Shiro Goto, Ryotaro Isobe, Shinya Kumashiro
This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where the rings are of minimal multiplicity and \operatorname{GGL} rings are analyzed.
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Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals
研究了由一般矩阵的子式定义的行列式环的初始代数关于其经典泛点的性质。这种方法导致非常短的证明结构性质的行列式环。此外，它允许我们分类他们的Cohen-Macaulay和Ulrich理想。
Ulrich modules over Cohen-Macaulay local rings with minimal multiplicity
设R是Cohen-Macaulay局部环.本文主要在R具有极小重数的情况下研究Ulrich $R$ -模的结构。我们研究了Ulrich R-模的生成，并阐明了Ulrich R-模何时恰是极大Cohen-Macaulay R-模的合合偶.我们还研究了作为正合范畴的Ulrich R-模的结构。
Non-existence of Ulrich modules over Cohen-Macaulay local rings
本文通过构造二维Cohen-Macaulay局部环族，否定回答了Ulrich关于是否总存在达到生成元数上界的模的问题，并给出Gorenstein正规整环与完全交整环的反例。
Cohen-Macaulayness of Rees Algebras of Modules
给出了模的Rees代数是Cohen-Macaulay的充分条件，Johnson-Ulrich和Goto-Nakamura-Nishida在理想的Rees代数的情况下证明了这一点.事实证明，从理想到模的推广不仅仅是一个常规的推广，而是需要大量的技术发展。利用Simis，Ulrich和Vasconcelos引入的一般Bourbaki理想的技巧，得到了模的Rees代数的Cohen-Macaulay性.
We show that a smooth projective non-degenerate arithmetically Cohen-Macaulay subvariety X of P^N infinite Cohen-Macaulay type becomes of finite Cohen-Macaulay type by removing Ulrich bundles if and only if N = 5 and X is a quartic scroll or the Segre product of a line and a plane. In turn, we give a complete and explicit classification of ACM bundles over these varieties.
In this paper, we show that almost Cohen-Macaulay algebras are solid and in this respect, we seek for some special situation when (a) an almost Cohen-Macaulay algebra will be phantom extension and (b) when it maps into a balanced big Cohen-Macaulay module.
About the Cohen-Macaulay defect and almost Cohen-Macaulay rings
本文揭示了几乎Cohen-Macaulay环与Cohen-Macaulay亏格的联系，并引入一种Serre型条件，该条件与亏格的关系类似于$(S_n)$条件与Cohen-Macaulay模的关系。
Ulrich modules : decomposition and relation with rings having minimal multiplicity
该研究探讨Cohen-Macaulay局部环中Ulrich模的基本性质，证明其可分解，并据此讨论具有极小重数环的若干刻画及作为模为Ulrich的理想的性质。
Ulrich ideals and almost Gorenstein rings
研究了Cohen-Macaulay局部环$R$中Ulrich理想$I$的复形$\operatorname{\mathbf{R}Hom}_R(R/I,R)$的结构.证明了在一维几乎Gorenstein但非Gorenstein局部环中，唯一可能的Ulrich理想是极大理想.当Ulrich理想具有相同的最小生成元数时，也研究了这一问题.
We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.
