Exercise 3, § 3

a) Let $A$ be a local noetherian ring, $p$ a prime ideal of $A$. Prove the inequality $\dim_p(A) \geq \operatorname{ht}(p) + \dim(A/p) - 1$. b) Prove that there exists a local noetherian ring $R$ of dimension 3 having a prime ideal $q$ such that $\operatorname{ht}(q) = 1$, $\dim(R/q) = 1$. (Use Exercise 16, p. 83, by taking $R = A[U]_{q_1}$ and $q = pR$.) Let $f \in m_R - q$ ; now prove that one has $\dim(R_f) = 2$ (use Exercise 1, p. 86) ; one thus has $\dim_q(R) = 2$ whereas $\operatorname{ht}(q) + \dim(R/q) - 1 = 1$.