Shiro Goto, Ryo Takahashi, Naoki Taniguchi
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
The notion of 2-almost Gorenstein local ring (2-AGL ring for short) is a generalization of the notion of almost Gorenstein local ring from the point of view of Sally modules of canonical ideals. In this paper, for further developments of the theory, we discuss three different topics on 2-AGL rings. The first one is to clarify the structure of minimal presentations of canonical ideals, and the second one is the study of the question of when certain fiber products, so called amalgamated duplications are 2-AGL rings. We also explore Ulrich ideals in 2-AGL rings, mainly two-generated ones.
This paper studies Ulrich ideals in hypersurface rings. A characterization of Ulrich ideals is given. Using the characterization, we construct a minimal free resolution of an Ulrich ideal concretely. We also explore Ulrich ideals in a hypersurface ring of the form R = k[[X, Y]]/(f).
This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where the rings are of minimal multiplicity and \operatorname{GGL} rings are analyzed.
The notion of $2$-almost Gorenstein ring is a generalization of the notion of almost Gorenstein ring in terms of Sally modules of canonical ideals. In this paper, we deal with two different topics related to $2$-almost Gorenstein rings. The purposes are to determine all the Ulrich ideals in $2$-almost Gorenstein rings and to clarify the structure of minimal free resolutions of $2$-almost Gorenstein rings.
Koszul homology of codimension 3 Gorenstein ideals
本文计算了余维为3的Gorenstein理想的Koszul同调.我们发现过滤的Koszul同调的模块与纯自由分解，并完全描述这些决议。我们还考虑Huneke-Ulrich偏差2理想。
Ulrich ideals on rational triple points of dimension two
本文证明了二维有理三重点的典范迹理想是 Ulrich 理想，并据此分类了有理三重点及重数至少为 4 的二维商奇点上的所有 Ulrich 理想。
Almost Gorenstein simplicial semigroup rings
本文给出单纯半群环具有几乎Gorenstein性质的判别准则，将Nari定理推广到高秩单纯半群，并据此分类了具有Ulrich元素的二维正规半群环。
The Ulrich ideals in the semigroup rings $k[[t^5, t^{11}]]$ and $k[[t^5,t^6,t^9]]$ are determined, by describing the normal forms of systems of generators, where $k[[t]]$ denotes the formal power series ring over a field $k$.
Ulrich ideals in numerical semigroup rings of small multiplicity are studied. If the semigroups are three-generated but not symmetric, the semigroup rings are Golod, since the Betti numbers of the residue class fields of the semigroup rings form an arithmetic progression; therefore, these semigroup rings are G-regular, possessing no Ulrich ideals. Nevertheless, even in the three-generated case, the situation is different, when the semigroups are symmetric. We shall explore this phenomenon, describing an explicit system of generators, that is the normal form of generators, for the Ulrich ideals…
Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals
研究了由一般矩阵的子式定义的行列式环的初始代数关于其经典泛点的性质。这种方法导致非常短的证明结构性质的行列式环。此外，它允许我们分类他们的Cohen-Macaulay和Ulrich理想。
Ulrich ideals and modules over two-dimensional rational singularities
本文的主要目的是从几何的角度对二维Gorenstein有理奇点（有理二重点）上的Ulrich理想和Ulrich模进行分类.为了达到这个目的，我们引入了关于理想的（弱）特殊Cohen-Macaulay模的概念，并研究了这些模与关于好理想的Ulrich模之间的关系.
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。
Ulrich modules : decomposition and relation with rings having minimal multiplicity
该研究探讨Cohen-Macaulay局部环中Ulrich模的基本性质，证明其可分解，并据此讨论具有极小重数环的若干刻画及作为模为Ulrich的理想的性质。
Ulrich ideals and modules
本文从不同的角度研究了Cohen-Macaulay局部环上的Ulrich理想和Ulrich模。确定了Ulrich模及其分次模的极小自由分解的结构，并对数值半群环和有限CM-表示型环的Ulrich理想进行了分类.
