Related papers: Hilbert Coefficients of Parameter Ideals

For an ideal $I$ of a Noetherian local ring $(R,\fm,k)$ we show that $\bt_1^R(I)-\bt_0^R(I)\geq -1$. It is demonstrated that some residual intersections of an ideal $I$ for which $\bt_1^R(I)-\bt_0^R(I)= -1\;\text{or}\;0$ are perfect. Some…

A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the…

Let $R$ be a commutative Noetherian ring, $I$ an ideal, $M$ and $N$ finitely generated $R$-modules. Assume $V(I)\cap Supp(M)\cap Supp(N)$ consists of finitely many maximal ideals and let ${\l}(\e^i(N/I^nN,M))$ denote the length of…

Let $I\subset S=\KK[x_1,...,x_n]$ be a lexsegment ideal, generated by monomials of degree $d$. The main aim of this paper is to characterize when the Hilbert depth of $I$ will be 1, in the standard graded case. In addition to this, we will…

For a standard graded algebra $R$, we consider embeddings of the the poset of Hilbert functions of quotients of $R$ into the poset of ideals of $R$, as a way of classification of Hilbert functions. There are examples of rings for which such…

Let $I ~(\ne A)$ be an ideal of a $d$-dimensional Noetherian local ring $A$ with $\operatorname{ht}_AI \ge 2$, containing a non-zerodivisor. The problem of when the ring $I:I=\operatorname{End}_AI$ is Gorenstein is studied, in connection…

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…

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…

The purpose of this paper is to present a characterization of sequentially Cohen-Macaulay modules in terms of its Hilbert coefficients with respect to distinguished parameter ideals. The formulas involve arithmetic degrees. Among…

We discuss a problem of parameter estimation for quantum two-level system, qubit system, in presence of unknown phase parameter. We analyze trade-off relations for mean-square errors when estimating relevant parameters with separable…

We give an explicit formula for the Hilbert-Poincar\'{e} series of the parity binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal generated by all $2\times 2$-permanents of a $2\times n$-matrix. It follows that the…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be…

Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero…

A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by…

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the…

The Goto number of a parameter ideal Q in a Noetherian local ring (R,m) is the largest integer q such that Q : m^q is integral over Q. The Goto numbers of the monomial parameter ideals of R = k[[x^{a_1}, x^{a_2},..., x_{a_{\nu}}]] are…

Given a numerical function $h:\mathbb Z_{\geq 0}\to\mathbb Z_{\geq 0}$ with $h(0)>0$, the Hilbert depth of $h$ is $\operatorname{hdepth}(h)=\max\{d\;:\;\sum\limits_{j=0}^k (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq d\}$; see…

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a…

Let R be a regular local ring of dimension d, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of Ext^i_R(R/I,H^j_I(M)) is finite for all i and j in the following cases: (1) dim…