In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.

The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I.

A field k has Krull dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.

There are several other ways that have been used to define the dimension of a ring. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings.

Explanation

We say that a chain of prime ideals of the form has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of to be the supremum of the lengths of all chains of prime ideals in .

Given a prime ideal in R, we define the height of , written , to be the supremum of the lengths of all chains of prime ideals contained in , meaning that .[1] In other words, the height of is the Krull dimension of the localization of R at . A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal.

In a Noetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension.[2] A ring is called catenary if any inclusion of prime ideals can be extended to a maximal chain of prime ideals between and , and any two maximal chains between and have the same length. A ring is called universally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.[3]

In a Noetherian ring, a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by n elements (Krull's height theorem and its converse).[4] It implies that the descending chain condition holds for prime ideals in such a way the lengths of the chains descending from a prime ideal are bounded by the number of generators of the prime.[5]

More generally, the height of an ideal I is the infimum of the heights of all prime ideals containing I. In the language of algebraic geometry, this is the codimension of the subvariety of Spec() corresponding to I.[6]

Schemes

It follows readily from the definition of the spectrum of a ring Spec(R), the space of prime ideals of R equipped with the Zariski topology, that the Krull dimension of R is equal to the dimension of its spectrum as a topological space, meaning the supremum of the lengths of all chains of irreducible closed subsets. This follows immediately from the Galois connection between ideals of R and closed subsets of Spec(R) and the observation that, by the definition of Spec(R), each prime ideal of R corresponds to a generic point of the closed subset associated to by the Galois connection.

Examples

Of a module

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module. That is, we define it by the formula:

where AnnR(M), the annihilator, is the kernel of the natural map R → EndR(M) of R into the ring of R-linear endomorphisms of M.

In the language of schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.

For non-commutative rings

The Krull dimension of a module over a possibly non-commutative ring is defined as the deviation of the poset of submodules ordered by inclusion. For commutative Noetherian rings, this is the same as the definition using chains of prime ideals.[10] The two definitions can be different for commutative rings which are not Noetherian.

See also

Notes

  1. ^ Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989
  2. ^ Eisenbud, D. Commutative Algebra (1995). Springer, Berlin. Exercise 9.6.
  3. ^ Matsumura, H. Commutative Algebra (1970). Benjamin, New York. Example 14.E.
  4. ^ Serre 2000, Ch. III, § B.2, Theorem 1, Corollary 4.
  5. ^ Eisenbud 1995, Corollary 10.3.
  6. ^ Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989
  7. ^ Krull dimension less or equal than transcendence degree?
  8. ^ Eisenbud 1995, Exercise 13.8
  9. ^ Hartshorne,Robin:"Algebraic Geometry", page 7,1977
  10. ^ McConnell, J.C. and Robson, J.C. Noncommutative Noetherian Rings (2001). Amer. Math. Soc., Providence. Corollary 6.4.8.

Bibliography