One-dimensional bad Noetherian domains

TL;DR

This paper provides a unified framework for understanding analytically ramified local Noetherian domains, especially their normalizations, using Kähler differentials, and classifies which DVRs can occur as such normalizations.

Contribution

It introduces a unified approach to classify analytically ramified local Noetherian domains and their normalizations via Kähler differentials, expanding understanding of their structure.

Findings

Classifies DVRs as normalizations of ramified domains

Parameterizes examples using Kähler differentials

Identifies which DVRs do not occur as normalizations

Abstract

Local Noetherian domains arising as local rings of points of varieties or in the context of algebraic number theory are analytically unramified, meaning their completions have no nontrivial nilpotent elements. However, looking elsewhere, many sources of analytically ramified local Noetherian domains have been exhibited over the last seventy five years. We give a unified approach to a number of such examples by describing classes of DVRs which occur as the normalization of an analytically ramified local Noetherian domain, as well as some that do not occur as such a normalization. We parameterize these examples, or at least large classes of them, using the module of K\"ahler differentials of a relevant field extension.