Completions of reduced local rings with prescribed minimal prime ideals

Abstract

Let $T$ be a complete local ring of Krull dimension at least one, and let ${\mathcal{C}}_1,{\mathcal{C}}_2,\dots ,{\mathcal{C}}_m$ 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 $Q_1,Q_2,\dots ,Q_m$, and such that, for every $i=1,2,\dots ,m$, the set of maximal elements of $\left\{P\in Spec\left(T\right)\mid P\cap A=Q_i\right\}$ is the set ${\mathcal{C}}_i$.