Jaffard families and localizations of star operations

TL;DR

This paper extends the concept of localization of star operations to flat overrings and explores representing star operations on a ring as a product over its overrings, with applications to Pr"ufer domains.

Contribution

It introduces a generalized localization method for star operations and analyzes their structure on Pr"ufer domains, including stable star operations and star-class groups.

Findings

Representation of star operations as products over overrings

Characterization of stable star operations on Pr"ufer domains

Analysis of star-class groups in the context of generalized localizations

Abstract

We generalize the concept of localization of a star operation to flat overrings; subsequently, we investigate the possibility of representing the set $\mathrm{Star}(R)$ of star operations on $R$ as the product of $\mathrm{Star}(T)$, as $T$ ranges in a family of overrings of $R$ with special properties. We then apply this method to study the set of star operations on a Pr\"ufer domain $R$, in particular the set of stable star operations and the star-class groups of $R$.