# Normal sheaf

An analogue to a normal bundle in sheaf theory. Let

$$( f, f ^ { \# } ): ( Y, {\mathcal O} _ {Y} ) \rightarrow ( X, {\mathcal O} _ {X} )$$

be a morphism of ringed spaces such that the homomorphism $f ^ { \# } : f ^ { * } {\mathcal O} _ {X} \rightarrow {\mathcal O} _ {Y}$ is surjective, and let ${\mathcal J} = \mathop{\rm Ker} f ^ { \# }$. Then ${\mathcal J} / {\mathcal J} ^ {2}$ is a sheaf of ideals in $f ^ { * } {\mathcal O} _ {X} / {\mathcal J} \cong {\mathcal O} _ {Y}$ and is, therefore, an ${\mathcal O} _ {Y}$- module. Here ${\mathcal N} _ {Y/X} ^ {*} = ( {\mathcal J} / {\mathcal J} ^ {2} )$ is called the conormal sheaf of the morphism and the dual ${\mathcal O} _ {Y}$- module ${\mathcal N} _ {Y/X} = \mathop{\rm Hom} _ { {\mathcal O} _ {Y} } ( {\mathcal N} _ {Y/X} ^ {*} , {\mathcal O} _ {Y} )$ is called the normal sheaf of the morphism $f$. These sheaves are, as a rule, examined in the following special cases.

1) $X$ and $Y$ are differentiable manifolds (for example, of class $C ^ \infty$), and $f: Y \rightarrow X$ is an immersion. There is an exact sequence of ${\mathcal O} _ {Y}$- modules

$$0 \rightarrow {\mathcal N} _ {Y/X} ^ {*} \mathop \rightarrow \limits ^ \delta \ f ^ { * } \Omega _ {X} ^ {1} \rightarrow \Omega _ {Y} ^ {1} \rightarrow 0,$$

where $\Omega _ {X} ^ {1}$ and $\Omega _ {Y} ^ {1}$ are the sheaves of germs of smooth $1$- forms on $X$ and $Y$, and $\delta$ is defined as differentiation of functions. The dual exact sequence

$$0 \rightarrow {\mathcal T} _ {Y} \rightarrow f ^ { * } {\mathcal T} _ {X} \rightarrow {\mathcal N} _ {Y/X} \rightarrow 0,$$

where ${\mathcal T} _ {X}$ and ${\mathcal T} _ {Y}$ are the tangent sheaves on $X$ and $Y$, shows that ${\mathcal N} _ {Y/X}$ is isomorphic to the sheaf of germs of smooth sections of the normal bundle of the immersion $f$. If $Y$ is an immersed submanifold, then ${\mathcal N} _ {Y/X}$ and ${\mathcal N} _ {Y/X} ^ {*}$ are called the normal and conormal sheaves of the submanifold $Y$.

2) $( X, {\mathcal O} _ {X} )$ is an irreducible separable scheme of finite type over an algebraically closed field $k$, $( Y, {\mathcal O} _ {Y} )$ is a closed subscheme of it and $f: Y \rightarrow X$ is an imbedding. Then ${\mathcal N} _ {Y/X}$ and ${\mathcal N} _ {Y/X} ^ {*}$ are called the normal and conormal sheaves of the subscheme $Y$. There is also an exact sequence of ${\mathcal O} _ {Y}$- modules

$$\tag{* } {\mathcal N} _ {Y/X} ^ {*} \mathop \rightarrow \limits ^ \delta \Omega _ {X} \otimes {\mathcal O} _ {Y} \rightarrow \ \Omega _ {Y} \rightarrow 0,$$

where $\Omega _ {X}$ and $\Omega _ {Y}$ are the sheaves of differentials on $X$ and $Y$. The sheaves ${\mathcal N} _ {Y/X} ^ {*}$ and ${\mathcal N} _ {Y/X}$ are quasi-coherent, and if $X$ is a Noetherian scheme, then they are coherent. If $X$ is a non-singular variety over $k$ and $Y$ is a non-singular variety, then ${\mathcal N} _ {Y/X} ^ {*}$ is locally free and the homomorphism $\delta$ in (*) is injective. In this case one obtains the dual exact sequence

$$0 \rightarrow {\mathcal T} _ {Y} \rightarrow {\mathcal T} _ {X} \otimes {\mathcal O} _ {Y} \rightarrow \ {\mathcal N} _ {Y/X} \rightarrow 0,$$

so that the normal sheaf ${\mathcal N} _ {Y/X}$ is locally free of rank $r = \mathop{\rm codim} Y$ corresponding to the normal bundle over $Y$. In particular, if $r = 1$, then ${\mathcal N} _ {Y/X}$ is the invertible sheaf corresponding to the divisor $Y$.

In terms of normal sheaves one can express the self-intersection $Y \cdot Y$ of a non-singular subvariety $Y \subset X$. Namely, $Y \cdot Y = f _ {*} c _ {r} ( {\mathcal N} _ {Y/X} )$, where $c _ {r}$ is the $r$- th Chern class and $f _ {*} : A ( Y) \rightarrow A ( X)$ is the homomorphism of Chow rings (cf. Chow ring) corresponding to the imbedding $f: Y \rightarrow X$.

3) $( X, {\mathcal O} _ {X} )$ is a complex space, $( Y, {\mathcal O} _ {Y} )$ is a closed analytic subspace of it and $f$ is the imbedding. Then ${\mathcal N} _ {Y/X}$ and ${\mathcal N} _ {Y/X} ^ {*}$ are called the normal and conormal sheaves of the subspace $Y$; they are coherent. If $X$ is an analytic manifold and $Y$ an analytic submanifold of it, then ${\mathcal N} _ {Y/X}$ is the sheaf of germs of holomorphic sections of the normal bundle over $Y$.

#### References

 [1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 [2] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001

If $X$ is a non-singular variety over $k$ and $Y$ is a subscheme of $X$ that is locally a complete intersection, then ${\mathcal N} _ {Y/X} ^ {*}$ is locally free.