Dimension Theory - Core Definitions

Dimension theory provides an algebraic measure of the "size" or "complexity" of rings and modules, fundamental to understanding geometric properties.

DefinitionKrull Dimension

The Krull dimension of a ring $R$ is the supremum of lengths of chains of prime ideals: $\dim(R) = \sup\{n : \mathfrak{p}_0 \subsetneq \mathfrak{p}_1 \subsetneq \cdots \subsetneq \mathfrak{p}_n\}$

For a prime ideal $\mathfrak{p}$ , the height $\text{ht}(\mathfrak{p})$ is $\dim(R_\mathfrak{p})$ , the length of the longest chain descending from $\mathfrak{p}$ to $(0)$ .

ExampleComputing Dimensions

$\dim(k) = 0$ for fields (only prime is $(0)$ )
$\dim(\mathbb{Z}) = 1$ (chains: $(0) \subsetneq (p)$ )
$\dim(k[x_1, \ldots, x_n]) = n$ (maximal chains have length $n$ )
$\dim(k[x,y]/(xy)) = 1$ (coordinate axes meet, dimension drops)
$\dim(k[[x,y]]) = 2$ (power series same dimension as polynomial)

DefinitionHeight of Ideal

The height of an ideal $I$ is: $\text{ht}(I) = \inf\{\text{ht}(\mathfrak{p}) : \mathfrak{p} \text{ minimal prime over } I\}$

This measures the "codimension" of the subvariety defined by $I$ .

ExampleHeights in Polynomial Rings

In $k[x,y,z]$ :

$\text{ht}((x)) = 1$ (one equation cuts out codimension 1)
$\text{ht}((x,y)) = 2$ (two equations, codimension 2)
$\text{ht}((x,y,z)) = 3$ (maximal ideal, codimension equals dimension)
$\text{ht}((xy)) = 1$ (principal ideal, height 1 by Krull's theorem)

DefinitionDimension of Module

For an $R$ -module $M$ , the dimension is: $\dim(M) = \dim(R/\text{ann}(M))$

where $\text{ann}(M) = \{r \in R : rM = 0\}$ is the annihilator. Equivalently, $\dim(M)$ is the dimension of the support $\text{Supp}(M)$ .

Remark

The dimension of an affine variety $V \subseteq \mathbb{A}^n$ over an algebraically closed field equals the Krull dimension of its coordinate ring $k[V]$ . This connects algebraic and geometric notions of dimension.

DefinitionTranscendence Degree

For a field extension $K/k$ , the transcendence degree $\text{trdeg}_k(K)$ is the maximum number of algebraically independent elements.

For affine domains, $\dim(R) = \text{trdeg}_k(\text{Frac}(R))$ when $R$ is finitely generated over a field $k$ .

ExampleFunction Fields

$\text{trdeg}_k(k(x,y)) = 2$ , matching $\dim(k[x,y]) = 2$
For an irreducible curve $C$ , $\text{trdeg}_k(k(C)) = 1$
For an irreducible surface $S$ , $\text{trdeg}_k(k(S)) = 2$

Transcendence degree captures the "geometric dimension" of function fields.

DefinitionCodimension

The codimension of a prime $\mathfrak{p}$ in $R$ is: $\text{codim}(\mathfrak{p}) = \dim(R) - \dim(R/\mathfrak{p}) = \text{ht}(\mathfrak{p})$

This measures how much dimension drops when restricting to the subvariety defined by $\mathfrak{p}$ .

Remark

In geometric terms, codimension-1 subvarieties are hypersurfaces (defined by one equation), codimension-2 are complete intersections of two equations (in good cases), and maximal ideals have codimension equal to the ambient dimension (points have codimension $n$ in $\mathbb{A}^n$ ).

Dimension theory bridges algebra and geometry, providing quantitative measures of complexity and relating ring-theoretic invariants to geometric intuition.