Completions of Uncountable Local Rings with Countable Spectra

TL;DR

This paper characterizes when complete local Noetherian rings can be realized as completions of uncountable local domains with countable spectra, revealing such domains are more prevalent than previously believed.

Contribution

It provides necessary and sufficient conditions for these completions and characterizes various classes of uncountable local domains with countable spectra.

Findings

Uncountable local domains with countable spectra are more common than previously thought.

Provides a complete characterization of when a complete local ring is a completion of such domains.

Identifies conditions involving the presence of rationals and properties like catenarity and UFD status.

Abstract

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that uncountable local domains with countable spectra are more common than previously thought. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and completions of uncountable noncatenary local unique factorization domains with countable spectra.