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 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 76.7.11.
\square
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