---
tag: C2R5Q2Q717
kind: definition
name: Dimension of an analytic set
msc: [32B10, 32C25]
defines: AN0BYRKNRA
uses: [VMDTGTMN9P, JPP8N6MZGP]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Definition

Let $X$ be a complex manifold and let $A$ be an [[analytic subset|VMDTGTMN9P]] of $X$.
Let $\operatorname{reg} A$ be the set of [[regular points|JPP8N6MZGP]] of $A$.
At a regular point $z$, the set $A$ is near $z$ a complex submanifold, and $\dim_z A$ is its complex dimension at $z$.
For an arbitrary point $a \in A$, the *dimension of $A$ at $a$* is
$$\dim_a A = \limsup_{z \to a,\ z \in \operatorname{reg} A} \dim_z A.$$
The *dimension of $A$* is $\dim A = \sup_{a \in A} \dim_a A$, with $\dim \emptyset = -\infty$.
The set $A$ has *pure dimension* $k$ if $\dim_a A = k$ for every $a \in A$.

## Remarks

The definition makes sense because $\operatorname{reg} A$ is dense in $A$; at a regular point $a$ it gives back the manifold dimension.
Example: let $A = \set{z_1 z_3 = 0,\ z_2 z_3 = 0}$ in $\C^3$, the union of the plane $P = \set{z_3 = 0}$ and the line $L = \set{z_1 = z_2 = 0}$.
The regular points are the points of $P \setminus \set{0}$, of dimension $2$, and the points of $L \setminus \set{0}$, of dimension $1$.
So $\dim_0 A = 2$, $\dim_a A = 1$ for $a \in L \setminus \set{0}$, and $\dim A = 2$.
Non-example of pure dimension: this set $A$ does not have pure dimension, since its local dimension takes the values $1$ and $2$.
It also shows that the dimension is not the number of variables minus the number of equations, which would be $1$ here.
There are equivalent forms of $\dim_a A$ when $X = \C^n$.
It is the Krull dimension of $\mathcal{O}_{n,a}/\mathcal{I}(A_a)$.
It is the largest of the integers $d$ given by the local parametrization theorem for the irreducible components of the germ $A_a$.
It is the smallest $k$ such that some complex affine subspace of codimension $k$ through $a$ meets $A$ in the isolated point $a$.