Lemma 29.31.2. Let $i : Z \to X$ be an immersion. The conormal sheaf of $i$ has the following properties:

Let $U \subset X$ be any open subscheme such that $i$ factors as $Z \xrightarrow {i'} U \to X$ where $i'$ is a closed immersion. Let $\mathcal{I} = \mathop{\mathrm{Ker}}((i')^\sharp ) \subset \mathcal{O}_ U$. Then

\[ \mathcal{C}_{Z/X} = (i')^*\mathcal{I}\quad \text{and}\quad i'_*\mathcal{C}_{Z/X} = \mathcal{I}/\mathcal{I}^2 \]For any affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$ such that $Z \cap U = \mathop{\mathrm{Spec}}(R/I)$ there is a canonical isomorphism $\Gamma (Z \cap U, \mathcal{C}_{Z/X}) = I/I^2$.

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