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Betti numbers of Z^n-graded modules

Morten Brun·Tim Roemer
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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^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.

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Morten Brun, Tim Roemer. 2004-09-27. Betti numbers of Z^n-graded modules. https://arxiv.org/abs/math/0303349

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