Codimension is a numerical measure comparing an ambient dimension to the dimension of a subobject; for a prime ideal p in a Noetherian ring R, coheight (codimension) is dim(R) - height(p), and for a subvariety it is the difference between ambient and subvariety dimensions.

Principle

Quantify how many independent conditions are imposed by passing from the ambient object to the subobject; codimension counts the drop in Krull dimension and is additive in expected exact geometric situations.

Demonstration

In R = k[x,y,z] (dim 3), the ideal of a plane defined by a single linear equation has codimension 1; the ideal of a line given by two independent linear equations has codimension 2, matching the dimension drop.

Misapplication

Assuming codimension equals the minimal number of generators of the defining ideal in all cases; in singular or noncomplete intersections the minimal generator count can exceed the codimension.

Consequence

Codimension predicts expected intersection dimensions and appears in dimension formulas, duality statements and counting parameters for families: a subobject of codimension c locally cuts down c dimensions from the ambient.

Reversal

The complementary viewpoint is height: instead of measuring how far an object is from the ambient, height measures how far it sits above the bottom of Spec(R); codimension can be seen as height measured from the top.

Boundary

Codimension requires a notion of ambient dimension; in rings without finite Krull dimension the naive difference may be undefined or infinite. For nonprime subschemes codimension can be scheme-theoretic and differ from naive topological codimension.

Semantic Tension

Codimension in algebraic geometry can conflict with linear-algebraic codimension (dimension of quotient vector space) when scheme structure or singularities are present; the algebraic codimension is Krull-theoretic rather than generator-count based.

Synthesis

Codimension measures the loss of Krull dimension when restricting to a subobject: a fundamental bookkeeping invariant for how many independent conditions or equations define that subobject relative to its ambient space.