Nonvanishing cohomology and classes of Gorenstein rings

AbstractWe give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer nM such that for all finitely generated Λ-modules N, if ExtΛi(M,N)=0 for all i≫0, then ExtΛi(M,N)=0 for all i⩾nM. 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