  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 Tachikawa Conjecture that
was considered by Dao-Kobayashi-Takahashi.
