Local properties of integral domains under extensions and pullback constructions
Hyungtae Baek, Jung Wook Lim, Omar Ouzzaouit., Ali Tamoussit
TL;DR
The paper develops a framework for transferring local properties of integral domains across extensions and pullbacks, focusing on locally X-domain and t-locally X-domain concepts within the language of star-operations. It proves that the locally X-property is preserved under flat overrings (and t-flat overrings for the t-version), and that stability under quotient extension lets one check locality at maximal (or maximal t-) ideals. It further shows that Nagata transforms, polynomial extensions, and certain quotient constructions preserve local properties under suitable hypotheses, and it provides a detailed analysis of pullback constructions, giving precise criteria for when a pullback is locally Krull-like. These results collectively facilitate classification of local properties in a broad range of extensions, including the nontrivial context of pullbacks and GK-domain-like situations. The findings have practical impact on understanding how local domain properties behave under standard algebraic constructions and offer tools for constructing examples with prescribed local behavior.
Abstract
For a property $\mathcal{X}$ of integral domains, an integral domain $D$ is said to be a {\it locally $\mathcal{X}$-domain} if $D_P$ has the property $\mathcal{X}$ for every prime ideal $P$ of $D$. In this paper, we study the transfer of local properties of integral domains under several extensions and constructions, including flat overrings, Nagata ideal transforms, polynomial rings and their quotient extensions, and pullback constructions.