Symmetry theorems for Ext vanishing
  It was proved by Avramov and Buchweitz that if A is a commutative local complete intersection ring with finitely generated modules M and N, then the Ext groups between M and N vanish from some step if and only if the Ext groups between N and M vanish from some step.   This paper shows that the same is true under the weaker conditions that A is Gorenstein and that M and N have finite complete intersection dimension. The result is also proved if A is Gorenstein and has finite Cohen-Macaulay type.   Similar results are given for two types of non-commutative rings: Frobenius algebras and complete semi-local algebras. 
Peter Jorgensen