---
tag: 64W1MJFRNT
kind: definition
name: Codimension of a singularity
msc: [58K40, 58K35, 58K60]
defines: B4FRQMWQKP
uses: [7SWSRGDT4A, 0YTCMFB4HC, 096Z9MP2ME]
sources:
  - url: https://en.wikipedia.org/wiki/Singularity_theory
---

## Definition

Let $\mathcal{E}_n$ be the ring of [[germs|7SWSRGDT4A]] at $0$ of smooth functions $\R^n \to \R$, and let $\mathfrak{m}$ be its maximal [[ideal|096Z9MP2ME]] of germs vanishing at $0$.
Let $f \in \mathcal{E}_n$ have a [[critical point|0YTCMFB4HC]] at $0$, and let $J(f) \subseteq \mathfrak{m}$ be the ideal generated by $\partial f / \partial x_1, \ldots, \partial f / \partial x_n$.
The *codimension* of $f$ is
$$\operatorname{codim}(f) = \dim_\R \mathfrak{m} / J(f),$$
which may be infinite.

## Remarks

Example: for $f = x^4$ on $\R$, $J(f) = (x^3)$ and $\mathfrak{m} / J(f)$ has basis the classes of $x$ and $x^2$, so $\operatorname{codim}(f) = 2$.
For a nondegenerate critical point $J(f) = \mathfrak{m}$, so the codimension is $0$.
Non-example: the definition needs a critical point at $0$; for $f = x$ one has $J(f) = \mathcal{E}_1 \not\subseteq \mathfrak{m}$, and a regular point is not given a codimension in this sense.
When it is finite, $\operatorname{codim}(f) = \dim_\R \mathcal{E}_n / J(f) - 1$.
For a holomorphic germ, the same quotient of the ring of convergent power series has dimension the Milnor number $\mu(f)$.
It equals the least number of parameters of a [[versal unfolding|KSAFWZVB8K]] of $f$, not counting an additive constant.
Conventions vary between authors; some count the additive constant as well and use $\dim_\R \mathcal{E}_n / J(f)$.