Non-existence of Ulrich modules over Cohen-Macaulay local rings
本文通过构造二维Cohen-Macaulay局部环族，否定回答了Ulrich关于是否总存在达到生成元数上界的模的问题，并给出Gorenstein正规整环与完全交整环的反例。
Full-Trace Modules
本研究在交换Noetherian局部环上引入全迹模概念，证明非正则非主理想环上剩余域的正合冲模均为全迹模，刻画了Cohen-Macaulay环极小重数与全迹Ulrich模的等价关系，并给出数值半群环上全迹Ulrich模的分解形式。
The Eisenbud-Huneke-Ulrich conjecture and bounds on minimal generators
本文在特征零下证明Eisenbud--Ulrich猜想，并推广至部分线性分解情形，通过高阶syzygy丛的整体生成结果，建立二次型生成及几乎线性表示理想的极小生成元下界，提出锐利下界猜想。
The Auslander-Reiten conjecture is a notorious open problem about the vanishing of Ext modules. In a Cohen-Macaulay complete local ring $R$ with a parameter ideal $Q$, the Auslander-Reiten conjecture holds for $R$ if and only if it holds for the residue ring $R/Q$. In the former part of this paper, we study the Auslander-Reiten conjecture for the ring $R/Q^\ell$ in connection with that for $R$, and prove the equivalence of them for the case where $R$ is Gorenstein and $\ell\le \dim R$. In the latter part, we generalize the result of the minimal multiplicity by J. Sally. Due to these two of our…
Constructing Homogeneous Gorenstein Ideals
在1983年Kustin和米勒介绍了建设Gorenstein理想在当地Gorenstein环，从较小的理想。我们在分次环的情形下回顾和修改了它们的构造，并在Gorenstein联络理论的框架下讨论了它。我们确定所构造理想的不变量。对于给定的Gorenstein理想何时能由构造法得到的问题，我们给出了一个必要条件，并证明了一个Gorenstein理想不能由构造法得到.
Higher syzygy bundles and the Eisenbud-Huneke-Ulrich conjecture
研究具有部分线性（虚拟）分解的多项式环中的理想，证明了Eisenbud-Huneke-Ulrich猜想的一个稍弱版本，并推广了Macaulay界和Eisenbud-Huneke-Ulrich定理。
Let $H\subseteq \mathbb{N}^d$ be a normal affine semigroup, $R=K[H]$ its semigroup ring over the field $K$ and $ω_R$ its canonical module. The Ulrich elements for $H$ are those $h$ in $H$ such that for the multiplication map by $\mathbf{x}^h$ from $R$ into $ω_R$, the cokernel is an Ulrich module. We say that the ring $R$ is almost Gorenstein if Ulrich elements exist in $H$. For the class of slim semigroups that we introduce, we provide an algebraic criterion for testing the Ulrich propery. When $d=2$, all normal affine semigroups are slim. Here we have a simpler combinatorial description of th…
The long standing Lech's conjecture in commutative algebra states that for a flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq e(S)$. In general the conjecture is wide open when $\dim R>3$, even in equal characteristic. In this paper, we prove Lech's conjecture in all dimensions, provided $(R,\mathfrak{m})$ is a standard graded ring over a perfect field localized at the homogeneous maximal ideal. We introduce the notions of lim Ulrich and weakly lim Ulrich sequences. Roughly speaking the…
We investigate the existence of ideals $I$ in a one-dimensional Gorenstein local ring $R$ satisfying $\mathrm{Ext}^{1}_{R}(I,I)=0$.
Gorenstein global dimension of an amalgamated duplication of a coherent ring along an ideal
本文研究了凝聚环的\emph{amalgamated duplication}沿正则主理想的Gorenstein整体维数.
Links of symbolic powers of prime ideals
本文对Polini和Ulrich提出的一个猜想给出了肯定的回答。
A proof of the Eisenbud-Huneke-Ulrich conjecture
本文证明了特征零下具有部分线性最小自由分解的m-准素理想幂的Eisenbud-Huneke-Ulrich猜想。
We provide the sufficient conditions for Rees algebras of modules to be Cohen-Macaulay, which has been proven in the case of Rees algebras of ideals by Johnson-Ulrich and Goto-Nakamura-Nishida. As it turns out the generalization from ideals to modules is not just a routine generalization, but requires a great deal of technical development. We use the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos to obtain the Cohen-Macaulayness of Rees Algebras of modules.
