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Regular embedding

In algebraic geometry, a closed immersion i : X ↪ Y {\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 X ∩ U {\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.

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

Non-examples

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

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

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

Local complete intersection morphisms and virtual tangent bundles

A morphism of finite type f : X → Y {\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 U → j V → g Y {\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 X → X × Y → Y {\displaystyle X\to X\times Y\to Y} 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 f : X → Y {\displaystyle f:X\to Y} be a local-complete-intersection morphism that admits a global factorization: it is a composition X ↪ i P → p Y {\displaystyle X{\overset {i}{\hookrightarrow }}P{\overset {p}{\to }}Y} where i {\displaystyle i} is a regular embedding and p {\displaystyle p} a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as:5

T f = [ i ∗ T P / Y ] − [ N X / P ] {\displaystyle T_{f}=[i^{*}T_{P/Y}]-[N_{X/P}]} ,

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

More generally, if f : X → Y {\displaystyle f\colon X\rightarrow Y} is a any local complete intersection morphism of schemes, its cotangent complex L X / Y {\displaystyle L_{X/Y}} is perfect of Tor-amplitude [-1,0]. If moreover f {\displaystyle f} is locally of finite type and Y {\displaystyle Y} 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 u : E → A {\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).7 Then a closed immersion X ↪ Y {\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.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 arises because the discussion of zero-divisors is tricky for non-Noetherian rings in that one cannot use the theory of associated primes.)

See also

Notes

References

  1. Sernesi 2006, D. Notes 2. - Sernesi, Edoardo (2006). Deformations of Algebraic Schemes. Physica-Verlag. ISBN 9783540306153. https://books.google.com/books?id=xkcpQo9tBN8C ↩

  2. Sernesi 2006, D.1. - Sernesi, Edoardo (2006). Deformations of Algebraic Schemes. Physica-Verlag. ISBN 9783540306153. https://books.google.com/books?id=xkcpQo9tBN8C ↩

  3. SGA 6 1971, Exposé VIII, Definition 1.1.; Sernesi 2006, D.2.1. - 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. https://doi.org/10.1007%2FBFb0066283 ↩

  4. EGA IV 1967, Definition 19.3.6, p. 196 - 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. MR 0238860. http://www.numdam.org/articles/PMIHES_1967__32__5_0 ↩

  5. Fulton 1998, Appendix B.7.5. - 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 https://mathscinet.ams.org/mathscinet-getitem?mr=1644323 ↩

  6. Illusie 1971, Proposition 3.2.6 , p. 209 - Illusie, Luc (1971), Complexe Cotangent et Déformations I, Lecture Notes in Mathematics 239 (in French), Berlin, New York: Springer-Verlag, ISBN 978-3-540-05686-7 ↩

  7. SGA 6 1971, Exposé VII. Definition 1.1. NB: We follow the terminology of the Stacks project.[1] - 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. https://doi.org/10.1007%2FBFb0066283 ↩

  8. SGA 6 1971, Exposé VII, Definition 1.4. - 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. https://doi.org/10.1007%2FBFb0066283 ↩

  9. SGA 6 1971, Exposé VIII, Definition 1.1. - 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. https://doi.org/10.1007%2FBFb0066283 ↩

  10. EGA IV 1967, § 16 no 9, p. 45 - 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. MR 0238860. http://www.numdam.org/articles/PMIHES_1967__32__5_0 ↩