Symmetry in the vanishing of Ext over stably symmetric algebras

A Frobenius algebra over a field k is called symmetric if the Nakayama automorphism is an inner automorphism. A stably symmetric algebra is defined to be a generalization of a symmetric k-algebra. In this paper we will study symmetry in the vanishing of Ext for such algebras R, namely, for all finitely generated R-modules M and N, Ext(R)(i) (M, N) = 0 for all i >> 0 if and only if Ext(R)(i) (N, M) = 0 for all i >> 0. We show that a certain class of noetherian stably symmetric Gorenstein algebras, such as the group algebra of a finite group and the exterior algebra Lambda(k(n)) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Lambda(k(n)), whether n is even or odd, has this symmetry for graded modules using Koszul duality. (c) 2005 Published by Elsevier Inc.