On fiber cones of ${\mathfrak m}$-primary ideals

A. V. Jayanthan; Tony J. Puthenpurakal; J. K. Verma

On fiber cones of ${\mathfrak m}$-primary ideals

Commutative Algebra 2007-05-23 v2

Authors: A. V. Jayanthan , Tony J. Puthenpurakal , J. K. Verma

Abstract

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

Keywords

fiber cones m-primary ideals multiplicity Cohen-Macaulay property Gorenstein property mixed multiplicity

Cite

@article{arxiv.math/0408346,
  title  = {On fiber cones of ${\mathfrak m}$-primary ideals},
  author = {A. V. Jayanthan and Tony J. Puthenpurakal and J. K. Verma},
  journal= {arXiv preprint arXiv:math/0408346},
  year   = {2007}
}

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17 Pages

On fiber cones of ${\mathfrak m}$-primary ideals

Abstract

Keywords

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