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Search: MSC category 13A30 ( Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics )

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1. CJM Online first

Miranda-Neto, Cleto B.
On special fiber rings of modules
We prove results concerning the multiplicity as well as the Cohen-Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^e$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen-Macaulay of dimension $1$ and that $E$ has finite colength in $R^e$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen-Macaulayness criterion and to detect a curious relation to the Buchsbaum-Rim multiplicity of $E$ in this setting. Further, we provide a Gorenstein-ness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen-Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

Keywords:special fiber ring, Rees algebra, reduction, reduction number, analytic spread, Hilbert-Samuel multiplicity, Cohen-Macaulay, Gorenstein, Buchsbaum-Rim multiplicity
Categories:13A30, 13H10, 13H15, 13A02, 13C15, 13D40, , 13E15

2. CJM 2013 (vol 66 pp. 1225)

Cortadellas Benítez, Teresa; D'Andrea, Carlos
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$
We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated with the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

Keywords:Rees Algebras, rational plane curves, minimal generators
Categories:13A30, 14H50

3. CJM 2009 (vol 61 pp. 762)

D'Cruz, Clare; Puthenpurakal, Tony J.
The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring
Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$. We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.

Keywords:fiber cone, $a$-invariant, Hilbert coefficients of fiber cone
Categories:13A30, 13D40

4. CJM 2007 (vol 59 pp. 109)

Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.
On Fiber Cones of $\m$-Primary Ideals
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
Categories:13H10, 13H15, 13A30, 13C15, 13A02

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