Betti numbers of Zn-graded modules
Abstract
Let S=K[X1,...,Xn] be the polynomial ring over a field K. For bounded below Zn-graded S-modules M and N we show that if TorSp(M,N) is nonzero, then for every i between 0 and p, the dimension of the K-vector space TorSi(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.
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