Hilbert Coefficients Research Articles

Unmixed torsions and Hilbert coefficients of d-sequences

In this paper, we define an unmixed torsion associated with a certain cohomological degree. We establish a connection between the Hilbert coefficients and the unmixed torsions of the module with respect to a parameter ideal generated by a d-sequence. As an application, we characterize the vanishing of the Hilbert coefficients through the depth of the module.

Vanishing and Non-negativity of the First Normal Hilbert Coefficient

Vanishing and Non-negativity of the First Normal Hilbert Coefficient

Hilbert coefficients of ideals under perturbation of an ideal

Hilbert coefficients of ideals under perturbation of an ideal

Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Let (A,m) be a Noetherian local ring and let M be a finitely generated Cohen Macaulay A module. Let G(A)=⨁n≥0mn/mn+1 be the associated graded ring of A and G(M)=⨁n≥0mnM/mn+1M be the associated graded module of M. If A is regular and if G(M) has a quasi-pure resolution then we show that G(M) is Cohen-Macaulay. If A is Gorenstein and G(A) is Cohen-Macaulay and if M has finite projective dimension then we give lower bounds on e0(M) and e1(M). Finally let A=Q/(f1,…,fc) be a strict complete intersection with ord(fi)=s for all i, where Q is a regular local ring. Let M be an Cohen-Macaulay module with complexity cxA(M)=r<c. We give lower bounds on e0(M) and e1(M).

Hilbert coefficients of good I-filtrations of modules

Let [Formula: see text] be a finitely generated module of dimension [Formula: see text] over a Noetherian local ring [Formula: see text] and [Formula: see text] an [Formula: see text]-primary ideal. Let be a pair of good [Formula: see text]-filtrations [Formula: see text] and [Formula: see text] of [Formula: see text]. We show that the Hilbert coefficients [Formula: see text] are bounded below and above in terms of [Formula: see text], [Formula: see text], and reduction numbers of [Formula: see text] and [Formula: see text], for all [Formula: see text].

Ratliff-Rush filtration, Hilbert coefficients and reduction number of integrally closed ideals

Let (R,m) be a Cohen-Macaulay local ring of dimension d≥3 and I an integrally closed m-primary ideal. We establish bounds for the third Hilbert coefficient e3(I) in terms of the lower Hilbert coefficients ei(I),0≤i≤2 and the reduction number of I. When d=3, the boundary cases of these bounds characterize certain properties of the Ratliff-Rush filtration of I. These properties, though weaker than depthG(I)≥1, guarantee that Rossi's bound for reduction number rJ(I) holds in dimension three. In that context, we prove that if depthG(I)≥d−3, then rJ(I)≤e1(I)−e0(I)+ℓ(R/I)+1+e2(I)(e2(I)−e1(I)+e0(I)−ℓ(R/I))−e3(I). We also discuss the signature of the fourth Hilbert coefficient e4(I).

On almost p-standard system of parameters of idealization and applications

Let (R,m) be a Noetherian local ring and M be a finitely generated R-module. In this paper, we construct almost p-standard systems of parameters (a strict subclass of d-sequences) of the idealization R⋉M. As applications, we give a class of Cohen-Macaulay Rees algebras for idealizations, Cohen-Macaulay Rees modules for unmixed modules, and give precise formulas computing all Hilbert coefficients of the idealization with respect to an almost p-standard system of parameters.

ON THE SECOND HILBERT COEFFICIENTS AND COHEN-MACAULAY RINGS

We explore the behavior of the second Hilbert coefficients of parameter ideals in Noetherian local rings. In this paper, we provide characterizations of a Cohen-Macaulay local ring in terms of the second Hilbert coefficients and the index of reducibility of parameter ideals.

Upper bounds on two Hilbert coefficients

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 Hilbert series as well as in terms of the Castelnuovo-Mumford regularity of the associated graded module.

Bounds for the reduction number of primary ideal in dimension three

Let ( R , m ) (R,\mathfrak {m}) be a Cohen-Macaulay local ring of dimension d ≥ 3 d\geq 3 and I I an m \mathfrak {m} -primary ideal of R R . Let r J ( I ) r_J(I) be the reduction number of I I with respect to a minimal reduction J J of I I . Suppose depth ⁡ G ( I ) ≥ d − 3 \operatorname {depth} G(I)\geq d-3 . We prove that r J ( I ) ≤ e 1 ( I ) − e 0 ( I ) + λ ( R / I ) + 1 + ( e 2 ( I ) − 1 ) e 2 ( I ) − e 3 ( I ) r_J(I)\leq e_1(I)-e_0(I)+\lambda (R/I)+1+(e_2(I)-1)e_2(I)-e_3(I) , where e i ( I ) e_i(I) are Hilbert coefficients. Suppose d = 3 d=3 and depth ⁡ G ( I t ) &gt; 0 \operatorname {depth} G(I^t)&gt;0 for some t ≥ 1 t\geq 1 . Then we prove that r J ( I ) ≤ e 1 ( I ) − e 0 ( I ) + λ ( R / I ) + t r_J(I)\leq e_1(I)-e_0(I)+\lambda (R/I)+t .

