Related papers: Depth of associated graded rings via Hilbert coeff…

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…

In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge…

By definition, an $\m$-primary ideal $I$ in a 2-dimensional regular local ring $(R, \m)$ is contracted if $I=R \cap IR[\m/x]$ for some $x \in \m \setminus \m^2$. Contracted ideals have been introduced by Zariski and used for proving the…

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 $AN_{d-2}^-$…

In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small first Hilbert coefficient in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. In particular, we…

In this paper we consider extremal and almost extremal bounds on the normal Hilbert coefficients of ${\mathfrak m}$-primary ideals of an analytically unramified Cohen-Macaulay ring $R$ of dimension $d>0$ and infinite residue field. In these…

We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with…

Criteria are given in terms of certain Hilbert coefficients for the fiber cone F(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R,m) so that it is Cohen-Macaulay or has depth at least dim(R)-1. A version of Huneke's fundamental…

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…

In a local Cohen-Macaulay ring $(A, \mathrm{m})$, we study the Hilbert function of an $\mathrm{m}$-primary ideal $I$ whose reduction number is two. It is a continuous work of the papers of Huneke, Ooishi, Sally, and Goto-Nishida-Ozeki. With…

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…

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p >…

The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched…

Let $k$ be a field. We determine the ideals $I$ in a finitely generated graded $k$-algebra $A$, whose associated graded rings are isomorphic to $A$. Also we compute the graded local cohomologies of the Rees rings $A[I t]$ and give the…

Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert…

Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an $R$-ideal. Inspired by the work of H\"ubl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ${\mathcal F}={\mathcal…

In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of…

Let $(R, m)$ be a $d$-dimensional Cohen-Macaulay local ring. In this note we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a $m$-primary ideal $I\subset R$ that improves all known upper…

We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a…

We investigate when the Rees algebra of an integrally closed $\mathfrak{m}$-primary ideal in a regular local ring is a Cohen-Macaulay normal domain. While this property always holds in dimension two, it fails in general in higher…