Auslander–Reiten annihilators

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.