Proposition 7

a) The height of a prime ideal p of A is the least upper bound of the lengths of chains of prime ideals $p_0 \subset ... \subset p_n$ such that $p_n = p$.

b) Let p be a prime ideal of A and a an ideal of A. Then one has $\dim(A_p/aA_p) = -\infty$ if a is not contained in p, and $\dim(A_p/aA_p) = \operatorname{codim}(V(p), V(a))$ if a is contained in p. In particular, if p is a prime ideal of A, one has $\dim(A_p) = \operatorname{ht}(p)$.

c) If a is an ideal of A, one has $\operatorname{ht}(a) = \inf_{p} \operatorname{ht}(p) = \inf_{p} \dim(A_p)$ where p runs through the set of prime ideals of A containing a.

Assertion a) is the translation of Definition 3, a) of No. 2. Assertion b) follows from the fact that the mapping $q \mapsto q(A_p/aA_p)$ is an increasing isomorphism of the set of prime ideals q of A such that $a \subset q \subset p$ onto the set of prime ideals of the local ring $A_p/aA_p$ (II, § 2, No. 5, Proposition 11). Let a be an ideal of A; the irreducible closed subsets of $V(a)$ are the sets $V(p)$, where p is a prime ideal of A containing a. Assertion c) therefore follows from Remark 1 of No. 2.