On modules with self Tor vanishing

TL;DR

This paper investigates the properties of modules with vanishing self Tor over local rings, establishing that the class of Tor-persistent rings remains stable under common ring-theoretic operations.

Contribution

It proves that Tor-persistent local rings are closed under standard procedures, advancing understanding of the structure of modules with self Tor vanishing.

Findings

Tor-persistent rings are closed under various ring-theoretic operations.

The study extends the understanding of modules with vanishing self Tor.

Provides new insights into the structure of Tor-persistent rings.

Abstract

The long-standing Auslander and Reiten Conjecture states that a finitely generated module over a finite-dimensional algebra is projective if certain Ext-groups vanish. Several authors, including Avramov, Buchweitz, Iyengar, Jorgensen, Nasseh, Sather-Wagstaff, and \c{S}ega, have studied a possible counterpart of the conjecture, or question, for commutative rings in terms of vanishing of Tor. This has led to the notion of Tor-persistent rings. Our main result shows that the class of Tor-persistent local rings is closed under a number of standard procedures in ring theory.