The Stacks project

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

  1. 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

    \[ \mathcal{C}_{Z/X} = (i')^*\mathcal{I}\quad \text{and}\quad i'_*\mathcal{C}_{Z/X} = \mathcal{I}/\mathcal{I}^2 \]

  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 $\Gamma (Z \cap U, \mathcal{C}_{Z/X}) = I/I^2$.

Proof. Mostly clear from the definitions. Note that given a ring $R$ and an ideal $I$ of $R$ we have $I/I^2 = I \otimes _ R R/I$. Details omitted. $\square$

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