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