The Stacks project

Lemma 37.7.10. Let $h : Z \to X$ be a formally unramified morphism of schemes over $S$. There is a canonical exact sequence

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

The first arrow is induced by $\text{d}_{Z'/S}$ where $Z'$ is the universal first order neighbourhood of $Z$ over $X$.

Proof. We know that there is a canonical exact sequence

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

see Morphisms, Lemma 29.32.15. Hence the result follows on applying Lemma 37.7.9. $\square$

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