Global perinormality in a generalized $D + M$ construction

TL;DR

This paper investigates the property of global perinormality in integral domains, demonstrating its preservation under a generalized $D+M$ pullback construction and providing conditions for transfer of flat overring localizations.

Contribution

It extends the understanding of global perinormality by showing its stability in a broad class of pullback constructions, including the classical $D+M$ case.

Findings

Global perinormality is preserved in a generalized $D+M$ pullback.

Conditions are identified for flat overrings to be localizations in the construction.

The results unify and extend previous knowledge on perinormal domains.

Abstract

A domain $R$ is \emph{perinormal} if every going-down overring is flat and a perinormal domain $R$ is \emph{globally perinormal} if every flat overring is a localization of $R$ [Epstein-Shapiro 2016]. I show that global perinormality is preserved in a pullback construction which encompasses a classical $D+M$ construction. In doing so, a result is given for the transfer of the property that every flat overring is a localization in the pullback construction considered.