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On Fiber Cones of $\m$-Primary Ideals |
Canad. J. Math. 59(2007), 109-126
Published:2007-02-01
Printed: Feb 2007
Printed: Feb 2007
- A. V. Jayanthan
- Tony J. Puthenpurakal
- J. K. Verma
Abstract
Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ 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 the reduction number of $I$ and lengths of certain ideals.
We also characterize the Cohen--Macaulay and Gorenstein properties of
fiber cones of $\m$-primary ideals with a $d$-generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$
| Keywords: | fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities |
| MSC Classifications: |
13H10 - Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13H15 - Multiplicity theory and related topics [See also 14C17] 13A30 - Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13C15 - Dimension theory, depth, related rings (catenary, etc.) 13A02 - Graded rings [See also 16W50] |