Symmetry in the vanishing of Ext over Gorenstein rings

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 $R$, $\mathrm{Ext}^i_R(M,N)=0$ for all $i\gg 0$ if and only if $\mathrm{Ext}^i_R(N,M)=0$ for all $i\gg 0$.