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


Authors: Craig Huneke, Paolo Mantero, Jason McCullough and Alexandra Seceleanu
Journal: Proc. Amer. Math. Soc. 143 (2015), 2365-2377
MSC (2010): Primary 13C14; Secondary 13H15, 13D40
DOI: https://doi.org/10.1090/S0002-9939-2015-12612-3
Published electronically: February 4, 2015
MathSciNet review: 3326019
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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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Additional Information

Craig Huneke
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
Email: huneke@virginia.edu

Paolo Mantero
Affiliation: Department of Mathematics, University of California Riverside, Riverside, California 92521
Email: mantero@math.ucr.edu

Jason McCullough
Affiliation: Department of Mathematics, Rider University, Lawrence Township, New Jersey 08648
Email: jmccullough@rider.edu

Alexandra Seceleanu
Affiliation: Department of Mathematics, University of Nebraska at Lincoln, Lincoln, Nebraska 68588
Email: aseceleanu2@math.unl.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12612-3
Received by editor(s): January 23, 2014
Published electronically: February 4, 2015
Additional Notes: The first author was partially supported by NSF grant DMS-1259142.
The second and third authors were partially supported by AMS-Simons Travel Grants.
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society