The Stacks project

Lemma 37.56.22. Let $f : X \to Y$ be a local complete intersection morphism of schemes. Then $f$ is unramified if and only if $f$ is formally unramified and in this case the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free on $X$.

Proof. The first assertion follows immediately from Lemma 37.6.8 and the fact that a local complete intersection morphism is locally of finite type. To compute the conormal sheaf of $f$ we choose, locally on $X$, a factorization of $f$ as $f = p \circ i$ where $i : X \to V$ is a Koszul-regular immersion and $V \to Y$ is smooth. By Lemma 37.11.13 we see that $\mathcal{C}_{X/Y}$ is a locally direct summand of $\mathcal{C}_{X/V}$ which is finite locally free as $i$ is a Koszul-regular (hence quasi-regular) immersion, see Divisors, Lemma 31.21.5. $\square$


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