Completions of Countable Excellent Local Rings in Equal Characteristic Zero

TL;DR

This paper characterizes which complete local rings in equal characteristic zero are completions of countable excellent local rings, providing insights into their prime ideal mappings and offering new characterization results.

Contribution

It introduces a characterization of complete local rings in equal characteristic zero that are completions of countable excellent local rings, including prime ideal correspondence.

Findings

Characterization of complete local rings as completions of countable excellent rings

Analysis of prime ideal mappings between rings

New characterization-style results in local ring theory

Abstract

We characterize which complete local (Noetherian) rings T containing the rationals are the completion of a countable excellent local ring S. We also discuss the possibilities for the map from the minimal prime ideals of T to the minimal prime ideals of S and we prove some characterization-style results.