On Sequentially Cohen-Macaulay Modules
本文给出了序列Cohen-Macaulay模的参数系刻画，推广了Cohen-Macaulay模和广义Cohen-Macaulay模的一些著名结果.研究了小嵌入维数的Stanley-Reisner环的序列Cohen-Macaulay性。
Perfect linkage of Cohen--Macaulay modules over Cohen--Macaulay rings
本文引入并研究了Cohen-Macaulay局部环上的Cohen-Macaulay模的完全模联结的概念，称之为完全联结.我们探讨了与合合子、最大Cohen-Macaulay近似和Yoshino-Isogawa连锁有关的完全连锁。恢复了Yoshino和Isogawa的一个定理，并分析了双完全连杆的结构。此外，我们还建立了两个余维为1的Cohen-Macaulay模完全连通的判别准则，并将其应用于理想的经典连通理论。我们还构建了模块和理想链接的各种示例。
Direct limits of Cohen-Macaulay rings
设$A$是Cohen-Macaulay环的直系的直极限.本文描述了$A$的Cohen-Macaulay性质。我们的结果表明，$A$不一定是Cohen-Macaulay。我们证明了$A$是Cohen-Macaulay在各种假设下。作为应用，我们研究了非仿射法线半群环的科恩-麦考莱性。
On sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules
本文引入序列序列和序列f-序列，刻画了序列Cohen-Macaulay模与序列广义Cohen-Macaulay模，并给出了Cohen-Macaulay模的新刻画。
Existence of almost Cohen-Macaulay algebras implies the existence of big Cohen-Macaulay algebras
在\cite{AB}中，将匕首闭包推广到Noether局部域$(R,\fm)$上的有限生成模上，并证明它是Dietz闭包.在这篇简短的笔记中，我们证明了它也满足\cite{R.G}的“代数公理”，这导致了本文的以下结果：对于完整的诺特局部域，如果它包含在几乎科恩-麦考利域中，则存在其上的平衡大科恩-麦考利代数。
Dual Bass Numbers and Co-Cohen Macaulay Modules
本文利用模的对偶Bass数的消失性质给出了余-Cohen-Macaulay模的一个刻画。另外，我们证明了在一定条件下，余-Cohen Macaulay模的余局部化保持了余-Cohen Macaulay性。
Local Homology, Cohen-Macaulayness and Cohen-Macaulayfications
设（R，m）是局部完备环，X是Noether维数为d的Artin R-模，x_1，…，x_d \in m是这样的：0：_X（x_1，.，x_d）R有有限长。则H^x_d（X）是一个有限R-模，从而对Tang提出的一个问题给出了肯定的回答.作为这个结果的第一个应用，推论1包含了有限模是CM的一个必要条件;其次，我们提出了Cohen-Macaulayization的概念，并证明了它的唯一性。3）;最后，我们证明了这个新的Cohen-Macaulayfication概念是Goto（th. 4）。
Computing families of Cohen-Macaulay and Gorenstein rings
刻画了由某些类型的凸体半群得到的Cohen-Macaulay环和Gorenstein环.给出了判定多边形半群或圆半群是Cohen-Macaulay/Gorenstein半群的数学方法。我们也给出了Cohen-Macaulay环和Gorenstein环的一些族。
We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.
The Cohen-Macaulay property of a graph arising from a poset has been studied by various authors. In this article, we study the Cohen-Macaulay property of a graph arising from a family of reflexive and antisymmetric relations on a set. We use this result to find classes of multipartite graphs which are Cohen-Macaulay.
On the discrete counterparts of Cohen-Macaulay algebras with straightening laws
研究了生成具有拉直律的Cohen-Macaulay代数（简称ASL）的偏序集的性质。证明了如果偏序集$P$生成Cohen-Macaulay ASL，则$P$是纯偏序集;如果$P$是Buchsbaum偏序集，则$P$是Cohen-Macaulay偏序集.还建立了关于由拉直闭理想定义的ASL的Rees代数的一些结果。证明了若$P$是具有唯一极小元的Cohen-Macaulay偏序集，且$Q$是$P$的偏序集理想，则$P\uplus Q$也是Cohen-Macaulay.
The third named author and Pérez proved that under certain conditions the test ideal of a module closure agrees with the trace ideal of the module closure. We use this fact to compute the test ideals of various rings with respect to the closures coming from their indecomposable maximal Cohen-Macaulay modules. We also give an easier way to compute the test ideal of a hypersurface ring in 3 variables coming from a module with a particular type of matrix factorization.
A finitely generated module $M$ over a commutative Noetherian ring $R$ is called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M), \] where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure of this class of modules. It is discovered that $I$-Cohen Macaulay modules enjoy many interesting properties which are analogous to those of Cohen Macaulay modules. Also, various characterizations of $I$-Cohen Macaulay modules are presented here.
Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type
本文给出了C. Huneke和G. Leuschke（2003年）：设R是分次可数Cohen-Macaulay表示型的标准分次Cohen-Macaulay环，且R有一个孤立奇点.那么R必然是分次有限Cohen-Macaulay表示型吗？特别地，这个问题对于标准分次非Gorenstein环以及标准分次最小重数Gorenstein环都有肯定的答案。在此基础上，我们得到了分次可数Cohen-Macaulay型的分次Cohen-Macaulay环的一个部分分类。
Cohen-Macaulay injective, projective, and flat dimension
我们定义了三个新的同调维度-科恩-麦考利内射，投射，平坦的维度-居住在一个理论类似的经典内射，投射，平坦的维度。具有对偶模的Cohen-Macaulay环的新维数的等价性。
On Cohen-Macaulayness and depth of ideals in invariant rings
研究了不变环中Cohen-Macaulay理想的存在性，证明了对应于$p$ -群的模表示的不变环的理想不是Cohen-Macaulay，除非不变环本身是.作为中间结果，我们得到了非Cohen-Macaulay阶乘环不含Cohen-Macaulay理想.对素数阶模循环群，证明了不变环模转移理想的商总是Cohen-Macaulay，推广了Fleischmann的一个结果.
Let $(A,\mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = \bigoplus_{n\geq 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$, the associated graded ring of $A$. As an application we give several examples of complete Cohen-Macaulay local rings $A$ with $G(A)$ Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules $M_n$ with $G(M_n)$ Cohen-Maca…
Let $R$ be a Cohen-Macaulay local ring, $I$ a strongly Cohen-Macaulay ideal of $R$. We show that $R/{\rm Ann}_R(I)$ is a maximal Cohen-Macaulay $R$-module by means of the delta-invariant. Also it is shown that there exists a Cohen-Macaulay ideal $J$ of $R$ such that the $δ_{R/J}$-invariant of all Koszul homologies of $R$ with respect $I$ is zero.
Super-stretched and graded countable Cohen-Macaulay type
定义了Cohen-Macaulay环的超拉伸性，证明了分次可数Cohen-Macaulay型的Cohen-Macaulay环是超拉伸的.我们利用这个结果证明了分次可数Cohen-Macaulay型正维数环有可能的h-向量（1），（1，n），或（1，n，1）。此外，一维标准分次Gorenstein环的分次可数型被证明是超曲面;这一结果是不知道在更高的维度。在非Gorenstein情形下，证明了维数大于2的分次可数Cohen-Macaulay型环具有极小重数.
Higher Cohen-Macaulay property of squarefree modules and simplicial posets
最近，G. Floystad研究了某些有限正则胞腔复形的“高阶Cohen-Macaulay性质”。本文将他的结果部分推广到无平方模、复曲面环和单纯偏序集。例如，我们证明了，如果一个单纯偏序集（的相应胞腔复形）是$l$ -Cohen-Macaulay，那么它的余维1骨架是$(l+1)$ -Cohen-Macaulay。
Cohen-Macaulay classes which are not conic
本文给出两个例子，证明了对于仿射半群环，Cohen-Macaulay类的集合可以大于锥类的集合。
Relative Cohen-Macaulay filtered modules with a view toward relative Cohen-Macaulay modules
设R是交换诺特环，a是R的真理想，M是有限R-模.证明了：若（R;m）是完全局部环，则在一定条件下a包含DR（Hca（M））上的正则元，其中c = cd（a;M）.给出了相对Cohen-Macaulay模的一个非零因子刻划.引入相对Cohen-Macaulay滤模的概念，研究了这类模的一些基本性质.特别地，我们给出了相对Cohen-Macaulay滤模的一个非零因子刻画.进一步，利用M的因子的相伴素理想刻画了M的上同调维数滤除.作为结果，我们提出了一个上同调维数过滤的零子模有一个准素分解。最后，我们给出了关于相对Cohen-Macaulay模的一些新结果.
In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster's celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that for any valuation domain $V$ of finite Krull dimension, $V[x]$ is Cohen-Macaulay in the sense of Hamilton-Marley.
