  Let S=K[X_1,...,X_n] be the polynomial ring over a field K. For bounded below
Z^n-graded S-modules M and N we show that if Tor^S_p(M,N) is nonzero, then for
every i between 0 and p, the dimension of the K-vector space Tor^S_i(M,N) is at
least as big as the binomial coefficient (p,i). In particular, we get lower
bounds for the total Betti numbers. These results are related to a conjecture
of Buchsbaum and Eisenbud.
