Regular embedding

In algebraic geometry, a closed immersion 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 is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor. 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 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 , is locally free (thus a vector bundle) and the natural map is an isomorphism: the normal cone coincides with the normal bundle. One non-example is a scheme which isn't equidimensional. For example, the scheme is the union of and . Then, the embedding isn't regular since taking any non-origin point on the -axis is of dimension while any non-origin point on the -plane is of dimension . A morphism of finite type 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 where j is a regular embedding and g is smooth. For example, if f is a morphism between smooth varieties, then f factors as where the first map is the graph morphism and so is a complete intersection morphism. Notice that this definition is compatible with the one in EGA IV for the special case of flat morphisms. Let be a local-complete-intersection morphism that admits a global factorization: it is a composition where is a regular embedding and a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as: where is the relative tangent sheaf of (which is locally free since is smooth) and is the normal sheaf (where is the ideal sheaf of in ), which is locally free since is a regular embedding.