Definition 31.19.1. Let $f : Z \to X$ be an immersion. The conormal algebra $\mathcal{C}_{Z/X, *}$ of $Z$ in $X$ or the conormal algebra of $f$ is the quasi-coherent sheaf of graded $\mathcal{O}_ Z$-algebras $\bigoplus _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1}$ described above.
31.19 The normal cone of an immersion
Let $i : Z \to X$ be a closed immersion. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the corresponding quasi-coherent sheaf of ideals. Consider the quasi-coherent sheaf of graded $\mathcal{O}_ X$-algebras $\bigoplus _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1}$. Since the sheaves $\mathcal{I}^ n/\mathcal{I}^{n + 1}$ are each annihilated by $\mathcal{I}$ this graded algebra corresponds to a quasi-coherent sheaf of graded $\mathcal{O}_ Z$-algebras by Morphisms, Lemma 29.4.1. This quasi-coherent graded $\mathcal{O}_ Z$-algebra is called the conormal algebra of $Z$ in $X$ and is often simply denoted $\bigoplus _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1}$ by the abuse of notation mentioned in Morphisms, Section 29.4.
Let $f : Z \to X$ be an immersion. We define the conormal algebra of $f$ as the conormal sheaf of the closed immersion $i : Z \to X \setminus \partial Z$, where $\partial Z = \overline{Z} \setminus Z$. It is often denoted $\bigoplus _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1}$ where $\mathcal{I}$ is the ideal sheaf of the closed immersion $i : Z \to X \setminus \partial Z$.
Thus $\mathcal{C}_{Z/X, 1} = \mathcal{C}_{Z/X}$ is the conormal sheaf of the immersion. Also $\mathcal{C}_{Z/X, 0} = \mathcal{O}_ Z$ and $\mathcal{C}_{Z/X, n}$ is a quasi-coherent $\mathcal{O}_ Z$-module characterized by the property
31.19.1.1
\begin{equation} \label{divisors-equation-conormal-in-degree-n} i_*\mathcal{C}_{Z/X, n} = \mathcal{I}^ n/\mathcal{I}^{n + 1} \end{equation}
where $i : Z \to X \setminus \partial Z$ and $\mathcal{I}$ is the ideal sheaf of $i$ as above. Finally, note that there is a canonical surjective map
31.19.1.2
\begin{equation} \label{divisors-equation-conormal-algebra-quotient} \text{Sym}^*(\mathcal{C}_{Z/X}) \longrightarrow \mathcal{C}_{Z/X, *} \end{equation}
of quasi-coherent graded $\mathcal{O}_ Z$-algebras which is an isomorphism in degrees $0$ and $1$.
Lemma 31.19.2. Let $i : Z \to X$ be an immersion. The conormal algebra of $i$ has the following properties:
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
\[ \mathcal{C}_{Z/X, *} = i^*\left(\bigoplus \nolimits _{n \geq 0} \mathcal{I}^ n\right) = i^{-1}\left( \bigoplus \nolimits _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1} \right) \]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. $\square$
Lemma 31.19.3. Let
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} & X' } \]
be a commutative diagram in the category of schemes. Assume $i$, $i'$ immersions. There is a canonical map of graded $\mathcal{O}_ Z$-algebras
\[ f^*\mathcal{C}_{Z'/X', *} \longrightarrow \mathcal{C}_{Z/X, *} \]
characterized by the following property: For every pair of affine opens $(\mathop{\mathrm{Spec}}(R) = U \subset X, \mathop{\mathrm{Spec}}(R') = U' \subset X')$ with $f(U) \subset U'$ such that $Z \cap U = \mathop{\mathrm{Spec}}(R/I)$ and $Z' \cap U' = \mathop{\mathrm{Spec}}(R'/I')$ the induced map
\[ \Gamma (Z' \cap U', \mathcal{C}_{Z'/X', *}) = \bigoplus \nolimits (I')^ n/(I')^{n + 1} \longrightarrow \bigoplus \nolimits _{n \geq 0} I^ n/I^{n + 1} = \Gamma (Z \cap U, \mathcal{C}_{Z/X, *}) \]
is the one induced by the ring map $f^\sharp : R' \to R$ which has the property $f^\sharp (I') \subset I$.
