{{distinguish|regular scheme}} In algebraic geometry, a 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 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.<ref>{{harvnb|Sernesi|2006|loc=D. Notes 2.}}</ref> If <math>\operatorname{Spec}B</math> is regularly embedded into a regular scheme, then ''B'' is a 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. 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>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 <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. <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, then ''f'' factors as <math>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.<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 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 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 <math>L_{X/Y}</math> is perfect 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.
== 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>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'').<ref>{{harvnb|SGA 6|1971|loc=Exposé VII. Definition 1.1.}} NB: We follow the terminology of the 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
== 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 | 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 | 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 | 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