Alberto Corso, Claudia Polini, Maria Vaz Pinto
We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with respect to a minimal reduction J of I.
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
The notion of 2-AGL ring in dimension one which is a natural generalization of almost Gorenstein local ring is posed in terms of the rank of Sally modules of canonical ideals. The basic theory is developed, investigating also the case where the rings considered are numerical semigroup rings over fields. Examples are explored.
We introduce the concept of $j$-stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched $\mathfrak{m}$-primary ideals of Sally and Rossi-Valla, as well as the concept of ideals of minimal and almost minimal $j$-multiplicity introduced by Polini-Xie. One of our main theorems states that, for a $j$-stretched ideal, the associated graded ring is Cohen-Macaulay if and only if two classical invariants of the ideal, the reduction number and the index of nilpotency, are equal. Our second main theorem, presenting numerical conditions w…
We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) To develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen-Macaulay rings to more general rings. (ii) To use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen-Macaulayness of the special fiber ring. (iii) To extend some of the results of (i) to two-dimension…
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.
Given a local Noetherian ring $(R, {\mathfrak m})$ of dimension $d>0$ and infinite residue field, we study the invariants $($dimension and multiplicity$)$ of the Sally module $S_J(I)$ of any ${\mathfrak m}$-primary ideal $I$ with respect to a minimal reduction $J$. As a by-product we obtain an estimate for the Hilbert coefficients of ${\mathfrak m}$ that generalizes a bound established by J. Elias and G. Valla in a local Cohen-Macaulay setting. We also find sharp estimates for the multiplicity of the special fiber ring ${\mathcal F}(I)$, which recover previous bounds established by C. Polini, …
The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. In this paper we give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. As a consequence, in this case we present a complete description of the H…
In this paper, we explore the structure of the normal Sally modules of rank one with respect to an $m$-primary ideal in a Nagata reduced local ring which is not necessary Cohen-Macaulay. As an application of this result, when the base ring is Cohen-Macaulay analytically unramified, the extremal bound on the first normal Hilbert coefficient leads to the Cohen-Macaulayness of the associated graded rings with respect to a normal filtration.
The Sally module of a Rees algebra $\BB$ relative to one of its Rees subalgebras $Å$ is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated graded ring for several types of filtrations. While originally devised to deal with filtrations of finite colength, here we treat aspects of these developments for filtrations in higher dimensions as well.
Given a finite module $M$ over a Noetherian local ring $(R, \m)$, we introduce the concept of $j$-stretched ideals on $M$. Thanks to a crucial specialization lemma, we show that this notion greatly generalizes (to arbitrary ideals, and with respect to modules) the classical definition of stretched $\m$-primary ideals of Sally and Rossi-Valla, as well as the notion of minimal and almost minimal $j$-multiplicity given recently by Polini-Xie. For $j$-stretched ideals $I$ on a Cohen-Macaulay module $M$, we show that ${\rm gr}_I(M)$ is Cohen-Macaulay if and only if two classical invariants of $I$, …
Numerical Semigroups of Sally Type
本文研究Sally类型的数值半群，推导其Frobenius数公式，并给出其为Gorenstein的条件，同时计算了半群环的基本生成元数目及Betti数。
The structure of Sally modules of $\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \m)$ satisfying the equality $\e_1(I)=\e_0(I)-\ell_A(A/I)+1$ is explored, where $\e_0(I)$ and $\e_1(I)$ denote the first two Hilbert coefficients of $I$.
In 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 $3$.
The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen-Macaulay local ring $A$ satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+\ell_A(I^2/QI)+1, $ where $Q$ is a minimal reduction of $I$, and $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the first two Hilbert coefficients of $I, $ respectively the multiplicity and the Chern number of $I$. This almost extremal value of $\mathrm{e}_1(I) $ …
The Betti numbers and Golodness of numerical semigroup rings of Sally type
本文确定了 Sally 型数值半群环的 Betti 数显式公式，推广了相关猜想，并证明固定重数下环的 Poincaré 级数相同但偏离 Serre 上界的时间可任意晚。
A complete structure theorem of Sally modules of $\fkm$-primary ideals $I$ in a Cohen-Macaulay local ring $(A, \m)$ satisfying the equality $\e_1(I)=\e_0(I)-\ell_A(A/I)+1$ is given, where $\e_0(I)$ and $\e_1(I)$ denote the first two Hilbert coefficients of $I$.