Proof. Let $\partial Z' = \overline{Z'} \setminus Z'$ and $\partial Z = \overline{Z} \setminus Z$. These are closed subsets of $X'$ and of $X$. Replacing $X'$ by $X' \setminus \partial Z'$ and $X$ by $X \setminus \big (g^{-1}(\partial Z') \cup \partial Z\big )$ we see that we may assume that $i$ and $i'$ are closed immersions.
The fact that $g \circ i$ factors through $i'$ implies that $g^*\mathcal{I}'$ maps into $\mathcal{I}$ under the canonical map $g^*\mathcal{I}' \to \mathcal{O}_ X$, see Schemes, Lemmas 26.4.6 and 26.4.7. Hence we get an induced map of quasi-coherent sheaves $g^*((\mathcal{I}')^ n/(\mathcal{I}')^{n + 1}) \to \mathcal{I}^ n/\mathcal{I}^{n + 1}$. Pulling back by $i$ gives $i^*g^*((\mathcal{I}')^ n/(\mathcal{I}')^{n + 1}) \to i^*(\mathcal{I}^ n/\mathcal{I}^{n + 1})$. Note that $i^*(\mathcal{I}^ n/\mathcal{I}^{n + 1}) = \mathcal{C}_{Z/X, n}$. On the other hand, $i^*g^*((\mathcal{I}')^ n/(\mathcal{I}')^{n + 1}) = f^*(i')^*((\mathcal{I}')^ n/(\mathcal{I}')^{n + 1}) = f^*\mathcal{C}_{Z'/X', n}$. This gives the desired map.
Checking that the map is locally described as the given map $(I')^ n/(I')^{n + 1} \to I^ n/I^{n + 1}$ is a matter of unwinding the definitions and is omitted. Another observation is that given any $x \in i(Z)$ there do exist affine open neighbourhoods $U$, $U'$ with $f(U) \subset U'$ and $Z \cap U$ as well as $U' \cap Z'$ closed such that $x \in U$. Proof omitted. Hence the requirement of the lemma indeed characterizes the map (and could have been used to define it). $\square$
Lemma 31.19.4. Let
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} & X' } \]
be a fibre product diagram in the category of schemes with $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X', *} \to \mathcal{C}_{Z/X, *}$ of Lemma 31.19.3 is surjective. If $g$ is flat, then it is an isomorphism.
Proof. Let $R' \to R$ be a ring map, and $I' \subset R'$ an ideal. Set $I = I'R$. Then $(I')^ n/(I')^{n + 1} \otimes _{R'} R \to I^ n/I^{n + 1}$ is surjective. If $R' \to R$ is flat, then $I^ n = (I')^ n \otimes _{R'} R$ and we see the map is an isomorphism. $\square$
Definition 31.19.5. Let $i : Z \to X$ be an immersion of schemes. The normal cone $C_ ZX$ of $Z$ in $X$ is
\[ C_ ZX = \underline{\mathop{\mathrm{Spec}}}_ Z(\mathcal{C}_{Z/X, *}) \]
see Constructions, Definitions 27.7.1 and 27.7.2. The normal bundle of $Z$ in $X$ is the vector bundle
\[ N_ ZX = \underline{\mathop{\mathrm{Spec}}}_ Z(\text{Sym}(\mathcal{C}_{Z/X})) \]
Thus $C_ ZX \to Z$ is a cone over $Z$ and $N_ ZX \to Z$ is a vector bundle over $Z$ (recall that in our terminology this does not imply that the conormal sheaf is a finite locally free sheaf). Moreover, the canonical surjection (31.19.1.2) of graded algebras defines a canonical closed immersion
31.19.5.1
\begin{equation} \label{divisors-equation-normal-cone-in-normal-bundle} C_ ZX \longrightarrow N_ ZX \end{equation}
of cones over $Z$.
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