The Stacks project

Lemma 37.5.3. Let $i : Z \to X$ be an immersion of schemes. Let $Z \subset Z'$ be the first order infinitesimal neighbourhood of $Z$ in $X$. Then the diagram

\[ \xymatrix{ Z \ar[r] \ar[d] & Z' \ar[d] \\ Z \ar[r] & X } \]

induces a map of conormal sheaves $\mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Z'}$ by Morphisms, Lemma 29.31.3. This map is an isomorphism.

Proof. This is clear from the construction of $Z'$ above. $\square$

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