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.

• Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, section B.7
• E. Sernesi: Deformations of algebraic schemes