Structure and Symmetry of Sally Type Semigroup Rings
本文研究Sally类型半群的结构与对称性，证明了特定条件下半群为Gorenstein，并构造了极小自由解析及Betti数计算。
Let $R$ be a Noetherian local ring. We define the minimal $j$-multiplicity and almost minimal $j$-multiplicity of an arbitrary $R$-ideal on any finite $R$-module. For any ideal $I$ with minimal $j$-multiplicity or almost minimal $j$-multiplicity on a Cohen-Macaulay module $M$, we prove that under some residual assumptions, the associated graded module ${\rm gr}_I(M)$ is Cohen-Macaulay or almost Cohen-Macaulay, respectively. Our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained by Sally, Rossi, Valla, Wang, Huckaba, Elias, Corso, Polini, and VazPinto…
In a local Cohen-Macaulay ring $(A, \mathrm{m})$, we study the Hilbert function of an $\mathrm{m}$-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 $\mathrm{e}_1(I)\ge \mathrm{e}_0(I) - \ell_A (A/I) + \mathrm{e}_2(I)$ 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.
In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most two. In this paper, having reduced the conjecture to the case of dimension three, if the ring is regular and local of dimension three, we explicitly describe a family of prime ideals of height two minimally generated by three elements. Weakening the hypothesis of regularity, we find that, to achieve the same end, we need to add extra hypotheses, such as complet…
Let $R$ be a Cohen-Macaulay local ring, and let $I\subset R$ be an ideal with minimal reduction $J$. In this paper we attach to the pair $I$, $J$ a non-standard bigraded module $Σ^{I,J}$. The study of the bigraded Hilbert function of $\SIJ$ allows us to prove a improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring $gr_I(R)$. The module $\SIJ$ can be considered as a refinement of the Sally's module previously introduced by W. Vasconcelos.
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.
On the core and adjoint of the product of complete ideals in two-dimensional regular local rings
该研究借助完全理想的联合约化，给出二维正则局部环中完全理想乘积的核心、伴随理想表达式及其余长度计算方法，强化了Rees与Sally对Briançon-Skoda定理的推广结果。
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.
Let $R$ be a $G$-graded ring. In this article, we introduce two new concepts on graded rings, namely, weakly graded rings and invertible graded rings, and we discuss the relations between these concepts and several properties of graded rings. Also, we study the concept of weakly crossed products and study some properties defined on weakly crossed product and give the relationship between this new concept and several properties of graded rings. Moreover, in this article, we give a generalization for the concept of graded essential submodules, and introduce the concept of graded semi-uniform mod…
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded classical and graded strongly classical 2-absorbing second submodules of graded modules over a graded commutative rings. A number of results concerning these classes of graded submodules and their homogeneous components are given.
Let $k$ be a field. We determine the ideals $I$ in a finitely generated graded $k$-algebra $A$, whose associated graded rings are isomorphic to $A$. Also we compute the graded local cohomologies of the Rees rings $A[I t]$ and give the condition for $A[I t]$ to be generalized Cohen-Macaulay.
This paper deals with the graded commutative rings in which every homogeneous prime ideal is contained in a unique homogeneous maximal ideal called Gelfand graded ring. The purpose is to establish some topological and algebraic characterizations of these rings, one of which is the algebraic analogue of the Urysohn's lemma. Finally we look at a special class of those graded rings called pm$^+$ graded rings which can be viewed as graded ring with a Gelfand strong property.
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…
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded primary-like submodules as a new generalization of graded primary ideals and give some basic results about graded primary-like submodules of graded modules. Special attention has been paid, when graded submodules satisfies the gr-primeful property, to find extra properties of these graded submodules.
Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field. Let $I$ be an ideal of $R$ that has analytic spread $\ell(I)=d$, satisfies the $G_d$ condition, the weak Artin-Nagata property $AN_{d-2}^-$ and depth$(R/I)\geq\min\lbrace 1,\dim R/I \rbrace$. In this paper, we show that if $j_1(I) = λ(I/J) +λ[R/(J_{d-1} :_{R} I+(J_{d-2} :_{R}I+I) :_R, \mathfrak{m}^\infty)]+1$, then depth$(G(I))\geq d -1$ and $r_J(I)\leq 2$, where $J$ is a general minimal reduction of $I$. In addition, we extend the result by Sally who has studied the depth of associated graded ri…
Almost Gorenstein standard graded rings and their localizations
本文证明了域上Cohen-Macaulay标准分次整环为几乎Gorenstein分次环，等价于其在分次极大理想处局部化的完备化为几乎Gorenstein局部环。
Hyperplane sections of non-standard graded rings
本文提出了一种将标准分级环上的伯特尼型定理推广到非标准分级环的方法。
The article proved the upper bound of leading coefficient of characteristic polynomial of graded ideal in a ring of generalized polynomials.
Let $R$ be a $G$-graded ring and M be a $G$-graded $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to be the set of all graded primary submodules $Q$ of M such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we define a topology on $\mathcal{PS}_G(M)$ having the Zariski topology on the graded prime spectrum $Spec_G(M)$ as a subspace topology, and investigate several topological properties of this topological space.
In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.
Matijevic-Roberts type theorems, Rees rings and associated graded rings
本文研究交换代数中提升问题、沿相伴分次环的上升/下降问题与Matijevic-Roberts型问题这三个显著问题之间的相互关联结构。
Goto rings
本文引入并发展 Goto 环理论，推广几乎 Gorenstein 环，通过扩展典范理想约化数至多 2 定义，并用 Sally 模刻画其结构、确定 Hilbert 函数，且给出多类例子。
We prove necessary and sufficient conditions for the existence of homogeneous prime elements in normal N-graded rings of dimension two, in terms of rational coefficient Weil divisors on projective curves.
In this article, we show that the depths of the associated graded ring and fiber cone of a lex-segment ideal in $K[x,y]$ are equal.
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity and $M$ a graded $R$-module. In this paper, we introduce the concept of graded $I_{e}$-prime submodule as a generalization of a graded prime submodule for $I=\oplus_{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning of these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded $I_{e}$-prime submodule of $M$ if whenever $% r_{g}\in h(R)$ and $m_{h}\in h(M)$ with $r_{g}m_{h}\in N-I_{e}N,$ then either $r_…
In this article, we give the explicit minimal free resolution of the associated graded ring of certain affine monomial curves in affine 4-space based on the standard basis theory. As a result, we give the minimal graded free resolution and compute the Hilbert function of the tangent cone of these families.
Lance Bryant noticed in his thesis that there was a flaw in our paper "Associated graded rings of one-dimensional analytically irreducible rings", J. Algebra 304 (2006), 349-358. It can be fixed by adding a condition, called the BF condition. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela. Finally we use our result to conclude when a semigroup ring in embedding dimension at m…
The Koszul Property for Truncations of Nonstandard Graded Polynomial Rings
本文证明非标准分次多项式环的截断是Herzog-Iyengar意义下的非标准Koszul模，从而将标准分次情形下截断具有线性消解的结论推广到非标准分次框架。
In this paper, we give the necessary and sufficient conditions for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroups using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also define the nice extension for simplicial affine semigroups, motivated by the notion of a nice extension of the numerical semigroups.
$\mathbb{Z}^r$-graded rings and their canonical modules
本研究将分次典范模的定义推广到多分次环，证明其局部化性质，发展多分次环与模的基础理论，并给出仿射正规幺半群环典范模相关定理的除子式证明。
Given a local Cohen-Macaulay ring $(R, {\mathfrak m})$, we study the interplay between the integral closedness -- or even the normality -- of an ${\mathfrak m}$-primary $R$-ideal $I$ and conditions on the Hilbert coefficients of $I$. We relate these properties to the depth of the associated graded ring of $I$.
We define what it means for a Cohen-Macaulay ring to to be super-stretched and show that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. We use this result to show that rings of graded countable Cohen-Macaulay type, and positive dimension, have possible h-vectors (1), (1,n), or (1,n,1). Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen-Macaulay type of dimension larger than 2 are shown to be …
In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay…
In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q, which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Finally, using these decompositions, we obtain results about the rationality of the Poincare series of Q.
