The Stacks project

Lemma 37.7.3. Let $i : Z \to X$ be an immersion of schemes. Then

  1. $i$ is formally unramified,

  2. the universal first order thickening of $Z$ over $X$ is the first order infinitesimal neighbourhood of $Z$ in $X$ of Definition 37.5.1, and

  3. the conormal sheaf of $i$ in the sense of Morphisms, Definition 29.31.1 agrees with the conormal sheaf of $i$ in the sense of Definition 37.7.2.

Proof. By Morphisms, Lemmas 29.35.7 and 29.35.8 an immersion is unramified, hence formally unramified by Lemma 37.6.8. The other assertions follow by combining Lemmas 37.5.2 and 37.5.3 and the definitions. $\square$

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Your email address will not be published.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 04F5. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 37.7.3 as pdf