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.

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