Estimates on the Depth of the Associated Graded Ring

Proceedings of the American Mathematical Society, 1995

Let (R, m) be a Cohen-Macaulay local ring of positive dimension d, let I be an m − m - primary ideal of R. In this paper we individuate some conditions on I that allow us to determine a lower bound for depth gr I ( R ) {\\text {gr}_I}(R) . It is proved that if J ⊆ I J \\subseteq I is a minimal reduction of I such that λ ( I 2 ∩ J / I J ) = 2 \\lambda ({I^2} \\cap J/IJ) = 2 and I n ∩ J = I n − 1 J {I^n} \\cap J = {I^{n - 1}}J for all n ≥ 3 n \\geq 3 , then depth gr I ( R ) ≥ d − 2 {\\text {gr}_I}(R) \\geq d - 2 ; let us remark that λ \\lambda denotes the length function.

arXiv: Commutative Algebra, 2017

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field. Let $I$ be an ideal of $R$ that has analytic spread $\ell(I)=d$, satisfies the $G_d$ condition, the weak Artin-Nagata property $AN_{d-2}^-$ and depth$(R/I)\geq\min\lbrace 1,\dim R/I \rbrace$. In this paper, we show that if $j_1(I) = \lambda (I/J) +\lambda [R/(J_{d-1} :_{R} I+(J_{d-2} :_{R}I+I) :_R, \mathfrak{m}^\infty)]+1$, then depth$(G(I))\geq d -1$ and $r_J(I)\leq 2$, where $J$ is a general minimal reduction of $I$. In addition, we extend the result by Sally who has studied the depth of associated graded rings and minimal reductions for an $,\mathfrak{m}$-primary ideals.

Journal of Pure and Applied Algebra, 2005

Given a local Cohen-Macaulay ring (R, m), we study the interplay between the integral closedness -or even the normality -of an m-primary R-ideal I and conditions on the Hilbert coefficients of I. We relate these properties to the depth of the associated graded ring of I.

In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences.

2017

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and $\sum\limits_{n=1}^\infty\lambda{({I^{n+1}\cap J_{d-1}})/({J{I^n} \cap J_{d-1}})=i}$ where i=0,1, then depth $G(I)\geq{d-i-1}$. Moreover, we prove that if $e_2(I) = \sum_{n=2}^\infty (n-1) \lambda (I^n/JI^{n-1})-2;$ or if $I$ is integrally closed and $e_2(I) = \sum_{n=2}^\infty (n-1)\lambda({I^{n}}/JI^{n-1})-i$ where $i=3,4$, then $e_1(I) = \sum_{n=1}^\infty \lambda(I^n / JI^{n-1})-1.$ In addition, we show that $r(I)$ is independent. Furthermore, we study the independence of $r(I)$ with some other conditions.

Journal of Algebra, 2004

Let R be a local Cohen-Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen-Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r + powers of I, where r denotes the Castelnuovo regularity of G and denotes the analytic spread of I.

Journal of Commutative Algebra, 2013

Communications in Algebra, 2011

The aim of this paper is to establish, among other results, the asymptotic stability of the depth of the graded pieces of a non-standard multigraded module. As a corollary we get the asymptotic stability of the depth of the graded pieces of the multigraded Rees algebra defined by a finite set of ideals and their associated multigraded rings.

Journal of Pure and Applied Algebra, 1983

Journal of the Mathematical Society of Japan, 1978

In this paper, we study a Noetherian graded ring $R$ and the category of graded R-modules. We consider injective objects of this category and we define the graded Cousin complex of a graded R-module $M$ . These concepts are essential in this paper