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. $\square$

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