A numerical characterization of reduction for arbitrary modules

Flow-up Bases for Generalized Spline Modules on Arbitrary Graphs

Let R be a commutative ring with identity. An edge labeled graph is a graph with edges labeled by ideals of R. A generalized spline over an edge labeled graph is a vertex labeling by elements of R, such that the labels of any two adjacent vertices agree modulo the label associated to the edge connecting them. The set of generalized splines forms a subring and module over R. Such a module it is called a generalized spline module. We show the existence of a flow-up basis for the generalized spline module on an edge labeled graph over a principal ideal domain by using a new method based on trails…

On reduction numbers and Castelnuovo-Mumford regularity of blowup rings and modules

We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing…

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 $3$.

Mixed multiplicities of arbitrary modules

Let $(R, \mathfrak m)$ be a Noetherian local ring. In this work we extend the notion of mixed multiplicities of modules, given in \cite{Kleiman-Thorup2} and \cite{Kirby-Rees1} (see also \cite{Bedregal-Perez}), to an arbitrary family $E,E_1,..., E_q$ of $R$-submodules of $R^p$ with $E$ of finite colength. We prove that these mixed multiplicities coincide with the Buchsbaum-Rim multiplicity of some suitable $R$-module. In particular, we recover the fundamental Rees's mixed multiplicity theorem for modules, which was proved first by Kirby and Rees in \cite{Kirby-Rees1} and recently also proved by…

任意模的系数模

On Coefficient Module of Arbitrary Modules

本文研究自由模任意子模的系数模，构造并证明满足长度与 Hilbert 多项式次数条件的唯一最大模链，给出结构定理及 Fitting 理想相关系数模的存在性。

On reduction number of products of ideals

Let $R$ be a standard graded algebra over an infinite field $\mathbb{K}$ and $M$ a finitely generated $\ZZ$-graded $R$-module. Let $I_1,\ldots I_m$ be graded ideals of $R$. The functions $r(M/I_1^{a_1}\ldots I_m^{a_m}M)$ and $r(I_1^{a_1}\ldots I_m^{a_m}M)$ are investigated and their asymptotical behaviours are given. Here $r(\bullet)$ stands for the reduction number of a finitely graded module $\bullet$.

Depth and Stanley depth of multigraded modules

We study the behavior of depth and Stanley depth along short exact sequences of multigraded modules and under reduction modulo an element.

k[x]-modules和核心-幂零态射

k[x]-modules and Core-Nilpotent endomorphisms

本文研究了向量空间上核心-幂零态射的性质，通过k[x]-module结构给出新的描述，并提出一种新的广义反演，扩展了Drazin逆的应用范围。

On residually S_2 ideals and projective dimension one modules

We prove that certain modules are faithful. This enables us to draw consequences about the reduction number and the integral closure of some classes of ideals.

任意模的半素子模与实子模是素子模与实素子模的交集

Semiprime and real submodules of arbitrary modules are intersections of prime and real prime submodules

本文证明任意交换环上任意模的半素子模与实子模，分别是包含它们的素子模与实素子模的交集，推广了此前仅适用于有限生成模的相关结论。

Reduction of quantum systems with arbitrary first class constraints and Hecke algebras

We propose a method for reduction of quantum systems with arbitrary first class constraints. An appropriate mathematical setting for the problem is homology of associative algebras. For every such an algebra $A$ and its subalgebra B with an augmentation e there exists a cohomological complex which is a generalization of the BRST one. Its cohomology is an associative graded algebra Hk^{*}(A,B) which we call the Hecke algebra of the triple (A,B,e). It acts in the cohomology space H^{*}(B,V) for every left A- module V. In particular the zeroth graded component Hk^{0}(A,B) acts in the space of B- …

矩形矩阵在交换环上的块对角化 I.（模的分解与支持的分解）

Block-diagonal reduction of matrices over commutative rings I. (Decomposition of modules vs decomposition of their support)

研究矩形矩阵在交换环上的块对角化条件，探讨模分解与支持分解的关系，提出平方矩阵分解的必要充分条件及线性行列式表示的应用。

Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements

We define {\bf primitive derivations} for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the {\bf primitive filtrations} of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration has already been studied, which generalizes the Hodge filtration introduced by K. Saito.

Multiplier Ideals and Modules on Toric Varieties

A simple formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given. Variants for the multiplier module and test ideals are also treated.

Totally reflexive modules constructed from smooth projective curves of genus g \geq 2

In this paper, from an arbitrary smooth projective curve of genus at least two, we construct a non-Gorenstein Cohen-Macaulay normal domain and a nonfree totally reflexive module over it.