M. Azeem Khadam
Let $\mathfrak{q}$ be an ideal of a Noetherian local ring $(A,\mathfrak{m})$ and $M$ a non-zero finitely generated $A$-module. We present a criterion of Cohen-Macaulayness of the form module $G_M(\mathfrak{q})$ in terms of (non-)vanishing of a variation of local cohomology introduced in \cite{KSch}.
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Cohen-Macaulayness of bipartite graphs is investigated by several mathematicians and has been characterized combinatorially. In this note, we give some different combinatorial conditions for a bipartite graph which are equal to Cohen-Macaulayness of the graphs. Conditions in the previous works are depending on an appropriate ordering on vertices of the graph. The conditions presented in this paper are not depending to any ordering. Finally, we present a fast algorithm to check Cohen-Macaulayness of a given bipartite graph.
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.
Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.
Let $R$ be a commutative Noetherian ring. The notion of regular sequences with respect to a Serre class of $R$-modules is introduced and some of their essential properties are given. Then in the local case, we explore a theory of Cohen-Macaulayness with respect to Serre classes.
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 study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.
There exist many characterizations of Noetherian Cohen-Macaulay rings in the literature. These characterizations do not remain equivalent if we drop the Noetherian assumption. The aim of this paper is to provide some comparisons between some of these characterizations in non Noetherian case. Toward solving a conjecture posed by Glaz, we give a generalization of the Hochster-Eagon result on Cohen-Macaulayness of invariant rings, in the context of non Noetherian rings.
We provide the sufficient conditions for Rees algebras of modules to be Cohen-Macaulay, which has been proven in the case of Rees algebras of ideals by Johnson-Ulrich and Goto-Nakamura-Nishida. As it turns out the generalization from ideals to modules is not just a routine generalization, but requires a great deal of technical development. We use the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos to obtain the Cohen-Macaulayness of Rees Algebras of modules.
A new proof of non-Cohen-Macaulayness of Bertin's example
本文通过环论性质给出了Bertin例子非Cohen-Macaulay性的新证明，替代了原有方法。
We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Grobner degenerates to the Stanley-Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen-Macaulay. Finally, we use Stanley-Reisner combinatorics and a result of Knutson-Lam-Speyer to show that all lower bound algebras are normal.
In this paper, we study the Cohen-Macaulayness of non-affine normal semigroups in $\mathbb{Z}^n$. We do this by establishing the following four statements each of independent interest: 1) a Lazard type result on $I$-supported elements of $\prod_{\mathbb{N}}\mathbb{Q}_{\geq0}$ for an index set $I\subset\mathbb{N}$; 2) a criterion of regularity of sequences of elements of the ring via projective dimension; 3) a direct limit of polynomial rings with toric maps; 4) any direct summand of rings of the third item is Cohen-Macaulay. To illustrate the idea, we give many examples.
Let $(R,\fm)$ be a commutative Noetherian local ring. Suppose that $M$ and $N$ are finitely generated modules over $R$ such that $M$ has finite projective dimension and such that $\Tor^R_i(M,N)=0$ for all $i>0$. The main result of this note gives a condition on $M$ which is necessary and sufficient for the tensor product of $M$ and $N$ to be a Cohen--Macaulay module over $R$, provided $N$ is itself a Cohen--Macaulay module.
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.
Vertex decomposability, shellability and Cohen-Macaulayness of graphs upon graph operations
该文研究经多种图运算构造所得图的顶点可分解性与可剥壳性，并利用图运算从已有图构造新的 Cohen-Macaulay 图类别。
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。
Let (R,m) be a local, complete ring, X an artinian R-module of Noetherian dimension d; let x_1,...,x_d\in m be such that 0:_X (x_1,...,x_d)R has finite length. Then H^x_d(X) is a finite R-module, providing a positive answer to a question posed by Tang. As a first application of this result corollory 1 contains a necessary condition for a finite module to be CM; secondly we propose a notion of Cohen-Macaulayfication and prove its uniqueness (th. 3); finally we show that this new notion of Cohen-Macaulayfication is a direct generalization of a notion of Cohen-Macaulayfication introduced by Goto …
Let $R$ be a commutative Noetherian ring, $\mathfrak a$ and $\mathfrak b$ ideals of $R$. In this paper, we study the finiteness dimension $f_{\mathfrak a}(M)$ of $M$ relative to $\mathfrak a$ and the $\mathfrak b$-minimum $\mathfrak a$-adjusted depth $λ_{\mathfrak a}^{\mathfrak b}(M)$ of $M$, where the underlying module $M$ is relative Cohen-Macaulay w.r.t $\mathfrak a$. Some applications of such modules are given.
