Thomas Polstra
It is shown in a local strongly $F$-regular ring there exits natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen-Macaulay module then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism contains a free summand. Consequently, the torsion subgroup of the divisor class group of a local strongly $F$-regular ring is finite.
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We study the Tate resolutions and the maximal Cohen-Macaulay approximations of Cohen-Macaulay modules over Gorenstein rings. One consequence is an extension of a well-known result about linkage of complete intersections.
The paper answers a question by Jonathan Wahl,giving examples of regular surfaces S (so their canonical ring is a Gorenstein graded ring) having the following properties: 1) their canonical divisor K_S = rL is a positive multiple of an ample divisor L 2) the graded ring R := R (X,L ) associated to L is not Cohen-Macaulay. In the appendix Wahl shows how these examples lead to the existence of Cohen-Macaulay singularities with K_X Q -Cartier which are not Q -Gorenstein, since their index one cover is not Cohen- Macaulay.
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.
By finding a p-adic obstruction, we construct many examples of positive characteristic complete noetherian local rings which do not admit any module-finite Cohen-Macaulay extension. These examples should be contrasted with a result of Hochster-Huneke: the union of all finite extensions is always Cohen-Macaulay.
Cohen-Macaulay multigraded modules
设S是局部环A上的标准N^r-分次分次代数，M是Z^r-分次生成的S-模.我们刻画了M的Cohen-Macaulayness的某些层上同调模的消失。因此，我们应用我们的结果来研究多Rees模的Cohen-Macaulay性（也称为Rees修改）。我们的工作扩展了以往关于多Rees代数的Cohen-Macaulay性的研究。
Bezout's theorem and Cohen-Macaulay modules
我们定义了模和投射子概型的非常恰当的交。证明了等维局部Cohen-Macaulay模非常恰当地相交当且仅当它们恰当地相交。我们证明了一个Bezout定理的模满足非常适当。进一步证明了对于等维子格式$X$和$Y$：如果它们在正维的算术Cohen-Macaulay子格式中适当相交，则$X$和$Y$是算术Cohen-Macaulay.这个结果的模版本意味着自反层的分裂准则。
A rational map whose source and image are projectively embedded varieties has an {\em Arithmetically Cohen-Macaulay graph} if the Rees algebra of one (hence any) of its base ideals is a Cohen-Macaulay ring. If the map is birational onto the image one considers how this property forces an upper bound on the degree of a representative of the map. In the plane case a complete description is given of the Cremona maps with Cohen-Macaulay graph, while in arbitrary dimension $n$ it is shown that a Cremona map with Cohen-Macaulay graph has degree at most $n^2$.
Remarks on the Small Cohen-Macaulay conjecture and new instances of maximal Cohen-Macaulay modules
本文证明了特定 3 维准-Gorenstein Buchsbaum 局部环的形变容许极大 Cohen-Macaulay 模，并在素特征下小 Cohen-Macaulay 猜想的启发下，探讨了 Frobenius 推前包含极大 Cohen-Macaulay 直和项的条件。
We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring $A$ is Cohen-Macaulay, and $a_1,\dots,a_n$ is any sequence of elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring $A$, by finding a Cohen-Macaulay DG-ring $B$ such that $\mathrm{H}^0(B) = A$, and using the Cohen-Macaula…
We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.
Integral perfectoid big Cohen-Macaulay algebras via André's theorem
本文通过安德烈定理证明混合特征下Noetherian局部域可映射到积分完美体大Cohen-Macaulay代数，并探讨了其绝对积分闭包的相关性质。
Rank two Cohen-Macaulay modules over singularities of type $x_1^3+x_2^3+x_3^3+x_4^3$
本文利用矩阵分解刻画了$x_1^3+x_2^3+x_3^3+x_4^3$型奇点上所有秩为2的极大Cohen-Macaulay模.
Maximal Cohen-Macaulay Modules over the Affine Cone of the Simple Node
给出了单结点（非孤立奇点）的仿射锥上所有秩为1和秩为2的分次极大Cohen-Macaulay模的具体刻画。为此，我们构造了一个算法，它提供了任意超曲面上MCM模的扩张。
Simple Cohen-Macaulay Codimension 2 Singularities
在这篇文章中，我们提供了一个完整的列表，简单的科恩-麦考利余维2奇点连同一个列表的邻接，这是完整的情况下，脂肪点和空间曲线奇点。
In this article, we compute Betti numbers for a Cohen-Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.
We give suffcient conditions for a standard graded Cohen-Macaulay ring, or equivalently, an arithmetically Cohen-Macaulay projective variety, to be Cohen-Macaulay wild in the sense of representation theory. In particular, these conditions are applied to hypersurfaces and complete intersections.
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.
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。
Let $A$ be a Cohen-Macaulay normal domain. A non commutative crepant resolution (NCCR) of $A$ is an $A$-algebra $Γ$ of the form $Γ= End_A(M)$, where $M$ is a reflexive $A$-module, $Γ$ is maximal Cohen-Macaulay as an $A$-module and $gldim(Γ)_P = \dim A_P $ for all primes $P$ of $A$. We give bountiful examples of equi-characteristic Cohen-Macaulay normal local domains and mixed characteristic Cohen-Macaulay normal local domains having NCCR. We also give plentiful examples of affine Cohen-Macaulay normal domains having NCCR.
