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In `F33f`_`[algebraic geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_geometry]`_`f, a `F33f`_`[closed immersion`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Closed_immersion]`_`f 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 `F33f`_`[regular sequence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regular_sequence]`_`f of length `*r`*. A regular embedding of codimension one is precisely an `F33f`_`[effective Cartier divisor`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Effective_Cartier_divisor]`_`f.

>>Contents

• `F0af`_`[Examples and usage`#examples-and-usage]`_`f

• `F0af`_`[Non-examples`#non-examples]`_`f

• `F0af`_`[Local complete intersection morphisms and virtual tangent bundles`#local-complete-intersection-morphisms-and-virtual-tangent-bundles]`_`f

• `F0af`_`[Non-Noetherian case`#non-noetherian-case]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Examples and usage

For example, if `*X`* and `*Y`* are `F33f`_`[smooth`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_morphism]`_`f 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.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] If Spec ⁡ ⁡ B {\\displaystyle \\operatorname {Spec} B} is regularly embedded into a `F33f`_`[regular scheme`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regular_scheme]`_`f, then `*B`* is a `F33f`_`[complete intersection ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complete_intersection_ring]`_`f.`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]

The notion is used, for instance, in an essential way in Fulton's approach to `F33f`_`[intersection theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Intersection_theory]`_`f. The important fact is that when `*i`* is a regular embedding, if `*I`* is the ideal sheaf of `*X`* in `*Y`*, then the `F33f`_`[normal sheaf`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_sheaf]`_`f, 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 `F33f`_`[normal cone`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_cone_(algebraic_geometry)]`_`f 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 `F33f`_`[smooth`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_morphism]`_`f. `:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] For example, if `*f`* is a morphism between `F33f`_`[smooth varieties`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_variety]`_`f, then `*f`* factors as X → → X × × Y → → Y {\\displaystyle X\\to X\\times Y\\to Y} where the first map is the `F33f`_`[graph morphism`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Graph_morphism_(algebraic_geometry)]`_`f and so is a complete intersection morphism. Notice that this definition is compatible with the one in EGA IV for the special case of `F33f`_`[flat morphisms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Flat_morphism]`_`f.`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]

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 `F33f`_`[Grothendieck group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Grothendieck_group]`_`f of vector bundles on `*X`* given as:`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]

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

where T P / Y = Ω Ω P / Y ∨ ∨ {\\displaystyle T_{P/Y}=\\Omega _{P/Y}^{\\vee }} is the relative tangent sheaf of p {\\displaystyle p} (which is `F33f`_`[locally free`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Locally_free_sheaf]`_`f 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 `F33f`_`[cotangent complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cotangent_complex]`_`f L X / Y {\\displaystyle L_{X/Y}} is `F33f`_`[perfect`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Perfect_complex]`_`f 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.`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f]

These notions are used for instance in the `F33f`_`[Grothendieck–Riemann–Roch theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Grothendieck–Riemann–Roch_theorem]`_`f.

>>Non-Noetherian case

`F33f`_`[SGA 6 Exposé VII`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Séminaire_de_géométrie_algébrique_du_Bois_Marie]`_`f 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 `F33f`_`[projective module`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Projective_module]`_`f `*E`* over a commutative ring `*A`*, an `*A`*-linear map u : E → → A {\\displaystyle u:E\\to A} is called `!Koszul-regular`! if the `F33f`_`[Koszul complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Koszul_complex]`_`f determined by it is `F33f`_`[acyclic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Acyclic_complex]`_`f in dimension > 0 (consequently, it is a resolution of the cokernel of `*u`*).`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f] 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.`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f]

It is this Koszul regularity that was used in SGA 6 `:cite-ref-9[`F5bf`_`[9`#cite-note-9]`_`f] 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.`:cite-ref-10[`F5bf`_`[10`#cite-note-10]`_`f]

(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

• `F33f`_`[Regular submanifold`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regular_submanifold]`_`f

>>Notes

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `F33f`_`[Sernesi 2006`#citerefsernesi2006]`_`f, D. Notes 2.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `F33f`_`[Sernesi 2006`#citerefsernesi2006]`_`f, D.1.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `F33f`_`[SGA 6 1971`#citerefsga-61971]`_`f, Exposé VIII, Definition 1.1.; `F33f`_`[Sernesi 2006`#citerefsernesi2006]`_`f, D.2.1.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `F33f`_`[EGA IV 1967`#citerefega-iv1967]`_`f, Definition 19.3.6, p. 196

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `F33f`_`[Fulton 1998`#citereffulton1998]`_`f, Appendix B.7.5.

`:cite-note-6`!6.`! `F0af`_`[↑`#cite-ref-6]`_`f `F33f`_`[Illusie 1971`#citerefillusie1971]`_`f, Proposition 3.2.6 , p. 209

`:cite-note-7`!7.`! `F0af`_`[↑`#cite-ref-7]`_`f `F33f`_`[SGA 6 1971`#citerefsga-61971]`_`f, Exposé VII. Definition 1.1. NB: We follow the terminology of the `F33f`_`[Stacks project`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stacks_project]`_`f.[1]

`:cite-note-8`!8.`! `F0af`_`[↑`#cite-ref-8]`_`f `F33f`_`[SGA 6 1971`#citerefsga-61971]`_`f, Exposé VII, Definition 1.4.

`:cite-note-9`!9.`! `F0af`_`[↑`#cite-ref-9]`_`f `F33f`_`[SGA 6 1971`#citerefsga-61971]`_`f, Exposé VIII, Definition 1.1.

`:cite-note-10`!10.`! `F0af`_`[↑`#cite-ref-10]`_`f `F33f`_`[EGA IV 1967`#citerefega-iv1967]`_`f, § 16 no 9, p. 45

>>References

• `:citerefsga-61971`a`F33f`_`[Berthelot, Pierre`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pierre_Berthelot_(mathematician)]`_`f; `F33f`_`[Alexandre Grothendieck`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Alexandre_Grothendieck]`_`f; `F33f`_`[Luc Illusie`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Luc_Illusie]`_`f, 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: `F33f`_`[Springer-Verlag`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer_Science+Business_Media]`_`f. xii+700. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/BFb0066283. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-05647-8. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0354655.

• `:citereffulton1998`a`F33f`_`[Fulton, William`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=William_Fulton_(mathematician)]`_`f (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: `F33f`_`[Springer-Verlag`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer-Verlag]`_`f, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-62046-4, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1644323, section B.7

• `:citerefega-iv1967`a`F33f`_`[Grothendieck, Alexandre`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Alexander_Grothendieck]`_`f; `F33f`_`[Dieudonné, Jean`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Jean_Dieudonné]`_`f (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". `*`F33f`_`[Publications Mathématiques de l'IHÉS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Publications_Mathématiques_de_l'IHÉS]`_`f`*. `!32`!: 5–361. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/bf02732123. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0238860., section 16.9, p. 46

• `:citerefillusie1971`a`F33f`_`[Illusie, Luc`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Luc_Illusie]`_`f (1971), `*Complexe Cotangent et Déformations I`*, Lecture Notes in Mathematics `!239`! (in French), Berlin, New York: `F33f`_`[Springer-Verlag`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer-Verlag]`_`f, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-05686-7

• `:citerefsernesi2006`aSernesi, Edoardo (2006). `*Deformations of Algebraic Schemes`*. Physica-Verlag. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9783540306153.

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