Exercise 26, § 4

Let A be an integral noetherian ring, X the topological space Spec(A), Nor(X) the subspace of X introduced in Exercise 25, Ā the integral closure of A. Suppose that there exists a nonzero element f of A such that the ring A[f^{-1}] is integrally closed, and that for every maximal ideal m of A, the ring Ā_m is a finite A_m-algebra. Consider Ā as the increasing filtered inductive limit of finite sub-A-algebras, Ā = \varinjlim (A_j)_{j \in J}. Put X_j = \operatorname{Spec}(A_j) and let G_j be the image in X of X_j − Nor(X_j). a) Nor(X_j) is open in X_j. (Use Exercise 25.) b) G_j is closed in X. c) There exists an index j_0 \in J such that G_{j_0} = \bigcap_{j \in J} G_j. d) Let p \in \operatorname{Spec}(A). Then for j \in J sufficiently large, G_j does not contain p. e) Ā is a finite A-algebra.