Regular embedding

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In algebraic geometry, a closed immersion [math]\displaystyle{ i: X \hookrightarrow Y }[/math] 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 [math]\displaystyle{ X \cap U }[/math] 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 [math]\displaystyle{ \operatorname{Spec}B }[/math] 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 [math]\displaystyle{ I/I^2 }[/math], is locally free (thus a vector bundle) and the natural map [math]\displaystyle{ \operatorname{Sym}(I/I^2) \to \oplus_0^\infty I^n/I^{n+1} }[/math] is an isomorphism: the normal cone [math]\displaystyle{ \operatorname{Spec}(\oplus_0^\infty I^n/I^{n+1}) }[/math] coincides with the normal bundle.

Non-examples

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

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

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

Local complete intersection morphisms and virtual tangent bundles

A morphism of finite type [math]\displaystyle{ f:X \to Y }[/math] 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 [math]\displaystyle{ U \overset{j}\to V \overset{g}\to Y }[/math] 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 [math]\displaystyle{ X \to X \times Y \to Y }[/math] where the first map is the graph morphism and so is a complete intersection morphism. Notice that this definition is compatible with the one in EGA IV for the special case of flat morphisms.[4]

Let [math]\displaystyle{ f: X \to Y }[/math] be a local-complete-intersection morphism that admits a global factorization: it is a composition [math]\displaystyle{ X \overset{i}\hookrightarrow P \overset{p}\to Y }[/math] where [math]\displaystyle{ i }[/math] is a regular embedding and [math]\displaystyle{ p }[/math] a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as:[5]

[math]\displaystyle{ T_f = [i^* T_{P/Y}] - [N_{X/P}] }[/math],

where [math]\displaystyle{ T_{P/Y}=\Omega_{P/Y}^{\vee} }[/math] is the relative tangent sheaf of [math]\displaystyle{ p }[/math] (which is locally free since [math]\displaystyle{ p }[/math] is smooth) and [math]\displaystyle{ N }[/math] is the normal sheaf [math]\displaystyle{ (\mathcal{I}/\mathcal{I}^2)^{\vee} }[/math] (where [math]\displaystyle{ \mathcal{I} }[/math] is the ideal sheaf of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ P }[/math]), which is locally free since [math]\displaystyle{ i }[/math] is a regular embedding.

More generally, if [math]\displaystyle{ f \colon X \rightarrow Y }[/math] is a any local complete intersection morphism of schemes, its cotangent complex [math]\displaystyle{ L_{X/Y} }[/math] is perfect of Tor-amplitude [-1,0]. If moreover [math]\displaystyle{ f }[/math] is locally of finite type and [math]\displaystyle{ Y }[/math] locally Noetherian, then the converse is also true.[6]

These notions are used for instance in the Grothendieck–Riemann–Roch theorem.

Non-Noetherian case

SGA 6 Exposé VII uses the following slightly weaker form of the notion of a regular embedding, which agrees with the one presented above for Noetherian schemes:

First, given a projective module E over a commutative ring A, an A-linear map [math]\displaystyle{ u: E \to A }[/math] 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).[7] Then a closed immersion [math]\displaystyle{ X \hookrightarrow Y }[/math] 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.[8]

It is this Koszul regularity that was used in SGA 6 [9] for the definition of local complete intersection morphisms; it is indicated there that Koszul-regularity was intended to replace the definition given earlier in this article and that had appeared originally in the already published EGA IV.[10]

(This questions arise because the discussion of zero-divisors is tricky for non-Noetherian rings in that one cannot use the theory of associated primes.)

See also

  • Regular submanifold

Notes

  1. ↑ Sernesi 2006, D. Notes 2.
  2. ↑ Sernesi 2006, D.1.
  3. ↑ SGA 6 1971, Exposé VIII, Definition 1.1.; Sernesi 2006, D.2.1.
  4. ↑ EGA IV 1967, Definition 19.3.6, p. 196
  5. ↑ Fulton 1998, Appendix B.7.5.
  6. ↑ Illusie 1971, Proposition 3.2.6 , p. 209
  7. ↑ SGA 6 1971, Exposé VII. Definition 1.1. NB: We follow the terminology of the Stacks project.[1]
  8. ↑ SGA 6 1971, Exposé VII, Definition 1.4.
  9. ↑ SGA 6 1971, Exposé VIII, Definition 1.1.
  10. ↑ EGA IV 1967, § 16 no 9, p. 45

References



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