Nonvanishing cohomology and classes of Gorenstein rings

Abstract We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ , there exists a positive integer n M such that for all finitely generated Λ -modules N , if Ext Λ i ( M , N )=0 for all i ≫0, then Ext Λ i ( M , N )=0 for all i ⩾ n M . Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.