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arXiv · 1401.6216

A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property

Craig Huneke·Paolo Mantero·Jason McCullough·Alexandra Seceleanu
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Abstract

Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.

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Craig Huneke, Paolo Mantero, Jason McCullough, Alexandra Seceleanu. 2014-01-23. A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property. https://arxiv.org/abs/1401.6216

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