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We find necessary and sufficient conditions for a complete local ring to be the completion of a noncatenary local (Noetherian) domain, as well as necessary and sufficient conditions for it to be the completion of a noncatenary local (Noetherian) unique factorization domain. We use our first result to demonstrate a large class of quasi-excellent domains that are not excellent, as well as a large class of catenary domains that are not universally catenary. We use our second result to find a larger class of noncatenary local UFDs than was previously known, and we show that there is no bound on how noncatenary a UFD can be.

Let T be a complete local ring and C a finite set of incomparable prime ideals of T . We find necessary and sufficient conditions for T to be the completion of an integral domain whose generic formal fiber is semilocal with maximal ideals the elements of C. In addition, if the characteristic of T is zero, we give necessary and sufficient conditions for T to be the completion of an excellent integral domain whose generic formal fiber is semilocal with maximal ideals the elements of C.

Let T be a complete local ring of Krull dimension at least one, and let C 1 ; C 2 ; : : : ; C m each be countable sets of prime ideals of T. We find necessary and sufficient conditions for T to be the completion of a reduced local ring A such that A has exactly m minimal prime ideals Q 1 ; Q 2 ; : : : ; Q m , and such that, for every i D 1; 2; : : : ; m, the set of maximal elements of fP 2 Spec.T / j P \ A D Q i g is the set C i .