Formal fibers at height one prime ideals

Abstract

Let (T,M) be a complete local (Noetherian) unique factorization domain with dimension at least two, |T/M|≥c where c is the cardinality of the real numbers, and p a nonmaximal prime ideal of T such that p intersected with the prime subring of T is the zero ideal. Furthermore, suppose F is a nonempty set of nonmaximal, incomparable prime ideals of T such that |F|<|T/M|, and for every q∈F, q⊄p, q intersected with the prime subring of T is the zero ideal, and ht p+1≥ ht q . Then there exists a local unique factorization domain A such that the completion of A is T, p∩A=(0), Q∩A≠(0) for all prime ideals Q of T such that ht Q> ht p , and A∩q=z qA for all q∈F where z q is a nonzero prime element of T. Moreover, if q,q′∈F then A∩q=A∩q′ if and only if q=q′. Therefore, the dimension of the generic formal fiber ring of A is equal to the height of p and the dimension of the formal fiber ring at the prime ideal z qA is greater than or equal to the height of q−1. We also show that this result leads to interesting examples of some easily describable generic formal fibers.