Stanley--Reisner rings with large multiplicities are Cohen--Macaulay
利用Alexander对偶证明了一类具有足够大重数的Stanley-Reisner环是Cohen-Macaulay环.
Cohen-Macaulayness of trivial extensions
我们的目标是确定当交换环的模平凡扩张是Hamilton和Marley意义下的Cohen-Macaulay。为此，我们将环的Cohen-Macaulay性概念推广到模。
On the structure of sequentially Cohen--Macaulay bigraded modules
设$K$是域，$S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$是$K$上的标准双阶多项式环.本文明确地刻画了关于$Q=(y_1,\ldots,y_n)$的双分次“序列Cohen-Macaulay”$S$ -模的结构.其次，我们给出了序列Cohen-Macaulay模关于$Q$的局部上同调模的刻画.考虑了相对于$Q$顺序地为Cohen-Macaulay的Cohen-Macaulay模.
Lyubeznik table of sequentially Cohen-Macaulay rings
证明了正特征的序列Cohen-Macaulay环和任意特征的序列Cohen-MacaulayStanley-Reisner环都有平凡Lyubeznik表.根据环的亏模，还提供了Lyubeznik表的一些其他配置。
Arithmetical rank of Cohen-Macaulay squarefree monomial ideals of height two
本文证明了一个高为2的无平方根单项式理想，其商环为Cohen-Macaulay，是集论完全交.
Local rings of countable Cohen--Macaulay type
本文证明了Schreyer猜想的一个极好的情形：具有可数Cohen-Macaulay型的局部环至多有一个一维奇异轨迹.进一步证明了可数CM型Cohen-Macaulay局部环的局部化也是可数CM型的。
Full-Trace Modules
本研究在交换Noetherian局部环上引入全迹模概念，证明非正则非主理想环上剩余域的正合冲模均为全迹模，刻画了Cohen-Macaulay环极小重数与全迹Ulrich模的等价关系，并给出数值半群环上全迹Ulrich模的分解形式。
More properties of almost Cohen-Macaulay rings
研究了几乎Cohen-Macaulay环的一些有趣的性质，并研究了与这类环有关的一个Serre型性质.
Cohen-Macaulayness of large powers of Stanley-Reisner ideals
证明了当m > 2时，Stanley-Reisner理想的m次符号幂是Cohen-Macaulay当且仅当单纯复形是拟阵.类似地，m次普通幂是Cohen-Macaulay，其中m > 2当且仅当复形是完全交。这些结果解决了Stanley-Reisner理想的普通幂和符号幂的Cohen-Macaulay性的几个公开问题。此外，它们对刻面理想和覆盖理想的象征力量的Cohen-Macaulayness有有趣的影响。
On two dimensional mixed characteristic rings of finite Cohen Macaulay type
本文给出了二维有限Cohen-Macaulay型混合特征环的大量例子。对于这些例子的一个大子类，我们给出了它的不可分解极大Cohen-Macaulay模的完整描述，并计算了它的AR-模。
Topics on sequentially Cohen-Macaulay modules
本文研究了与序列Cohen-Macaulay模相关的两个不同的主题。问题是环的序列Cohen-Macaulay性质何时保持局部化和模有限扩张。
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}.
Classes of sequentially Cohen-Macaulay squarefree monomial ideals
我们计算了完全无平方lexsegment理想的最小准素分解。我们证明了临界无平方根单项理想是序列Cohen-Macaulay理想。作为应用，我们给出了序列Cohen-Macaulay的完全无平方lexsegment理想的一个完整刻画，并导出了这类理想的一些同调不变量的计算公式.
On approximately Cohen-Macaulay binomial edge ideal
文[4]引入了图G的二项边理想IG。他们发现了一些类的图G的性质，即IG是一个科恩-麦考利理想。这可能只发生在少数几类图上。科恩-麦考利的一种推广，称为近似科恩-麦考利，是由S。Goto在[3]中。研究了一类二项式边理想近似Cohen-Macaulay的图。此外，我们使用一些同调的方法来计算它们的希尔伯特级数。
Sequentially Cohen-Macaulay monomial ideals of embedding dimension four
设$I$是域$K$上多项式环$S=K[x_1,...,x_4]$的单项理想.然后$S/I$是顺序的Cohen-Macaulay当且仅当$S/I$是非常干净的。特别地，如果$S/I$是顺序Cohen-Macaulay，则$I$是Stanley理想。
We investigate the behavior of Cohen-Macaulay defect undertaking tensor product with a perfect module. Consequently, we study the perfect defect of a module. As an application, we connect to associated prime ideals of tensor products.
