TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, codimension is a basic geometric idea that applies to subspaces in
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, to
submanifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s in
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s, and suitable
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s of
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
. For
affine Affine (pronounced /əˈfaɪn/) relates to connections or affinities. It may refer to: *Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology *Affine cipher, a special case of the more general substitution cipher *Aff ...
and projective algebraic varieties, the codimension equals the
height Height is measure of vertical distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can ...
of the defining
ideal Ideal may refer to: Philosophy * Ideal (ethics) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
. For this reason, the height of an ideal is often called its codimension. The dual concept is
relative dimension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.

# 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :$\operatorname\left(W\right) = \dim\left(V\right) - \dim\left(W\right).$ 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 An ambient space or ambient configuration space is the space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, al ...
''V:'' :$\dim\left(W\right) + \operatorname\left(W\right) = \dim\left(V\right).$ Similarly, if ''N'' is a submanifold or subvariety in ''M'', then the codimension of ''N'' in ''M'' is :$\operatorname\left(N\right) = \dim\left(M\right) - \dim\left(N\right).$ Just as the dimension of a submanifold is the dimension of the
tangent bundle Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differen ...

(the number of dimensions that you can move ''on'' the submanifold), the codimension is the dimension of the
normal bundle In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral ...
(the number of dimensions you can move ''off'' the submanifold). More generally, if ''W'' is a
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of a (possibly infinite dimensional)
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''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 The cokernel of a linear mapping In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...

of the inclusion. For finite-dimensional vector spaces, this agrees with the previous definition :$\operatorname\left(W\right) = \dim\left(V/W\right) = \dim \operatorname \left( W \to V \right) = \dim\left(V\right) - \dim\left(W\right),$ and is dual to the relative dimension as the dimension of the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
. Finite-codimensional subspaces of infinite-dimensional spaces are often useful in the study of
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an Abstra ...
s.

# Additivity of codimension and dimension counting

The fundamental property of codimension lies in its relation to
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
: 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 (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is ...
.

# Dual interpretation

In terms of the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, it is quite evident why dimensions add. The subspaces can be defined by the vanishing of a certain number of
linear functional In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s, which if we take to be
linearly independent In the theory of vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are con ...
, 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 In the theory of vector spaces, a set of vectors is said to be if at least one of the vectors in the set can be defined as a linear combinationIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...
: 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 In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is ...
, 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 A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified wh ...

s to adjust (i.e. we have ''N''
degrees of freedom Degrees of Freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical ...
), and a constraint means we have to 'consume' a parameter to satisfy it, then the codimension of the
solution set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
solution, which is therefore discounted). The second is a matter of geometry, on the model of
parallel lines In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...

; it is something that can be discussed for linear problems by methods of linear algebra, and for non-linear problems in
projective space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, over the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

field.

# In geometric topology

Codimension also has some clear meaning in
geometric topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
: 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 the mathematical field of topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...
. 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 In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
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.