  We describe new classes of noetherian local rings $R$ whose finitely
generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$
implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg
0$ implies that $M$ has finite projective dimension or finite injective
dimension.
