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
p 0 ⊊ p 1 ⊊ … ⊊ p n {\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \ldots \subsetneq {\mathfrak {p}}_{n}}
has length n {\displaystyle n} . That is, the length is the number of strict inclusions, not the number of primes; these differ by 1 {\displaystyle 1} . We define the Krull dimension of R {\displaystyle R} to be the supremum of the lengths of all chains of prime ideals in R {\displaystyle R} . Given a prime ideal p {\displaystyle {\mathfrak {p}}} in R {\displaystyle R} , we define the height of p {\displaystyle {\mathfrak {p}}} , written ht ( p ) {\displaystyle \operatorname {ht} ({\mathfrak {p}})} , to be the supremum of the lengths of all chains of prime ideals contained in p {\displaystyle {\mathfrak {p}}} , meaning that p 0 ⊊ p 1 ⊊ … ⊊ p n = p {\displaystyle {\mathfrak {p}}_{0}\subsetneq {\mathfrak {p}}_{1}\subsetneq \ldots \subsetneq {\mathfrak {p}}_{n}={\mathfrak {p}}} . In other words, the height of p {\displaystyle {\mathfrak {p}}} is the Krull dimension of the localization of R {\displaystyle R} at p {\displaystyle {\mathfrak {p}}} . 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. A ring is called catenary if any inclusion p ⊂ q {\displaystyle {\mathfrak {p}}\subset {\mathfrak {q}}} of prime ideals can be extended to a maximal chain of prime ideals between p {\displaystyle {\mathfrak {p}}} and q {\displaystyle {\mathfrak {q}}} , and any two maximal chains between p {\displaystyle {\mathfrak {p}}}
and q {\displaystyle {\mathfrak {q}}} 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. In a Noetherian ring, a prime ideal has height at most n {\displaystyle n} if and only if it is a minimal prime ideal over an ideal generated by n {\displaystyle n} elements (Krull's height theorem and its converse). 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. More generally, the height of an ideal I {\displaystyle I} is the infimum of the heights of all prime ideals containing I {\displaystyle I} . In the language of algebraic geometry, this is the codimension of the subvariety of S p e c ( R ) {\displaystyle \mathrm {Spec} (R)} corresponding to I {\displaystyle I} .
Schemes
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