Auslander-Reiten annihilators

TL;DR

This paper investigates the annihilators of Ext-modules related to the Auslander-Reiten Conjecture, extending known results to various classes of rings and providing evidence for a related Tachikawa Conjecture variant.

Contribution

It formulates a generalized Auslander-Reiten Conjecture involving Ext-module annihilators and proves it for high syzygies over several classes of rings.

Findings

Proved the generalized conjecture for analytically unramified Arf rings

Established results for 2D local normal domains with rational singularities

Extended findings to Gorenstein isolated singularities and canonical modules

Abstract

The Auslander-Reiten Conjecture for commutative Noetherian rings predicts that a finitely generated module is projective when certain Ext-modules vanish. But what if those Ext-modules do not vanish? We study the annihilators of these Ext-modules and formulate a generalisation of the Auslander-Reiten Conjecture. We prove this general version for high syzygies of modules over several classes of rings including analytically unramified Arf rings, 2-dimensional local normal domains with rational singularities, Gorenstein isolated singularities of Krull dimension at least 2 and more. We also prove results for the special case of the canonical module of a Cohen-Macaulay local ring. These results both generalise and also provide evidence for a version of the Tachikawa Conjecture that was considered by Dao-Kobayashi-Takahashi.