  We prove two theorems on the vanishing of Ext over commutative Noetherian
local rings. Our first theorem shows that there are no Burch ideals which are
rigid over non-regular local domains. Our second theorem reformulates a
conjecture of Huneke-Wiegand in terms of the vanishing of Ext, and highlights
its relation with the celebrated Auslander-Reiten conjecture. We also discuss
several consequences of our results, for example, about the rigidity of the
Frobenius endomorphism in prime characteristic p and a generalization of a
result of Araya.
