Conditions on the Koszul complex of a noetherian local ring R guarantee that TorRi(M,N) is non-zero for infinitely many i, when M and N are finitely generated R-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.
Avramov, Luchezar L., Srikanth B. Iyengar, Saeed Nasseh, Sean Sather-Wagstaff.
"Homology Over Trivial Extensions of Commutative DG Algebras."
arXIV.org Repository: 1-14.