Symmetry in the vanishing of Ext over Gorenstein rings

TL;DR

This paper explores symmetry properties in the vanishing of Ext modules over Gorenstein rings, introducing a new class called AB rings where such symmetry holds for finitely generated modules.

Contribution

The paper defines AB rings and proves that over these rings, vanishing of Ext in high degrees is symmetric between modules, extending understanding of Ext behavior in Gorenstein contexts.

Findings

Introduces AB rings as a new class of Gorenstein rings.

Establishes symmetry in Ext vanishing for modules over AB rings.

Provides conditions under which Ext vanishing is symmetric in high degrees.

Abstract

We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $Ext^i_R(M,N)=0$ for all $i >> 0$ if and only if $Ext^i_R(N,M)=0$ for all $i >> 0$.