Auslander–Reiten Conjecture for Gorenstein rings

Determine whether the Auslander–Reiten Conjecture holds for all Gorenstein commutative Noetherian local rings; specifically, ascertain if for every Gorenstein local ring R and finitely generated R‑module M, the vanishing Ext_R^i(M, M ⊕ R) = 0 for all i ≥ 1 implies that M is projective.

Background

Despite substantial progress on the Auslander–Reiten Conjecture for many classes of rings (e.g., complete intersections, certain Cohen–Macaulay normal rings), its validity for Gorenstein local rings remains a central unresolved case.

The authors discuss recent advances and related generalizations, highlighting that even in the Gorenstein setting the conjecture has not yet been settled.