Exercise 1, § 3

Soit $A$ a ring, $U$ an open subset of $\mathrm{Spec}(A)$. a) Now prove that the mapping $p \mapsto V(p) \cap U$ induces a bijection from $U$ onto the set of irreducible closed subsets of $U$. b) Now prove that $\dim(U)$ is the supremum of the set of lengths of chains of prime ideals of $A$ belonging to $U$. c) Let $\Phi$ be the set of minimal prime ideals of $A$ belonging to $U$. Now prove that one has

d) We now suppose $A$ local and $U \neq \mathrm{Spec}(A)$. If $A$ is noetherian, establish the equality $\dim(U) = \sup_{p \in \Phi}(\dim(A/p) - 1)$. e) Let $h$ be an integer such that $0 < h < \dim(A)$. Let $E_h$ be the set of prime ideals $p \in U$ such that $\mathrm{ht}(p) = h$ and $\dim(A/p) = \dim(A) - h$. If $A$ is noetherian, show that $E_h$ is infinite. Show that $E_h$ can be finite when $A$ is not noetherian (take for $A$ a valuation ring of height 2, for $U$ the open subset $\mathrm{Spec}(A) - \{ m_A \}$, and choose $h = 1$).

EXERCISES

AC VIII.87