By definition, an $\m$-primary ideal $I$ in a 2-dimensional regular local ring $(R, \m)$ is contracted if $I=R \cap IR[\m/x]$ for some $x \in \m \setminus \m^2$. Contracted ideals have been introduced by Zariski and used for proving the unique factorization theorem for complete (i.e. integrally closed) ideals. Any complete ideal is contracted but not the other way round. While the associated graded rings to complete ideals are always Cohen-Macaulay, this is not the case for contracted ideals. Our goal is to study depth, Hilbert function and defining equations of the graded rings of homogeneous…
Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and $I$ an $\mathcal{I}$-primary ideal of $A$. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring $A$ to be passed onto the associated graded ring of filtration. Let $\mathcal{I}$ denote an $I$-good filtration. We prove that if $A$ is Buchsbaum and the I-invariant, $I(A)$ and $I(G(\mathcal{I}))$, coincide then the associated graded ring $G(\mathcal{I})$ is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions…
For a graded domain $R=k[X_0,...,X_m]/J$ over an arbitrary domain $k$, it is shown that the ideal generated by elements of degree $\geq mA$, where $A$ is the least common multiple of the weights of the $X_i$, is a normal ideal.
In this paper, we present several algorithms for dealing with graded components of Laurent polynomial rings. To be more precise, let $S$ be the Laurent polynomial ring $k[x_1,...,x_{r},x_{r+1}^{\pm 1},..., x_n^{\pm 1}]$, $k$ algebraicaly closed field of characteristic 0. We define the multigrading of $S$ by an arbitrary finitely generated abelian group $A$. We construct a set of fans compatible with the multigrading and use this fans to compute the graded components of $S$ using polytopes. We give an algorithm to check whether the graded components of $S$ are finite dimensional. Regardless of …
Let (R, m) be the semigroup ring associated to a numerical semigroup S. In this paper we study the property of its associated graded ring G(m) to be Complete Intersection. In particular, we introduce and characterise beta-rectangular and gamma-rectangular Apéry sets, which will be the fundamental concepts of the paper and will provide, respectively, a sufficient condition and a characterisation for G(m) to be Complete Intersection. Then we use these notions to give four equivalent conditions for G(m) in order to be Complete Intersection.
Let $\Bbb Z_2:=\Bbb Z/2\Bbb Z$ be the additive group with two elements. In this article, we focus only on $\Bbb Z_2$-graded commutative ring i.e commutative ring $R$ such that $R=R_0\oplus R_1$ as Abelian group and $R_iR_j\subseteq R_{i+j}$ for all $i,j\in \Bbb Z_2$. Our main goals is to establish a strong relation between $\Bbb Z_2$-graded prime ( maximal ) ideals of $R$ and prime ( maximal) ideals of $R_0$, for instance, it is showed that, the $\Bbb Z_2$-graded spectrum of $R$ is homeomorphic to the spectrum of $R_0$ with respect to the Zariski topologies.
Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the Hilbert-Samuel multiplicities for quotient modules of S. As an application, we get formulas for the mixed multiplicities of ideals that covers the main result of Trung-Verma 2007 [TV].
On Frobenius Betti numbers of graded rings of finite Cohen-Macaulay type
针对环的Frobenius Betti数显式计算场景极少的问题，本文聚焦于有限Cohen-Macaulay型Cohen-Macaulay分次环，开展其Frobenius Betti数的显式计算研究，拓展该奇异性不变量的计算范围。
In this paper we give an effective characterization of Hilbert functions and polynomials of standard algebras over an Artinian equicharacteristic local ring; the cohomological properties of such algebras are also studied. We describe algorithms to check the admissibility of a given function or polynomial as a Hilbert function or polynomial, and to produce a standard algebra with a given Hilbert function.
Comparing generalized Gorenstein properties in semi-standard graded rings
本文研究半标准分次环这一标准分次环的推广对象，比较其中的广义Gorenstein性质，分析两类分次环的共通性并揭示半标准分次环独有的特性。
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.
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。