Definition 76.6.1. Let $i : Z \to X$ be an immersion. The conormal algebra $\mathcal{C}_{Z/X, *}$ of $Z$ in $X$ or the conormal algebra of $i$ is the quasi-coherent sheaf of graded $\mathcal{O}_ Z$-algebras $\bigoplus _{n \geq 0} \mathcal{I}^ n/\mathcal{I}^{n + 1}$ described above.
76.6 The normal cone of an immersion
Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the corresponding quasi-coherent sheaf of ideals, see Morphisms of Spaces, Lemma 67.13.1. 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 of Spaces, Lemma 67.14.1. 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 of Spaces, Section 67.14.
In case $i : Z \to X$ is a (locally closed) immersion we define the conormal algebra of $i$ as the conormal algebra of the closed immersion $i : Z \to X \setminus \partial Z$, see Morphisms of Spaces, Remark 67.12.4. 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
76.6.1.1
\begin{equation} \label{spaces-more-morphisms-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
76.6.1.2
\begin{equation} \label{spaces-more-morphisms-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 76.6.2. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. Let $\varphi : U \to X$ be an étale morphism where $U$ is a scheme. Set $Z_ U = U \times _ X Z$ which is a locally closed subscheme of $U$. Then canonically and functorially in $U$.
\[ \mathcal{C}_{Z/X, *}|_{Z_ U} = \mathcal{C}_{Z_ U/U, *} \]
Proof. Let $T \subset X$ be a closed subspace such that $i$ defines a closed immersion into $X \setminus T$. Let $\mathcal{I}$ be the quasi-coherent sheaf of ideals on $X \setminus T$ defining $Z$. Then the lemma follows from the fact that $\mathcal{I}|_{U \setminus \varphi ^{-1}(T)}$ is the sheaf of ideals of the immersion $Z_ U \to U \setminus \varphi ^{-1}(T)$. This is clear from the construction of $\mathcal{I}$ in Morphisms of Spaces, Lemma 67.13.1. $\square$
Lemma 76.6.3. Let $S$ be a scheme. Let be a commutative diagram of algebraic spaces over $S$. Assume $i$, $i'$ immersions. There is a canonical map of graded $\mathcal{O}_ Z$-algebras
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} & X' } \]
\[ f^*\mathcal{C}_{Z'/X', *} \longrightarrow \mathcal{C}_{Z/X, *} \]
Proof. First find open subspaces $U' \subset X'$ and $U \subset X$ such that $g(U) \subset U'$ and such that $i(Z) \subset U$ and $i(Z') \subset U'$ are closed (proof existence omitted). Replacing $X$ by $U$ and $X'$ by $U'$ we may assume that $i$ and $i'$ are closed immersions. Let $\mathcal{I}' \subset \mathcal{O}_{X'}$ and $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaves of ideals associated to $i'$ and $i$, see Morphisms of Spaces, Lemma 67.13.1. Consider the composition
\[ g^{-1}\mathcal{I}' \to g^{-1}\mathcal{O}_{X'} \xrightarrow {g^\sharp } \mathcal{O}_ X \to \mathcal{O}_ X/\mathcal{I} = i_*\mathcal{O}_ Z \]
Since $g(i(Z)) \subset Z'$ we conclude this composition is zero (see statement on factorizations in Morphisms of Spaces, Lemma 67.13.1). Thus we obtain a commutative diagram
\[ \xymatrix{ 0 \ar[r] & \mathcal{I} \ar[r] & \mathcal{O}_ X \ar[r] & i_*\mathcal{O}_ Z \ar[r] & 0 \\ 0 \ar[r] & g^{-1}\mathcal{I}' \ar[r] \ar[u] & g^{-1}\mathcal{O}_{X'} \ar[r] \ar[u] & g^{-1}i'_*\mathcal{O}_{Z'} \ar[r] \ar[u] & 0 } \]
The lower row is exact since $g^{-1}$ is an exact functor. By exactness we also see that $(g^{-1}\mathcal{I}')^ n = g^{-1}((\mathcal{I}')^ n)$ for all $n \geq 1$. Hence the diagram induces a map $g^{-1}((\mathcal{I}')^ n/(\mathcal{I}')^{n + 1}) \to \mathcal{I}^ n/\mathcal{I}^{n + 1}$. Pulling back (using $i^{-1}$ for example) to $Z$ we obtain $i^{-1}g^{-1}((\mathcal{I}')^ n/(\mathcal{I}')^{n + 1}) \to \mathcal{C}_{Z/X, n}$. Since $i^{-1}g^{-1} = f^{-1}(i')^{-1}$ this gives maps $f^{-1}\mathcal{C}_{Z'/X', n} \to \mathcal{C}_{Z/X, n}$, which induce the desired map. $\square$
Lemma 76.6.4. Let $S$ be a scheme. Let be a cartesian square of algebraic spaces over $S$ with $i$, $i'$ immersions. Then the canonical map $f^*\mathcal{C}_{Z'/X', *} \to \mathcal{C}_{Z/X, *}$ of Lemma 76.6.3 is surjective. If $g$ is flat, then it is an isomorphism.
\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ f & X \ar[d]^ g \\ Z' \ar[r]^{i'} & X' } \]
Proof. We may check the statement after étale localizing $X'$. In this case we may assume $X' \to X$ is a morphism of schemes, hence $Z$ and $Z'$ are schemes and the result follows from the case of schemes, see Divisors, Lemma 31.19.4. $\square$
We use the same conventions for cones and vector bundles over algebraic spaces as we do for schemes (where we use the conventions of EGA), see Constructions, Sections 27.7 and 27.6. In particular, a vector bundle is a very general gadget (and not locally isomorphic to an affine space bundle).
Definition 76.6.5. Let $S$ be a scheme. Let $i : Z \to X$ be an immersion of algebraic spaces over $S$. The normal cone $C_ ZX$ of $Z$ in $X$ is see Morphisms of Spaces, Definition 67.20.8. The normal bundle of $Z$ in $X$ is the vector bundle
\[ C_ ZX = \underline{\mathop{\mathrm{Spec}}}_ Z(\mathcal{C}_{Z/X, *}) \]
\[ N_ ZX = \underline{\mathop{\mathrm{Spec}}}_ Z(\text{Sym}(\mathcal{C}_{Z/X})) \]
Thus $C_ ZX \to Z$ is a cone over $Z$ and $N_ ZX \to Z$ is a vector bundle over $Z$. Moreover, the canonical surjection (76.6.1.2) of graded algebras defines a canonical closed immersion
76.6.5.1
\begin{equation} \label{spaces-more-morphisms-equation-normal-cone-in-normal-bundle} C_ ZX \longrightarrow N_ ZX \end{equation}
of cones over $Z$.
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