Remark 3

By Proposition 7, b) and Proposition 4 of No. 2, the ring A is catenary if and only if, for every triple (p, q, r) of prime ideals of A such that $r \subset q \subset p$, one has dim(A_p/qA_p) + dim(A_q/rA_q) = dim(A_p/rA_p). If A is an integral domain and of finite dimension, then A is catenary if and only if one has ht(q) + dim(A_p/qA_p) = ht(p) for every pair (p, q) of prime ideals of A such that q ⊂ p. In fact, the topological space Spec(A) is then irreducible and of finite dimension, and one applies the Corollary to Proposition 4 of No. 2.