let $Z \to Z'$ be the universal first order thickening of $Z$ over $X$. From the proof of Lemma 76.15.13 we see that our sequence is identified with the sequence

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

Since $Z \to S$ is formally smooth we can ├ętale locally on $Z'$ find a left inverse $Z' \to Z$ over $B$ to the inclusion map $Z \to Z'$. Thus the sequence is ├ętale locally split, see Lemma 76.7.11. $\square$

Comments (2)

Comment #7422 by Sungwoo on

Is in the lemma typo?

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