A Schreier type property for modules

Maximal depth property of finitely generated modules

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\mathfrak{p}$ of $M$ such that depth $M=\dim R/\mathfrak{p}$. In this paper, we study finitely generated modules with maximal depth. It is shown that the maximal depth property is preserved under some important module operations. Generalized Cohen--Macaulay modules with maximal depth are classified. Finally, the attached primes of $H^i_{\mathfrak{m}}(M)$ are considered for $i<\mathrm{dim} M$.

The duals of annihilator conditions for modules

Let R be a commutative ring with identity and let M be an R-module. The purpose of this paper is to introduce and investigate the submodules of an R-module M which satisfy the dual of Property A, the dual of strong Property A, and the dual of proper strong Property A. Moreover, a submodule N of M which satisfy Property SJ (N) and Property IM J (N) will be introduced and investigated.

Higher Cohen-Macaulay property of squarefree modules and simplicial posets

Recently, G. Floystad studied "higher Cohen-Macaulay property" of certain finite regular cell complexes. In this paper, we partially extend his results to squarefree modules, toric face rings, and simplicial posets. For example, we show that if (the corresponding cell complex of) a simplicial poset is $l$-Cohen-Macaulay then its codimension one skeleton is $(l+1)$-Cohen-Macaulay.

A Property Of Local Cohomology Modules Of Polynomial Rings

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ and let $I=(f_1,...,f_s)$ be an ideal of $R.$ We prove that every associated prime $P$ of $H^i_I(R)$ satisfies $\text{dim}R/P\geqslant n-\sum\text{deg}f_i.$ In characteristic 0 the question is open.

On the generators of the canonical module of a Hibi ring: a criterion of level property and the degrees of generators

In this paper, we study the minimal generating system of the canonical module of a Hibi ring. Using the results, we state a characterization of a Hibi ring to be level. We also give a characterization of a Hibi ring to be of type 2. Further, we show that the degrees of the elements of the minimal generating system of the canonical module of a Hibi ring form a set of consecutive integers.

A rigidity property of local cohomology modules

The relationships between the invariants and the homological properties of $I$, ${\rm Gin}(I)$ and $I^{\rm lex}$ have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid behaviours of their Betti numbers. In this work we establish a local cohomology counterpart of their theorem. To this end, we make use of properties of sequentially Cohen-Macaulay modules and we study a generalization of such concept by introducing what we call partially sequentially Cohen-Macaulay modules, which might be of interest by themselves.

Cohen-Macaulay环的典范模的迹的根性质

Radical property of the traces of the canonical modules of Cohen-Macaulay rings

本文提出介于Gorenstein与Cohen-Macaulay之间的新环性质，证明其在多种环运算下保持，并表明对Schubert循环及某些图环接近Gorenstein。

On the Cohen-Macaulay property of the Rees algebra of the module of differentials

Let $R$ be an algebra essentially of finite type over a field $k$ and let $Ω_k(R)$ be its module of Kähler differentials over $k$. If $R$ is a homogeneous complete intersection and $\mathrm{char}(k)=0$, we prove that $Ω_k(R)$ is of linear type whenever its Rees algebra is Cohen-Macaulay and locally at every homogeneous prime $\mathfrak{p}$ the embedding dimension of $R_{\mathfrak{p}}$ is at most twice its dimension.

Ascent and descent of the Golod property along algebra retracts

We study ascent and descent of the Golod property along an algebra retract. We characterise trivial extensions of modules, fibre products of rings to be Golod rings. We present a criterion for a graded module over a graded affine algebra of characteristic zero to be a Golod module.

Revisiting G-Dedekind domains

Let $R$ be an integral domain with $qf(R)=K$ and let $F(R)$ be the set of nonzero fractional ideals of $R.$ Call $R$ a dually compact domain (DCD) if for each $I\in F(R)$ the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori and Krull domains. In addition we show that a Schreier DCD is a GCD domain with the property that for each $A\in F(R)$ the ideal $A_{v}$ is principal. We show that a domain $R$ is G-Dedekind domain (i.e. has…

Systems of Parameters and the Cohen-Macaulay Property

Let $R$ be a commutative, Noetherian, local ring and $M$ an $R$-module. Consider the module of homomorphisms $\operatorname{Hom}_R(R/\mathfrak{a},M/\mathfrak{b} M)$ where $\mathfrak{b}\subseteq\mathfrak{a}$ are parameter ideals of $M$. When $M=R$ and $R$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is always isomorphic to $R/\mathfrak{a}$, and in particular, a free module over $R/\mathfrak{a}$ of rank one. In this work, we study the structure of such modules of homomorphisms for general $M$.

