A Note on the Multiplicities of Gorenstein Algebras

In this note we give a formula for the multiplicities of homogenous Gorenstein algebras. Herzog, Huneke, and Srinivasan have conjectured bounds for the multiplicities of homogeneous Cohen–Macaulay algebras. Herzog and Srinivasan have proved this conjecture for C-M algebras with quasi-pure resolutions. Using this formula for the multiplicities of Gorenstein algebras, we prove a stronger bound for the multiplicities of Goerenstein algebras with quasi-pure resolutions.