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 Examples

One non-example is a scheme which isn't equidimensional. For example, the scheme

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

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

Virtual tangent bundle

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

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

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 $u:E\to A$ 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

• 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. {{cite book}}: Unknown parameter |nopp= ignored (|no-pp= suggested) (help)

• 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