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-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).[4]

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.[5]

(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. ↑ SGA 6, Expo VII. Definition 1.1. NB: We follow the terminology of the Stacks project.
  5. ↑ 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.
  • 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
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