Krull dimension and height

Define dim R as the supremum of lengths of strict chains p₀⊊⋯⊊p_r in Spec R. Define ht p by chains ending at p; it equals dim R_p because localization retains precisely the primes contained in p. The quotient R/p instead retains primes containing p. Thus localization and quotient inspect opposite portions of the inclusion structure. A field has dimension zero, while k[x] has the strict chain (0)⊊(x) and no longer chain, giving dimension one.

For a finite module M, its dimension is the dimension of Supp M, equivalently dim R/Ann M. Nilpotent thickening does not change the prime spectrum or its chain lengths. Passing from k[x] to k[x,ε]/(ε²) retains dimension one even though the ring acquires nilpotents and a larger module structure. Counting generators of an algebra gives only a possible presentation and bound, not necessarily the dimension: relations can reduce it, while redundant generators can enlarge a presentation without changing the ring.

Krull’s principal ideal theorem says that in a Noetherian ring, each prime minimal over a proper principal ideal has height at most one. The generalized height theorem replaces one generator by r generators and gives height at most r for primes minimal over that ideal. Minimality is crucial: a maximal ideal containing a principal ideal can have much larger height. These are theorem statements; their proofs require dimension-theoretic arguments beyond the elementary chain calculations here. They do not say every ideal generated by r elements has every overlying prime of height at most r.

A Noetherian local ring has finite dimension because its maximal ideal is finitely generated and is the unique prime minimal over itself. A general Noetherian ring need not have a uniform finite Krull dimension for the whole ring; finite heights at individual primes need not have a common bound. Finite-type algebras over a field do have finite dimension. For a finitely generated k-domain, Noether normalization and integral dimension preservation give dimension equal to the transcendence degree of its fraction field. These additional hypotheses are what justify the familiar finite-coordinate picture.

For a Noetherian ring, dimension is the maximum of the dimensions of its quotients by minimal primes. This lets a reducible algebra be analyzed component by component. In k[x,y,z]/(xy), the components x=0 and y=0 are both polynomial planes, so the ring has dimension two despite being presented with three generators. Embedded associated primes carry extra module information but do not create additional irreducible components or increase the dimension beyond this minimal-component calculation.

One often expects ht p+dim R/p=dim R, and it is valid in many well-behaved polynomial and finite-type settings under appropriate conditions. It is not a universal identity for arbitrary Noetherian rings, especially when chain lengths behave unevenly. Catenarity and dimension formulas require their own hypotheses. A safe computation proves a lower bound with an explicit chain and an upper bound with a justified theorem, rather than subtracting the number of written equations from the number of written variables without checking independence or minimality.