# 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

## 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*(in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.**225**)

- 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*