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We prove two theorems on the vanishing of Ext over commutative Noetherianlocal rings. Our first theorem shows that there are no Burch ideals which arerigid over non-regular local domains. Our second theorem reformulates aconjecture of Huneke-Wiegand in terms of the vanishing of Ext, and highlightsits relation with the celebrated Auslander-Reiten conjecture. We also discussseveral consequences of our results, for example, about the rigidity of theFrobenius endomorphism in prime characteristic p and a generalization of aresult of Araya.

Two theorems on the vanishing of Ext