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Codimension
Codimension is a term used in a number of algebraic and geometric contexts to indicate the difference between the dimension of certain objects
and the dimension of a smaller object contained in
it. This rough definition applies to vector spaces
(the codimension of the subspace 
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]](/web/20140328164811im_/http://mathworld.wolfram.com/images/equations/Codimension/Inline10.gif)
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]](/web/20140328164811im_/http://mathworld.wolfram.com/images/equations/Codimension/NumberedEquation3.gif) |
(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.
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