In algebraic geometry, a closed immersion
i
:
X
↪
Y
of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of
X
∩
U
is generated by a regular sequence of length r.
For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If
Spec
B
is regularly embedded into a regular scheme, then B is a complete intersection ring.
The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when i is a regular embedding, if I is the ideal sheaf of X in Y, then the normal sheaf, the dual of
I
/
I
2
, is locally free (thus a vector bundle) and the natural map
Sym
(
I
/
I
2
)
→
⊕
0
∞
I
n
/
I
n
+
1
is an isomorphism: the normal cone
Spec
(
⊕
0
∞
I
n
/
I
n
+
1
)
coincides with the normal bundle.
A flat morphism of finite type
f
:
X
→
Y
is called a (local) complete intersection morphism if each point x in X has an open affine neighborhood U so that f |U factors as
U
→
j
V
→
g
Y
where j is a regular embedding and g is smooth. For example, if f is a morphism between smooth varieties, then f factors as
X
→
X
×
Y
→
Y
where the first map is the graph morphism and so is a complete intersection morphism.
