The Stacks project

Lemma 76.15.6. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. 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 76.12.1,

  3. the conormal sheaf of $i$ in the sense of Definition 76.5.1 agrees with the conormal sheaf of $i$ in the sense of Definition 76.15.5.

Proof. An immersion of algebraic spaces is by definition a representable morphism. Hence by Morphisms, Lemmas 29.35.7 and 29.35.8 an immersion is unramified (via the abstract principle of Spaces, Lemma 65.5.8). Hence it is formally unramified by Lemma 76.14.7. The other assertions follow by combining Lemmas 76.12.2 and 76.12.3 and the definitions. $\square$

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