Equivalence of dimension notions for Noetherian local ring

Statement

For a Noetherian local ring $(A,\mathfrak{m})$, the following notions of dimension are equivalent:

• The Krull dimension of the ring, i.e. the maximum possible length of a strictly descending chain of prime ideals
• The degree of the length polynomial for the Noetherian local ring (this is the variant of the Hilbert-Samuel polynomial that measures the length of the quotient modules $A/{\mathfrak{m}}^d$
• The minimum possible length of a system of parameters for $\mathfrak{m}$

Proof

Proof outline

The proof rests on some basic observations:

• If the ring has Krull dimension at least one, i.e. is not a local Artinian ring, then we can find an element in the maximal ideal that is not in any minimal prime. This uses basic facts about Noetherian rings, and the prime avoidance lemma For full proof, refer: Noetherian local ring of positive dimension has element in maximal ideal outside minimal primes
• Every time we quotient out by such an element, the degree of the length polynomial goes down (because the leading terms of the polynomials cancel).