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.
The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties
本文证明射影空间中具有 Cohen-Macaulay 分次坐标环的约化闭子概形几乎都是野 Cohen-Macaulay 型，并对少数例外情形给出完全分类。
On Sequentially Cohen-Macaulay Modules
本文给出了序列Cohen-Macaulay模的参数系刻画，推广了Cohen-Macaulay模和广义Cohen-Macaulay模的一些著名结果.研究了小嵌入维数的Stanley-Reisner环的序列Cohen-Macaulay性。
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 this article, we study Cohen-Macaulay modules over non-reduced curve singularities. We prove that the rings $k[[x,y,z]]/(xy, y^q -z^2)$ have tame Cohen-Macaulay representation type. For the singularity $k[[x,y,z]]/(xy, z^2)$ we give an explicit description of all indecomposable Cohen--Macaulay modules and apply the obtained classification to construct explicit families of indecomposable matrix factorizations of $(xy)^2 \in k[[x,y]]$.
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.
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。
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.
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
设（R，m）是局部完备环，X是Noether维数为d的Artin R-模，x_1，…，x_d \in m是这样的：0：_X（x_1，.，x_d）R有有限长。则H^x_d（X）是一个有限R-模，从而对Tang提出的一个问题给出了肯定的回答.作为这个结果的第一个应用，推论1包含了有限模是CM的一个必要条件;其次，我们提出了Cohen-Macaulayization的概念，并证明了它的唯一性。3）;最后，我们证明了这个新的Cohen-Macaulayfication概念是Goto（th. 4）。
On Cohen-Macaulay modules over surface singularities
给出了T型尖点曲面奇点和单模超曲面奇点上的Cohen-Macaulay模的刻划。证明了在极小椭圆奇点及其分支中，只有单椭圆奇点和尖点奇点是驯服的，其它奇点都是野性的（相对于Cohen-Macaulay模的分类）.
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 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.
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.
By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DG-modules of finite injective dimension over commutative local DG-rings, complementing results of Jørgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natu…
On Cohen--Macaulay modules over the Weyl algebra
本文提出了 Weyl 代数上 Cohen--Macaulay 模的定义，并给出了 GKZ $A$-超几何 $D$-模满足 Cohen--Macaulay 性质的充分条件。
On the discrete counterparts of Cohen-Macaulay algebras with straightening laws
研究了生成具有拉直律的Cohen-Macaulay代数（简称ASL）的偏序集的性质。证明了如果偏序集$P$生成Cohen-Macaulay ASL，则$P$是纯偏序集;如果$P$是Buchsbaum偏序集，则$P$是Cohen-Macaulay偏序集.还建立了关于由拉直闭理想定义的ASL的Rees代数的一些结果。证明了若$P$是具有唯一极小元的Cohen-Macaulay偏序集，且$Q$是$P$的偏序集理想，则$P\uplus Q$也是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.
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
我们定义了三个新的同调维度-科恩-麦考利内射，投射，平坦的维度-居住在一个理论类似的经典内射，投射，平坦的维度。具有对偶模的Cohen-Macaulay环的新维数的等价性。
Vector bundles and Cohen-Macaulay modules
本文介绍了G. M.Greuel和作者关于代数曲线上的向量丛和关于曲面奇点上的Cohen-Macaulay模。主要研究分类问题，特别是驯服-野性二分法以及驯服情形下的向量丛和Cohen-Macaulay模的刻画。
To reduce to resolving Cohen-Macaulay singularities, Faltings initiated the program of "Macaulayfying" a given Noetherian scheme $X$. For a wide class of $X$, Kawasaki built the sought Cohen-Macaulay modifications, with a crucial drawback that his blowups did not preserve the locus $\mathrm{CM}(X) \subset X$ where $X$ is already Cohen-Macaulay. We extend Kawasaki's methods to show that every quasi-excellent, Noetherian scheme $X$ has a Cohen-Macaulay $\widetilde{X}$ with a proper map $\widetilde{X} \rightarrow X$ that is an isomorphism over $\mathrm{CM}(X)$. This completes Faltings' program, r…
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.
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…
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.
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 …
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
本文给出两个例子，证明了对于仿射半群环，Cohen-Macaulay类的集合可以大于锥类的集合。
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…
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
利用Alexander对偶证明了一类具有足够大重数的Stanley-Reisner环是Cohen-Macaulay环.
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.
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.
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.
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
本文证明了Schreyer猜想的一个极好的情形：具有可数Cohen-Macaulay型的局部环至多有一个一维奇异轨迹.进一步证明了可数CM型Cohen-Macaulay局部环的局部化也是可数CM型的。
Some interesting properties of almost Cohen-Macaulay rings are investigated and a Serre type property connected with this class of rings is studied.
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.
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