Formal fibers with countably many maximal elements

Abstract

Let $T$ be a complete local (Noetherian) ring. Let $C$ be a countable set of pairwise incomparable nonmaximal prime ideals of $T$. We find necessary and sufficient conditions for $T$ to be the completion of a local integral domain whose generic formal fiber has maximal elements precisely the elements of $C$. Furthermore, if the characteristic of $T$ is zero, we provide necessary and sufficient conditions for $T$ to be the completion of an \textit{excellent} local integral domain whose generic formal fiber has maximal elements precisely the elements of $C$. In addition, for a positive integer $k$, we construct local integral domains that contain a prime ideal of height~$k$ whose formal fiber has countably many maximal elements.