Krull dimension

Krull dimension
Name	Krull dimension
Field	Commutative algebra
Introduced	1928
Introduced by	Wolfgang Krull
Related	Noetherian ring, Spec, prime ideal, Zariski topology

Krull dimension Krull dimension is an invariant of commutative rings measuring the maximal length of chains of prime ideals; it serves as a primary notion of "dimension" in commutative algebra and algebraic geometry. Developed in the early 20th century by Wolfgang Krull, it connects to the structure theory of David Hilbert's ideals, the Hilbert's nullstellensatz, and to notions used by Oscar Zariski, Pierre Samuel, and Jean-Pierre Serre. Krull dimension appears in classical results of Emmy Noether, Irving Kaplansky, and in modern treatments by Matsumura and Atiyah and Macdonald.

Definition

For a commutative ring R with unity, the Krull dimension is defined as the supremum of the lengths n of chains P_0 ⊂ P_1 ⊂ ... ⊂ P_n of distinct prime ideals of R. This notion was formalized by Wolfgang Krull and is central to many theorems by Oscar Zariski, Shreeram Abhyankar, and Alexander Grothendieck. In geometric language via the spectrum functor Spec, the Krull dimension of R equals the topological dimension of the Zariski topology - the maximum number of strict inclusions among irreducible closed subsets studied by André Weil and David Mumford. The definition extends to schemes in the work of Alexander Grothendieck and plays a role in the coherence results of Jean-Pierre Serre.

Examples and basic properties

Principal examples include fields, Dedekind domains, polynomial rings, and coordinate rings. A field has Krull dimension 0, a principal ideal domain like Ernst Kummer-inspired examples with unique factorization have dimension 1 if non-field, and a polynomial ring k[x_1,...,x_n] over a field k has dimension n by results of Krull and Hilbert. For a Noetherian local ring, the Krull dimension equals the minimal number of generators of an ideal needed in chains studied by Auslander and Buchsbaum. The dimension of a Dedekind domain such as the ring of integers of a number field (studied by Richard Dedekind and Hilbert) is 1. Dimension behaves monotonically with respect to inclusion of rings in many classical contexts studied by Emmy Noether and counterexamples were investigated by Nagata.

Key properties: the Krull dimension of a finite product R × S is the maximum of the dimensions of R and S, a fact used in work by Claude Chevalley and Jean-Pierre Serre. Localization at a prime ideal p yields a local ring whose Krull dimension equals the height of p, a concept related to the pioneering studies of Krull and refined by Matsumura. The dimension of an integral extension preserves equality under hypotheses explored by Noether and Krull.

Computation techniques and invariants

Computing Krull dimension uses heights, depths, transcendence degree, and chain conditions. The height (or codimension) of a prime ideal is the supremum of lengths of chains ending at that prime; this is tied to dimension theory developed by Krull and Zariski. For finitely generated k-algebras, the Krull dimension equals the transcendence degree of their field of fractions over k, a principle applied in the work of Hilbert and Chevalley. The Auslander–Buchsbaum formula and the New Intersection Theorem by Auslander and Buchsbaum link projective dimension and depth to dimension. For Noetherian rings, Krull dimension can be bounded via Hilbert polynomials and multiplicity theory from David Hilbert and Serre; Castelnuovo–Mumford regularity and homological invariants studied by Eisenbud provide computational handles. Computational algebra systems implement algorithms relying on primary decomposition from the work of Krull, Noether, and Macaulay.

Behavior under ring constructions

Krull dimension interacts predictably under localization, completion, quotienting, polynomial extension, and integral extension. Localization at a multiplicative set often reduces dimension to the supremum of heights of primes disjoint from the set, a technique used in Matsumura and Atiyah and Macdonald. Completion with respect to an ideal, as in Grothendieck's formal schemes, preserves dimension under excellent and Noetherian hypotheses examined by Nagata and Matsumura. For polynomial rings, dim R[x] = dim R + 1 for Noetherian rings by results tracing to Krull and Hermann. Integral extensions preserve equality of dimensions under lying-prime correspondence, a theme in Dedekind and Krull theory. Fiber product and pushout constructions in scheme theory, elaborated by Grothendieck in the EGA series, show subtler behavior connecting dimension to flatness and fibers studied by Serre and Zariski.

Relation to dimension theory in algebraic geometry

In algebraic geometry, Krull dimension of coordinate rings equals the geometric dimension of corresponding affine varieties as in Hilbert's nullstellensatz and the foundational work of Zariski and Weil. For schemes, the dimension of a scheme equals the supremum of dimensions of local rings at its points, a notion formalized by Grothendieck in EGA and used extensively by Hartshorne. The connection to transcendence degree over base fields is central in birational geometry studied by Zariski, Weil, and Mumford. The dimension theory interacts with cohomological dimension in theorems by Serre and with intersection theory developed by Fulton.

Applications and significance in commutative algebra

Krull dimension underpins structural classification, homological conjectures, and dimension-sensitive invariants. It appears in theorems on primary decomposition, chain conditions by Noether, and in the formulation of depth and Cohen–Macaulayness investigated by Serre, Grothendieck, and Rees. Invariant theory, singularity theory, and resolution of singularities studied by Hironaka rely on dimension arguments. Krull dimension informs arithmetic geometry via the dimension of rings of integers in number fields studied by Dedekind and Weil, and guides algorithmic strategies in computational algebra by Buchberger and Macaulay. It remains a central tool in modern research across algebraic geometry, number theory, and commutative algebra influenced by Grothendieck, Serre, and Eisenbud.

Category:Commutative algebra