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.

Examples and usage

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 and the natural map is an isomorphism: the normal cone coincides with the normal bundle.
A morphism of finite type is called a 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.

Non Examples

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.

Virtual tangent bundle

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:
The notion is used for instance in the Riemann–Roch-type theorem.

Non-noetherian case

uses the following weakened form of the notion of a regular embedding, that agrees with the usual one for Noetherian schemes.
First, given a projective module E over a commutative ring A, an A-linear map is called Koszul-regular if the Koszul complex determined by it is acyclic in dimension > 0.
Then a closed immersion is called Koszul-regular if the ideal sheaf determined by it is such that, locally, there are a finite free A-module E and a Koszul-regular surjection from E to the ideal sheaf.