Searcharxiv⌕ Search

arXiv subjects

Shiro Goto

Publications and source records attributed to Shiro Goto.

At least 55 records · Page 3Linked to original sources
arXiv2014

Almost Gorenstein rings - towards a theory of higher dimension -

The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fröberg 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-dimensional notion and develop the basic theory. The graded version is also posed and explored.

math.AC↗
arXiv2014

The second Hilbert coefficients and the homological torsions of parameters

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 parameters and the homological torsions of modules.

math.AC↗
arXiv2014

Sectional genera of parameter ideals

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 ${\mathrm{T}}_Q^1(M)$ respectively denote the sectional genus, the multiplicity, the first Hilbert coefficient, and the Homological torsion of $M$ with respect to $Q$.

math.AC↗
arXiv2014

The first Euler characteristics versus the homological degrees

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 $χ_1(Q;M)=\operatorname{hdeg}_Q(M)-\mathrm{e}_Q^0(M)$, where $χ_1(Q;M)$, $\operatorname{hdeg}_Q(M)$, and $\mathrm{e}_Q^0(M)$ respectively denote the first Euler characteristic, the homological degree, and the multiplicity of $M$ with respect to $Q$. We also study homological torsions of $M$ and give a criterion for a certain equality of the first Hilbert coefficients of parameters and the homological torsions of $M$.

math.AC↗
arXiv2014

The Chern Numbers and Euler Characteristics of Modules

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 the ring case. By developing a robust setting to treat these coefficients for unmixed rings and modules, the case of modules is analyzed in a more transparent manner. Another series of integers arise from partial Euler characteristics and are shown to carry similar properties of the module. The technology of homological degree theory is also introduced in order to derive bounds for these two sets of numbers.

math.AC↗
arXiv2014

Hilbert polynomials of j-transforms

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 properties. We focus on deriving the Hilbert functions of $\mathbf{j}$-transforms and studying the significance of the vanishing of some of its coefficients.

math.AC↗
arXiv2014

Non-complete intersection prime ideals in dimension 3

We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.

math.AC↗
arXiv2013

Huneke-Wiegand conjecture and change of rings

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 an $R$-module, once $I\otimes_RI^{\vee}$ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher dimensional assertion is also discussed.

math.AC↗
arXiv2013

Ulrich ideals and modules over two-dimensional rational singularities

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 modules and Ulrich modules with respect to good ideals.

math.AC↗
arXiv2013

Ulrich ideals and modules

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.

math.AC↗
arXiv2012

J. Sally's question and a conjecture of Y. Shimoda

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 completeness, infiniteness of the residue field and the multiplicity of the ring being at most three. In the second part of the paper we turn our attention to the category of standard graded algebras. A geometrical approach via a double use of a Bertini Theorem, together with a result of A. Simis, B. Ulrich and W.V. Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of M. Miller on prime Bourbaki ideals in local rings, we detail some more technical results concerning the existence in standard graded algebras of homogeneous prime ideals with an "excessive" number of generators.

math.AC↗
arXiv2012

On the cofiniteness of generalized local cohomology modules

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 then $H^j_I(M,N)$ is $I$-cofinite for all $M, N$ and all $j$. Secondly, let $t$ be a non-negative integer such that $\dim\Supp(H^j_I(M,N))\le 1 \text{for all} j<t.$ Then $H^j_I(M,N)$ is $I$-cofinite for all $j<t$ and $\Hom(R/I,H^t_I(M,N))$ is finitely generated. Finally, we show that if $\dim(M)\le 2$ or $\dim(N)\le 2$ then $H^j_I(M,N)$ is $I$-cofinite for all $j$.

math.AC↗
arXiv2012

Hilbert coefficients and sequentially Cohen-Macaulay modules

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 bound for the first Hilbert coefficient.

math.AC↗
arXiv2011

Almost Gorenstein rings

The notion of almost Gorenstein ring given by Barucci and Fr{ö}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 full generality, where $\m$ denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.

math.AC↗
arXiv2008

Quasi-socle ideals in local rings with Gorenstein tangent cones

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 of a system $x_1, x_2, ..., x_d$ of elements in $A$ such that $(x_1, x_2, ..., x_d)$ is a reduction of $\mathfrak{m}$. The questions of when $I$ is integral over $Q$ and of when the graded rings $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ and $\mathrm{F}(I) = \bigoplus_{n \ge 0}I^n/\mathfrak{m} I^n$ are Cohen-Macaulay are answered. Criteria for $\mathrm{G} (I)$ and $\mathcal{R} (I) = \bigoplus_{n \geq 0}I^n$ to be Gorenstein rings are given.

math.AC↗
arXiv2008

On vanishing of certain Ext modules

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. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

math.AC↗
arXiv2008

Quasi-socle ideals in Gorenstein numerical semigroup rings

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal ideal in the base local ring, and $q \geq 1$ is an integer. The problems of when $I$ is integral over $Q$ and of when the associated graded ring $\mathrm{G}(I) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ of $I$ is Cohen-Macaulay are studied. The problems are rather wild; examples are given.

math.AC↗