Related papers: The second Hilbert coefficient of modules with alm…

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…

Let $M$ be a finitely generated module over a Noetherian local ring. This paper gives, for a given parameter ideal $Q$ for $M$, bounds for the second Hilbert coefficients ${\mathrm{e}}_Q^2(M)$ in terms of the homological degrees and…

Let $M$ be a finitely generated module of dimention d over a Noetherian local ring (A,m) and I an m-primary ideal. Let be a pair of good I-filtrations F and F' of M. We show that the Hilbert coefficients e_i(F) are bounded below and above…

Let $R$ be a polynomial ring over a field and $M= \bigoplus_n M_n$ a finitely generated graded $R$-module, minimally generated by homogeneous elements of degree zero with a graded $R$-minimal free resolution $\mathbf{F}$. A Cohen-Macaulay…

This paper gives an upper bound of the second Hilbert coefficients ${\mathrm{e}}_2(I)$ of $\mathfrak{m}$-primary ideals $I$ in a Cohen-Macaulay local ring $(A,{\mathfrak{m}})$ in terms of the sectional genera 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…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and depth R$\geq d-1$. Let $Q$ be a parameter ideal of $R$. In this paper, we derive uniform lower and upper bounds for the Hilbert coefficient $e_i(Q)$ under certain…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\mathfrak{p}$ of $M$ such that depth $M=\dim R/\mathfrak{p}$. In this paper, we study…

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 $AN_{d-2}^-$…

Let $M$ be a finitely generated module of dimension $d$ and depth $t$ over a Noetherian local ring ($A, {\mathfrak m}$) and $I$ an ${\mathfrak m}$-primary ideal. In the main result it is shown that the last $t$ Hilbert coefficients…

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…

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…

New upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring are given. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of…

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$ be a finitely generated Cohen Macaulay $A$ module. Let $G(A)=\bigoplus_{n\geq 0}\mathfrak{m}^n/\mathfrak{m}^{n+1}$ be the associated graded ring of $A$ and $G(M)=\bigoplus_{n\geq…

If $(A,\mathfrak{m})$ is a hypersurface ring of dimension $d$ with $e(A)=3$. Let $M$ be an MCM $A$-module with $\mu(M)=4$ then we prove that $\depth{G(M)}\geq d-3$.

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…

Let $(A,\mathfrak{m})$ be a complete intersection ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $M$ be a maximal \CM \ $A$-module. For $i = 0,1,\cdots,d$, let $e_i^I(M)$ denote the $i^{th}$ Hilbert -coefficient…

Given a local Cohen-Macaulay ring $(R, {\mathfrak m})$, we study the interplay between the integral closedness -- or even the normality -- of an ${\mathfrak m}$-primary $R$-ideal $I$ and conditions on the Hilbert coefficients of $I$. We…

We study Hilbert functions of maximal Cohen-Macaulay(=CM) modules over CM local rings. We show that if $A$ is a hypersurface ring with dimension $d > 0$ then the Hilbert function of $M$ \wrt $\m$ is non-decreasing. If $A = Q/(f)$ for some…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an integrally closed $\mathfrak{m}$-primary ideal. We establish bounds for the third Hilbert coefficient $e_3(I)$ in terms of the lower Hilbert…