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On Fiber Cones of $\text{m}$-Primary Ideals

Published online by Cambridge University Press:  20 November 2018

A. V. Jayanthan ,
Tony J. Puthenpurakal  and
J. K. Verma
A. V. Jayanthan
Affiliation:
Department of Mathematics, Indian Institute of Technology, Chennai, India 600036 e-mail: [email protected]
Tony J. Puthenpurakal
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 40076, India e-mail: [email protected], [email protected]
J. K. Verma
Affiliation:
Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 40076, India e-mail: [email protected], [email protected]
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Abstract

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Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$-primary ideal of a $d$-dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$, the multiplicity $e\left( I \right)$, and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$.

Keywords

13H1013H1513A3013C1513A02fiber conesmixed multiplicitiesjoint reductionsCohen–Macaulay fiber conesGorenstein fiber conesideals having minimal and almost minimal mixed multiplicities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 1 , 01 February 2007 , pp. 109 - 126
Copyright
Copyright © Canadian Mathematical Society 2007

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