Shiro Goto, Naoyuki Matsuoka, Ryo Takahashi
This paper explores the structure of quasi-socle ideals I=Q:m^2 in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problem of when Q is a reduction of I and when the associated graded ring G(I) = \bigoplus_{n \geq 0}I^n/I^{n+1} is Cohen-Macaulay. Wild examples are explored.
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The purpose of this paper is to give some equivalent conditions to the socle and Bass numbers' conjectures which raised by C. Huneke in (Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 1992, pp. 93-108). In addition, some results about certain Gorenstein ideals are included.
The weak Lefschetz property for Standard graded, Artinian Gorenstein algebras of embedding dimension four and socle degree three
该研究针对嵌入维数4、基座次数3的标准分次Artinian Gorenstein代数的弱Lefschetz性质展开，证明除特征2下的唯一特例外均满足该性质，明确了例外的特征条件。
Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal ideal in the base local ring, and $q \geq 1$ is an integer. The problems of when $I$ is integral over $Q$ and of when the associated graded ring $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are studied. The problems are rather wild; examples are given.
Koszulness of Gorenstein quadratic algebras of small socle degree is studied. In this note, we construct non-Koszul Gorenstein quadratic toric ring such that its socle degree is more than 3 by using stable set polytopes.
Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in a Noetherian local ring $(A, \mathfrak{m})$ with the Gorenstein tangent cone $\mathrm{G}(\mathfrak{m}) = \bigoplus_{n \geq 0}{\mathfrak{m}}^n/{\mathfrak{m}}^{n+1}$ are explored, where $q \geq 1$ is an integer and $Q$ is a parameter ideal of $A$ generated by monomials of a system $x_1, x_2, ..., x_d$ of elements in $A$ such that $(x_1, x_2, ..., x_d)$ is a reduction of $\mathfrak{m}$. The questions of when $I$ is integral over $Q$ and of when the graded rings $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ and $…
There is given a characterization for the Rees algebras of parameters in a Gorenstein local ring to be almost Gorenstein graded rings. A characterization is also given for the Rees algebras of socle ideals of parameters. The latter one shows almost Gorenstein Rees algebras rather rarely exist for socle ideals, if the dimension of the base local ring is greater than two.
Let $K$ be an algebraically closed field of characteristic $0$, and let $A$ be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, with maximal ideal $M$. In the present paper we prove a structure theorem describing such kind of $K$--algebras satisfying $M^4=0$. We use this result in order to prove that such a $K$--algebra $A$ has rational Poincaré series and it is always smoothable in any embedding dimension, if $\dim_K M^2/M^3 \le 4$. We also prove that the generic Artinian Gorenstein local $K$--algebra with socle degree three has rational Poincaré series, in spite of the f…
Artinian Gorenstein algebras of embedding dimension four and socle degree three over an arbitrary field
本文研究任意域上嵌入维数为四、基座阶为三的Artinian Gorenstein代数，通过Macaulay逆系统给出其极小预解式的显式公式，并依据3乘3对称矩阵SM的秩完全刻画核的生成元个数及结构。
Artinian Gorenstein algebras of embedding dimension four and socle degree three
本文证明了在特征零代数闭域的多项式环中，特定Gorenstein齐次理想可由三级完美理想通过加倍构造，并给出了相应的分次极小自由分解的完整描述。
We classify all possible $h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and codimension $\leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $h$-vector is $(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a long-standing open question in this area. All of our results are characteristi…
In this paper, we explore a relationship between Hilbert functions and the irreducible decompositions of ideals in local rings. Applications are given to characterize the regularity, Gorensteinness, Cohen-Macaulayness and sequentially Cohen-Macaulayness of local rings.
In this paper we study the O-sequences of the local (or graded) $K$-algebras of socle degree $4.$ More precisely, we prove that an O-sequence $h=(1, 3, h_2, h_3, h_4)$, where $h_4 \geq 2,$ is the $h$-vector of a local level $K$-algebra if and only if $h_3\leq 3 h_4.$ We also prove that $h=(1, 3, h_2, h_3, 1)$ is the $h$-vector of a local Gorenstein $K$-algebra if and only if $h_3 \leq 3$ and $h_2 \leq \binom{h_3+1}{2}+(3-h_3).$ In each of these cases we give an effective method to construct a local level $K$-algebra with a given $h$-vector. Moreover we refine a result by Elias and Rossi by sho…
We introduce classes of rings which are close to being Gorenstein. These rings arise naturally as specializations of rings of countable CM type. We study these rings in detail, and along the way generalize an old result of Teter which characterized Artinian rings which are Gorenstein rings modulo their socle.
