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.[1] If 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 , is locally free (thus a vector bundle) and the natural map is an isomorphism: the normal cone coincides with the normal bundle.

A morphism of finite type 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 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 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:[4]

.

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

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 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).[5]

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.[6]

(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

Notes

  1. Sernesi, D. Notes 2.
  2. Sernesi, D.1.
  3. Sernesi, D.2.1.
  4. Fulton, Appendix B.7.5.
  5. SGA 6, Expo VII. Definition 1.1. NB: We follow the terminology of the Stacks project.
  6. SGA 6, Expo VII. Definition 1.4.

References

  • Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.
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