On a Theorem of Macaulay on Colons of Ideals

Classes of sequentially Cohen-Macaulay squarefree monomial ideals

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.

Cohen-Macaulay test ideals over rings of finite and countable Cohen-Macaulay type

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.

Cohen-Macaulay Polymatroidal Ideals

All Cohen--Macaulay polymatroidal ideals are classified. The Cohen--Macaulay polymatroidal ideals are precisely the principal ideals, the Veronese ideals, and the squarefree Veronese ideals.

On Cohen-Macaulayness and depth of ideals in invariant rings

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.

Arithmetical rank of Cohen-Macaulay squarefree monomial ideals of height two

In this paper, we prove that a squarefree monomial ideal of height 2 whose quotient ring is Cohen-Macaulay is set-theoretic complete intersection.

Cohen-Macaulayness of large powers of Stanley-Reisner ideals

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.

Sequentially Cohen-Macaulay monomial ideals of embedding dimension four

Let $I$ be a monomial ideal of the polynomial ring $S=K[x_1,...,x_4]$ over a field $K$. Then $S/I$ is sequentially Cohen-Macaulay if and only if $S/I$ is pretty clean. In particular, if $S/I$ is sequentially Cohen-Macaulay then $I$ is a Stanley ideal.

无混且序列 Cohen–Macaulay 的斜表理想

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。

Tensor Products of Cohen-Macaulay Rings. Solution to a Problem of Grothendieck

In this paper we solve a problem, originally raised by Grothendieck, on the transfer of Cohen-Macaulayness to tensor products of algebras over a field. As a prelude to this, we investigate the grade for some specific types of ideals that play a primordial role within the ideal structure of such constructions.

A study of strongly Cohen-Macaulay ideals by delta invariant

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.

几乎Cohen-Macaulay 3生成理想在矩阵理论中的结构

The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory

研究通过矩阵理论描述了codimension为2的几乎Cohen-Macaulay 3生成理想结构，基于自由分解和潜在数据，引入了水平矩阵概念，并给出其存在条件。

Classification of Cohen-Macaulay $t$-spread lexsegment ideals via simplicial complexes

We study the minimal primary decomposition of completely $t$-spread lexsegment ideals via simplicial complexes. We determine some algebraic invariants of such a class of $t$-spread ideals. Hence, we classify all $t$-spread lexsegment ideals which are Cohen-Macaulay.

On approximately Cohen-Macaulay binomial edge ideal

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.

Big Cohen-Macaulay test ideals in equal characteristic zero via ultraproducts

Utilizing ultraproducts, Schoutens constructed a big Cohen-Macaulay algebra $\mathcal{B}(R)$ over a local domain $R$ essentially of finite type over $\mathbb{C}$. We show that if $R$ is normal and $Δ$ is an effective $\mathbb{Q}$-Weil divisor on $\operatorname{Spec} R$ such that $K_R+Δ$ is $\mathbb{Q}$-Cartier, then the BCM test ideal $τ_{\hat{\mathcal{B}(R)}}(\hat{R},\hatΔ)$ of $(\hat{R},\hatΔ)$ with respect to $\hat{\mathcal{B}(R)}$ coincides with the multiplier ideal $\mathcal{J}(\hat{R},\hatΔ)$ of $(\hat{R},\hatΔ)$, where $\hat{R}$ and $\hat{\mathcal{B}(R)}$ are the $\mathfrak{m}$-adic com…

加权形式的初始理想与局部Cohen-Macaulay曲线的亏格

Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

研究局部Cohen-Macaulay曲线的最大亏格问题，证明当曲线度数d等于s或d≥2s-1时，预测的亏格最大值成立，基于对非标准格雷度多项式环中特定齐次形式的初始理想猜想的证明。

Vanishing of local cohomology and set-theoretically Cohen-Macaulay ideals

We generalize some known results on the relation between the cohomological and projective dimension. Then we examine the set-theoretically Cohen-Macaulay ideals to find some cohomological characterization of these kind of ideals.

Strongly Cohen-Macaulay ideals of small second analytic deviation

We characterize the strongly Cohen-Macaulay ideals of second analytic deviation one in terms of depth properties of the powers of the ideal in the `standard range.' This provides an explanation of the behaviour of certain ideals that have appeared in the literature.

Cohen-Macaulay binomial edge ideals of cactus graphs

We classify the Cohen-Macaulay binomial edge ideals of cactus and bicyclic graphs.

Lexsegment ideals are sequentially Cohen-Macaulay

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.

A note on almost Cohen-Macaulay monomial ideals

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.

