Regular embedding
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
- ↑ Sernesi 2006, D. Notes 2.
- ↑ Sernesi 2006, D.1.
- ↑ SGA 6 1971, Exposé VIII, Definition 1.1.; Sernesi 2006, D.2.1.
- ↑ EGA IV 1967, Definition 19.3.6, p. 196
- ↑ Fulton 1998, Appendix B.7.5.
- ↑ Illusie 1971, Proposition 3.2.6 , p. 209
- ↑ SGA 6 1971, Exposé VII. Definition 1.1. NB: We follow the terminology of the Stacks project.[1]
- ↑ SGA 6 1971, Exposé VII, Definition 1.4.
- ↑ SGA 6 1971, Exposé VIII, Definition 1.1.
- ↑ EGA IV 1967, § 16 no 9, p. 45
References
- Berthelot, Pierre, ed (1971) (in fr). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225). Berlin; New York: Springer-Verlag. pp. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8.
- 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, section B.7
- Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS 32: 5–361. doi:10.1007/bf02732123. http://www.numdam.org/articles/PMIHES_1967__32__5_0., section 16.9, p. 46
- Illusie, Luc (1971) (in fr), Complexe Cotangent et Déformations I, Lecture Notes in Mathematics 239, Berlin, New York: Springer-Verlag, ISBN 978-3-540-05686-7
- Sernesi, Edoardo (2006). Deformations of Algebraic Schemes. Physica-Verlag. ISBN 9783540306153. https://books.google.com/books?id=xkcpQo9tBN8C.
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