In
algebraic geometry, a
closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formal ...
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
is generated by a
regular sequence
In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a complete intersection.
Definitions
F ...
of length ''r''. A regular embedding of codimension one is precisely an
effective Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumf ...
.
Examples and usage
For example, if ''X'' and ''Y'' are
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
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. If
is regularly embedded into a
regular scheme
In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth..
For an example of a regular ...
, then ''B'' is a
complete intersection ring In commutative algebra, a complete intersection ring is a commutative ring similar to the coordinate rings of varieties that are complete intersections. Informally, they can be thought of roughly as the local rings that can be defined using the "mi ...
.
The notion is used, for instance, in an essential way in Fulton's approach to
intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
. 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
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannia ...
, the dual of
, is locally free (thus a vector bundle) and the natural map
is an isomorphism: the
normal cone
In algebraic geometry, the normal cone C_XY of a subscheme X of a scheme Y is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
Definition
The normal cone or C_ of an embedding , defined by some sheaf of i ...
coincides with the normal bundle.
Non Examples
One non-example is a scheme which isn't equidimensional. For example, the scheme
:
is the union of
and
. Then, the embedding
isn't regular since taking any non-origin point on the
-axis is of dimension
while any non-origin point on the
-plane is of dimension
.
Local complete intersection morphisms and virtual tangent bundles
A morphism of finite type
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
where ''j'' is a regular embedding and ''g'' is
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
.
For example, if ''f'' is a morphism between
smooth varieties, then ''f'' factors as
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 morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ ...
s.
Let
be a local-complete-intersection morphism that admits a global factorization: it is a composition
where
is a regular embedding and
a smooth morphism. Then the virtual tangent bundle is an element of the
Grothendieck group
In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
of vector bundles on ''X'' given as:
:
cotangent complex In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If f: X \to Y is a morphism of geometric or algebraic ob ...
L_ is
perfect
Perfect commonly refers to:
* Perfection, completeness, excellence
* Perfect (grammar), a grammatical category in some languages
Perfect may also refer to:
Film
* Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama
* Perfect (2018 f ...
of Tor-amplitude
1,0
Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
If moreover
f is locally of finite type and
Y locally Noetherian, then the converse is also true.
These notions are used for instance in the
Grothendieck–Riemann–Roch theorem
In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
.
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
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
''E'' over a commutative ring ''A'', an ''A''-linear map
u: E \to A is called Koszul-regular if the
Koszul complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ...
determined by it is
acyclic in dimension > 0 (consequently, it is a resolution of the cokernel of ''u'').
Then a closed immersion
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.
It is this Koszul regularity that was used in SGA 6
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.
(This questions arise 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
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
Notes
References
*, section 16.9, p. 46
*
*
*{{Citation , last1=Fulton , first1=William , author1-link=William Fulton (mathematician) , title=Intersection theory , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics
esults 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
*E. Sernesi:
Deformations of algebraic schemes'
Theorems in algebraic geometry
Morphisms of schemes