Flat ring epimorphisms and universal localisations of commutative rings

TL;DR

This paper investigates criteria for when flat ring epimorphisms of commutative noetherian rings are universal localisations, linking these conditions to properties like factoriality, Krull dimension, and divisor class groups.

Contribution

It provides new criteria to identify universal localisations among flat ring epimorphisms using support theory and ring invariants.

Findings

All flat ring epimorphisms are universal localisations for locally factorial or Krull dimension one rings.

The structure of the Picard group influences whether universal localisations are classical rings of fractions.

Divisor class groups determine when flat ring epimorphisms are universal localisations in normal rings.

Abstract

We study different types of localisations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localisation in the sense of Cohn and Schofield; and (b) when such universal localisations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialisation closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localisations. Moreover, it turns out that an answer to the question of when universal localisations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localisation. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.