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

1. $i$ is locally of finite presentation,

2. the conormal sheaf $\mathcal{C}_{Z/X}$ is finite locally free, and

3. 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. $\square$

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