One-dimensional stable rings

TL;DR

This paper characterizes one-dimensional stable local rings, including their integral closures and completions, revealing two main types: Bass rings and bad stable domains, with implications for module theory and ring classifications.

Contribution

It provides multiple characterizations of one-dimensional stable local rings, especially distinguishing between Bass rings and bad stable domains based on integral closure properties.

Findings

Reduced stable rings are either Bass rings or bad stable domains.

Bass rings have finitely generated integral closure and two-generated ideals.

Bad stable domains have non-finitely generated integral closure.

Abstract

A commutative ring $R$ is stable provided every ideal of $R$ containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen-Macaulay rings of multiplicity at most $2$, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations. We characterize one-dimensional stable local rings in several ways. The characterizations involve the integral closure ${\bar{R}}$ of $R$ and the completion of $R$ in a relevant ideal-adic topology. For example, we show: If $R$ is a reduced stable ring, then there are exactly two possibilities for $R$: (1) $R$ is a {\it Bass ring}, that is, $R$ is a reduced Noetherian local ring such that $\bar{R}$ is finitely generated over $R$ and every ideal of $R$ is generated by two elements; or (2) $R$ is a {\it bad stable domain}, that is, $R$ is a one-dimensional stable local domain such that $\bar{R}$ is not a finitely generated $R$-module.