In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Fröberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring. This unexpected result has been extended to compressed level K-algebras of socle degree 3 by A. De Stefani. In this paper we end the investigation proving that extremal Artin Gorenstein…
The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
本文证明了基座度数为d且Sperner数至多为d+1的artinian Gorenstein代数均满足弱Lefschetz性质，且与余维数无关，并指出该界限在一般情况下是严格的。
Secants, Socles and Stability
该研究将d次对称张量射影空间重新解释为基座在d次的有限分次Gorenstein环射影空间，通过奇偶次不同的显式稳定性条件按Harder-Narasimhan滤过型划分空间，显式推导三维低次实例并与Gorenstein环Betti表比较。
Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever $(1,h_1,\dots,h_i,\dots,h_{e-i},\dots,h_{e-1},1)$ and $(1,h_1,\dots,h_i+α,\dots,h_{e-i}+α,\dots,h_{e-1},1)$ are both Gorenstein Hilbert functions for some $α\geq 2$, then $(1,h_1,\dots,h_i+β,\dots,h_{e-i}+β,\dots,h_{e-1},1)$ is also Gorenstein, for all $β=1,2,\dots,α-1$. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such ba…
Let $A$ be an Artinian Gorenstein algebra over an infinite field $k$ with either $\hbox{char}(k)=0$ or $\hbox{char}(k)>ν$, where $ν$ is the socle degree of $A$. To every such algebra and a linear projection $π$ on its maximal ideal ${\mathfrak m}$ with range equal to the socle $\hbox {Soc}(A)$ of $A$, one can associate a certain algebraic hypersurface $S_π\subset{\mathfrak m}$, which is the graph of a polynomial map $P_π:\hbox{ker}\,π\to \hbox{Soc}(A)\simeq k$. Recently, the author and his collaborators have obtained the following surprising criterion: two Artinian Gorenstein algebras $A$, $\t…
We introduce a family of standard bigraded binomial Artinian Gorenstein algebras, whose combinatoric structure characterizes the ones presented by quadrics. These algebras provide, for all socle degree grater than two and in sufficiently large codimension with respect to the socle degree, counter-examples to Migliore-Nagel conjectures, see \cite{MN1} and \cite{MN2}. One of them predicted that Artinian Gorenstein algebras presented by quadratics should satisfy the weak Lefschetz property. We also prove a generalization of a Hessian criterion for the Lefschetz properties given by Watanabe, see \…
For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $\mathrm{Soc}(I)$, whose graded components give us the socle of the powers of $I$. It is observed that $\mathrm{Soc}(I)$ is a finitely generated module over the fiber cone of $I$. In the case that $S$ is the polynomial ring and all powers of $I\subseteq S$ have linear resolution, we define the module $\mathrm{Soc}^*(I)$ which is a module over the Rees ring of $I$. For the edge ideal of a graph and for classes of polymatroidal ideals we study the …
In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay homogeneous ring $R$, we give a sufficient condition for $R$ to be almost Gorenstein in terms of the $h$-vector of $R$ and we also characterize almost Gorenstein homogeneous domains with small socle degrees in terms of the $h$-vector of $R$. Moreover, we also provide the examples of almost Gorenstein homogeneous domains arising from lattice polytopes.
Generic local rings on a spectrum between Golod and Gorenstein
本文研究了局部环商在Golod与Gorenstein之间的分类，证明了socle秩为2的分级商环的代数结构呈现平凡扩张的泛型行为，且该行为由socle多项式的阶数与次数之差控制。
Let $R$ be a formal power series ring over a field, with maximal ideal $\mathfrak m$, and let $I$ be an ideal of $R$ such that $R/I$ is Artinian. We study the iterated socles of $I$, that is the ideals which are defined as the largest ideal $J$ with $J\mathfrak m^s\subset I$ for a fixed positive integer $s$. We are interested in these ideals in connection with the notion of integral dependence of ideals. In this article we show that the iterated socles are integral over $I$, with reduction number one, provided $s \leq \text{o}(I_1(φ_d))-1$, where $\text{o}(I_1(φ_d))$ is the order of the ideal …
Let R be a polynomial ring in r variables and D a dual ring upon which R acts as partial differential operators (classical apolarity). For a type two graded level Artinian algebras A=R/I, of socle degree j we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to a unique 2-dimensional vector space W(A)=<F,G> of degree-j homogeneous elements (forms) in D, and each such AG quotient of A corresponds to an element $F_λ=F+λG$ in W(A), up to non-zero constant multiple. Our main result is a lower bound for the Hilbert functi…
We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the {\em Interval Conjecture} (IC): If, for some positive integer $α$, $(1,h_1,...,h_i,...,h_e)$ and $(1,h_1,...,h_i+α,...,h_e)$ are both level $h$-vectors, then $(1,h_1,...,h_i+β,...,h_e)$ is also level for each integer $β=0,1,..., α.$ In the Gorenstein case, i.e. when $h_e=1$, we also supply the {\em Gorenstein Interval Conjecture} (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetr…
On posets, monomial ideals, Gorenstein ideals and their combinatorics
本文从组合角度比较单项式理想的 socle 元素与 Macaulay 逆系统，建立相关偏序集的闭包性质，推导 Γ-分次环的代数命题及其组合推论，并对零维 Gorenstein 理想进行直接计算，揭示多项式环与局部环之间的关键差异。
Let R be a local ring with maximal ideal m admitting a non-zero element a\in\fm for which the ideal (0:a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M, with particular attention to the case when m^4=0. Let e denote the minimal number of generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely generated R-module M. In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.
