Lemma 31.19.2. Let $i : Z \to X$ be an immersion. The conormal algebra of $i$ has the following properties:

1. Let $U \subset X$ be any open such that $i(Z)$ is a closed subset of $U$. Let $\mathcal{I} \subset \mathcal{O}_ U$ be the sheaf of ideals corresponding to the closed subscheme $i(Z) \subset U$. Then

$\mathcal{C}_{Z/X, *} = i^*\left(\bigoplus \nolimits _{n \geq 0} \mathcal{I}^ n\right) = i^{-1}\left( \bigoplus \nolimits _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1} \right)$
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 $\Gamma (Z \cap U, \mathcal{C}_{Z/X, *}) = \bigoplus _{n \geq 0} I^ n/I^{n + 1}$.

Proof. Mostly clear from the definitions. Note that given a ring $R$ and an ideal $I$ of $R$ we have $I^ n/I^{n + 1} = I^ n \otimes _ R R/I$. Details omitted. $\square$

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