Symmetry in the vanishing of Ext over stably symmetric algebras

AbstractA 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, ExtRi(M,N)=0 for all i≫0 if and only if ExtRi(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 Λ(kn) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Λ(kn), whether n is even or odd, has this symmetry for graded modules using Koszul duality