What is codim?

ension in algebraic or geometric sense.

The codimension of a geometric object, such as a submanifold or subspace, is a measure of how far away it is from being the entire space. It is defined as the difference between the dimension of the space and the dimension of the object. For example, a line in three-dimensional space has codimension 1, a point has codimension 2, and the entire space has codimension 0.

In algebraic geometry, the codimension of an ideal in a polynomial ring is similar to the geometric notion, but is defined in terms of the degrees of the polynomials involved. Specifically, the codimension of an ideal is the minimum number of variables needed to generate it plus one. For example, the ideal generated by the polynomials x^2-y^2 and xz-y has codimension 2 in the polynomial ring k[x,y,z]. The codimension is an important invariant of the ideal, and is related to its complexity and its ability to define a geometric object.