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.
Sequential 1-Cohen-Macaulayness for direct sums of modules
本文研究诺特局部环上有限生成模直和的序列1-Cohen-Macaulay性，给出充要条件：M是序列1-Cohen-Macaulay当且仅当每个Mi都是序列1-Cohen-Macaulay。
Cohen-Macaulayness of Powers of Edge Ideals of Weighted Oriented Graphs
本文研究了加权有向图边理想的幂的Cohen-Macaulay性质，给出了符号幂和普通幂为Cohen-Macaulay的充要条件，并提供了幂等性判定准则。
Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an $R$-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ${\mathcal F}={\mathcal R}/{\mathfrak m}{\mathcal R}$ of $I$, where ${\mathcal R}$ denotes the Rees algebra of $I$. Our key idea is to require `good' intersection properties as well as `few' homogeneous generating relations in low degrees. In particular, if $I$ is a strongly Cohen-Macaulay $R$-ideal with $G_{\ell}$ and the expected reduction number, we conclude that ${\mathcal F}$ is alwa…
Let $I$ be a monomial ideal $I$ in a polynomial ring $R = k[x_1,...,x_r]$. In this paper we give an upper bound on $\overline{\dstab} (I)$ in terms of $r$ and the maximal generating degree $d(I)$ of $I$ such that $\depth R/\overline{I^n}$ is constant for all $n\geqslant \overline{\dstab}(I)$. As an application, we classify the class of monomial ideals $I$ such that $\overline{I^n}$ is Cohen-Macaulay for some integer $n\gg 0$.
The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient $e_1(Q)$ is solved affirmatively, where $Q$ is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which $e_1(Q)$ vanishes are derived. The invariance of $e_1(Q)$ for parameter ideals $Q$ and its relationship to Buchsbaum rings are studied.
A criterion for sequential Cohen-Macaulayness
本文证明了分次模的序列Cohen-Macaulay性判别准则，即其算术次数与特定商模的算术次数相等，从而肯定地回答了Lu和Yu于2016年提出的猜想。
Grade and Cohen-Macaulayness for DG-modules
本文研究了DG模的梯度与Cohen-Macaulay性关系，引入完美DG模概念，并证明了相关条件下的Cohen-Macaulay性判定。
In this paper we give necessary and sufficient conditions for the Cohen-Macaulayness of the tangent cone of a monomial curve in the 4-dimensional affine space. We study particularly the case where $C$ is a Gorenstein non-complete intersection monomial curve.
Cohen-Macaulayness of powers of edge ideals of edge-weighted graphs
本文研究了边加权图的边理想幂的Cohen-Macaulay性，针对非常良覆盖图的二次幂以及具有悬挂边完美匹配的树或连通图的普通幂给出了完整刻画。
We graph-theoretically characterize triangle-free Gorenstein graphs $G$. As an application, we classify when $I(G)^2$ is Cohen-Macaulay.
Cohen-Macaulayness of formal fibers and dimension of local cohomology modules
本文研究了局部环中形式纤维的Cohen-Macaulay性质与局部上同调模维数的关系，证明了在特定条件下环的维数与模的支撑关系。
Cohomological Cohen--Macaulayness in Non-Noetherian Rings
本文研究非诺特环的上同调Cohen--Macaulay性，证明局部有限维CCM环必为局部HMCM环，给出满足条件的环类实例，并建立特征p>0诺特环与其完备化的相关等价关系。
In this paper we study the finitely generated bigraded modules over a standard bigraded polynomial ring which are relative Cohen-Macaulay or relative unmixed with respect to one of the irrelevant bigraded ideals. A generalization of Reisner's criterion for Cohen-Macaulay simplicial complexes is considered.