For a finite subset $M\subset [x_1,\ldots,x_d]$ of monomials, we describe how to constructively obtain a monomial ideal $I\subseteq R = K[x_1,\ldots,x_d]$ such that the set of monomials in $\text{Soc}(I)\setminus I$ is precisely $M$, or such that $\overline{M}\subseteq R/I$ is a $K$-basis for the the socle of $R/I$. For a given $M$ we obtain a natural class of monomials $I$ with this property. This is done by using solely the lattice structure of the monoid $[x_1,\ldots,x_d]$. We then present some duality results by using anti-isomorphisms between upsets and downsets of $(\mathbb Z^d,\preceq)$…
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \in S$ belonging to the socle $\Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u \not\in I^{k}$ and $ux_{i} \in I^{k}$ for $1 \leq i \leq n$. We prove that if a monomial $x_1^{a_1}\cdots x_n^{a_n}$ belongs to $\Soc(S/I^{k})$, then $a_i\leq k-1$ for all $1 \leq i \leq n$. We then discuss squarefree monomial ideals $I \subset S$ for which $x_{[n]}^{k-1} \in \Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}\cdots x_{n}$. Furtherm…
In 1983 Kustin and Miller introduced a construction of Gorenstein ideals in local Gorenstein rings, starting from smaller such ideals. We review and modify their construction in the case of graded rings and discuss it within the framework of Gorenstein liaison theory. We determine invariants of the constructed ideal. Concerning the problem of when a given Gorenstein ideal can be obtained by the construction, we derive a necessary condition and exhibit a Gorenstein ideal that can not be obtained using the construction.
The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the sphere $g$-conjecture implies all enumerative consequences of its far reaching generalization (due to Kalai) to manifolds. A special case of Kalai's manifold $g$-conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.
Macaulay's inverse system is an effective method to construct Artinian K-algebras with additional properties like, Gorenstein, level, more generally with any socle type. Recently, Elias and Rossi gave the structure of the inverse system of $d$-dimensional Gorenstein K-algebras for any $d>0$. In this paper we extend their result by establishing a one-to-one correspondence between $d$-dimensional level K-algebras and certain submodules of the divided power ring. We give several examples to illustrate our result.
We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.
How many ideals whose quotient rings are Gorenstein exist?
本文研究Gorenstein环中商环为Gorenstein的非主理想数量问题，证明对称数值半群环中此类分次理想个数等于半群导子，并给出完整列表。
In this paper, we study the Gorenstein global dimension of an \emph{amalgamated duplication} of a coherent ring along a regular principal ideal.
The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.
Characterization of almost Gorenstein rings in terms of the trace ideal
本文用迹理想刻画一维几乎Gorenstein环，并应用于研究$\mathbb{Z}_2$-分次环的几乎Gorenstein性质。
Level and pseudo-Gorenstein binomial edge ideals
本文证明正则度2的level二项式边理想与正则度3的伪Gorenstein二项式边理想均为锥并完全描述之，同时刻画二部图的相应理想。
This note proves that if S is an unramified regular local ring and I and J are proper ideals of height at least two, then S/IJ is never Gorenstein.
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.
Given a Noetherian local ring (R,m) it is shown that there exists an integer l such that R is Gorenstein if and only if some system of parameters contained in m^l generates an irreducible ideal. We obtain as a corollary that R is Gorenstein if and only if every power of the maximal ideal contains an irreducible parameter ideal.
This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra $\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di}$ is Gorenstein for every parameter ideal $\mathfrak{q}$ that is a reduction of the maximal ideal in a $d$-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.
Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K \cong A/{\mathfrak m}$ and $I$ is a $p_g$-ideal; (2) $(A,{\mathfrak m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity and $I={\mathfrak m}^{\ell}$ for all $\ell \ge 1$; (3) $(A,{\mathfrak m})$ is a regular local ring of dimension $d \ge 2$ and $I={\mathfrak m}…
Via the theory of reverse lexicographic squarefree initial ideals of toric ideals, we give a new class of Gorenstein Fano polytopes (reflexive polytopes) arising from a pair of stable set polytopes of perfect graphs.
The main achievement of this paper is to provide a structure theorem for Artinian, Gorenstein local rings with the property that the square of the maximal ideal is generated by two elements. The moduli problem for this class of local algebras is also discussed. Finally, upper and lower bounds for the minimal number of generators of perfect ideals are given.
Structure theorems for Gorenstein ideals of codimension four with small number of generators
本文用表示论方法研究余维数为四的Gorenstein理想的极小自由消解，引入高阶结构映射族，并证明由六个元素极小生成的该类理想必为余维数为三的Gorenstein理想的超平面截面，同时提出七、八个生成元情形的猜想。
Let $R$ be a Noetherian local ring. We prove that $R$ is regular of dimension at most four if, and only if, every prime ideal, defining a Gorenstein quotient ring, is syzygetic. We deduce a characterization of these rings in terms of the André-Quillen homology.
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
In this paper we study the isomorphism classes of Artinian Gorenstein local rings with socle degree three by means of Macaulay's inverse system. We prove that their classification is equivalent to the projective classification of the hypersurfaces of $\mathbb P ^{n }$ of degree three. This is an unexpected result because it reduces the study of this class of local rings to the homogeneous case. The result has applications in problems concerning the punctual Hilbert scheme $Hilb_d (\mathbb P^n)$ and in relation to the problem of the rationality of the Poincaré series of local rings.
Let $R$ be a standard graded algebra over a field $k$ and $I$ be a homogeneous ideal of $R$. We study the question whether there is a constant $c$ such that $\Soc(H^{j}_{\fm}(R/I^t))_{<-ct}=0$ for all $t\geq 1$ and a variation of this question. We also draw a connection between this question and the notion of gauge-boundedness in prime characteristic.
Gorenstein Endomorphism Rings on Curve Singularities
本文研究曲线奇点上分式理想自同态环的 Gorenstein 性，利用理想稳定性与值半群理想条件给出刻画，并分类相关 Gorenstein 代数型曲线。
The question of when the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring is explored, where $R$ is a two-dimensional regular local ring and $I$ a contracted ideal of $R$. It is known that ${\mathcal R} (I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present paper show that if $I$ is a contracted ideal with $\mathrm{o}(I) \le 2$, then ${\mathcal R} (I)$ is an almost Gorenstein graded ring, while if $\mathrm{o}(I) \ge 3$, then ${\mathcal R} (I)$ is not necessarily an almost Gorenstei…
In this paper, we introduce an invariant of Cohen-Macaulay local rings in terms of the reduction number of canonical ideals. The invariant can be defined in arbitrary Cohen-Macaulay rings and it measures how close to being Gorenstein. First, we clarify the relation between almost Gorenstein rings and nearly Gorenstein rings by using the invariant in dimension one. We next characterize the idealization of trace ideals over Gorenstein rings in terms of the invariant. It provides better prospects for a result of the almost Gorenstein property of idealization.
Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra P padded with a non-zero graded vector space on which P_{\ge 1} acts trivially. We explicitly construct an infinite family of such rings.
Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $ν(I_p)$ of $I$ that are in degree $p$ is bounded above by $ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $ν(I_p)=ν_0$.
Let $I_G$ be the binomial edge ideal on the generic 2 x n - Hankel matrix associated with a closed graph $G$ on the vertex set [n]. We characterize the graphs $G$ for which $I_G$ has maximal regularity and is Gorenstein.
Let $k$ be a field and $G \subseteq Gl_n(k)$ be a finite group with $|G|^{-1} \in k$. Let $G$ act linearly on $A = k[X_1, \ldots, X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional representation of $G$ over $k$. Then we show that if $Q$ is a $G$-invariant homogeneous ideal of $A$ such that $A/Q$ is a Gorenstein ring then $A^G/Q^G$ is also a Gorenstein ring.
Let R be a local Noetherian commutative ring. We prove that R is an Artinian Gorenstein ring if and only if every ideal in R is a trace ideal. We discuss when the trace ideal of a module coincides with its double annihilator.
Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$ such that the ideal $I$ is the sum of images of all $R$-linear maps $S\to R$.
On generalized Gorenstein local rings
本文提出广义戈林斯坦局部环的概念，并探讨其与Cohen-Macaulay环相关代数结构的联系，同时提供了多种实例。
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