Linquan Ma
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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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。
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。
In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen-Macaulay modules over Cohen--Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen-Macaulay approximations and Yoshino-Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen-Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of lin…
Let $A$ be a direct limit of a direct system of Cohen-Macaulay rings. In this paper, we describe the Cohen-Macaulay property of $A$. Our results indicate that $A$ is not necessarily Cohen-Macaulay. We show $A$ is Cohen-Macaulay under various assumptions. As an application, we study Cohen-Macaulayness of non-affine normal semigroup rings.
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。
In \cite{AB}, the dagger closure is extended over finitely generated modules over Noetherian local domain $(R,\fm)$ and it is proved to be a Dietz closure. In this short note we show that it also satisfies the `Algebra axiom' of \cite{R.G} and this leads to the following result of this paper: For a complete Noetherian local domain, if it is contained in an almost Cohen-Macaulay domain, then there exists a balanced big Cohen-Macaulay algebra over it.
In this paper, we give a characterization of co-Cohen Macaulay modules by vanishing properties of the dual Bass numbers of modules. In addition, we show that the co-localization of co-Cohen Macaulay modules preserves co-Cohen Macaulayness under a certain condition.
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）。
We characterize Cohen-Macaulay and Gorenstein rings obtained from certain types of convex body semigroups. Algorithmic methods to check if a polygonal or circle semigroup is Cohen-Macaulay/Gorenstein are given. We also provide some families of Cohen-Macaulay and Gorenstein rings.
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.
In this paper we show a partial answer the a question of C. Huneke and G. Leuschke (2003): Let R be a standard graded Cohen-Macaulay ring of graded countable Cohen-Macaulay representation type, and assume that R has an isolated singularity. Is R then necessarily of graded finite Cohen-Macaulay representation type? In particular, this question has an affirmative answer for standard graded non-Gorenstein rings as well as for standard graded Gorenstein rings of minimal multiplicity. Along the way, we obtain a partial classification of graded Cohen-Macaulay rings of graded countable Cohen-Macaulay…
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。
We investigate the presence of Cohen-Macaulay ideals in invariant rings and show that an ideal of an invariant ring corresponding to a modular representation of a $p$-group is not Cohen-Macaulay unless the invariant ring itself is. As an intermediate result, we obtain that non-Cohen-Macaulay factorial rings cannot contain Cohen-Macaulay ideals. For modular cyclic groups of prime order, we show that the quotient of the invariant ring modulo the transfer ideal is always Cohen-Macaulay, extending a result of Fleischmann.
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.
We define what it means for a Cohen-Macaulay ring to to be super-stretched and show that Cohen-Macaulay rings of graded countable Cohen-Macaulay type are super-stretched. We use this result to show that rings of graded countable Cohen-Macaulay type, and positive dimension, have possible h-vectors (1), (1,n), or (1,n,1). Further, one dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen-Macaulay type of dimension larger than 2 are shown to be …
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.
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。
Let R be a commutative Noetherian ring, a a proper ideal of R and M a finite R-module. It is shown that, if (R;m) is a complete local ring, then under certain conditions a contains a regular element on DR(Hc a(M)), where c = cd(a;M). A non-zerodivisor characterization of relative Cohen-Macaulay modules w.r.t a is given. We introduce the concept of relative Cohen-Macaulay filtered modules w.r.t a and study some basic properties of such modules. In paticular, we provide a non-zerodivisor characterization of relative Cohen-Macaulay filtered modules w.r.t a. Furthermore, a characterization of coho…
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.
Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen-Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen-Macaulayness of rings to modules.
Let $K$ be a field and $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be the standard bigraded polynomial ring over $K$. In this paper, we explicitly describe the structure of finitely generated bigraded "sequentially Cohen--Macaulay" $S$-modules with respect to $Q=(y_1,\ldots,y_n)$. Next, we give a characterization of sequentially Cohen--Macaulay modules with respect to $Q$ in terms of local cohomology modules. Cohen--Macaulay modules that are sequentially Cohen--Macaulay with respect to $Q$ are considered.
We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.
In this paper, we prove that a squarefree monomial ideal of height 2 whose quotient ring is Cohen-Macaulay is set-theoretic complete intersection.
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。
Some interesting properties of almost Cohen-Macaulay rings are investigated and a Serre type property connected with this class of rings is studied.
We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal is Cohen-Macaulay if and only if the simplicial complex is a matroid. Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only if the complex is a complete intersection. These results solve several open questions on the Cohen-Macaulayness of ordinary and symbolic powers of Stanley-Reisner ideals. Moreover, they have interesting consequences on the Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.
