Auslander–Reiten Conjecture (commutative Noetherian local rings)

Prove the Auslander–Reiten Conjecture for commutative Noetherian local rings: Establish that for every commutative Noetherian local ring R and every finitely generated R-module M, if Ext_R^i(M, M ⊕ R) = 0 for all integers i ≥ 1, then M is projective (equivalently, free).

Background

The Auslander–Reiten Conjecture originated from representation theory of Artin algebras and has a commutative algebra formulation widely studied over the last decades. It predicts that a module must be projective if certain Ext groups vanish.

The paper introduces new annihilator-based approaches related to the conjecture and develops evidence in various ring classes, but the general conjecture remains unresolved.