Cohen-Macaulay chordal graphs

Bi-Cohen-Macaulay graphs

In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General bi-Cohen-Macaulay graphs are classified up to separation. The inseparable bi-Cohen-Macaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.

Minimal Cohen-Macaulay Simplicial Complexes

We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal Cohen-Macaulay. We show that many interesting examples of Cohen-Macaulay complexes in combinatorics are minimal, including Rudin's ball, Ziegler's ball, the dunce hat, and recently discovered non-partitionable Cohen-Macaulay complexes. We further provide various ways to construct such complexes.

Cohen-Macaulay binomial edge ideals of cactus graphs

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

Cohen-Macaulayness of triangular graphs

We study the Cohen-Macaulay property of triangular graphs $T_n$. We show that $T_2$, $T_3$ and $T_5$ are Cohen-Macaulay graphs, and that $T_4$, $T_6$, $T_8$ and $T_n$ are not Cohen-Macaulay graphs, for $n\geq 10$. Finally, we prove that over fields of characteristic zero $T_7$ and $T_9$ are Cohen-Macaulay.

Sequentially Cohen-Macaulay mixed product ideals

We classify the ideals of mixed products that are sequentially 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.

On Sequentially Cohen-Macaulay Complexes and Posets

The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra.

Cohen-Macaulay lexsegment complexes in arbitrary codimension

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension $t$ if and only if it is the join of a lexsegment pure disconnect…

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

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

Almost Cohen-Macaulay bipartite graphs and connected in codimension two

In this paper we study almost Cohen-Macaulay bipartite graphs. Furthermore, we prove that if $G$ is almost Cohen-Macaulay bipartite graph with at least one vertex of positive degree, then there is a vertex of $°(v) \leq 2$. In particular, if $G$ is an almost Cohen-Macaulay bipartite graph and $u$ is a vertex of degree one of $G$ and $v$ its adjacent vertex, then $G\setminus\{v\}$ is almost Cohen-Macaulay. Also, we show that an unmixed Ferrers graph is almost Cohen-Macaulay if and only if it is connected in codimension two. Moreover, we give some examples.

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).

On the Hilbert series of vertex cover algebras of Cohen-Macaulay bipartite graphs

We study the Hilbert function and the Hilbert series of the vertex cover algebra $A(G)$, where $G$ is a Cohen-Macaulay bipartite graph.

具有大围长的Cohen-Macaulay有向图

Cohen-Macaulay oriented graphs with large girth

该研究针对底图围长不小于5的带权有向图，完成了Cohen-Macaulay类型的分类工作。

SCMAlgebras：用于序列Cohen-Macaulay性的Macaulay2包

SCMAlgebras: a Macaulay2 package for sequentially Cohen-Macaulayness

介绍Macaulay2包SCMAlgebras，提供计算亏格模和滤子理想的函数以检查模或理想的序列Cohen-Macaulay性，并实现其他Cohen-Macaulay型条件和无混合性的研究。

Cohen-Macaulay $r$-partite graphs with minimal clique cover

In this note, we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay. It is proved that if $G$ is an $r$-partite Cohen-Macaulay graph which is covered by some disjoint cliques of size $r$, then the clique cover is unique.

Cohen Macaulay Hybrid Graphs

We introduce a new family of graphs, namely, hybrid graphs. There are infinitely many hybrid graphs associated to a single graph. We show that every hybrid graph associated to a given graph is Cohen Macaulay. Furthermore, we show that every CohenMacaulay chordal graph is a hybrid graph.

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.

Cohen-Macaulay Circulant Graphs

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,...,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C_n(1,2,...,d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, t…

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.

On Virtually Cohen-Macaulay Simplicial Complexes

We examine virtual resolutions of Stanley-Reisner ideals for a product of projective spaces. In particular, we provide sufficient conditions for a simplicial complex to be virtually Cohen-Macaulay (to have a virtual resolution with length equal to its codimension). We also show that all balanced simplicial complexes are virtually Cohen-Macaulay.

Cohen-Macaulay-ness in codimension for bipartite graphs

Let $G$ be an unmixed bipartite graph of dimension $d-1$. Assume that $K_{n,n}$, with $n\ge 2$, is a maximal complete bipartite subgraph of $G$ of minimum dimension. Then $G$ is Cohen-Macaulay in codimension $d-n+1$. This generalizes a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and a result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph $G$ which is Cohen-Macaulay in codimension $t$, is obtained from a Cohen-Macaulay graph by replacing certain edges of $G$ with complete bipartite graphs. We provide some examples.

A Characterization of Triangle-free Gorenstein graphs and Cohen-Macaulayness of second powers of edge ideals

We graph-theoretically characterize triangle-free Gorenstein graphs $G$. As an application, we classify when $I(G)^2$ is Cohen-Macaulay.

