  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 $i\in\mathbb{Z}$.
