Tony J. Puthenpurakal
In this paper we introduce a new technique to study associated graded modules. Let $(A,\m)$ be a Noetherian local ring with $\depth A \geq 2$. Our techniques gives a necessary and sufficient condition for $\depth G_{\m^n}(A) \geq 2$ for all $n \gg 0$. Other applications are also included; most notable is an upper bound regarding the Ratliff-Rush filtration.
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This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal $I$ of a domain $R$ is the ideal given by $\tilde{I}:=\bigcup(I^{n+1}:_{R}I^{n})$ and an ideal $I$ is said to be a Ratliff-Rush ideal if $\tilde{I}=I$. We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush closure of an ideal in a valuation domain.
We establish new classes of Ratliff-Rush closed ideals and some pathological behavior of the Ratliff-Rush closure. In particular, Ratliff-Rush closure does not behave well under passage modulo superficial elements, taking powers of ideals, associated primes, leading term ideals, and the minimal number of generators. In contrast, the reduction number of the Ratliff-Rush filtration behaves better: it preserves some information on the reduction number of the ideal.
Let $I$ be a regular $\mathfrak m$-primary ideal in $(R,\mathfrak m,k)$. Then the Ratliff-Rush ideal associated to $I$ is denoted by $\bar I$ and is defined as the largest ideal containing $I$ with the same Hilbert polynomial as $I$. In this paper we present a method to compute Ratliff-Rush ideals for a certain class of monomial ideals in the rings $k[x,y]$ and $k[[x,y]]$. We find an upper bound for the Ratliff-Rush reduction number for an ideal in this class. Moreover, we establish some new characterizations of when all powers of $I$ are Ratliff-Rush.
We prove that the initial ideal of the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers is Ratliff-Rush closed.
In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra, which generalizes the result of S. Goto and N. Matsuoka. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.
Let $R$ be a Cohen-Macaulay local ring with maximal ideal $\max$. In this paper we present a procedure for computing the Ratllif-Rush closure of a $\max-$primary ideal $I \subset R$.
Let I\subset K[x,y] be a <x,y>-primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I=<m_0,...,m_{n}> whenever m_{i} is contained in the integral closure of the ideal <x^{a_{n}},y^{b_0}>. This generalizes of the work of Crispin \cite{Cri}. Also, it provides generalizations and answers for some questions given in \cite{HJLS}, and enables us to construct infinite families of Ratliff-Rush ideals.
Powers of (monomial) ideals is a subject that still calls attraction in various ways. In this paper we present a nice presentation of high powers of ideals in a certain class in $\mathbb K[x_1, \ldots, x_n]$ and $\mathbb K[[x_1, \ldots, x_n]]$. As an interesting application it leads to an algorithm for computation of the Ratliff--Rush operation on ideals in that class. The Ratliff--Rush operation itself has several applications, for instance, if $I$ is a regular $\mathfrak m$-primary ideal in a local ring $(R,m)$, then the Ratliff--Rush associated ideal $\tilde I$ is the unique largest ideal c…
Starting from \cite{Ayy2} we compute the Groebner basis for the defining ideal, P, of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P, inP. The first result of this paper introduces a procedure for generating infinite families of Ratliff-Rush ideals, in polynomial rings with multivariables, from a Ratliff-Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff-Rush. The proof is through applying the first result of this paper combin…
Let $(R,m)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an m-primary ideal and $J$ a minimal reduction of $I$. In this paper, we show that $\widetilde{r_J(I)}\leq r_J(I)$. This answer to a question that made by M.E. Rossi and I. Swanson in [\ref{Rs}, Question 4.6].
On the Ratliff-Rush closure of an ideal of a one-dimensional ring
本文在一维Noetherian环中研究理想的渐近Ratliff-Rush数，通过其与约化数的关系，寻找决定该数取得极值的条件。
Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r \geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extend…
Let $R$ be a commutative Noetherian ring, $E$ a non-zero finitely generated $R$-module and $I$ an ideal of $R$. One purpose of this paper is to show that the sequences $\Ass_RE/ \widetilde{I_E^n}$ and $\Ass_R\widetilde{I^n _E}/\widetilde{I^{n+1}_E}$, $n = 1,2, \dots$, of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff \cite[Theorem 3.1]{MR}. In addition, a characterization concerning the set $\widetilde{A^*}(I,E)$ is included. A second purpose of this paper is to prove that $I$ has linear growth primary decompositions for …
We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjectu…
Let $(R,\fm)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\fm$-primary ideal of $R$ and $J$ be a minimal reduction of $I$. In this paper we show that if $\widetilde{I^k}=I^k$ and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde{I^n}=I^n$ for all $n\geq k$. As a consequence, we can deduce that if $r_J(I)=2$, then $\widetilde{I}=I$ if and only if $\widetilde{I^n}=I^n$ for all $n\geq 1$. Moreover, we recover some main results [\ref{Cpv}] and [\ref{G}]. Finally, we give a counter example for question 3 of [\ref{P1}].
Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $I$ a proper ideal of $R$. In this paper we introduce and analyze some properties of $r(I, M)=\bigcup_{k\geqslant 1} (I^{k+1}M: I^kM)$, {\it the Ratliff-Rush ideal associated with $I$ and $M$}. When $M= R$ (or more generally when $M$ is projective) then $r(I, M)= \widetilde{I}$, the usual Ratliff-Rush ideal associated with $I$. If $I$ is a regular ideal and $\ann M=0$ we show that $\{r(I^n,M) \}_{n\geqslant 0}$ is a stable $I$-filtration. If $M_{\p}$ is free for all ${\p}\in \spec R\setminus \mspec R,$ then under…
Ratliff-Rush filtration, Hilbert coefficients and the reduction number of integrally closed ideals
本文在 Cohen-Macaulay 局部环中建立了整闭理想第三 Hilbert 系数的界，利用边界情形刻画了 Ratliff-Rush 滤链性质，并在特定深度条件下证明了约化数满足 Rossi 的界。
A note on Ratliff-Rush filtration, reduction number and postulation number of $\mathfrak m$-primary ideals
研究了Cohen-Macaulay局部环中m-primary理想的相关性质，证明了在不同维数下，reduction数与postulation数的关系，并推广了Marley关于Hilbert-Samuel函数与多项式关系的结果。
Bounds on the Ratliff-Rush Index and the Castelnuovo-Mumford Regularity
本文基于Hilbert系数和减少数，建立了理想Ratliff-Rush指数的上界，并推导了关联graded环的Castelnuovo-Mumford正则度的界，特别研究了二维Buchsbaum环中的情况。
Quasi-Hilbert rings and Ratliff-Rush filtrations
本文研究了准希尔伯特环的性质，引入了准希尔伯特环的概念，并探讨了其在理想闭包和环完成中的应用。
Bounding reduction number and the Hilbert coefficients of filtration
本文研究了Cohen-Macaulay局部环上滤链的第三Hilbert系数的界，证明了在特定边界条件下Ratliff-Rush滤链的相伴分次环性质、约化数界及正则指数等结论。
We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing…
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let $\widetilde{I} = I_d$ denote the Ratliff-Rush closure of $A$. Let $G = G_I(A)$ be the associated graded ring of $I$. We show that if $\dim H^j_{G_+}(G)^\vee \leq j -1$ for $1 \leq j \leq i \leq d-1$ then $(I^n)_{d-i} = \widetilde{I^n}$ for all $n \geq 1$. In particular if $G$ is generalized Cohen-Macaulay then $(I^n)_1 = \widetilde{I^n}$ for all $n \geq 1$. As …
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