On the Betti numbers of some Gorenstein ideals

Abstract

Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge 3$ and initial degree $p\ge 2$. We prove that the number of minimal generators $\nu (I_p)$ of $I$ that are in degree $p$ is bounded above by ${\nu }_0=\left(\genfrac{}{}{0ex}{}{p+g-1}{g-1}\right)-\left(\genfrac{}{}{0ex}{}{p+g-3}{g-1}\right)$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $\nu (I_p)={\nu }_0$.