Shiro Goto

Shiro Goto
Meiji University · Department of Mathematics

Honorary Doctorate

About

200
Publications
12,253
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3,545
Citations
Citations since 2016
39 Research Items
1217 Citations
2016201720182019202020212022050100150200
2016201720182019202020212022050100150200
2016201720182019202020212022050100150200
2016201720182019202020212022050100150200
Additional affiliations
April 1990 - March 2016
Meiji University
Position
  • Professor

Publications

Publications (200)
Article
Let I (≠A) be an ideal of a d-dimensional Noetherian local ring A with htAI≥2, containing a non-zerodivisor. The problem of when the ring I:I=EndAI is Gorenstein is studied, in connection with the problem of the Gorensteinness in Rees algebras RA(Qd) for certain parameter ideals Q of A, that was closely explored by the preceding paper [8 Goto, S.,...
Preprint
The weakly Arf $(S_2)$-ification of a commutative Noetherian ring $R$ is considered to be a birational extension which is good next to the normalization. The weakly Arf property (WAP for short) of $R$ was introduced in 1971 by J. Lipman with his famous paper [12], and recently rediscovered by [4], being closely explored with further developments. T...
Preprint
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>1. The main result implies that, for example, the Rees algebra of Q^d is Gorenstein for any parameter ideal Q that is a reduction of the maximal ideal in a d-dimensional Buc...
Preprint
Let $M$ be a finitely generated module over a ring $\Lambda$. With certain mild assumptions on $\Lambda$, it is proven that $M$ is a reflexive $\Lambda$-module, once $M \cong M^{**}$ as a $\Lambda$-module.
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In 1971, Lipman (Am J Math 93:649–685, 1971) introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns constr...
Preprint
Let $I ~(\ne A)$ be an ideal of a $d$-dimensional Noetherian local ring $A$ with $\operatorname{ht}_AI \ge 2$, containing a non-zerodivisor. The problem of when the ring $I:I=\operatorname{End}_AI$ is Gorenstein is studied, in connection with the problem of the Gorensteinness in Rees algebras $\mathcal{R}_A(Q^d)$ for certain parameter ideals $Q$ of...
Preprint
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 $k$.
Preprint
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....
Preprint
In 1971, J. Lipman introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns construction of the closure, and...
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The notion of strict closedness of rings was given by J. Lipman in connection with a conjecture of O. Zariski. The present purpose is to give a practical method of construction of strictly closed rings. It is also shown that the Stanley-Reisner rings of simplicial complexes (resp. F-pure rings satisfying the condition (S_2) of Serre) are strictly c...
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In this paper we investigate the question of when a Noetherian F-pure ring is weakly Arf. Under mild conditions we provide an affirmative answer for the question. In addition, we prove that every Stanley-Reisner algebra over a field satisfying the Serre's (S2) condition is a weakly Arf ring.
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In this paper, we introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf ri...
Article
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 minim...
Article
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 o...
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This note aims at finding explicit and efficient generation of ideals in subalgebras R of the polynomial ring S = k[t] (k a field) such that t^c S ⊆ R for some integer c > 0. The class of these subalgebras which we call cores of S includes the semigroup rings k[H] of numerical semigroups H, but much larger than the class of numerical semigroup ring...
Preprint
This note aims at finding explicit and efficient generation of ideals in subalgebras $R$ of the polynomial ring $S=k[t]$ ($k$ a field) such that $t^{c_0}S \subseteq R$ for some integer $c_0 > 0$. The class of these subalgebras which we call cores of $S$ includes the semigroup rings $k[H]$ of numerical semigroups $H$, but much larger than the class...
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The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product of Cohen-Macaulay local rings R, S of the same dimension d > 0 over a regular local ring T with dim T = d − 1 is an almost Gorenstein ring if and only...
Preprint
The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local rings $R$, $S$ of the same dimension $d>0$ over a regular local ring $T$ with $\dim T=d-1$ is an almost Gorenst...
