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.

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.

Local complete intersection morphisms and virtual tangent bundles

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. 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

and is the normal sheaf
, which is locally free since
is a regular embedding.
More generally,
if is a any local complete intersection morphism of schemes, its
cotangent complex is perfect of Tor-amplitude .
If moreover is locally of finite type and locally Noetherian, then the converse is also true.
These notions are used for instance in the Grothendieck–Riemann–Roch theorem.

Non-Noetherian case

SGA 6 Exposé VII uses the following slightly weaker form of the notion of a regular embedding, which agrees with the one presented above 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.
It is this Koszul regularity that was used in SGA 6
for the definition of local complete intersection morphisms; it is indicated there that Koszul-regularity was intended to replace the definition given earlier in this article and that had appeared originally in the already published EGA IV.