Structure of spectra of precompletions

Abstract

Let $T$ be a complete local (Noetherian) ring, and let $A$ be a local subring of $T$ such that the completion of $A$ with respect to its maximal ideal is $T$. We investigate the possible structures of the partially ordered set $Spec\left(A\right)$. Specifically, we explore the minimal prime ideals of $A$ and their formal fibers, the maximal chains of prime ideals in $A$, and the number of prime ideals in $A$ containing combinations of minimal prime ideals of $A$.