Lemma 74.19.13. 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

$0 \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/B} \to \Omega _{Z/B} \to 0$

of Lemma 74.15.13 is short exact.

Proof. Let $Z \to Z'$ be the universal first order thickening of $Z$ over $X$. From the proof of Lemma 74.15.13 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 étale locally on $Z'$ find a left inverse $Z' \to Z$ over $B$ to the inclusion map $Z \to Z'$. Thus the sequence is étale locally split, see Lemma 74.7.11. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).