Lemma 37.62.23. Let $Z \to Y \to X$ be formally unramified morphisms of schemes. Assume that $Z \to Y$ is a local complete intersection morphism. The exact sequence
of Lemma 37.7.12 is short exact.
Proof. The question is local on $Z$ hence we may assume there exists a factorization $Z \to \mathbf{A}^ n_ Y \to Y$ of the morphism $Z \to Y$. Then we get a commutative diagram
As $Z \to Y$ is a local complete intersection morphism, we see that $Z \to \mathbf{A}^ n_ Y$ is a Koszul-regular immersion. Hence by Divisors, Lemma 31.21.6 the sequence
is exact and locally split. Note that $i^*\mathcal{C}_{Y/X} = (i')^*\mathcal{C}_{\mathbf{A}^ n_ Y/\mathbf{A}^ n_ X}$ by Lemma 37.7.7 and note that the diagram
is commutative. Hence the lower horizontal arrow is a locally split injection. This proves the lemma. $\square$
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