Proof that the polynomial ring over a field is catenary. Let $K$ be a field, $A=K[X_1,\dots,X_n]$. The following well known theorem shows $A$ is catenary; > _Every Cohen-Macaulay ring is catenary.

Proof that the polynomial ring over a field is catenary. Let $K$ be a field, $A=K[X_1,\dots,X_n]$. The following well known theorem shows $A$ is catenary; > _Every Cohen-Macaulay ring is catenary._ However, I want to know if we could prove that $A$ is catenary without theory of Cohen-Macaulay rings.

There is a proof in Chapter 13 of Eisenbud's Commutative Algebra with a view toward algebraic geometry. The first part is a slight strengthening of the Noether Normalisation Lemma (Eisenbud's Theorem 13.3), but then to show that all maximal chains of primes have the same length, one also needs the Going-Down Theorem (Eisenbud's Theorem 13.9).