Sequentially Cohen-Macaulay mixed product ideals

We classify the ideals of mixed products that are sequentially Cohen-Macaulay.

Construction of Cohen-Macaulay binomial edge ideals

We discuss algebraic and homological properties of binomial edge ideals associated to graphs which are obtained by gluing of subgraphs and the formation of cones.

The structure of chains of Ulrich ideals in Cohen-Macaulay local rings of dimension one

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.

顶点可分裂单项式理想的Cohen-Macaulay性质

Cohen-Macaulayness of vertex splittable monomial ideals

本文提出顶点可分裂理想Cohen-Macaulay性质的新判据，基于Betti分裂提供归纳检验方法，并据此刻画了Gorenstein等特殊理想及双Cohen-Macaulay图族。

Cohen-Macaulay binomial edge ideals

We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen--Macaulay.

Cohen-Macaulay Monomial Ideals of Codimension 2

We give a structure theorem for Cohen Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common.

Cohen-Macaulay Weighted Oriented Edge Ideals and its Alexander Dual

The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$ with underlying graph $G$ started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave for edge ideals of simple graphs. We use this construction to classify all the Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We show that the conjecture on Cohen-Macaulayness of $I(D_{G})$, proposed by Pitones et al. (2019), holds for $I(D_{C_{n}})$, where $C_{n}$ denotes the cycle of length $n$. Miller generalized the concept of…

Integral Closures of Cohen-Macaulay monomial ideals

The purpose of this paper is to present a family of Cohen-Macaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.

Sequentially Cohen-Macaulay binomial edge ideals of closed graphs

In this paper we provide a full combinatorial characterization of sequentially Cohen-Macaulay binomial edge ideals of closed graphs. In addition, we show that a binomial edge ideal of a closed graph is approximately Cohen-Macaulay if and only if it is almost Cohen-Macaulay.

Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.

Cohen-Macaulay binomial edge ideals of small deviation

We classify all binomial edge ideals that are complete intersection and Cohen-Macaulay almost complete intersection. We also describe an algorithm and provide an implementation to compute primary decomposition of binomial edge ideals.

An algorithm for computing the Hilbert--Samuel multiplicities and reductions of zero-dimensional ideals of Cohen--Macaulay local rings

In this paper, we present an algorithm for computing the minimal reductions of $\mathfrak{m}$-primary ideals of Cohen--Macaulay local rings. Using this algorithm, we are able to compute the Hilbert--Samuel multiplicities and solve the membership problem for the integral closure of $\mathfrak{m}$-primary ideals.

Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich ideals.

Sequentially Cohen-Macaulay matroidal ideals

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $J$ be a matroidal ideal of degree $d$ in $R$. In this paper, we study the class of sequentially Cohen-Macaulay matroidal ideals. In particular, all sequentially Cohen-Macaulay matroidal ideals of degree $2$ are classified. Furthermore, we give a classification of sequentially Cohen-Macaulay matroidal ideals of degree $d\geq 3$ in some special cases.

具有图围长的二项式边理想的 Cohen–Macaulay 性质

Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs

本文利用分次理想与其无平方因子初始理想的深度相等性，研究 Cohen–Macaulay 二项式边理想，给出基于 whisker、双连通图和图块的刻画条件，并证明其图围长必小于 5 或为无穷大。

Cohen-Macaulay residual intersections and their Castelnuovo-Mumford Regularity

In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained through studying such a complex is to prove the Cohen-Macaulayness of residual intersections of a wide class of ideals. For example we show that, in a Cohen-Macaulay local ring, any geometric residual inter…

余维数二的 Cohen–Macaulay 理想与平面点的几何

Cohen--Macaulay ideals of codimension two and the geometry of plane points

本文通过分析 Hilbert–Burch 矩阵研究余维数二 Cohen–Macaulay 理想及平面点几何，并证明超平面排列 Jacobian 理想猜想。

Cohen-Macaulay Binomial edge ideals in terms of blocks with whiskers

For a graph $G$, Bolognini et al. have shown $J_{G}$ is strongly unmixed $\Rightarrow$ $J_{G}$ is Cohen-Macaulay $\Rightarrow$ $G$ is accessible, where $J_{G}$ denotes the binomial edge ideals of $G$. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all $G$ with Cohen-Macaulay $J_{G}$. We show that accessible and strongly unmixed properties of $G$ depend only on the corresponding properties of its blocks with whiskers and vice versa. Also, we give an infinite class of graphs whose bin…

The projective dimension of sequentially Cohen-Macaulay monomial ideals

In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal cardinality of a minimal vertex cover of its facet complex. This in particular gives a formula for the projective dimension of facet ideals of these classes of ideals, which are known to be sequentially Cohen-Macaulay: graph trees and simplicial trees, chordal graphs and some cycles, chordal clutters and graphs, and some path ideals to mention a few. Since polarizat…

Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals

We give a generalization of Hochster's formula for local cohomologies of square-free monomial ideals to monomial ideals, which are not necessarily square-free. Using this formula, we give combinatorial characterizations of generalized Cohen-Macaulay monomial ideals. We also give other applications of the generalized Hochster's formula.

Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals

Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay). As an application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum, generalized Cohen-Macaulay).

Cohen-Macaulayness of generically complete intersection monomial ideals

In this paper we discuss the problem of characterizing the Cohen-Macaulay property of certain families of monomial ideals with fixed radical. More precisely, we consider generically complete intersection monomial ideals whose radical corresponds to special classes of simplicial complexes.

具有线性分解的2-子式素理想与Cohen-Macaulay理想

Prime and Cohen-Macaulay ideals of 2-minors with linear resolution

本文分类了变量矩阵2-子式理想中具有线性分解的素理想与Cohen-Macaulay理想。

Invariants of Cohen-Macaulay rings associated to their canonical ideals

The purpose of this paper is to introduce new invariants of Cohen-Macaulay local rings. Our focus is the class of Cohen-Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers--the type of R, the reduction number of C--that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers--the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C.

On Hilbert coefficients of parameter ideals and Cohen-Macaulayness

Let $(R, \mathfrak m)$ be an unmixed Noetherian local ring, Q a parameter ideal and $K$ an $\mathfrak m$-primary ideal of $R$ containing $Q$. We give a necessary and sufficient condition for $R$ to be Cohen-Macaulay in terms of $g_0(Q)$ and $g_1(Q)$, the Hilbert coefficients of $Q$ with respect to $K$. As a consequence, we obtain a result of Ghezzi et al. which settles the negativity conjecture of W. V. Vasconcelos [15] in unmixed local rings.

Cohen-Macaulay Properties of Square-Free Monomial Ideals

In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.

A sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity

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…

Absolute bounds on the number of generators of Cohen-Macaulay ideals of height at most 2

For a Noetherian local domain $A$, there exists an upper bound $N_τ(A)$ on the minimal number of generators of any height two ideal $I$ for which $A/I$ is Cohen-Macaulay of type $τ$. More precisely, we may take $N_τ(A):=(τ+1)e_{\text{h}}(A)$, where $e_{\text{h}}(A)$ is the homological multiplicity of $A$.

Regularity of powers of edge ideals of Cohen-Macaulay weighted oriented forests

In this paper, we explicitly give combinatorial formulas for the regularity of powers of edge ideals, $\reg(I(D)^k)$, of weighted oriented unmixed forests $D$ whose leaves are sinks ($V^+(D)$ are sinks). This combinatorial formula is a piecewise linear function of $k$, for $k \geq 1$.

Some Cohen-Macaulay and unmixed binomial edge ideals

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen-Macaulay and unmixed. So that we generalize the results of Ene, Herzog and Hibi on block graphs. Moreover, we study unmixedness and Cohen-Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs'.

Cohen-Macaulay 纤维锥的定义理想

Defining ideals of Cohen-Macaulay fiber cones

本文针对交换 Noether 局部环中的理想，给出了 Cohen-Macaulay 纤维锥的定义理想的构造方法。

Cohen-Macaulay edge ideal whose height is half of the number of vertices

We consider a class of graphs $G$ such that the height of the edge ideal $I(G)$ is half of the number $\sharp V(G)$ of the vertices. We give Cohen-Macaulay criteria for such graphs.

B-sequence and approximations of generalized Cohen-Macaulay ideals

We introduce the notion of b-sequence for finitely generated modules over Noetherian rings, which characterizes long Bourbaki sequences. Our main concern is an application of this notion to generalized Cohen-Macaulay approximation, which we introduced lately. We will show how we can construct long Bourbaki sequences of non-trivial type characterizing generalized Cohen-Macaualy ideals by finding suitable b-sequences.

When is a Specht ideal Cohen-Macaulay?

For a partition $λ$ of $n$, let $I^{\rm Sp}_λ$ be the ideal of $R=K[x_1, \ldots, x_n]$ generated by all Specht polynomials of shape $λ$. We show that if $R/I^{\rm Sp}_λ$ is Cohen--Macaulay then $λ$ is of the form either $(a, 1, \ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${\rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{\rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${\rm char}(K)=2$.