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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 minim...
Preprint
Full-text available
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 minim...
Article
The Cohen-Macaulay type of idealizations of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings is explored. There are two extremal cases, one of which is closely related to the theory of Ulrich modules \cite{BHU, GOTWY1, GOTWY2, GTT2}, and the other one is closely related to the theory of residually faithful modules and the theory of cl...
Article
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.
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In this paper we study generalized Gorenstein Arf rings; a class of one-dimensional Cohen-Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we gen...
Article
Let (A,m) be a Cohen-Macaulay local ring, and let I be an ideal of A. We prove that the Rees algebra R(I) is an almost Gorenstein ring in the following cases: (1) (A, m) is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field K ≅ A/m, and I is a pg-ideal; (2) (A, m) is a two-dimensional almost Gorenstein local rin...
Preprint
The behavior of trace ideals and modules is explored in connection with the structure of the base ring and the ambient module. Firstly, over a commutative Noetherian ring, a characterization of a module for which every submodule is a trace module is given. Secondly, over an arbitrary commutative ring, correspondences between three sets, the set of...
Article
Let M be a finitely generated module over a Noetherian local ring R. The sequential polynomial type sp(M) of M was recently introduced by Nhan, Dung and Chau, which measures how far the module M is from the class of sequentially Cohen–Macaulay modules. The present paper purposes to give a parametric characterization for M to have sp(M) = s, where s...
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This paper is a sequel to an article where we introduced an invariant--canonical degree--of Cohen-Macaulay local rings that admit a canonical ideal. Here to each such ring with a canonical ideal, we attach a different invariant, i.e., bicanonical degree. The minimal values of these functions characterize specific classes of Cohen-Macaulay rings. Ou...
Article
A conjecture of C. Huneke and R. Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely-generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for...
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The structure of the defining ideal of the semigroup ring $k[H]$ of a numerical semigroup $H$ over a field $k$ is described, when the pseudo-Frobenius numbers of $H$ are multiples of a fixed integer.
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The notion of generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. In this article, there is given a characterization of GGL rings in terms of their canonical ideals and related invariants.
Article
Let $(R, \mathfrak{m}) $ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a Cohen-Macaulay ring of dimension $d$. There is given a complete answer to the question of when the idealization $A = R \ltimes I$ of $I$ over $R$ is an almost Gorenstein local ring.
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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 mini...
Article
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.
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The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide valuable metrics to express the deviation of R fr...
Article
This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov– Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa co...
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In this paper, we generalize the main result in [7] via three different approaches as follows. Let $(A,{\mathfrak m})$ be a two-dimensional excellent normal local domain over an algebraically closed field. If $I \subset A$ is a $p_g$-ideal in the sense of [15], then the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gor...
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We refine a well-known theorem of Auslander and Reiten about the extension closedness of n-th syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre's condition (S_n) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of...
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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 $R$.
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Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots, a_r)$. It is shown that if the Rees algebra ${\mathcal R}(Q)$ of $Q$ is an almost Gorenstein graded ring, then...
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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 sam...
Article
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 rin...
Article
Let $(R,\mathfrak{m})$ be a two-dimensional regular local ring with infinite residue class field. Then the Rees algebra $\mathcal{R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring in the sense of Goto-Takahashi-Taniguchi for every $\mathfrak{m}$-primary integrally closed ideal $I$ in $R$.
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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-Macaula...
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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 con...
Article
Let $M$ be a finitely generated module over a Noetherian local ring. This paper gives, for a given parameter ideal $Q$ for $M$, bounds for the second Hilbert coefficients ${\mathrm{e}}_Q^2(M)$ in terms of the homological degrees and torsions of modules. We also report a criterion for a certain equality of the second Hilbert coefficients of paramete...
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Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality ${\mathrm{g}}_s(Q;M)=\operatorname{hdeg}_Q(M)-{\mathrm{e}}_Q^0(M)-{\mathrm{T}}_Q^1(M)$, where ${\mathrm{g}}_s(Q;M)$, ${\mathrm{e}}_Q^0(M)$, ${\mathrm{e}}_Q^1(M)$, and ${\mathrm{T}}_Q^1(M)$ re...
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Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality $\chi_1(Q;M)=\operatorname{hdeg}_Q(M)-\mathrm{e}_Q^0(M)$, where $\chi_1(Q;M)$, $\operatorname{hdeg}_Q(M)$, and $\mathrm{e}_Q^0(M)$ respectively denote the first Euler characteristic, the homo...
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We study transformations of finite modules over Noetherian local rings that attach to a module $M$ a graded module $H^{0}_{\mathfrak{m}}( \mathrm{gr}_{I}(M))$ defined via partial systems of parameters of $M$. Despite the generality of the process, which are called $\mathbf{j}$-transforms, in numerous cases they have interesting cohomological proper...
Article
The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fr\"oberg for one-dimensional Noetherian local rings which are analytically unramified has been generalized by S. Goto, N. Matsuoka and T. T. Phuong to one-dimensional Cohen-Macaulay local rings, possessing canonical ideals. The present purpose is to propose a higher-dimensi...
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The set of the first Hilbert coefficients of parameter ideals relative to a module—its Chern coefficients—over a local Noetherian ring codes for considerable information about its structure-noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studied...
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In connection with a conjecture of Shimoda, we describe prime ideals of height 2 minimally generated by three elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3.
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In 2007, Shimoda, in connection with a long-standing question of 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 2. In this paper, having reduced the conjecture to the case of dimension 3, if the ring is regular and local of dimensi...
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The main aim of this paper is to classify Ulrich ideals and Ulrich modules over two-dimensional Gorenstein rational singularities (rational double points) from a geometric point of view. To achieve this purpose, we introduce the notion of (weakly) special Cohen-Macaulay modules with respect to ideals, and study the relationship between those module...
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The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in...
Article
Let $R$ be a Cohen-Macaulay local ring of dimension one with a canonical module $\rm{K_R}$. Let $I$ be a faithful ideal of $R$. We explore the problem of when $I \otimes_RI^{\vee}$ is torsionfree, where $I^{\vee} = \operatorname{Hom_R(I, \rm{K_R})}$. We prove that if $R$ has multiplicity at most $6$, then $I$ is isomorphic to $R$ or $\rm{K_R}$ as a...
Article
Let R be a sequentially Cohen-Macaulay local ring and assume that dim R 2 or that dim R = 1 and e(R) > 1. Then the equality I-2 = qI holds true for every good parameter ideal q of R contained in a sufficiently high power of the maximal ideal m, where I = q :(R) m. The structure of the graded rings gr(I) = circle plus(n >= 0) I-n/I-n+1,I- R(I) = cir...
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For a Noetherian local ring, we analyze conjectural relationships between the first Hilbert coefficient of a parameter ideal and the first partial Euler characteristic of its Koszul complex. Given their similar role as predictors of the Cohen-Macaulay property, we consider a direct comparison between them. For parameter ideals generated by d-sequen...
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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\displaystyle H^j_I(M,N)=\dlim\Ext^j_R(M/I^nM,N)$ of $M$ and $N$ with respect to $I$. We first prove that if $I$ is a principal ideal the...
Article
The purpose of this paper is to present a characterization of sequentially Cohen-Macaulay modules in terms of its Hilbert coefficients with respect to distinguished parameter ideals. The formulas involve arithmetic degrees. Among corollaries of the main result we obtain a short proof of Vasconcelos Vanishing Conjecture for modules and an upper boun...
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In this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.
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In this paper we survey the results on the negativity of the Chern number of parameter ideals.
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For a Noetherian local ring ( R , m ) (\mathbf {R}, \mathfrak {m}) , the first two Hilbert coefficients, e 0 e_0 and e 1 e_1 , of the I I -adic filtration of an m \mathfrak {m} -primary ideal I I are known to code for properties of R \mathbf {R} , of the blowup of Spec ⁡ ( R ) \operatorname {Spec}(\mathbf {R}) along V ( I ) V(I) , and even of their...
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The set of the first Hilbert coefficients of parameter ideals relative to a module--its Chern coefficients--over a local Noetherian ring codes for considerable information about its structure--noteworthy properties such as that of Cohen-Macaulayness, Buchsbaumness, and of having finitely generated local cohomology. The authors have previously studi...
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A problem posed by Wolmer V. Vasconcelos on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring is studied. Affirmative answers are given and counterexamples are explored as well.
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The notion of almost Gorenstein ring given by Barucci and Fr{\"o}berg \cite{BF} in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra $\m : \m$ of $\m$ is a Gorenstein ring is solved in...
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Let A be a Noetherian local ring with d=dimA>0. This paper shows that the Hilbert coefficients {e Q i (A)} ≤i≤d of parameter ideals Q have uniform bounds if and only if A is a generalized Cohen-Macaulay ring. The uniform bounds are huge; the sharp bound for e ( 2 A) in the case where A is a generalized Cohen-Macaulay ring with dimA≥3 is given.
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Let R be an analytically unramified local ring with maximal ideal m and d = dimR > 0. If R is unmixed, then e1I (R) = 0 for every m-primary ideal I in R, where e1I (R) denotes the first coefficient of the normal Hilbert polynomial of R with respect to I. Thus the positivity conjecture on e1I(R) posed by Wolmer V. Vasconcelos is settled affirmativel...
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Quasi-socle ideals, that is, ideals of the form I = Q: m q (q ≥ 2), with Q parameter ideals in a Buchsbaum local ring (A,m), are explored in connection to the question of when I is integral over Q and when the associated graded ring G(I) ⊕ n≥0In/In+1 of I is Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after neces...
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Sally modules of m-primary ideals in a generalized Cohen–Macaulay and/or Buchsbaum local ring (A,m) are investigated. The structure theorems of Sally modules of rank one given by Goto et al. (2008) and in the case where A is a Cohen–Macaulay local ring are generalized.
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Let $(A,\mathfrak{m})$ be a Noetherian local ring with $d=\operatorname{dim}A\ge 2$ . Then, if $A$ is a Buchsbaum ring, the first Hilbert coefficients $\mathrm{e}_Q^1(A)$ of $A$ for parameter ideals $Q$ are constant and equal to $-\sum_{i=1}^{d-1}\binom{d-2}{i-1}h^i(A)$ , where $h^i(A)$ denotes the length of the ith local cohomology module $\mathrm...
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Goto numbers g (Q) = max {q ∈ Z {divides} Q : mq is integral over Q} for certain parameter ideals Q in a Noetherian local ring (A, m) with Gorenstein associated graded ring G (m) = {N-ary circled plus operator}n ≥ 0 mn / mn + 1 are explored. As an application, the structure of quasi-socle ideals I = Q : mq (q ≥ 1) in a one-dimensional local complet...
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This gives an alternate proof of the Theorem by the authors that shows the first Hilbert coefficient of parameter ideals in an unmixed Noetherian local ring is always negative unless the ring is Cohen--Macaulay.
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We prove that parametric decomposition of powers of parameter ideals characterizes, under some additional conditions, sequentially Cohen-Macaulayness in modules together with a certain good property of corresponding systems of parameters.
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The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$ vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and its relationship to Buchsbaum rings are stu...
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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 $I$.
Article
Quasi-socle ideals, that is the ideals I of the form I=Q:mq in Gorenstein numerical semigroup rings over fields are explored, where Q is a parameter ideal, and m is the maximal ideal in the base local ring, and q⩾1 is an integer. The problems of when I is integral over Q and of when the associated graded ring G(I)=⊕n⩾0In/In+1 of I is Cohen–Macaulay...
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Let (S,𝔫) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I* be the leading ideal of I in the associated graded ring gr𝔫(S), and set R = S/I and 𝔪 = 𝔫/I. In Goto et al. (20073. Goto , S. , Heinzer , W. , Kim , M.-K. ( 2007 ). The leading ideal of a complete interse...
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This paper explores the structure of quasi-socle ideals I=Q:m2 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 problems as to when Q is a reduction of I and when the associated graded ring is Cohen–Macaulay. Wild examples are explored.
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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...
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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 $I$.
Article
In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations...
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Let (S, 𝔫) be an s-dimensional regular local ring with s > 2, and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. As in [22. Goto , S. , Heinzer , W. , Kim , M.-K. ( 2006 ). The leading ideal of a complete intersection of height two . J. Algebra 298 : 238 – 247 . View all references, 33. Goto , S. , Heinzer...
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Let R be a Noetherian local ring with the maximal ideal m and dim R=1. In this paper, we shall prove that the module Ext^1_R(R/Q,R) does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m^2 kills the 0th local cohomology module H_m^0(R). The assertion is no longer true unless v(R) \leq 4....
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We introduce a class of Stanley-Reisner ideals called generalized complete intersection, which is characterized by the property that all the residue class rings of powers of the ideal have FLC. We also give a combinatorial characterization of such ideals.
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Let $A$ be a Buchsbaum local ring with the maximal ideal $\mathfrak{m}$ and let $\mathrm{e}(A)$ denote the multiplicity of $A$ . Let $Q$ be a parameter ideal in $A$ and put $I=Q$ : $\mathfrak{m}$ . Then the equality $I^{2}=QI$ holds true, if $\mathrm{e}(A)=2$ and $\mathop{depth} A>0$ . The assertion is no longer true, unless $\mathrm{e}(A)=2$ . Cou...
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There is given a characterization of Noetherian local rings $A$ with $d = \dim A \geq 2$, in which the equality $(a_i \mid 1 \leq i \leq d)^n=\underset{\alpha}{\bigcap} (a_1^{\alpha_1}, a_2^{\alpha_2}, \cdots, a_d^{\alpha_d})$ holds true for all systems $a_1,a_2, \cdots, a_d$ of parameters and integers $n \geq 1$, where the suffix $\alpha$ runs ove...
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Let A be a Buchsbaum local ring with the maximal ideal m and let e(A) denote the multiplicity of A. Let Q be a parameter ideal in A and put I=Q:m. Then the equality I^2=QI holds true, if e(A)=2 and depthA>0. The assertion is no longer true, unless e(A)=2. Counterexamples are given.
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Let A be a Noetherian local ring with the maximal ideal and I an -primary ideal. The purpose of this paper is to generalize Northcott's inequality on Hilbert coefficients of I given in Northcott (J. London Math. Soc. 35 (1960) 209), without assuming that A is a Cohen–Macaulay ring. We will investigate when our inequality turns into an equality. It...
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Let A be a noetherian local ring with the maximal ideal m and d=dimA. Let Q be a parameter ideal in A. Let I=Q:m. The problem of when the equality I2=QI holds true is explored. When A is a Cohen-Macaulay ring, this problem was completely solved by A. Corso, C. Huneke, C. Polini, and W. Vasconcelos [CHV, CP, CPV], while nothing is known when A is no...
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Let A be a Noetherian local ring with the maximal ideal m and d = dim A. The set ΧA of Gorenstein m-primary integrally closed ideals in A is explored in this paper. If k = A/m is alge- braically closed and d≥2, then ΧA is infinite. In contrast, for each field k which is not algebraically closed and for each integer d ≥ 0, there exists a Noetherian...

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Research on local cohomology modules, generalized local cohomology modules, finite properties of some class of modules
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Classification of non-Gorenstein Cohen-Macaulay graded/local rings \R in terms of a certain embedding of \R into its canonical module \K_R