# Regular embedding

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{{distinguish|regular scheme}}
In [algebraic geometry](/source/algebraic_geometry), a [closed immersion](/source/closed_immersion) <math>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>X \cap U</math> is generated by a [regular sequence](/source/regular_sequence) of length ''r''. A regular embedding of codimension one is precisely an [effective Cartier divisor](/source/effective_Cartier_divisor).

== Examples and usage ==
For example, if ''X'' and ''Y'' are [smooth](/source/smooth_morphism) 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.<ref>{{harvnb|Sernesi|2006|loc=D. Notes 2.}}</ref> If <math>\operatorname{Spec}B</math> is regularly embedded into a [regular scheme](/source/regular_scheme), then ''B'' is a [complete intersection ring](/source/complete_intersection_ring).<ref>{{harvnb|Sernesi|2006|loc=D.1.}}</ref>

The notion is used, for instance, in an essential way in Fulton's approach to [intersection theory](/source/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](/source/normal_sheaf), the dual of <math>I/I^2</math>, is locally free (thus a vector bundle) and the natural map <math>\operatorname{Sym}(I/I^2) \to \oplus_0^\infty I^n/I^{n+1}</math> is an isomorphism: the [normal cone](/source/normal_cone_(algebraic_geometry)) <math>\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>
X = \text{Spec}\left( \frac{\mathbb{C}[x,y,z]}{(xz,yz)}\right)
</math>
is the union of <math>\mathbb{A}^2</math> and <math>\mathbb{A}^1</math>. Then, the embedding <math>X \hookrightarrow \mathbb{A}^3</math> isn't regular since taking any non-origin point on the <math>z</math>-axis is of dimension <math>1</math> while any non-origin point on the <math>xy</math>-plane is of dimension <math>2</math>.

== Local complete intersection morphisms and virtual tangent bundles ==
A morphism of finite type <math>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'' |<sub>''U''</sub> factors as <math>U \overset{j}\to V \overset{g}\to Y</math> where ''j'' is a regular embedding and ''g'' is [smooth](/source/smooth_morphism).
<ref>{{harvnb|SGA 6|1971|loc=Exposé VIII, Definition 1.1.}}; {{harvnb|Sernesi|2006|loc=D.2.1.}}</ref> 
For example, if ''f'' is a morphism between [smooth varieties](/source/smooth_variety), then ''f'' factors as <math>X \to X \times Y \to Y</math> where the first map is the [graph morphism](/source/graph_morphism_(algebraic_geometry)) 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 morphism](/source/flat_morphism)s.<ref>{{harvnb|EGA IV|1967|loc=Definition 19.3.6, p. 196}}</ref>

Let <math>f: X \to Y</math> be a local-complete-intersection morphism that admits a global factorization: it is a composition <math>X \overset{i}\hookrightarrow P \overset{p}\to Y</math> where <math>i</math> is a regular embedding and <math>p</math> a smooth morphism. Then the '''virtual tangent bundle''' is an element of the [Grothendieck group](/source/Grothendieck_group) of vector bundles on ''X'' given as:<ref>{{harvnb|Fulton|1998|loc=Appendix B.7.5.}}</ref>
:<math>T_f = [i^* T_{P/Y}] - [N_{X/P}]</math>,
where <math>T_{P/Y}=\Omega_{P/Y}^{\vee}</math> is the relative tangent sheaf of <math>p</math> 
(which is [locally free](/source/locally_free_sheaf) since <math>p</math> is smooth) 
and <math>N</math> is the normal sheaf <math>(\mathcal{I}/\mathcal{I}^2)^{\vee}</math>
(where <math>\mathcal{I}</math> is the ideal sheaf of <math>X</math> in <math>P</math>), which is locally free since
<math>i</math> is a regular embedding.
 
More generally, 
if <math>f \colon X \rightarrow Y</math> is a ''any'' local complete intersection morphism of schemes, its 
[cotangent complex](/source/cotangent_complex) <math>L_{X/Y}</math> is [perfect](/source/perfect_complex) of Tor-amplitude [-1,0].
If moreover <math>f</math> is locally of finite type and <math>Y</math> locally Noetherian, then the converse is also true.<ref>{{harvnb|Illusie|1971|loc=Proposition 3.2.6 , p. 209}}</ref>
 
These notions are used for instance in the [Grothendieck–Riemann–Roch theorem](/source/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem).

== Non-Noetherian case ==
[SGA 6 Exposé VII](/source/S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique_du_Bois_Marie) 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](/source/projective_module) ''E'' over a commutative ring ''A'', an ''A''-linear map <math>u: E \to A</math> is called '''Koszul-regular''' if the [Koszul complex](/source/Koszul_complex) determined by it is [acyclic](/source/acyclic_complex) in dimension > 0 (consequently, it is a resolution of the cokernel of ''u'').<ref>{{harvnb|SGA 6|1971|loc=Exposé VII. Definition 1.1.}} NB: We follow the terminology of the [Stacks project](/source/Stacks_project).[https://stacks.math.columbia.edu/tag/061T]</ref>
Then a closed immersion <math>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.<ref>{{harvnb|SGA 6|1971|loc=Exposé VII, Definition 1.4.}}</ref> 

It is this Koszul regularity that was used in SGA 6
<ref>{{harvnb|SGA 6|1971|loc=Exposé VIII, Definition 1.1.}}</ref> 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.<ref>{{harvnb|EGA IV|1967|loc=§ 16 no 9, p. 45}}</ref>

(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 ==
*[Regular submanifold](/source/Regular_submanifold)

== Notes ==
{{Reflist}}

== References ==
{{sfn whitelist|CITEREFEGA_IV1967}}
*{{cite book
 | editor-last = Berthelot
 | editor-first = Pierre
 | editor-link = Pierre Berthelot (mathematician)
 | editor2=Alexandre Grothendieck
 | editor2-link=Alexandre Grothendieck
 | editor3=Luc Illusie
 | editor3-link=Luc Illusie
 | title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''')
 | year = 1971
 | publisher = [Springer-Verlag](/source/Springer_Science%2BBusiness_Media)
 | location = Berlin; New York
 | language = fr
 | pages = xii+700
 | no-pp = true
 |doi=10.1007/BFb0066283
 |isbn= 978-3-540-05647-8 
 | mr = 0354655
 |ref = {{sfnref|SGA 6|1971}}
}}
*{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | series=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] | isbn=978-3-540-62046-4|mr=1644323 | year=1998 | volume=2}}, section B.7
*{{EGA|book=4-4| pages = 5–361 |ref={{sfnref|EGA IV|1967}} }}, section 16.9, p. 46
*{{Citation|last1=Illusie | first1=Luc | author1-link=Luc Illusie | title=Complexe Cotangent et Déformations I | publisher =[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | language=fr | series=Lecture Notes in Mathematics '''239''' | isbn=978-3-540-05686-7 | year=1971}}
* {{cite book |last=Sernesi |first=Edoardo |title=Deformations of Algebraic Schemes |url={{GBurl|xkcpQo9tBN8C}} <!--&hl=en Deformations of algebraic schemes--> |date=2006 |publisher=Physica-Verlag |isbn=9783540306153}}

Category:Theorems in algebraic geometry
Category:Morphisms of schemes

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