Regular embedding

In algebraic geometry, a closed immersion i: X \hookrightarrow 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 \cap 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.[1] If \operatorname{Spec}B is regularly embedded into a regular scheme, then B is a complete intersection ring.[2]

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 \operatorname{Sym}(I/I^2) \to \oplus_0^\infty I^n/I^{n+1} is an isomorphism: the normal cone \operatorname{Spec}(\oplus_0^\infty I^n/I^{n+1}) coincides with the normal bundle.

A flat morphism of finite type f:X \to 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 \overset{j}\to V \overset{g}\to Y where j is a regular embedding and g is smooth.[3] For example, if f is a morphism between smooth varieties, then f factors as X \to X \times Y \to Y where the first map is the graph morphism and so is a complete intersection morphism.

References

  1. Sernesi, D. Notes 2.
  2. Sernesi, D.1.
  3. Sernesi, D.2.1.


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