We describe the simplicial complex $Δ$ such that the initial ideal of $J_G$ is the Stanley-Reisner ideal of $Δ$. By $Δ$ we show that if $J_G$ is $(S_2)$ then $G$ is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen-Macaulay are all and only the accessible ones.
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.
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.
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.
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.
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…
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.
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.
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.
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.
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.
设G是一个n阶简单无向图，I（G）\subseteq R = k[x_1，.，x_n]表示其相关的边理想。我们证明了所有弦图G都是序列Cohen-Macaulay图;我们的证明依赖于证明I（G）的Alexander对偶是分量线性的。我们的结果补充了Faridi的定理，即单纯树的小面理想是顺序Cohen-Macaulay，并暗示了Herzog，Hibi和Zheng的定理，弦图是Cohen-Macaulay当且仅当它的边理想是unmixed。我们还刻画了序列Cohen-Macaulay圈，并给出了一些非弦序列Cohen-Macaulay图的例子。
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.
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.
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'.
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.
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.
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.
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…
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.
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.
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.
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.
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.
