Symmetry in the vanishing of Ext over stably symmetric algebras

Abstract

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 Λ ( k n ) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Λ ( k n ) , whether n is even or odd, has this symmetry for graded modules using Koszul duality.