Integrally Closed Ideals of Reduction Number Three

In a Cohen-Macaulay local ring [Formula: see text], we study the Hilbert function of an integrally closed [Formula: see text]-primary ideal [Formula: see text] whose reduction number is three. With a mild assumption we give an inequalityi [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th Hilbert coefficient and [Formula: see text] denotes a minimal reduction of [Formula: see text]. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore, this inequality becomes an equality if and only if the depth of the associated graded ring of [Formula: see text] is larger than or equal to [Formula: see text]. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

Graded filtrations and ideals of reduction number 2

AbstractIn this paper, we give a way to construct graded filtrations of graded modules. We then apply it to the Sally module, which describes a correction term of the Hilbert function. As a result, we obtain the inequality of the Hilbert coefficients for ideals of reduction number 2 or 3.

Tight Hilbert polynomial and F-rational local rings

Let $$(R,\mathfrak {m})$$ be a Noetherian local ring of prime characteristic p and Q be an $$\mathfrak {m}$$ -primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function $$H^*_Q(n)=\ell (R/(Q^n)^*)$$ and the coefficient $$e_1^*(Q)$$ of the tight Hilbert polynomial $$P^*_Q(n)=\sum _{i=0}^d(-1)^ie_i^*(Q)\left( {\begin{array}{c}n+d-1-i\\ d-i\end{array}}\right) .$$ We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalizes a result of Goto and Nakamura. We show that if $$\dim R=2 $$ , the Hochster–Huneke graph of R is connected and this lower bound is achieved, then R is F-rational. Craig Huneke asked if the F-rationality of unmixed local rings may be characterized by the vanishing of $$e_1^*(Q).$$ We construct examples to show that without additional conditions, this is not possible. Let R be an excellent, reduced, equidimensional Noetherian local ring and Q be generated by parameter test elements. We find formulas for $$e_1^*(Q), e_2^*(Q), \ldots , e_d^*(Q)$$ in terms of Hilbert coefficients of Q, lengths of local cohomology modules of R, and the length of the tight closure of the zero submodule of $$H^d_{\mathfrak {m}}(R).$$ Using these, we prove: R is F-rational $$\iff e_1^*(Q)=e_1(Q) \iff {\text {depth}}R\ge 2$$ and $$e_1^*(Q)=0.$$

On Hilbert coefficients and sequentially generalized Cohen–Macaulay modules

This paper shows that if [Formula: see text] is a homomorphic image of a Cohen–Macaulay local ring, then [Formula: see text]-module [Formula: see text] is sequentially generalized Cohen–Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all distinguished parameter ideals of [Formula: see text] are bounded.

Cohen-Macaulay local rings with e2 = e1 − e + 1

In this paper we study Cohen-Macaulay local rings of dimension d, multiplicity e and second Hilbert coefficient e2 in the case e2=e1−e+1. Let h=μ(m)−d. If e2≠0 then in our case we can prove that type(A)≥e−h−1. If type(A)=e−h−1 then we show that the associated graded ring G(A) is Cohen-Macaulay. In the next case when type(A)=e−h we determine all possible Hilbert series of A. In this case we show that depthG(A) completely determines the Hilbert Series of A.

NORMAL HILBERT COEFFICIENTS AND ELLIPTIC IDEALS IN NORMAL TWO-DIMENSIONAL SINGULARITIES

AbstractLet $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and the strongly elliptic ideals of A with the aim to characterize elliptic and strongly elliptic singularities, according to the definitions given by Wagreich and Yau. In analogy with the rational singularities, in the main result, we characterize a strongly elliptic singularity in terms of the normal Hilbert coefficients of the integrally closed $\mathfrak m$ -primary ideals of A. Unlike $p_g$ -ideals, elliptic ideals and strongly elliptic ideals are not necessarily normal and necessary, and sufficient conditions for being normal are given. In the last section, we discuss the existence (and the effective construction) of strongly elliptic ideals in any two-dimensional normal local ring.

On Hilbert coefficients and sequentially Cohen-Macaulay rings

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 its Hilbert coefficients of non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients of non-parameter ideals.

The First Hilbert Coefficient of Stretched Ideals

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 explore the structure of stretched ${\mathfrak m}$ -primary ideals satisfying the equality e1(I) = e0(I) − ℓA(A/I) + 4, where e0(I) and e1(I) denote the multiplicity and the first Hilbert coefficient, respectively.

Bounds for Hilbert Coefficients

Let $(A,\mathfrak{m})$ be a noetherian local ring and $J$ an $\mathfrak{m}$-primary ideal. Elias [3] proved that $\operatorname{depth}(G(J^k))$ is constant for $k \gg 0$ and denoted this number by $\sigma(J)$. In this paper, we prove the non-positivity for the Hilbert coefficients $e_i(J)$ under some conditions for $\sigma(J)$. In case of $J = Q$ is a parameter ideal, we establish bounds for the Hilbert coefficients of $Q$ in terms of the dimension and the first Hilbert coefficient $e_1(Q)$.

Ideals of reduction number two

In a local Cohen-Macaulay ring (A, m), we study the Hilbert function of an 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 some conditions, we show the inequality e1(I) ≥ e0(I) − ℓA(A/I) + e2(I) of the Hilbert coefficients, which is the converse inequality of Sally and Itoh. We also study relations between the Hilbert coefficients and the depth of the associated graded ring.