The Stacks project

Lemma 29.3.2. Let $h : Z \to X$ be an immersion. If $h$ is quasi-compact, then we can factor $h = i \circ j$ with $j : Z \to \overline{Z}$ an open immersion and $i : \overline{Z} \to X$ a closed immersion.

Proof. Note that $h$ is quasi-compact and quasi-separated (see Schemes, Lemma 26.23.8). Hence $h_*\mathcal{O}_ Z$ is a quasi-coherent sheaf of $\mathcal{O}_ X$-modules by Schemes, Lemma 26.24.1. This implies that $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to h_*\mathcal{O}_ Z)$ is a quasi-coherent sheaf of ideals, see Schemes, Section 26.24. Let $\overline{Z} \subset X$ be the closed subscheme corresponding to $\mathcal{I}$, see Lemma 29.2.3. By Schemes, Lemma 26.4.6 the morphism $h$ factors as $h = i \circ j$ where $i : \overline{Z} \to X$ is the inclusion morphism. To see that $j$ is an open immersion, choose an open subscheme $U \subset X$ such that $h$ induces a closed immersion of $Z$ into $U$. Then it is clear that $\mathcal{I}|_ U$ is the sheaf of ideals corresponding to the closed immersion $Z \to U$. Hence we see that $Z = \overline{Z} \cap U$. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 29.3: Immersions

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Your email address will not be published.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 01QV. Beware of the difference between the letter 'O' and the digit '0'.