Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module $M$ over an Artin algebra $\mathrm{\Lambda }$, there exists a positive integer $n_M$ such that for all finitely generated $\mathrm{\Lambda }$-modules $N$, if ${\text{Ext}}_{\mathrm{\Lambda }}^i(M,N)=0$ for all $i\gg 0$, then ${\text{Ext}}_{\mathrm{\Lambda }}^i(M,N)=0$ for all $i\ge n_M$. 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.