Symmetry in vanishing of Tate cohomology over Gorenstein rings

TL;DR

This paper proves a symmetry property in the vanishing of Tate cohomology for finitely generated modules over Gorenstein local rings, establishing an if-and-only-if condition for the vanishing of Tate Ext groups.

Contribution

It establishes a new symmetry result in Tate cohomology over Gorenstein rings, linking the vanishing of Ext groups in both directions.

Findings

Vanishing of Tate Ext from M to N implies the same from N to M.

Symmetry holds for all integer degrees in Tate cohomology.

Provides a characterization of vanishing conditions in Gorenstein local rings.

Abstract

We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$.