Two theorems on the vanishing of Ext

TL;DR

This paper establishes two theorems on the conditions for the vanishing of Ext over commutative Noetherian local rings, linking ideal properties with homological conjectures and implications for algebraic structures.

Contribution

It proves the non-existence of Burch ideals that are rigid over non-regular local domains and reformulates the Huneke-Wiegand conjecture via Ext vanishing, connecting it to the Auslander-Reiten conjecture.

Findings

No Burch ideals are rigid over non-regular local domains.

Reformulation of Huneke-Wiegand conjecture in terms of Ext vanishing.

Implications for the rigidity of Frobenius endomorphism and generalizations of Araya's results.

Abstract

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.