Chains of Rings with Local Formal Fibers

Abstract

Let (T, M) be a complete local domain containing the integers. Let p 1 ⊆ p 2 ⊆ ··· ⊆ p n be a chain of nonmaximal prime ideals of T such that T p n is a regular local ring. We construct a chain of excellent local domains A n ⊆ A n−1 ⊆ ··· ⊆ A 1 such that for each 1 ≤ i ≤ n, the completion of A i is T, the generic formal fiber of A i is local with maximal ideal p i , and if I is a nonzero ideal of A i then A i /I is complete. We then show that if Q is a nonmaximal prime ideal of T and 1 ≤ h = ht T Q, then there is a chain of excellent local domains B 0 ⊆ B 1 ⊆ ··· ⊆ B h ⊆ T such that for every i = 0, 1, 2,…, h we have ht(Q ∩ B i ) = i, the completion of B i is isomorphic to T[[X 1, X 2,…, X i ]] where the X j 's are indeterminants, and the formal fiber of Q ∩ B i is local.