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We describe new classes of noetherian local rings $R$ whose finitelygenerated 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 $i\gg0$ implies that $M$ has finite projective dimension or finite injectivedimension.

Persistence of homology over commutative noetherian rings