The category of maximal Cohen--Macaulay modules as a ring with several objects

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.

Local Homology, Cohen-Macaulayness and Cohen-Macaulayfications

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 …

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.

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.

On the discrete counterparts of Cohen-Macaulay algebras with straightening laws

We study properties of a poset generating a Cohen-Macaulay algebra with straightening laws (ASL for short). We show that if a poset $P$ generates a Cohen-Macaulay ASL, then $P$ is pure and, if $P$ is moreover Buchsbaum, then $P$ is Cohen-Macaulay. Some results concerning a Rees algebra of an ASL defined by a straightening closed ideal are also established. And it is shown that if $P$ is a Cohen-Macaulay poset with unique minimal element and $Q$ is a poset ideal of $P$, then $P\uplus Q$ is also Cohen-Macaulay.

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.

$I$-Cohen Macaulay modules

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.

Non-Gorenstein isolated singularities of graded countable Cohen-Macaulay type

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

We define three new homological dimensions - Cohen-Macaulay injective, projective, and flat dimension - which inhabit a theory similar to that of classical injective, projective, and flat dimension. Finiteness of the new dimensions characterizes Cohen-Macaulay rings with dualizing modules.

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.

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…

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.

Super-stretched and graded countable Cohen-Macaulay type

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 …

Higher Cohen-Macaulay property of squarefree modules and simplicial posets

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

We provide two examples which show that for an affine semigroup ring the set of Cohen-Macaulay classes can be larger than the set of conic classes.

Relative Cohen-Macaulay filtered modules with a view toward relative Cohen-Macaulay modules

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…

A note on Non-Noetherian Cohen-Macaulay rings

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

We prove that certain class of Stanley--Reisner rings having sufficiently large multiplicities are Cohen--Macaulay using Alexander duality.

Cohen-Macaulayness of trivial extensions

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.

On the structure of sequentially Cohen--Macaulay bigraded 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.

Lyubeznik table of sequentially Cohen-Macaulay rings

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.

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.

Local rings of countable Cohen--Macaulay type

We prove (the excellent case of) Schreyer's conjecture that a local ring with countable Cohen--Macaulay type has at most a one-dimensional singular locus. Furthermore we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type.

More properties of almost Cohen-Macaulay rings

Some interesting properties of almost Cohen-Macaulay rings are investigated and a Serre type property connected with this class of rings is studied.

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.

On two dimensional mixed characteristic rings of finite Cohen Macaulay type

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.

Topics on sequentially Cohen-Macaulay modules

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.

Residually faithful modules and the Cohen-Macaulay type of idealizations

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}.

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.

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.

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.

A note on Cohen-Macaulay descent

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.

Algebraic Approximation of Cohen-Macaulay Algebras

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.

Some Characterizations of Relative Sequentially Cohen-Macaulay and Relative Cohen-Macaulay Modules

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…

Cohen-Macaulay local rings with $e_1 = e + 2$

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 $e_1 = e + 2$.

On Hilbert coefficients and sequentially generalized Cohen-Macaulay modules

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.

Eventually index of reducibility on sequentially Cohen-Macaulay modules

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.

Cohen-Macaulay Loci of modules

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.

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。

Uniform bounds in generalized Cohen-Macaulay rings

We establish a uniform bound for the Castelnuovo-Mumford regularity of associated graded rings of parameter ideals in a generalized Cohen-Macaulay ring. As consequences, we obtain uniform bounds for the relation type and the postulation number. Moreover, we show that generalized Cohen-Macaulay rings can be characterized by the existence of such uniform bounds.

Maximal Cohen-Macaulay modules over certain Segre products

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.

Big Cohen-Macaulay Algebras and Seeds

We delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke's "weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic local ring (R,m) maps to a bi…

The structure of balanced big Cohen-Macaulay modules over Cohen-Macaulay rings

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…

Maximal Cohen-Macaulay tensor products and vanishing of Ext modules

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.

An Alexandrov Topology for Maximal Cohen-Macaulay Modules

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

Let R = k[[x_0,...,x_d]]/(f), where k is a field and f is a non-zero non-unit of the formal power series ring k[[x_0,...,x_d]]. We investigate the question of which rings of this form have bounded Cohen--Macaulay type, that is, have a bound on the multiplicities of the indecomposable maximal Cohen--Macaulay modules. As with finite Cohen--Macaulay type, if the characteristic is different from two, the question reduces to the one-dimensional case: The ring R has bounded Cohen--Macaulay type if and only if R is isomorphic to k[[x_0,...,x_d]]/(g+x_2^2+...+x_d^2), where g is an element of k[[x_0,x_…

A new characterization of Cohen-Macaulay rings

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.

The Cohen-Macaulay property of separating invariants of finite groups

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…

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.

Ranks of indecomposable modules over rings of infinite Cohen-Macaulay type

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…