Sequentially Cohen-Macaulay Edge Ideals

Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R = k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some ex…

边加权r-路径理想的Cohen-Macaulay型

The Cohen-Macaulay type of edge-weighted r-path ideals

本文对任意正整数r，用组合方法刻画了边加权r-路径悬挂的Cohen-Macaulay型，并将边加权悬挂情形统一为r=1的特例。

Equivalent condition for approximately Cohen-Macaulay complexes

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen-Macaulay. Namely it is approximately Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and generated in two consecutive degrees. This completes the result of Herzog and Hibi who proved that a simplicial complex is sequentially Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear.

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。

On the h-vectors of Cohen-Macaulay Flag Complexes

Starting from an unpublished conjecture of Kalai and from a conjecture of Eisenbud, Green and Harris, we study several problems relating h-vectors of Cohen-Macaulay, flag simplicial complexes and face vectors of simplicial complexes.

虚拟Cohen-Macaulay单纯复形的条件

Conditions for Virtually Cohen--Macaulay Simplicial Complexes

本文给出单纯复形成为虚拟Cohen-Macaulay单纯复形的充分条件，引入虚拟可剥单纯复形这一可剥复形的推广，并证明其具有虚拟Cohen-Macaulay性质，推广了可剥复形是Cohen-Macaulay的结论。

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.

Cohen-Macaulay graphs with large girth

We classify Cohen-Macaulay graphs of girth at least $5$ and planar Gorenstein graphs of girth at least $4$. Moreover, such graphs are also vertex decomposable.

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.

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.

Bipartite $S_2$ graphs are Cohen-Macaulay

In this paper we show that if the Stanley-Reisner ring of the simplicial complex of independent sets of a bipartite graph $G$ satisfies Serre's condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi carries over this family of bipartite graphs. We check that the equivalence of Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs and we classify cyclic graphs with respect to the condition $S_2$.

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'.

Reciprocal domains and Cohen-Macaulay $d$-complexes in $R^d$

We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the issue of whether a Cohen-Macaulay complex of dimension d embedded piecewise-linearly in d-space is necessarily a d-ball. This is observed to be true for d at most 3, but false for d=4.

Shellable and Cohen-Macaulay complete t-partite graphs

Let G be a simple undirected graph. We find the number of maximal independent sets in complete t-partite graphs. We will show that vertex decomposability and shellability are equivalent in this graphs. Also, we obtain an equivalent condition for Cohen-Macaulay in complete t-partite graphs.

Graded Cohen-Macaulay domains and lattice polytopes with short $h$-vector

Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that if $h_2^* \leq h_1^*$, then $P$ is IDP. More generally, we show the corresponding statements for semi-standard graded Cohen-Macaulay domains over algebraically closed fields.

Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity

Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially Cohen-Macaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the Castelnuovo-Mumford regularity of R/I(G) can be determined from the invariants of G.

Cohen-Macaulay graphs arising from digraphs

In this paper we show a correspondence between directed graphs and bipartite undirected graphs with a perfect matching, that allows to study properties of directed graphs through the properties of the corresponding undirected graphs. In particular it is shown that a directed graph is transitive iff a corresponding undirected graph is Cohen-Macaulay.

When almost Cohen-Macaulay algebras map into big Cohen-Macaulay modules

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.

关于Cohen-Macaulay亏格与几乎Cohen-Macaulay环

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。

Extremal Betti numbers of some Cohen-Macaulay binomial edge ideals

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

In this paper we present characterizations of sequentially Cohen-Macaulay modules in terms of systems of parameters, which are generalizations of well-known results on Cohen-Macaulay and generalized Cohen-Macaulay modules. The sequentially Cohen-Macaulayness of Stanley-Reisner rings of small embedding dimension are also examined.

Perfect linkage of Cohen--Macaulay modules over Cohen--Macaulay rings

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…

Direct limits of Cohen-Macaulay rings

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.

关于序列Cohen-Macaulay模与序列广义Cohen-Macaulay模

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

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.

Dual Bass Numbers and Co-Cohen Macaulay Modules

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.

Syzygies of Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings

We study syzygies of (maximal) Cohen-Macaulay modules over one dimensional Cohen-Macaulay local rings. We compare these modules to Cohen-Macaulay modules over the endomorphism ring of the maximal ideal. After this comparison, we give several characterizations of almost Gorenstein rings in terms of syzygies of Cohen-Macaulay modules.

Computing families of Cohen-Macaulay and Gorenstein rings

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.

Krull--Gabriel dimension of Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type

We calculate the Krull--Gabriel dimension of the functor category of the (stable) category of maximal Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type. We show that the Krull--Gabriel dimension is $0$ if the hypersurface is of finite Cohen--Macaulay representation type and that is $2$ if the hypersurface is of countable but not finite Cohen--Macaulay representation type.

Cohen-Macaulay homological dimensions

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.

Certain Classes of Cohen-Macaulay Multipartite Graphs

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.