Andrew R. Kustin, Claudia Polini, Bernd Ulrich
We give an explicit formula for the Hilbert Series of an algebra defined by a linearly presented, standard graded, residual intersection of a grade three Gorenstein ideal.
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In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field $K$, we give a new constructive proof of the existence of the almost revlex ideal $J\subset K[x_1,\dots,x_n]$, with the same Hilbert function as a complete intersection defined by $n$ forms of degrees $d_1\leq \dots \leq d_n$. Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We…
We derive the duality relation for the Hilbert series H(d^m;z) of almost symmetric numerical semigroup S(d^m) combining it with its dual H(d^m;z^{-1}). On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F_j, generators d_i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and efficient conditions for minimal set …
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 circumstances we show that the associated graded ring of the normal filtration of the ideal is either Cohen-Macaulay or almost Cohen-Macaulay.
Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals.
CI-sequences and almost complete intersections
本文研究几乎完全交阿廷代数的希尔伯特函数与分次贝蒂数，证明完全交的希尔伯特函数均可由几乎完全交实现，并在余维数3情形下分类了可能的贝蒂数与极小自由消解中的消去。
The second Hilbert coefficient of modules with almost maximal depth
本文针对深度至少为维数减一的模，推广了第二 Hilbert 系数的上下界结果，并给出了等号成立与相伴分次模深度相关的条件。
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is …
Let A = S/J be a standard artinian graded algebra over the polynomial ring S. A theorem of Macaulay dictates the possible growth of the Hilbert function of A from any degree to the next, and if this growth is the maximal possible then strong consequences have been given by Gotzmann. It can be phrased in terms of the base locus of the linear system defined by the relevant component(s) of J. If J is the artinian reduction of the ideal of a finite set of points in projective space then this maximal growth for A was shown by Bigatti, Geramita and the second author to imply strong geometric consequ…
The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results characterizing the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen-Macaulay local ring.
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 $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+4$ where $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the multiplicity and the first Hilbert coefficient respectively.
We study almost reverse lexicographic ideals in a polynomial ring over a field of arbitrary characteristic. We give a criterion for a given sequence of nonnegative integers to be the Hilbert function of an almost reverse lexicographic ideal in the polynomial ring. Then it will be shown that every Fröberg sequence satisfies this criterion.
In this paper we examine the role of four Hilbert functions in the determination of the defining relations of the Rees algebra of almost complete intersections of finite colength. Because three of the corresponding modules are Artinian, some of these relationships are very effective, opening up tracks to the determination of the equations and also to processes of going from homologically defined sets of equations to higher degrees ones assembled by resultants.
Almost Noetherian rings and modules
本文研究了近诺特环和模的性质，提出了多个重要定理，并在特定条件下解决了相关问题。
We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the Weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miró-Roig, and Nagel.
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