Consecutive cancellations in Tor modules over local rings

Let M, N be finite modules over a Noetherian local ring R  . We show that the bigraded Hilbert series of gr(TorR(M,N))gr(TorR(M,N)) is obtained from that of Torgr(R)(gr(M),gr(N))Torgr(R)(gr(M),gr(N)) by negative consecutive cancellations, thus extending a theorem of Rossi and Sharifan.