Regular embedding#Local complete intersection morphisms and virtual tangent bundles

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.{{harvnb|Sernesi|2006|loc=D. Notes 2.}} If $\backslash operatorname\{Spec\}B$ is regularly embedded into a regular scheme, then B is a complete intersection ring.{{harvnb|Sernesi|2006|loc=D.1.}}

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 $I/I^2$, is locally free (thus a vector bundle) and the natural map $\backslash operatorname\{Sym\}(I/I^2)\; \backslash to\; \backslash oplus\_0^\backslash infty\; I^n/I^\{n+1\}$ is an isomorphism: the normal cone $\backslash operatorname\{Spec\}(\backslash oplus\_0^\backslash infty\; I^n/I^\{n+1\})$ coincides with the normal bundle.

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

:$$

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

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

A morphism of finite type $f:X\; \backslash to\; Y$ 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 $U\; \backslash overset\{j\}\backslash to\; V\; \backslash overset\{g\}\backslash to\; Y$ where j is a regular embedding and g is smooth.

{{harvnb|SGA 6|1971|loc=Exposé VIII, Definition 1.1.}}; {{harvnb|Sernesi|2006|loc=D.2.1.}}

For example, if f is a morphism between smooth varieties, then f factors as $X\; \backslash to\; X\; \backslash times\; Y\; \backslash to\; Y$ 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.{{harvnb|EGA IV|1967|loc=Definition 19.3.6, p. 196}}

Let $f:\; X\; \backslash to\; Y$ be a local-complete-intersection morphism that admits a global factorization: it is a composition $X\; \backslash overset\{i\}\backslash hookrightarrow\; P\; \backslash overset\{p\}\backslash to\; Y$ where $i$ is a regular embedding and $p$ a smooth morphism. Then the virtual tangent bundle is an element of the Grothendieck group of vector bundles on X given as:{{harvnb|Fulton|1998|loc=Appendix B.7.5.}}

:$T\_f\; =\; [i^*\; T\_\{P/Y\}]\; -\; [N\_\{X/P\}]$,

where $T\_\{P/Y\}=\backslash Omega\_\{P/Y\}^\{\backslash vee\}$ is the relative tangent sheaf of $p$

(which is locally free since $p$ is smooth)

and $N$ is the normal sheaf $(\backslash mathcal\{I\}/\backslash mathcal\{I\}^2)^\{\backslash vee\}$

(where $\backslash mathcal\{I\}$ is the ideal sheaf of $X$ in $P$), which is locally free since

$i$ is a regular embedding.

More generally,

if $f\; \backslash colon\; X\; \backslash rightarrow\; Y$ is a any local complete intersection morphism of schemes, its

cotangent complex $L\_\{X/Y\}$ is perfect of Tor-amplitude [-1,0].

If moreover $f$ is locally of finite type and $Y$ locally Noetherian, then the converse is also true.{{harvnb|Illusie|1971|loc=Proposition 3.2.6 , p. 209}}

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 $u:\; E\; \backslash to\; A$ 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).{{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]

Then a closed immersion $X\; \backslash hookrightarrow\; Y$ 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.{{harvnb|SGA 6|1971|loc=Exposé VII, Definition 1.4.}}

It is this Koszul regularity that was used in SGA 6

{{harvnb|SGA 6|1971|loc=Exposé VIII, Definition 1.1.}} 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.{{harvnb|EGA IV|1967|loc=§ 16 no 9, p. 45}}

(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

{{Reflist}}

{{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}} |date=2006 |publisher=Physica-Verlag |isbn=9783540306153}}

Category:Theorems in algebraic geometry

Category:Morphisms of schemes