Related papers: A Northcott type inequality for Buchsbaum-Rim coef…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ with infinite residue field. Let $M$ be a finitely generated proper $R$-submodule of a free $R$-module $F$ with $\ell (F/M) < \infty$ and having rank $r$. In this article,…

A complete structure theorem of Sally modules of $\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \m)$ satisfying the equality $\e_1(I)=\e_0(I)-\ell_A(A/I)+1$ is given, where $\e_0(I)$ and $\e_1(I)$ denote the first two Hilbert…

Let $(A,\mathfrak m)$ be a Noetherian local ring of dimension $d>0$ with infinite residue field and $I$ an $\mathfrak{m}$-primary ideal. Let $\mathcal I$ be an $I$-good filtration. We study an equality of Hilbert coefficients, first given…

Let $R$ be a Cohen-Macaulay local ring of dimension $d$ with infinite residue field. Let $I$ be an $R$-ideal that has analytic spread $\ell(I)=d$, $G_d$ condition and the Artin-Nagata property $AN^-_{d-2}$. We provide a formula relating the…

The structure of Sally modules of $\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \m)$ satisfying the equality $\e_1(I)=\e_0(I)-\ell_A(A/I)+1$ is explored, where $\e_0(I)$ and $\e_1(I)$ denote the first two Hilbert…

In this article, we prove that the Buchsbaum-Rim function $\ell_A(\S_{\nu+1}(F)/N^{\nu+1})$ of a parameter module $N$ in $F$ is bounded above by $e(F/N) \binom{\nu+d+r-1}{d+r-1}$ for every integer $\nu \geq 0$. Moreover, it turns out that…

Let $(R, \mathfrak m)$ be an unmixed Noetherian local ring, Q a parameter ideal and $K$ an $\mathfrak m$-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and…

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…

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a…

We give a partial characterization for when the difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ is independent of the choice of parameter ideal $\mathfrak{q}\subseteq R$ in an excellent equidimensional local ring $(R,\mathfrak{m})$ of…

A Noetherian local ring $(R,\mathfrak{m})$ is called Buchsbaum if the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q})$, where $\mathfrak{q}$ is an ideal generated by a system of parameters, is a constant independent of $\mathfrak{q}$.…

In this article we prove that the Buchsbaum-Rim multiplicity $e(F/N)$ of a parameter module $N$ in a free module $F=A^r$ is bounded above by the colength $\ell_A(F/N)$. Moreover, we prove that once the equality $\ell_A(F/N)=e(F/N)$ holds…

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…

Given a local Noetherian ring $(R, {\mathfrak m})$ of dimension $d>0$ and infinite residue field, we study the invariants $($dimension and multiplicity$)$ of the Sally module $S_J(I)$ of any ${\mathfrak m}$-primary ideal $I$ with respect to…

Over a regular local ring of dimension two with maximal ideal m, we study the Buchsbaum-Rim multiplicity of a finitely generated module M of finite colength in a free module F. First, we investigate the colength of an m-primary ideal and…

Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. A famous result of Northcott says that if $M$ is Cohen-Macaulay, then the index of reducibility of parameter ideals on $M$ is an…

The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$ vanishes…

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…

The Chern numbers of the title are the first coefficients (after the multiplicities) of the Hilbert functions of various filtrations of ideals of a local ring $(R, \mathfrak{m})$. For a Noetherian (good) filtration $\mathcal{A}$ of…

Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality $\chi_1(Q;M)=\operatorname{hdeg}_Q(M)-\mathrm{e}_Q^0(M)$, where $\chi_1(Q;M)$,…