9.2 Krull dimension and its properties

Krull dimension measures a ring's complexity by finding the longest chain of prime ideals. It's a key concept in commutative algebra, helping us understand ring structure and properties.

Krull dimension has important applications in algebraic geometry and number theory. It's used to analyze polynomial rings, local rings, and quotient rings, providing insights into their algebraic and geometric properties.

Fundamentals of Krull Dimension

Definition of Krull dimension

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• Krull dimension measures the "complexity" of a ring by determining the maximum length of chains of prime ideals
• Chain of prime ideals forms a sequence of strictly increasing prime ideals $P_0 \subset P_1 \subset \cdots \subset P_n$
• Notation $\dim(R)$ denotes the Krull dimension for a ring $R$
• Possible values include non-negative integers or infinity for certain rings (polynomial rings with infinitely many variables)

Properties of Krull dimension

• Localization invariance ensures $\dim(R) \geq \dim(S^{-1}R)$ for a multiplicative set $S$ in $R$, with equality when $S$ avoids minimal prime ideals
• Quotient operation yields $\dim(R/I) \leq \dim(R)$ for an ideal $I$ in $R$, with equality possible for certain ideals (prime ideals)
• Dimension of integral domains equals the transcendence degree of its field of fractions
• Noetherian rings possess finite Krull dimension due to ascending chain condition
• Artinian rings have Krull dimension zero as they contain only maximal ideals

Applications and Advanced Concepts

Computation of Krull dimension

• Polynomial rings over a field $k$ have $\dim(k[x_1, \ldots, x_n]) = n$, proved using induction and transcendence degree
• Local rings maintain dimension, $\dim(R_m) = \dim(R)$ for a maximal ideal $m$ (localization at maximal ideal)
• Product of rings follows $\dim(R \times S) = \max(\dim(R), \dim(S))$
• Specific examples include $\dim(\mathbb{Z}) = 1$ (prime ideals $(0) \subset (p)$) and $\dim(k[x,y]/(xy-1)) = 1$ (hyperbola)

Krull dimension vs prime ideals

• Height of a prime ideal $P$ measures the maximum length of chain of prime ideals contained in $P$, satisfying $\text{ht}(P) \leq \dim(R)$
• Depth of a prime ideal $P$ determines the minimum length of maximal chain of prime ideals containing $P$
• Relationship between height and depth yields $\dim(R) = \max\{\text{ht}(P) + \text{depth}(P) : P \text{ is prime}\}$
• Codimension defined as $\text{codim}(P) = \dim(R) - \dim(R/P)$, equals height for Noetherian rings

Krull's Principal Ideal Theorem

• Statement asserts in a Noetherian ring $R$, any minimal prime ideal over a principal ideal has height at most 1
• Proof outline involves Noetherian induction, constructing a chain of ideals, and applying Nakayama's lemma
• Consequences include determining dimension of hypersurfaces and bounding the number of generators for prime ideals
• Generalization extends to $n$ elements, where minimal primes have height at most $n$
• Applications encompass dimension theory of affine varieties and characterization of Cohen-Macaulay rings