FORMAL FIBERS WITH COUNTABLY MANY MAXIMAL ELEMENTS

摘要

Let T be a complete local (Noetherian) ring. Let C be a countable set of pairwise incomparable non-maximal 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 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.