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.

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.

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} .

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.

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.)

• Regular submanifold

• 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], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323, 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. MR 0238860., section 16.9, p. 46
• 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
• Sernesi, Edoardo (2006). Deformations of Algebraic Schemes. Physica-Verlag. ISBN 9783540306153.