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

Huneke, Craig; Mantero, Paolo; McCullough, Jason; Seceleanu, Alexandra

doi:10.1090/S0002-9939-2015-12612-3

A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property
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by Craig Huneke, Paolo Mantero, Jason McCullough and Alexandra Seceleanu

Proc. Amer. Math. Soc. 143 (2015), 2365-2377

DOI: https://doi.org/10.1090/S0002-9939-2015-12612-3

Published electronically: February 4, 2015

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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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