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.
