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Shinya Kumashiro

Publications and source records attributed to Shinya Kumashiro.

35 records · Page 2Linked to original sources
arXiv2022

The set of trace ideals of curve singularities

This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to birational extensions, reflexive ideals, and reflexive Ulrich modules. Special attention is given in the case of numerical semigroup rings of non-gap four. We then obtain a criterion for a ring to have a finite number of reflexive ideals up to isomorphism. Non-domains arising from fiber products are also explored.

math.AC↗
arXiv2022

The reduction number of canonical ideals

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.

math.AC↗
arXiv2021

Graded filtrations and ideals of reduction number two

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 $3$.

math.AC↗
arXiv2021

The irreducible multiplicity and Ulrich modules

In this paper, we give a relation between the Hilbert multiplicity and the irreducible multiplicity. As an application, we characterize Ulrich modules in term of the irreducible multiplicity.

math.AC↗
arXiv2021

The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one

We study one-dimensional Cohen-Macaulay rings whose trace ideal of the canonical module is as small as possible. In this paper we call such rings far-flung Gorenstein rings. We investigate far-flung Gorenstein rings in relation with the endomorphism algebras of the maximal ideals and numerical semigroup rings. We show that the solution of the Rohrbach problem in additive number theory provides an upper bound for the multiplicity of far-flung Gorenstein numerical semigroup rings. Reflexive modules over far-flung Gorenstein rings are also studied.

math.AC↗
arXiv2021

Integrally closed ideals of reduction number three

In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge \mathrm{e}_0(I) - \mathrm{e}_1(I) + \dfrac{\mathrm{e}_2(I) + \ell_A(I^2/QI)}{2}$, where $\mathrm{e}_i(I)$ denotes the $i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of $I$ is larger than or equal to $\dim A-1$. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.

math.AC↗
arXiv2021

Certain monomial ideals whose numbers of generators of powers descend

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $μ(I^2)\ge 9$ for the number of minimal generators of $I^2$ with $μ(I)\geq 6$. Recently, Gasanova constructed monomial ideals such that $μ(I)>μ(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $μ(I)>μ(I^2)>\cdots >μ(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $μ(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends.

math.AC↗
arXiv2021

Graded Bourbaki ideals of graded modules

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal.

math.AC↗
arXiv2020

Ideals of reduction number two

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.

math.AC↗
arXiv2020

Reducibility index and sum-reducibility index

Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$. On the other hand, for an Artinian $R$-module $A$, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$. In the former part of this paper, we compute the reducibility index of $S\otimes_R M$, where $R\to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$, that of the completion of $M$, and the sum-reducibility index of the Matlis dual of $M$.

math.AC↗
arXiv2018

Correspondence between trace ideals and birational extensions with application to the analysis of the Gorenstein property of rings

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 module is given. The notion of anti-stable rings is introduced, exploring their basic properties.

math.AC↗
arXiv2018

Residually faithful modules and the Cohen-Macaulay type of idealizations

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 closed ideals \cite{BV}.

math.AC↗
arXiv2017

Characterization of generalized Gorenstein rings

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.

math.AC↗
arXiv2017

When is $R \ltimes I$ an almost Gorenstein local ring?

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.

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↗