A study of the cohomological rigidity property

In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersecti…

maximal depth property of bigraded modules

Let $S=K[x_1, \dots, x_m, y_1, \dots, y_n]$ be the standard bigraded polynomial ring over a field $K$. Let $M$ be a finitely generated bigraded $S$-module and $Q=(y_1, \dots, y_n)$. We say $M$ has maximal depth with respect to $Q$ if there is an associated prime $\pp$ of $M$ such that $\grade(Q, M)=\cd(Q, S/\pp)$. In this paper, we study finitely generated bigraded modules with maximal depth with respect to $Q$. It is shown that sequentially Cohen--Macaulay modules with respect to $Q$ have maximal depth with respect to $Q$. In fact, maximal depth property generalizes the concept of sequentiall…

Constructing non-proxy small test modules for the complete intersection property

A local ring $R$ is regular if and only if every finitely generated $R$-module has finite projective dimension. Moreover, the residue field $k$ is a test module: $R$ is regular if and only if $k$ has finite projective dimension. This characterization can be extended to the bounded derived category $\mathsf{D}^f(R)$, which contains only small objects if and only if $R$ is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: $R$ is a complete intersection if and only if every object in $\mathsf{D…

Artinian Gorenstein代数的中心单模

The central simple modules of Artinian Gorenstein algebras

设A是特征为零的域上的标准分次Artin代数，z是A中的线性型.我们为每个这样的对（A，z）定义中心单模。假设A是Gorenstein。然后证明了A具有强Lefschetz性质当且仅当A中存在一个线性型z使得（A，z）的所有中心单模都具有强Lefschetz性质.在证明过程中，我们需要将强Lefschetz性质的定义推广到分次Artin代数上的有限分次模上，而这一性质以前只定义在标准分次Artin代数上。

凝聚环的Auslander-Bridger公式和Gorenstein性质

The Auslander-Bridger formula and the Gorenstein property for coherent rings

研究了Auslander和Bridger关于诺特环上的有限生成模的Gorenstein维数的概念，并在凝聚环上的有限表示模的背景下讨论了Gorenstein维数. Auslander-Bridger公式的一个推广的建立，并作为一个基石的发展理论的连贯Gorenstein环。

A Class of Hilbert Series and the Strong Lefschetz Property

We determine the class of Hilbert series H so that if M is a finitely generated zero-dimensional R-graded module with the strong Lefschetz property, then the tensor product of M and k[y]/(y^m) has the strong Lefschetz property for y an indeterminate and all positive integers m if and only if the Hilbert series of M is in H. Given two finite graded R-modules M and N with the strong Lefschetz property, we determine sufficient conditions in order that the tensor product of M and N has the strong Lefschetz property.

On graded local cohomology modules defined by a pair of ideals

Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H^{i}_{R_{+},J}(M)_{n}$. Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H^{i}_{R_{+},J}(M)$.

上局部上同调与有限生成模的非混合支撑的链性

Top local cohomology and the catenaricity of the unmixed support of a finitely generated module

设$(R,\m)$是Noether局部环，$M$是由$\dim M=d.$生成的$R-$模。本文研究了顶局部上同调模$H^d_\m(M)$的如下性质：$$\Ann (0:_{H^d_\m(M)}\p)=\p\ \text{for all prime ideals} \p\supseteq\Ann H^d_\m(M).$$在本文中，我们将证明这一性质等价于由$\Usupp M=\Supp M/U_M(0)$定义的$M$的非混合支集$\Usupp M$的链状性，其中$U_M(0)$是$M$的维数小于$d.$的最大子模.给出了这一性质的参数系刻画以及$M$与$\m$-adic完备$\hat M$的非混合支集之间的关系.

A Property of the Frobenius Map of a Polynomial Ring

Let R be a ring of polynomials in a finite number of variables over a perfect field k of characteristic p>0 and let F:R\to R be the Frobenius map of R, i.e. F(r)=r^p. We explicitly describe an R-module isomorphism Hom_R(F_*(M),N)\cong Hom_R(M,F^*(N)) for all R-modules M and N. Some recent and potential applications are discussed.

The trace property in preenveloping classes

We develop the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given preenvelope. As a consequence we identify new examples of trace ideals and trace modules, and characterize several classes of rings with a focus on the Gorenstein and regular properties.

仿射半群环K[C]上C-分次模的注记

Notes on C-graded modules over an affine semigroup ring K[C]

设$C \subset {\bf N}^d$是仿射半群，$R=K[C]$是它的半群环.本文是关于“$C$ -分次”$R$ -模，特别是单项式理想的各种结果的集合.例如，我们证明如下：如果$R$是正规的，并且$I$是根单项理想（即，$R/I$是Stanley-Reisner环的推广），则$R/I$的序贯Cohen-Macaulay性质是与$I$相关的胞腔复形的“几何实现”的拓扑性质.此外，我们可以给出这个结果的一个无平方模/可构造层版本。

Arf 环、简单奇点与自反模

Arf rings, simple singularities and reflexive modules

本文研究交换 Noether 局部环中自对偶性质，探讨 Arf 环在更高 Krull 维数下的推广，基于 Dao 关于 Arf 环自反模有限性与自对偶性的结果。

多重托模块：刚性与梅耶-维蒂里斯谱序列

Multiple Tor modules: rigidity and Mayer-Vietoris spectral sequences

本文研究多重理想下的托模块性质扩展，提出新的复形并探讨其同调关系，揭示托独立性在多格下的支撑区域关联。

Modules over some group rings having d-generator property

For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.

