In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, codimension is a basic geometric idea that applies to
subspaces in
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, to
submanifolds in
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, and suitable
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of
algebraic varieties.
For
affine and
projective algebraic varieties, the codimension equals the
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is ab ...
of the defining
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
. For this reason, the height of an ideal is often called its codimension.
The dual concept is
relative dimension.
Definition
Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space.
If ''W'' is a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions:
:
It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the
ambient space ''V:''
:
Similarly, if ''N'' is a submanifold or subvariety in ''M'', then the codimension of ''N'' in ''M'' is
:
Just as the dimension of a submanifold is the dimension of the
tangent bundle (the number of dimensions that you can move ''on'' the submanifold), the codimension is the dimension of the
normal bundle (the number of dimensions you can move ''off'' the submanifold).
More generally, if ''W'' is a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of a (possibly infinite dimensional)
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' then the codimension of ''W'' in ''V'' is the dimension (possibly infinite) of the
quotient space ''V''/''W'', which is more abstractly known as the
cokernel of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition
:
and is dual to the relative dimension as the dimension of the
kernel.
Finite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of
topological vector spaces.
Additivity of codimension and dimension counting
The fundamental property of codimension lies in its relation to
intersection: if ''W''
1 has codimension ''k''
1, and ''W''
2 has codimension ''k''
2, then if ''U'' is their intersection with codimension ''j'' we have
:max (''k''
1, ''k''
2) ≤ ''j'' ≤ ''k''
1 + ''k''
2.
In fact ''j'' may take any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
value in this range. This statement is more perspicuous than the translation in terms of dimensions, because the
RHS is just the sum of the codimensions. In words
:''codimensions (at most) add''.
:If the subspaces or submanifolds intersect
transversally (which occurs
generically), codimensions add exactly.
This statement is called dimension counting, particularly in
intersection theory.
Dual interpretation
In terms of the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
, it is quite evident why dimensions add. The subspaces can be defined by the vanishing of a certain number of
linear functionals, which if we take to be
linearly independent, their number is the codimension. Therefore, we see that ''U'' is defined by taking the
union of the sets of linear functionals defining the ''W''
i. That union may introduce some degree of
linear dependence: the possible values of ''j'' express that dependence, with the RHS sum being the case where there is no dependence. This definition of codimension in terms of the number of functions needed to cut out a subspace extends to situations in which both the ambient space and subspace are infinite dimensional.
In other language, which is basic for any kind of
intersection theory, we are taking the union of a certain number of
constraints. We have two phenomena to look out for:
# the two sets of constraints may not be independent;
# the two sets of constraints may not be compatible.
The first of these is often expressed as the principle of counting
constraints: if we have a number ''N'' of
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s to adjust (i.e. we have ''N''
degrees of freedom), and a constraint means we have to 'consume' a parameter to satisfy it, then the codimension of the
solution set
In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.
For example, for a set of polynomials over a ring ,
the solution set is the subset of on which the polynomials all vanish (evaluate t ...
is ''at most'' the number of constraints. We do not expect to be able to find a solution if the predicted codimension, i.e. the number of ''independent'' constraints, exceeds ''N'' (in the linear algebra case, there is always a ''trivial'',
null vector solution, which is therefore discounted).
The second is a matter of geometry, on the model of
parallel lines; it is something that can be discussed for
linear problems by methods of linear algebra, and for non-linear problems in
projective space, over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
field.
In geometric topology
Codimension also has some clear meaning in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originate ...
: on a manifold, codimension 1 is the dimension of topological disconnection by a submanifold, while codimension 2 is the dimension of
ramification and
knot theory. In fact, the theory of high-dimensional manifolds, which starts in dimension 5 and above, can alternatively be said to start in codimension 3, because higher codimensions avoid the phenomenon of knots. Since
surgery theory requires working up to the middle dimension, once one is in dimension 5, the middle dimension has codimension greater than 2, and hence one avoids knots.
This quip is not vacuous: the study of embeddings in codimension 2 is knot theory, and difficult, while the study of embeddings in codimension 3 or more is amenable to the tools of high-dimensional geometric topology, and hence considerably easier.
See also
*
Glossary of differential geometry and topology
References
*{{Springer, id=C/c022870, title=Codimension
Algebraic geometry
Geometric topology
Linear algebra
Dimension