Let $K$ be a field, $S=K[x_1,\ldots,x_m, y_1,\ldots,y_n]$ be a standard bigraded polynomial ring and $M$ a finitely generated bigraded $S$-module. In this paper we study sequentially Cohen--Macaulayness of $M$ with respect to $Q=(y_1,\ldots,y_n)$. We characterize the sequentially Cohen--Macaulayness of $L\tensor_KN$ with respect to $Q$ as an $S$-module when $L$ and $N$ are non-zero finitely generated graded modules over $K[x_1, \dots, x_m]$ and $K[y_1, \dots, y_n]$, respectively. All hypersurface rings that are sequentially Cohen--Macaulay with respect to $Q$ are classified.
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.
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.
We give the Castelnuovo-Mumford regularity of arrangements of (n-2)-planes in P^n whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen-Macaulay.
Let $C({\bf n})$ be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve $C({\bf n}+w{\bf v})$, where $w>0$ is an integer and ${\bf v} \in \mathbb{N}^{4}$. Also we investigate the Cohen-Macaulayness of the tangent cone of $C({\bf n}+w{\bf v})$.
Let $\mathscr{L}\subset \mathbb{Z}^n$ be a lattice, $I$ its corresponding lattice ideal, and $J$ the toric ideal arising from the saturation of $\mathscr{L}$. We produce infinitely many examples, in every codimension, of pairs $I,J$ where one of these ideals is Cohen--Macaulay but the other is not.
In the present paper we investigate a question stemming from a long-standing conjecture of Vasconcelos: given a generically a complete intersection perfect ideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) is Cohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trung and Tuan and - very recently - Rinaldo Terai and Yoshida, already considered this question and gave a positive answer for special classes of ideals. We give a positive answer for some classes of ideals, however, we also exhibit prime ideals in regular local rings and homogeneous level idea…
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).
Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\vpl}_nH^i_{\fm}(M/\fa^n M)$, $i\geq 0$. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $\dim R\leq 2$ or either $\fa$ is principal or $\dim R/\fa\leq 1$, then $\Tor_j^R(R/\fa,{\vpl}_nH^i_{\fm}(M/\fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $\fgrad…
Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves
本文研究 4 维空间中 Gorenstein 非完全交单项式曲线的切锥 Cohen-Macaulay 性，利用定义理想的极小生成元集给出判定准则，并刻画了相应理想极小生成元个数的可能取值。
Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety. When the Rees algebra and the Symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.
We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).
Let $H$ be a positive semigroup in $\mathbb{Z}^d$ generated by $A$, and let $K[H]$ be the associated semigroup ring over a field $K$. We investigate heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side we show that for every $H$ there exist generating sets $A$ for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on $A$-hypergeometric Euler-Koszul homology. As a…
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。
We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers a…
Let S be a standard N^r-graded algebra over a local ring A, and let M be a finitely generated Z^r-graded S-module. We characterize the Cohen-Macaulayness of M in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen-Macaulayness of multi-Rees modules (also called Rees modification). Our work extends previous studies on the Cohen-Macaulayness of multi-Rees algebras.
In this paper, we explore a relationship between Hilbert functions and the irreducible decompositions of ideals in local rings. Applications are given to characterize the regularity, Gorensteinness, Cohen-Macaulayness and sequentially Cohen-Macaulayness of local rings.
