Semilocal formal fibers of principal prime ideals

TL;DR

This paper characterizes when a complete local ring can be realized as the completion of an integral domain with a specific semilocal formal fiber structure, focusing on prime ideals and their properties.

Contribution

It provides necessary and sufficient conditions for constructing integral domains with prescribed semilocal formal fibers over prime ideals.

Findings

Conditions for T to be the completion of such an integral domain are established.

The structure of formal fibers over prime ideals is characterized.

The results connect properties of prime ideals with the formal fiber structure.

Abstract

Let (T,m) be a complete local (Notherian) ring, C a finite set of pairwise incomparable nonmaximal prime ideals of T, and p a nonzero element. We provide necessary and sufficient conditions for T to be the completion of an integral domain A containing the prime ideal pA whose formal fiber is semilocal with maximal ideals the elements of C.