In this paper we give a bountiful number of examples of two dimensional mixed characteristic rings of finite Cohen Macaulay type. For a large sub-class of these examples we give a complete description of its indecomposable maximal Cohen-Macaulay modules and we also compute its AR-quiver.
In this paper, we study the two different topics related to sequentially Cohen-Macaulay modules. The questions are when the sequentially Cohen-Macaulay property preserve the localization and the module-finite extension of rings.
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}.
We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.
Binomial edge ideals IG of a graph G were introduced by [4]. They found some classes of graphs G with the property that IG is a Cohen-Macaulay ideal. This might happen only for few classes of graphs. A certain generalization of being Cohen-Macaulay, named approximately Cohen-Macaulay, was introduced by S. Goto in [3]. We study classes of graphs whose binomial edge ideal are approximately Cohen-Macaulay. Moreover we use some homological methods in order to compute their Hilbert series.
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.
It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the Gorensteinness, Cohen-Macaulayness of local rings in term of eventually index of reducibility for distinguished parameter ideals.
Over a commutative local Cohen--Macaulay ring, we view and study the category of maximal Cohen--Macaulay modules as a ring with several objects. We compute the global dimension of this category and thereby extend a result of Leuschke to the case where the ring has arbitrary (as opposed to finite) CM-type.
The Cohen-Macaulay locus of any finite module over a noetherian local ring $A$ is studied and it is shown that it is a Zariski-open subset of $\Spec A$ in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen-Macaulay are studied.
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。
Big Cohen-Macaulay Algebras and Seeds
我们深入研究代数所拥有的属性，我们称之为种子，映射到大科恩-麦考利代数。我们将证明在一个正特征的完备局部区域上，任意两个大Cohen-Macaulay代数映射到一个公共的大Cohen-Macaulay代数。我们也将加强Hochster和Huneke的“弱函子”存在性结果大Cohen-Macaulay代数的种子性质是稳定的基础下的变化之间的完整的局部区域的正特征。我们还证明了正特征局部环（R，m）上的每一个种子映射到一个大的Cohen-Macaulay R-代数，该代数是一个绝对整闭的m-根分离的拟局部整环.
Over a Cohen-Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a "structure theorem" for balanced big CM modules. (2) Every finitely generated module has a preenvelope with respect to the class of fi…
In this paper, we investigate the maximal Cohen-Macaulay property of tensor products of modules, and then give criteria for projectivity of modules in terms of vanishing of Ext modules. One of the applications shows that the Auslander-Reiten conjecture holds for Cohen-Macaulay normal rings.
Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.
Hypersurfaces of bounded Cohen--Macaulay type
令R = k[[x_0，.，x_d]]/（f），其中k是域，f是形式幂级数环k[[x_0，.，x_d]]。我们调查的问题，这种形式的环有界科恩-麦考利型，也就是说，有一个限制的重不可分解的极大科恩-麦考利模块。与有限Cohen-Macaulay型一样，如果特征不同于2，则问题简化为一维情况：环R具有有界Cohen-Macaulay型当且仅当R同构于k[[x_0，...，x_d]]/（g+x_2^2+.+ x_d^2），其中g是k[[x_0，x_1]]中的元素，且k[[x_0，x_1]]/（g）具有有界Cohen-Macaulay型.本文确定了形式为k[[x_0，x_1]]/（g）的环具有有界的Cohen-Macaulay型.
The purpose of this article is to provide a new characterization of Cohen-Macaulay local rings. As a consequence we deduce that a local (Noetherian) ring $R$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible.
In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the…
The associated primes of an arbitrary lexsegment ideal $I\subset S=K[x_1,...,x_n]$ are determined. As application it is shown that $S/I$ is a pretty clean module, therefore, $S/I$ is sequentially Cohen-Macaulay and satisfies Stanley's conjecture.
Let $R = k[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $k$ and let $I$ be a monomial ideal of $R$. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost Cohen-Macaulay polymatroidal Veronese type and transversal polymatroidal ideals and furthermore we give some examples.
Let (R,m,k) be a one-dimensional analytically unramified local ring with minimal prime ideals P_1,...,P_s. Our ultimate goal is to study the direct-sum behavior of maximal Cohen-Macaulay modules over R. Such behavior is encoded by the monoid C(R) of isomorphism classes of maximal Cohen-Macaulay R-modules: the structure of this monoid reveals, for example, whether or not every maximal Cohen-Macaulay module is uniquely a direct sum of indecomposable modules; when uniqueness does not hold, invariants of this monoid give a measure of how badly this property fails. The key to understanding the mono…
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