Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Order m

Abstract

The authors in Harima et al. (2003) characterize the Hilbert function of algebras with the Lefschetz property. We extend this characterization to algebras with the Lefschetz property m times. We also give upper bounds for the Betti numbers of Artinian algebras with a given Hilbert function and with the Lefschetz property m times and describe the cases in which these bounds are reached.

Key Words:

• Betti numbers
• Distraction matrix
• Hilbert function
• Lefschetz property

2000 Mathematics Subject Classification:

• Primary 13F20
• Secondary 13E10

ACKNOWLEDGMENTS

The author wishes to thank his advisor, Prof. Aldo Conca, for his encouragement, for suggesting this problem and for his very helpful remarks on preliminary versions of this article.

After this article was written, we were informed that Tadahito Harima and Akihito Wachi have independently obtained some of the results in this article by different methods (see Harima and Wachi, 2007).

Notes

Communicated by R. H. Villarreal