Gluing Minimal Prime Ideals in Local Rings

Abstract

Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose that $B$ contains the rationals, $B/M$ is uncountable and $|B|=|B/M|$. Let the minimal prime ideals of $B$ be partitioned into $m\ge 1$ subcollections $C_1,\dots ,C_m$. We show that there is a reduced local ring $S\subseteq B$ with maximal ideal $S\cap M$ such that the completion of $S$ with respect to its maximal ideal is isomorphic to the completion of $B$ with respect to its maximal ideal and such that, if $P$ and $Q$ are prime ideals of $B$, then $P\cap S=Q\cap S$ if and only if $P$ and $Q$ are in $C_i$ for some $i=1,2,\dots ,m$.

9 pages