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  is 3, the Krull dimension of the quotient ring

(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.