Gluing Minimal Prime Ideals in Local Rings

TL;DR

This paper constructs a reduced local ring within a given ring that preserves the completion and groups minimal prime ideals according to specified partitions, enhancing understanding of ring structures.

Contribution

It introduces a method to embed a reduced local ring into a larger ring, controlling the intersection of prime ideals to match a given partition, while preserving the completion.

Findings

Existence of a reduced local ring with prescribed prime ideal intersections

Preservation of the ring's completion in the embedding

Control over prime ideal groupings within the constructed ring

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 $m \geq 1$ subcollections $C_1, \ldots ,C_m$. 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 $i = 1,2, \ldots ,m$.