Tachikawa conjecture (commutative algebra version): vanishing of Ext of the canonical module implies Gorenstein

Prove that if R is a Noetherian local ring or a positively graded k-algebra that is Cohen–Macaulay with canonical module ω_R and satisfies Ext^i_R(ω_R, R) = 0 for all i > 0, then R is Gorenstein.

Background

The paper recalls a commutative algebra version of the famous Tachikawa conjecture, formulated for Cohen–Macaulay local rings or positively graded k-algebras. The conjecture predicts that vanishing of all positive Ext-groups of the canonical module forces the ring to be Gorenstein.

Known cases include generically Gorenstein rings, certain graded settings, and small-invariant situations (e.g., μ_R(ω_R) ≤ 2 or m^3 = 0). Despite these advances, the conjecture remains unresolved in general and serves as a central motivator for studying annihilators of homologically defined objects and their influence on trace ideals.