This paper shows that Cohen-Macaulay algebras can be algebraically approximated in such a way that their Cohen-Macaulayness and minimal Betti numbers are preserved. This is achieved by showing that finitely generated modules over power series rings can be algebraically approximated in a manner that preserves their diagrams of initial exponents and their minimal Betti numbers. These results are also applied to obtain an approximation result for flat homomorphisms from rings of power series to Cohen-Macaulay algebras.
Let $M$ be an $R$-module over a Noetherian ring $R$ and $\mathfrak{a}$ be an ideal of $R$ with $c={\rm cd}(\mathfrak{a},M)$. First, we prove that $M$ is finite $\mathfrak{a}$-relative Cohen-Macaulay if and only if ${\rm H}_i(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M)))=0$ for all $i\neq c$ and ${\rm H}_c(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M))) \cong \widehat{M}^{\mathfrak{a}}$. Next, over an $\mathfrak{a}$-relative Cohen-Macaulay local ring $(R,\mathfrak{m})$, we provide a characterization of $\mathfrak{a}$-relative sequentially Cohen-Macaulay modules $M$ in terms of $\mathfrak{a}$-re…
In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.
This paper shows that if $R$ is a homomorphic image of a Cohen-Macaulay local ring, then $R$-module $M$ is sequentially generalized Cohen-Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all distinguished parameter ideals of $M$ are bounded.
Eventually index of reducibility on sequentially Cohen-Macaulay modules
证明了一个模是序列Cohen-Macaulay模当且仅当它的可区别参数理想的约化指数最终为具有特殊值的常数。作为主要定理的推论，我们给出了局部环的Gorensteinness性，Cohen-Macaulay性的特征，它们是用区别参数理想的最终约化指数来刻画的.
The category of maximal Cohen--Macaulay modules as a ring with several objects
在交换局部Cohen-Macaulay环上，我们将极大Cohen-Macaulay模范畴看作多对象环来研究。我们计算这个类别的整体尺寸，从而延长Leuschke的结果的情况下，环具有任意（而不是有限）CM型。
Cohen-Macaulay Loci of modules
研究了Noether局部环$A$上任意有限模的Cohen-Macaulay轨迹，证明了在某些情形下它是$\Spec A$的Zerkiki-开子集.在这方面，环的形式纤维在某些素理想是Cohen-Macaulay的研究。
Unmixed and sequentially Cohen-Macaulay skew tableau ideals
本文研究由正整数填充斜 Ferrers 图所定义的斜表理想，完成了对无混与序列 Cohen–Macaulay 情形的分类，并据此得到 Cohen–Macaulay、Buchsbaum 和广义 Cohen–Macaulay 情形的完整分类。
Uniform bounds in generalized Cohen-Macaulay rings
建立了广义Cohen-Macaulay环中参数理想的相伴分次环的Castelnuovo-Mumford正则性的一致界.作为结果，我们得到了关系类型和假设数的统一界。此外，我们还证明了广义Cohen-Macaulay环可以用这种一致界的存在性来刻画。
Modules of minimal multiplicity over one-dimensional Cohen-Macaulay local rings
本文研究了一维Cohen-Macaulay局部环上最小乘数模的性质，通过刻画跟踪理想和反射理想的作用，揭示了极小乘数模与Burch和Ulrich模的关系。
We prove some results on the non-existence of rank one maximal Cohen-Macaulay modules over certain Segre product rings. As an application we show that over these Segre product rings there do not exist maximal Cohen-Macaulay modules with multiplicity less than or equal to the parameter degree of the ring, which disproves a conjecture of Schoutens.
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