Exercise 14, § 4

Let $A$ be a ring, $X = \operatorname{Spec}(A)$ its spectrum and $P$ the set of minimal prime ideals of $A$.

(a) Let $R \{ p, p' \}$ be the symmetric and reflexive relation: "there exists a prime ideal $p$ of $A$ containing $p + p'$" between the elements $p, p'$ of $P$; let $S$ be the equivalence relation whose graph is the smallest of the graphs of equivalences containing the graph of $R$ (Set Theory, Chapter II, § 6, Exercise 10). Show that, if $I$ is an equivalence class with respect to $S$, the set $V_I = \bigcup_{p \in I} V(p)$ is connected.

(b) Show that, if $P$ is finite (and in particular if $A$ is Noetherian), the $V_I$ are the connected components of $X$. Does this result extend to the case where $P$ is infinite (cf. Exercise 12)?