同调维数、Gorenstein 性质与若干猜想的特殊情形

Homological dimensions, the Gorenstein property, and special cases of some conjectures

本文研究 Noetherian 环上有限模的同调维数有限性判据及其在局部 Cohen-Macaulay 环中的影响，并利用反典范模刻画 Gorenstein 性质，进而证实 Vasconcelos 正规模猜想等若干猜想的特殊情形。

通过塞尔深度具有可提升局部上同调的广义科恩 - 麦考利模与环的\(S_r\)性质

$S_r$ properties of generalized Cohen--Macaulay modules and rings having liftable local cohomology via Serre depth

研究通过局部上同调模的马蒂斯对偶探讨塞尔深度，关联模与正则元商的塞尔深度，刻画广义科恩 - 麦考利模的\(S_r\)性质，证明在特定假设下该性质可降至约化结构。

Descent of restricted flat Mittag-Leffler modules and generalized vector bundles

A basic question for any property of quasi--coherent sheaves on a scheme $X$ is whether the property is local, that is, it can be defined using any open affine covering of $X$. Locality follows from the descent of the corresponding module property: for (infinite dimensional) vector bundles and Drinfeld vector bundles, it was proved by Kaplansky's technique of dévissage already in \cite[II.\S3]{RG}. Since vector bundles coincide with $\aleph_0$-restricted Drinfeld vector bundles, a question arose in \cite{EGPT} of whether locality holds for $κ$-restricted Drinfeld vector bundles for each infini…

Hartshorne-Rao模块在$\mathbb P^3$中曲线的弱和强Lefschetz性质

Weak and strong Lefschetz properties for Hartshorne-Rao modules of curves in $\mathbb P^3$

本文研究了Hartshorne-Rao模块是否满足弱和强Lefschetz性质，通过分析曲线的几何特性，证明了由一般斜线组成的联合体在特定条件下具有最大秩，并探讨了特殊几何配置对弱Lefschetz性质的影响，同时展示了平滑不可约曲线在不同情况下的结果。

零化子乘法模：环的刻画与构造

Annihilator Multiplication Modules: Ring Characterizations and Constructions

本文刻画了忠实模或所有模为零化子乘法模的环（分别为拟Frobenius环和Artin主理想环），并给出约化环上的等价条件、长度界及具体构造，揭示了该性质与环结构及模理论的深刻联系。

Decompositions of modules lacking zero sums

A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always does holds for free modules over the max-plus semiring and related semirings. In such situations, the direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated projective module is a finite direct sum of summands of $R$ (assuming the mild assumption that $1$ is a finite sum of orthogonal primitive idempotents…

Cohen-Macaulay Property of pinched Veronese Rings

In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen-Macaulay and we the study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

Stable polynomial division and essential normality of graded Hilbert modules

The purpose of this paper is to initiate a new attack on Arveson's resistant conjecture, that all graded submodules of the $d$-shift Hilbert module $H^2$ are essentially normal. We introduce the stable division property for modules (and ideals): a normed module $M$ over the ring of polynomials in $d$ variables has the stable division property if it has a generating set $\{f_1, ..., f_k\}$ such that every $h \in M$ can be written as $h = \sum_i a_i f_i$ for some polynomials $a_i$ such that $\sum \|a_i f_i\| \leq C\|h\|$. We show that certain classes of modules have this property, and that the s…

On the Weak Lefschetz Property for Vector Bundles on $\mathbb P^2$

Let $R=\mathbb K[x,y,z]$ be a standard graded polynomial ring where $\mathbb K$ is an algebraically closed field of characteristic zero. Let $M = \oplus_j M_j$ be a finite length graded $R$-module. We say that $M$ has the Weak Lefschetz Property if there is a homogeneous element $L$ of degree one in $R$ such that the multiplication map $\times L : M_j \rightarrow M_{j+1}$ has maximal rank for every $j$. The main result of this paper is to show that if $\mathcal E$ is a locally free sheaf of rank 2 on $\mathbb P^2$ then the first cohomology module of $\mathcal E$, $H^1_*(\mathbb P^2, \mathcal E…

The ascent-descent property for $2$-term silting complexes

We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.

关于不可分解模由合成因子决定的Artin代数

On Artin algebras whose indecomposable modules are determined by composition factors

本文研究Artin代数中不可分解模由合成因子决定的性质，证实了该性质蕴含有限表示型的猜想在交换、遗传及根平方为零等特殊情形下成立。

On the Gorenstein Property of the Fiber Cone to Filtrations

Let $(A, \mathfrak{m})$ be a Noetherian local ring and $\mathfrak{F}=(I_{n})_{n\geq 0}$ a filtration. In this paper, we study the Gorenstein properties of the fiber cone $F(\mathfrak{F})$, where $\mathfrak{F}$ is a Hilbert filtration. Suppose that $F(\mathfrak{F})$ and $G(\mathfrak{F})$ are Cohen-Macaulay. If in addition, the associated graded ring $G(\mathfrak{F})$ is Gorenstein; similarly to the $I$-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal number of generators of ideals, for Gorensteiness of the fiber cone. Moreover, we find a description of …