Regular embedding

Not to be confused with regular scheme.

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. 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 $\operatorname {Spec} B$ 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 $I/I^{2}$, 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 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. 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.

Non-noetherian case

SGA 6 Expo VII 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 $u:E\to A$ is called Koszul-regular if the Koszul complex determined by it is acyclic in dimension > 0 (consequently, it is a resolution of the cokernel of u).

Then a closed immersion $X\hookrightarrow Y$ 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.

(This complication is because the discussion of a zero-divisor is tricky for Non-noetherian rings in that one cannot use the theory of associated primes.)

See also

• regular submanifold

References

• 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], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, section B.7
• E. Sernesi: Deformations of algebraic schemes