An Improved Multiplicity Conjecture for Codimension 3 Gorenstein Algebras

Abstract

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen–Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case.

Key Words:

• Betti numbers
• Gorenstein algebras
• Hilbert functions
• Level algebras
• Minimal free resolution
• Multiplicity conjecture

2000 Mathematics Subject Classification:

• Primary 13H15
• Secondary 13D02, 13D40, 13E10

ACKNOWLEDGMENTS

The research contained in this article was performed during the third author's visit to the first author at the University of Notre Dame, and that visit was supported by a grant of the Vetenskåpsradet (Swedish Research Council) and a grant of the Department of Mathematics of the University of Notre Dame. Moreover, the third author was funded by the Göran Gustafsson Foundation.