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On the Betti numbers of some Gorenstein ideals

Matthew Miller·Rafael H. Villarreal
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Abstract

Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $ν(I_p)$ of $I$ that are in degree $p$ is bounded above by $ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, 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 $ν(I_p)=ν_0$.

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Matthew Miller, Rafael H. Villarreal. 1994-06-20. On the Betti numbers of some Gorenstein ideals. https://arxiv.org/abs/math/9406208

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