Remarks on torsionfreeness and its applications

TL;DR

This paper characterizes torsionfreeness of modules relative to semidualizing modules using Serre's condition and applies these results to characterize certain Cohen-Macaulay rings, addressing conjectures in ring theory.

Contribution

It introduces a new characterization of torsionfreeness via Serre's condition and applies it to problems in Cohen-Macaulay rings and longstanding conjectures.

Findings

Characterization of torsionfreeness in terms of Serre's condition (S_n)

Criteria for Cohen-Macaulay rings with Gorenstein localizations

Partial solutions to Tachikawa and Auslander-Reiten conjectures

Abstract

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre's condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_p is Gorenstein for all prime ideals p of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.