Lemma 37.59.22. 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: