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Naoyuki Matsuoka

Publications and source records attributed to Naoyuki Matsuoka.

At least 19 recordsLinked to original sources
arXiv2026

The structure of almost symmetric almost complete intersection numerical semigroups

We prove a structure theorem for numerical semigroups H that are almost symmetric and almost complete intersections. Specifically, we show that a row-factorization (RF) matrix of H must possess a highly regular structure, which we call a cascade matrix. Consequently, the defining ideal I_H of the associated semigroup ring k[H] also exhibits a highly regular structure, derived from this cascade matrix. Moreover, both the RF-matrix and the binomial minimal generating set of I_H are unique. Conversely, we show that this structure completely characterizes almost symmetric almost complete intersection numerical semigroups: to every cascade matrix M we associate a monoid H and, whenever this is a numerical semigroup, we prove that it is pseudo-symmetric, almost complete intersection, and has M as RF-matrix. As a consequence of our study, we obtain several additional key results. 1) A rigidity theorem: if an almost complete intersection semigroup is almost symmetric, then it is forced to have odd embedding dimension and to be pseudo-symmetric. This result can be regarded as the ``next step'' after Kunz's theorem, which states that an almost complete intersection semigroup is never symmetric. 2) Cascade polynomials: for each odd positive integer e, we construct a multivariate squarefree polynomial P_e with integer coefficients, arising from a cascade matrix of variables. We provide an enumerative interpretation of its coefficients, thereby proving their non-negativity. 3) Herzog--Watanabe question: en route to proving the main theorem, we prove that every minimal relation of an arbitrary numerical semigroup H can be obtained by subtracting two rows in some RF-matrix of H, affirmatively answering a 2019 question by Herzog and Watanabe.

math.AC↗
arXiv2026

Normality of ideals beyond the standard graded setting: families from numerical semigroup rings

Let $k$ be an arbitrary field and let $S=k[x,y,z]$ be the polynomial ring with three variables $x,y,z$. We study integrally closed $(x,y,z)$-primary ideals of $S$ that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup $H$ of embedding dimension three, we obtain such ideals as inverse images $I_h=φ_H^{-1}(t^hk[t]\cap k[H])$. If $\ell$ is the least degree of a defining relation of $k[H]$, then $I_\ell$ is monomial, whereas $I_{\ell+1}$ has a binomial generator in the cases considered here. We determine $I_{\ell+1}$ for numerical semigroups of embedding dimension three and multiplicity three or four. The six-generated cases arising in multiplicity three and in the symmetric multiplicity-four case form two explicit families. For every ideal $I$ in these families, we prove that its Rees algebra is a Cohen--Macaulay normal domain. In the non-symmetric multiplicity-four case, $I_{\ell+1}$ is seven-generated; for $H=\langle4,9,15\rangle$, we prove that its Rees algebra is again a Cohen--Macaulay normal domain.

math.AC↗
arXiv2026

Normality of monomial ideals in three variables

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three and higher, this is no longer true in general, including monomial ideals in polynomial rings. In this paper, we study the normality of integrally closed monomial ideals in the polynomial ring $k[x,y,z]$ over a field $k$. We prove that every such ideal with at most seven minimal monomial generators is normal, thereby giving a sharp bound for normality in this setting. The proof is based on a detailed case-by-case analysis, combined with valuation-theoretic and combinatorial methods via Newton polyhedra.

math.AC↗
arXiv2025

Pseudo-Frobenius numbers and defining ideals in stretched numerical semigroup rings

The pseudo-Frobenius numbers of a numerical semigroup $H$ are deeply connected to the structure of the defining ideal of its semigroup ring $k[H]$. In this paper, we resolve a certain conjecture related to this connection under the assumption that $k[H]/(t^a)$ is stretched, where $a$ is the multiplicity of $H$. Furthermore, we provide numerical conditions for the tangent cone of $k[H]$ to be Cohen-Macaulay.

