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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Mixed multiplicities of filtrations


Authors: Steven Dale Cutkosky, Parangama Sarkar and Hema Srinivasan
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 13H15; Secondary 14C17
DOI: https://doi.org/10.1090/tran/7745
Published electronically: January 16, 2019
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Abstract: In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $ m_R$-primary ideals in a Noetherian local ring $ R$, generalizing the classical theory for $ m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $ R$ is less than the dimension of $ R$, which holds, for instance, if $ R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $ m_R$-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and of Rees and Sharp.


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Additional Information

Steven Dale Cutkosky
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: cutkoskys@missouri.edu

Parangama Sarkar
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: parangamasarkar@gmail.com

Hema Srinivasan
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: srinivasanh@missouri.edu

DOI: https://doi.org/10.1090/tran/7745
Received by editor(s): May 3, 2018
Received by editor(s) in revised form: October 25, 2018
Published electronically: January 16, 2019
Additional Notes: The first author was partially supported by NSF grant DMS-1700046.
The second author was supported by IUSSTF, SERB Indo-U.S. Postdoctoral Fellowship 2017/145, and DST-INSPIRE India
Article copyright: © Copyright 2019 American Mathematical Society