We show the Cohen-Macaulayness and describe the canonical module of residual intersections $J=\mathfrak{a}\colon_R I$ in a Cohen-Macaulay local ring $R$, under sliding depth type hypotheses. For this purpose, we construct and study, using a recent article of Hassanzadeh and the second named author, a family of complexes that contains important informations on a residual intersection and its canonical module. We also determine several invariants of residual intersections as the graded canonical module, the Hilbert series, the Castelnuovo-Mumford regularity and the type. Finally, whenever $I$ is…
We call a (simple) graph G codismantlable if either it has no edges or else it has a codominated vertex x, meaning that the closed neighborhood of x contains that of one of its neighbor, such that G-x codismantlable. We prove that if G is well-covered and it lacks induced cycles of length four, five and seven, than the vertex decomposability, codismantlability and Cohen-Macaulayness for G are all equivalent. The rest deals with the computation of Castelnuovo-Mumford regularity of codismantlable graphs. Note that our approach complements and unifies many of the earlier results on bipartite, cho…
This paper purposes to characterize Noetherian local rings $(R, \mathfrak{m})$ such that the Chern numbers of certain $\mathfrak{m}$-primary ideals in $R$ bounded above or range among only finitely many values. Consequently, we characterize the Gorensteinness, Cohen-Macaulayness, generalized Cohen-Macaulayness of local rings in terms of the behavior of its Chern numbers.
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.
Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $\mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(\mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(\mathcal{D})$ when the underlying graph of $\mathcal{D}$ is a bipartite graph. When $I(\mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(\mathcal{D})$ is shown to be sequentially Cohen-Macaulay.
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.
Given a finite module $M$ over a Noetherian local ring $(R, \m)$, we introduce the concept of $j$-stretched ideals on $M$. Thanks to a crucial specialization lemma, we show that this notion greatly generalizes (to arbitrary ideals, and with respect to modules) the classical definition of stretched $\m$-primary ideals of Sally and Rossi-Valla, as well as the notion of minimal and almost minimal $j$-multiplicity given recently by Polini-Xie. For $j$-stretched ideals $I$ on a Cohen-Macaulay module $M$, we show that ${\rm gr}_I(M)$ is Cohen-Macaulay if and only if two classical invariants of $I$, …
Let $\k$ be a field and let $A$ be a standard $\mathbb{N}$-graded $\k$-algebra. Using numerical information of some invariants in the primary decomposition of $0$ in $A$, namely the so called dimension filtration, we associate a bivariate polynomial $\BW(A;t,w)$, that we call the Björner--Wachs polynomial, to $A$. It is shown that the Björner--Wachs polynomial is an algebraic counterpart of the combinatorially defined $h$-triangle of finite simplicial complexes introduced by Björner \& Wachs. We provide a characterisation of sequentially Cohen--Macaulay algebras in terms of the effect of the r…
Cohen--Macaulayness of $\mathrm{Inc}(\mathbb{N})$-Invariant Chains of Edge Ideals
本文针对五类经典图族，研究Inc(ℕ)-不变边理想链对应的商环的Cohen--Macaulay性，给出线图的完整分类，证明三类图的链无条件Cohen--Macaulay，确定循环图的Cohen--Macaulay范围并提出完整分类猜想。
An affine oriented matroid is a combinatorial abstraction of an affine hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a squarefree monomial ideal in a polynomial ring, called an oriented matroid ideal, and got beautiful results. Developing their theory, we will show the following. (1) If an oriented matroid ideal is Cohen-Macaulay, then the bounded complex (a regular CW complex associated with it) of the corresponding affine oriented matroid is a contractible homology manifold with boundary. This is closely related to Dong's theorem, which used to be "Zaslavsky's c…
Given a partition $λ$ of n, consider the subspace $E_λ$ of $C^n$ where the first $λ_1$ coordinates are equal, the next $λ_2$ coordinates are equal, etc. In this paper, we study subspace arrangements $X_λ$ consisting of the union of translates of $E_λ$ by the symmetric group. In particular, we focus on determining when $X_λ$ is Cohen-Macaulay. This is inspired by previous work of the third author coming from the study of rational Cherednik algebras and which answers the question positively when all parts of $λ$ are equal. We show that $X_λ$ is not Cohen-Macaulay when $λ$ has at least 4 distinct…
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.
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