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

16 pages