Betti numbers of Z(n)-graded modules

Abstract

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(p)(s) (M, N) not equal 0, then for 0 less than or equal to i less than or equal to p, the dimension of the K-vector space Tor(i)(s) (M, N) is at least ((p)(i)). In particular, we get lower bounds for the total Betti numbers of such modules. These results are related to a conjecture of Buchsbaum and Eisenbud.