Codimension is a term used in a number of algebraic and geometric contexts to indicate the difference between the dimension of a certain object
and the dimension of a smaller object contained in
it. This rough definition applies to vector spaces
(the codimension of the subspace spanned
by the vector 
in 
is 
) and to topological spaces
(with respect to the Euclidean topology and
the Zariski topology, the codimension of a sphere in 
is 
).
The first example is a particular case of the formula
 |
(1)
|
which gives the codimension of a subspace 
of a finite-dimensional abstract
vector space 
.
The second example has an algebraic counterpart in ring
theory. A sphere in the three-dimensional
real Euclidean space is defined by the following
equation in Cartesian
coordinates
 |
(2)
|
where the point 
is the center and
is the radius.
The Krull dimension of the polynomial
ring ![R[x,y,z]](/images/equations/Codimension/Inline10.svg)
is 3, the Krull dimension of the quotient
ring
![R[x,y,z]/[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2]](/images/equations/Codimension/NumberedEquation3.svg) |
(3)
|
is 2, and the difference 
is also called the codimension of the ideal
 |
(4)
|
According to Krull's principal ideal theorem, its height is also equal
to 1. On the other hand, it can be shown that for every proper
ideal 
in a polynomial ring over a field,
. This is a consequence
of the fact that these rings are all Cohen-Macaulay
rings. In a ring not fulfilling this assumption, only
the inequality 
is true in general.
