Lemma 31.21.5. Let $i : Z \to X$ be an immersion of schemes. Then $i$ is a quasi-regular immersion if and only if the following conditions are satisfied
$i$ is locally of finite presentation,
the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free, and
the map (31.19.1.2) is an isomorphism.
Proof. An open immersion is locally of finite presentation. Hence we may replace $X$ by an open subscheme $U \subset X$ such that $i$ identifies $Z$ with a closed subscheme of $U$, i.e., we may assume that $i$ is a closed immersion. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the corresponding quasi-coherent sheaf of ideals. Recall, see Morphisms, Lemma 29.21.7 that $\mathcal{I}$ is of finite type if and only if $i$ is locally of finite presentation. Hence the equivalence follows from Lemma 31.20.4 and unwinding the definitions. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)