Codimension

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In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties.

If W is a linear subspace of a vector space V then the codimension of W in V is the dimension of the quotient space V/W. If V is finite-dimensional then

The fundamental property of codimension lies in its relation to intersection: if W₁ has codimension k₁, and W₂ has codimension k₂, then if U is their intersection with codimension j we have

max (k₁, k₂) ≤ j ≤ k₁ + k₂.

In fact j may take any integer value in this range. This statement is more perspicuous than the translation in terms of dimensions, because the RHS is just the sum of the codimensions. In words

codimensions (at most) add. If the subspaces or submanifolds intersect transversally, codimensions add exactly.

In terms of the dual space, it is quite evident why that is. The subspaces being defined by the vanishing of a certain number of linear functionals, which we can take to be linearly independent, that number is the codimension. Therefore we see that U is defined by taking the union of the sets of linear functionals defining the W_i. That union may introduce some degree of linear dependence: the possible values of j express that dependence, with the RHS sum being the case where there is no dependence. This definition of codimension in terms of the number of functions needed to cut out a subspace extends to situations in which both the ambient space and subspace are infinite dimensional.

In other language, which is basic for any kind of intersection theory, we are taking the union of a certain number of constraints. We have two phenomena to look out for:

1. the two sets of constraints may not be independent;
2. the two sets of constraints may not be compatible.

The first of these is often expressed as the principle of counting constraints: if we have a number N of parameters to adjust (i.e. we have N degrees of freedom), and a constraint means we have to 'consume' a parameter to satisfy it, then the codimension of the solution set is at most the number of constraints. We do not expect to be able to find a solution if the predicted codimension, i.e. the number of independent constraints, exceeds N (in the linear algebra case, there is always a trivial, null vector solution, which is therefore discounted).

The second is a matter of geometry, on the model of parallel lines; it is something that can be discussed for linear problems by methods of linear algebra, and for non-linear problems in projective space, over the complex number field.

Codimension also has some clear meaning in geometric topology: on a manifold (real) codimension 1 is the dimension of topological disconnection by a submanifold, while codimension 2 is the dimension of ramification and knot theory.

See also: glossary of differential geometry and topology.