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 maximum possible length of a regular sequence inside $\mathfrak{m}$

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 a nonzerodivisor in the maximal ideal. This uses basic facts about Noetherian rings, and the prime avoidance lemma For full proof, refer: Noetherian local ring of positive dimension has nonzerodivisor in maximal ideal and outside minimal primes
• Every time we quotient out by a nonzerodivisor, the degree of the length polynomial goes down (because the leading terms of the polynomials cancel).