Openness of various loci over Noetherian rings

TL;DR

This paper investigates the openness of various loci related to module properties over Noetherian rings, establishing conditions under which these loci are open and providing a module version of Nagata's criterion.

Contribution

It introduces a module-based Nagata criterion and proves the openness of FID-loci over acceptable rings for several important properties.

Findings

FID-loci over acceptable rings are open

Module version of Nagata's criterion established

Openness results for properties like Gor, CM, MCM, (S_n), and (T_n)

Abstract

In this paper, we consider the openness of the P-locus of a finitely generated module over a commutative noetherian ring in the case where P is each of the properties FID, Gor, CM, MCM, (S_n), and (T_n). One of the main results asserts that FID-loci over an acceptable ring are open. We give a module version of the Nagata criterion, and prove that it holds for all of the aforementioned properties.