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 (4,-1,10) in R^3 is 3-1=2) and to topological spaces (with respect to the Euclidean topology and the Zariski topology, the codimension of a sphere in R^3 is 3-2=1).
The first example is a particular case of the formula
| codimW=dimV-dimW, |
(1)
|
which gives the codimension of a subspace W of a finite-dimensional abstract vector space V. 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
| (x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2=0, |
(2)
|
where the point C(x_0,y_0,z_0) is the center and r is the radius. The Krull dimension of the polynomial ring R[x,y,z] 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] |
(3)
|
is 2, and the difference 3-2=1 is also called the codimension of the ideal
| <(x-x_0)^2+(y-y_0)^2+(z-z_0)^2-r^2>. |
(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 I in a polynomial ring over a field, codimI=heightI. 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.
See also
Bifurcation, Coheight, Dimension, Krull DimensionPortions of this entry contributed by Todd Rowland
Portions of this entry contributed by Margherita Barile
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Cite this as:
Barile, Margherita; Rowland, Todd; and Weisstein, Eric W. "Codimension." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Codimension.html