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

Comments

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.