Betti Numbers of ℤⁿ-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 , then for 0 ≤ i ≤ p, the dimension of the K-vector space is at least . 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.

Key Words:

• Lower bounds
• Betti numbers
• Koszul homology

Mathematics Subject Classification:

• Primary 13D07
• Secondary 13D02, 18G15

Acknowledgments

Morten Brun was supported by the DFG-project VO 166/3–1.

Notes

^#Communicated by Winfried Bruns.