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.
Gorenstein rings and irreducible parameter ideals
给定一个Noether局部环（R，m），证明了存在整数l使得R是Gorenstein当且仅当m^l中的某个参数系生成一个不可约理想.作为推论，我们得到R是Gorenstein当且仅当R的每个极大理想的幂都包含一个不可约参数理想。
Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings
本文研究了一维Cohen-Macaulay局部环上最小乘数模的性质，通过刻画跟踪理想和反射理想的作用，揭示了极小乘数模与Burch和Ulrich模的关系。
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.
Structure theorems for certain Gorenstein ideals
本文的主要成果是给出了具有极大理想的平方由两个元素生成性质的Artinian，Gorenstein局部环的一个结构定理。讨论了这类局部代数的模问题。最后，给出了完全理想生成元的最小个数的上界和下界。
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 the present paper we investigate reflexive modules over the endomorphism algebras of reflexive trace ideals in a one-dimensional Cohen-Macaulay local ring. The main theorem generalizes both of the results of S. Goto, N. Matsuoka, and T. T. Phuong and T. Kobayashi concerning the endomorphism algebra of its maximal ideal. We also explore the question of when the category of reflexive modules is of finite type, i.e., the base ring has only finitely many isomorphism classes of indecomposable reflexive modules. We show that, if the category is of finite type, the ring is analytically unramified …
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.
Socle finiteness of local cohomology modules and Gorenstein ideals
本文给出了C. Huneke在（局部上同调问题，自由分解交换代数和代数几何，Res.笔记数学。1992年，页。93-108）。此外，还得到了一些关于Gorenstein理想的结果.
On the Betti numbers of some Gorenstein ideals
设$R$是域上的多项式环，$I$是余维为$g\ge3$且初度为$p\ge2$的齐次Gorenstein理想.证明了$I$的在度$p$中的极小生成元$ν(I_p)$的个数由$ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$上有界，$ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$是余维$g$和初度$p$的极值Gorenstein代数的定义理想的极小生成元个数.此外，如果$ν(I_p)=ν_0$，则$I$本身是极值的。
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.
Quasi-socle ideals in a Gorenstein local ring
本文研究Gorenstein局部环A中的拟柱理想I=Q：m^2的结构，其中Q是参数理想，m是A中的极大理想.目的是回答当Q是I的约化时，当伴随的分次环G（I）= \bigoplus_{n \geq 0} I^n/I^{n+1}是Cohen-Macaulay的问题.野生的例子进行了探索。
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.
Quasi-socle ideals in Gorenstein numerical semigroup rings
研究域上Gorenstein数值半群环中的拟socle理想，即$I= Q : \mathfrak{m}^q$形式的理想$I$，其中$Q$是参数理想，$\mathfrak{m}$是基局部环中的极大理想，$q \geq 1$是整数.研究了$I$何时是$Q$上的积分环以及$I$的分次环$\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$何时是Cohen-Macaulay环的问题.问题是相当疯狂的;例子。
Additions to a theorem of Morey-Ulrich
本文针对满足条件G_{d-1}但不满足G_d的完美理想，扩展了Morey-Ulrich定理的表述。
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 $S\to R$.
Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace modul…
Fitting ideals and the Gorenstein property
设p是素数，G是有限交换群，使得p^{2}不整除G的阶.本文证明了对群环Z_{p} [G]上的每个有限模M，不等式#MPDMATHABMN # Z_{p} [G]/Fit_{ Z_{p} [G]}（M）成立.这里，Fit_{ Z_{p} [G]}（M）是M的Z_{p} [G]拟合理想。
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.
Annihilation of cohomology over one dimensional almost Gorenstein rings
本文研究一维几乎Gorenstein环的上同调湮灭理想，给出其与稳定范畴湮灭理想相等的充分条件，并证明其与导子理想一致。
We show that the ideal generated by the $(n-2)$ minors of a general symmetric $n$ by $n$ matrix has an initial ideal that is the Stanley-Reisner ideal of the boundary complex of a simplicial polytope and has the same Betti numbers.
Tight closure and linkage classes in Gorenstein rings
研究了理想的紧闭包与其环类中所有理想之和之间的关系。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。