math.AC↗
arXiv2024

Almost Gorenstein simplicial semigroup rings

We give a criterion for almost Gorenstein property for semigroup rings associated with simplicial semigroups. We extend Nari's theorem for almost symmetric numerical semigroups to simplicial semigroups with higher rank. By this criterion, we determine $2$-dimensional normal semigroup rings which have ``Ulrich elements'' defined in [Herzog-Jafari-Stamate].

math.AC↗
arXiv2024

Remarks on almost Gorenstein rings

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once $R$ is an almost Gorenstein graded ring, the localization $R_M$ of $R$ at the graded maximal ideal $M$ is almost Gorenstein as a local ring. The converse does not hold true in general. However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well.

math.AC↗
arXiv2023

Nearly Gorenstein local rings defined by maximal minors of a $2 \times n$ matrix

We investigate the nearly Gorenstein property of a local ring defined by the maximal minors of a specific $2 \times n$ matrix with entries in the formal power series ring $k[[X_1, X_2, \ldots , X_n]]$ over a field $k$. Our findings allow us to present numerous concrete examples, such as nearly Gorenstein rings that are not almost Gorenstein and vice versa.

math.AC↗
arXiv2023

On the ubiquity of Arf rings

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 rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.

math.AC↗
arXiv2023

Rings with q-torsionfree canonical modules

Let A be a Noetherian local ring with canonical module K. We characterize A when K is a torsionless, reflexive, or q-torsionfree module. If A is a Cohen-Macaulay ring, H.-B. Foxby proved in 1974 that the A-module K is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby's result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasi-normal rings, introduced by W. V. Vasconcelos. We provide several examples as well.

math.AC↗
arXiv2022

On the weakly Arf $(S_2)$-ifications of Noetherian rings

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. The present paper aims at constructing, for a given Noetherian ring $R$ which satisfies certain mild conditions, the smallest module-finite birational extension of $R$ which satisfies WAP and the condition $(S_2)$ of Serre. We shall call this extension the weakly Arf $(S_2)$-ification, and develop the basic theory, including some existence theorems.

math.AC↗
arXiv2021

When are the rings $I:I$ Gorenstein?

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 $A$, that was closely explored by the preceding paper of the second and third authors. Examples are given.

math.AC↗
arXiv2021

Ulrich ideals in the ring $k[[t^5,t^{11}]]$

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 $k$.

math.AC↗
arXiv2019

Efficient generation of ideals in core subalgebras of the polynomial ring k[t] over a field k

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 of numerical semigroup rings. For $R=k[H]$ and $M \in \operatorname{Max}R$, our result eventually shows that $μ_{R}(M) \in \{1,2,μ(H)\}$ where $μ_{R}(M)$ (resp. $μ(H)$) stands for the minimal number of generators of $M$ (resp. $H$), which covers in the specific case the classical result of O. Forster-R. G. Swan.

math.AC↗
arXiv2017

Almost Gorenstein Rees algebras of $p_g$-ideals, good ideals, and powers of the maximal ideals

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 $I={\mathfrak m}^{d-1}$. Conversely, if ${\mathcal R}({\mathfrak m}^{\ell})$ is an almost Gorenstein graded ring for some $\ell \ge 2$ and $d \ge 3$, then $\ell=d-1$.

math.AC↗
arXiv2017

Sally modules of canonical ideals in dimension one and 2-AGL rings

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.

math.AC↗
arXiv2016

On the almost Gorenstein property in Rees algebras of contracted ideals

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 Gorenstein graded ring, even though $I$ is a contracted stable ideal. Thus both affirmative answers and negative answers are given.

math.AC↗
arXiv2016

Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein* simplicial complexes

In this paper, we study simplicial complexes whose Stanley-Reisner rings are almost Gorenstein and have $a$-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen-Macaulay simplicial complexes. A $d$-dimensional simplicial complex $Δ$ is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet $F$ of $Δ$, the simplicial complex $Δ\setminus\{F\}$ is Cohen-Macaulay of dimension $d$. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen-Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension $\leq 2$.

math.AC↗