Dimensions of Formal Fibers of Height One Prime Ideals

Abstract

Let T be a complete local (Noetherian) ring with maximal ideal M, P a nonmaximal ideal of T, and C = {Q 1, Q 2,…} a (nonempty) finite or countable set of nonmaximal prime ideals of T. Let {p 1, p 2,…} be a set of nonzero regular elements of T, whose cardinality is the same as that of C. Suppose that p i ∈ Q j if and only if i = j. We give conditions that ensure there is an excellent local unique factorization domain A such that A is a subring of T, the maximal ideal of A is M ∩ A, the (M ∩ A)-adic completion of A is T, and so that the following three conditions hold: (1) p i ∈ A for every i; (2) A ∩ P = (0), and if J is a prime ideal of T with J ∩ A = (0), then J ⊆ P or J ⊆ Q i for some i; (3) for each i, p i A is a prime ideal of A, Q i ∩ A = p i A, and if J is a prime ideal of T with J ⊈ Q i , then J ∩ A ≠ p i A.