Regular embedding

In algebraic geometry, a closed immersion ${\displaystyle i:X\hookrightarrow Y}$ 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 ${\displaystyle X\cap U}$ is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor.

Not to be confused with regular scheme.

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 ${\displaystyle \operatorname {Spec} B}$ 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 ${\displaystyle I/I^{2}}$, is locally free (thus a vector bundle) and the natural map ${\displaystyle \operatorname {Sym} (I/I^{2})\to \oplus _{0}^{\infty }I^{n}/I^{n+1}}$ is an isomorphism: the normal cone ${\displaystyle \operatorname {Spec} (\oplus _{0}^{\infty }I^{n}/I^{n+1})}$ coincides with the normal bundle.

A morphism of finite type ${\displaystyle f:X\to Y}$ 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 ${\displaystyle U{\overset {j}{\to }}V{\overset {g}{\to }}Y}$ 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 ${\displaystyle X\to X\times Y\to Y}$ 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

${\displaystyle X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)}$

is the union of ${\displaystyle \mathbb {A} ^{2}}$ and ${\displaystyle \mathbb {A} ^{1}}$. Then, the embedding ${\displaystyle X\hookrightarrow \mathbb {A} ^{3}}$ isn't regular since taking any non-origin point on the ${\displaystyle z}$-axis is of dimension ${\displaystyle 1}$ while any non-origin point on the ${\displaystyle xy}$-plane is of dimension ${\displaystyle 2}$.

Virtual tangent bundle

Let ${\displaystyle f:X\to Y}$ be a local-complete-intersection morphism that admits a global factorization: it is a composition ${\displaystyle X{\overset {i}{\hookrightarrow }}P{\overset {p}{\to }}Y}$ where ${\displaystyle i}$ is a regular embedding and ${\displaystyle p}$ a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as:[4]

${\displaystyle T_{f}=[i^{*}T_{P/Y}]-[N_{X/P}]}$.

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 ${\displaystyle u:E\to A}$ 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).[5]

Then a closed immersion ${\displaystyle X\hookrightarrow Y}$ 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.[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

• regular submanifold

Notes

1. Sernesi, D. Notes 2. harvnb error: no target: CITEREFSernesi (help)
2. Sernesi, D.1. harvnb error: no target: CITEREFSernesi (help)
3. Sernesi, D.2.1. harvnb error: no target: CITEREFSernesi (help)
4. Fulton, Appendix B.7.5. harvnb error: no target: CITEREFFulton (help)
5. SGA 6, Expo VII. Definition 1.1. harvnb error: no target: CITEREFSGA_6 (help) NB: We follow the terminology of the Stacks project.
6. SGA 6, Expo VII. Definition 1.4. harvnb error: no target: CITEREFSGA_6 (help)

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