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. $\square$

Comments (0)

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04FC. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 37.7.10 as pdf