7.1 Krull dimension and geometric dimension

Krull dimension and geometric dimension are key concepts in algebraic geometry. They measure the complexity of algebraic structures and spaces. Understanding these ideas helps us grasp the fundamental properties of rings, varieties, and their relationships.

These concepts connect abstract algebra with geometry. By studying prime ideals and chains, we can determine the dimension of algebraic objects. This knowledge is crucial for analyzing the structure and behavior of algebraic varieties and their coordinate rings.

Krull dimension for rings

Definition and properties

• The Krull dimension of a ring $R$ is the supremum of the lengths of all chains of prime ideals in $R$, where the length of a chain is the number of strict inclusions
  • For example, if $R = k[x, y]$ and $P_0 \subsetneq P_1 \subsetneq P_2$ is a chain of prime ideals, then the length of this chain is 2
• For an integral domain $R$, the Krull dimension is equal to the transcendence degree of its field of fractions over its prime subfield
  • For instance, the field of fractions of $\mathbb{Q}[x, y]$ is $\mathbb{Q}(x, y)$, which has transcendence degree 2 over $\mathbb{Q}$, so the Krull dimension of $\mathbb{Q}[x, y]$ is 2
• The Krull dimension of a noetherian ring is finite, and it coincides with the maximum length of a chain of prime ideals in the ring
  • A ring is noetherian if it satisfies the ascending chain condition on ideals, meaning that any ascending chain of ideals stabilizes (becomes constant) after finitely many steps

Relationship to algebraic varieties

• The Krull dimension of a finitely generated algebra over a field $k$ is equal to the dimension of the corresponding algebraic variety defined over $k$
  • For example, the polynomial ring $k[x, y, z]/(x^2 + y^2 - z^2)$ corresponds to the affine variety $V(x^2 + y^2 - z^2)$ in $\mathbb{A}^3$, which is a quadric surface of dimension 2
• For an affine algebraic variety $V$, the Krull dimension of its coordinate ring is equal to the dimension of $V$ as a topological space
  • The coordinate ring of an affine variety $V \subseteq \mathbb{A}^n$ is the quotient ring $k[x_1, \ldots, x_n]/I(V)$, where $I(V)$ is the ideal of polynomials vanishing on $V$

Computing Krull dimension

Common rings

• The Krull dimension of a field is 0, as the only prime ideal is the zero ideal
  • For instance, $\dim_{\text{Krull}}(\mathbb{Q}) = 0$ and $\dim_{\text{Krull}}(\mathbb{C}) = 0$
• The Krull dimension of the polynomial ring $k[x_1, \ldots, x_n]$ over a field $k$ is $n$, as the longest chain of prime ideals corresponds to the chain of irreducible subvarieties
  • For example, $\dim_{\text{Krull}}(\mathbb{R}[x, y, z]) = 3$
• The Krull dimension of the ring of integers $\mathbb{Z}$ is 1, as the prime ideals are $(0)$ and $(p)$ for each prime number $p$
  • The chain $(0) \subsetneq (p)$ has length 1, and there are no longer chains of prime ideals in $\mathbb{Z}$
• The Krull dimension of a product of rings $R_1 \times \cdots \times R_n$ is the maximum of the Krull dimensions of the individual rings $R_i$
  • For instance, $\dim_{\text{Krull}}(\mathbb{Z} \times \mathbb{Q}[x]) = \max\{1, 1\} = 1$

Affine varieties

• For an affine variety $V$ defined by a prime ideal $I$ in $k[x_1, \ldots, x_n]$, the Krull dimension of $V$ is equal to $n - \operatorname{ht}(I)$, where $\operatorname{ht}(I)$ is the height of the ideal $I$
  • The height of a prime ideal $I$ is the supremum of the lengths of chains of prime ideals contained in $I$
  • For example, if $V = V(xy - 1) \subseteq \mathbb{A}^2$, then $I(V) = (xy - 1)$ has height 1, so $\dim_{\text{Krull}}(V) = 2 - 1 = 1$

Krull dimension properties

Localization and quotients

• For a ring $R$ and a multiplicative subset $S$, the Krull dimension of the localization $S^{-1}R$ is equal to the Krull dimension of $R$
  • This follows from the correspondence between prime ideals in $R$ and prime ideals in $S^{-1}R$ that do not intersect $S$
  • For example, $\dim_{\text{Krull}}(\mathbb{Z}_{(p)}) = \dim_{\text{Krull}}(\mathbb{Z}) = 1$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at the prime ideal $(p)$
• For a ring $R$ and an ideal $I$, the Krull dimension of the quotient ring $R/I$ is less than or equal to the Krull dimension of $R$
  • This inequality becomes an equality if $I$ is a minimal prime ideal
  • For instance, $\dim_{\text{Krull}}(\mathbb{Z}/6\mathbb{Z}) = 0 \leq \dim_{\text{Krull}}(\mathbb{Z}) = 1$

Noetherian integral domains

• If $R$ is a finitely generated algebra over a field $k$ and $P$ is a prime ideal of $R$, then $\dim(R/P) + \operatorname{ht}(P) = \dim(R)$, where $\dim$ denotes the Krull dimension
  • This property relates the dimension of a quotient ring to the height of the corresponding prime ideal
• For a noetherian integral domain $R$, the Krull dimension of $R$ is equal to the supremum of the heights of maximal ideals in $R$
  • A maximal ideal is a proper ideal that is not contained in any other proper ideal

Krull dimension vs transcendence degree

Finitely generated field extensions

• For a finitely generated field extension $K/k$, the transcendence degree of $K$ over $k$ is equal to the Krull dimension of any finitely generated $k$-subalgebra $R$ of $K$ such that the field of fractions of $R$ is $K$
  • The transcendence degree of a field extension $K/k$ is the maximum number of algebraically independent elements in $K$ over $k$
  • For example, if $K = k(x, y)$ and $R = k[x, y]$, then $\dim_{\text{Krull}}(R) = \operatorname{tr.deg}_k(K) = 2$

Integral domains and function fields

• If $R$ is an integral domain with field of fractions $K$, then the Krull dimension of $R$ is equal to the transcendence degree of $K$ over the prime subfield of $R$
  • The prime subfield of a ring is the smallest subfield contained in the ring (e.g., $\mathbb{Q}$ for $\mathbb{Z}$ or $\mathbb{Q}[x]$)
• For an affine variety $V$ over a field $k$, the dimension of $V$ is equal to the transcendence degree of the function field $k(V)$ over $k$
  • The function field $k(V)$ is the field of rational functions on $V$, which consists of quotients of polynomials in the coordinate ring of $V$

Polynomial rings and rational function fields

• The Krull dimension of a finitely generated algebra $R$ over a field $k$ is equal to the transcendence degree of the field of fractions of $R$ over $k$
  • For instance, if $R = k[x, y, z]/(x^2 + y^2 - z^2)$, then its field of fractions has transcendence degree 2 over $k$, so $\dim_{\text{Krull}}(R) = 2$
• The Krull dimension of a polynomial ring $k[x_1, \ldots, x_n]$ is equal to the transcendence degree of the field of rational functions $k(x_1, \ldots, x_n)$ over $k$
  • The field of rational functions $k(x_1, \ldots, x_n)$ consists of quotients of polynomials in $k[x_1, \ldots, x_n]$