Cohen-Macaulay Property Research Articles

Almost Cohen-Macaulay and almost regular algebras via almost flat extensions

The present paper deals with various aspects of the notion of almost Cohen- Macaulay property, which was introduced and studied by Roberts, Singh and Srinivas. We employ the definition of almost zero modules as defined by a value map, which is different from the version of Gabber-Ramero. We prove that, if the local cohomology modules of an algebra T of certain type over a local Noetherian ring are almost zero, T maps to a big Cohen-Macaulay algebra. Then we study how the almost Cohen-Macaulay property behaves under almost faithfully flat extension. As a consequence, we study the structure of F-coherent rings of positive characteristic in terms of almost regularity.

The homology of parameter ideals

For a Noetherian local ring, we analyze conjectural relationships between the first Hilbert coefficient of a parameter ideal and the first partial Euler characteristic of its Koszul complex. Given their similar role as predictors of the Cohen–Macaulay property, we consider a direct comparison between them. For parameter ideals generated by d-sequences these numbers are related in an explicit formula. We then turn to study of families of parameter ideals that have the same Hilbert function.

Torsion in equivariant cohomology and Cohen-Macaulay G-actions

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.

Higher Cohen-Macaulay property of squarefree modules and simplicial posets

Recently, G. Fløystad 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 l -Cohen-Macaulay, then its codimension one skeleton is ( l + 1 ) (l+1) -Cohen-Macaulay.

Spherical complexes attached to symplectic lattices

To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g-2)-spheres. This, in turn, implies that the rational homology of moduli space of (unmarked) principal polarized abelian varieties of genus g modulo the decomposable ones vanishes in degree g-2 or lower. Another application is an improved stability range for the homology of the symplectic groups over Euclidean rings. But the original motivation comes from envisaged applications to the homology of groups of Torelli type. The proof of our main result rests on a refined nerve theorem for posets that may have an interest in its own right.

Sequentially $S_{r}$ simplicial complexes and sequentially $S_{2}$ graphs

We introduce sequentially S r modules over a commutative graded ring and sequentially S r simplicial complexes. This generalizes two properties for modules and simplicial complexes: being sequentially Cohen-Macaulay, and satisfying Serre's condition S r . In analogy with the sequentially Cohen-Macaulay property, we show that a simplicial complex is sequentially S r if and only if its pure i-skeleton is S r for all i. For r = 2, we provide a more relaxed characterization. As an algebraic criterion, we prove that a simplicial complex is sequentially S r if and only if the minimal free resolution of the ideal of its Alexander dual is componentwise linear in the first r steps. We apply these results for a graph, i.e., for the simplicial complex of the independent sets of vertices of a graph. We characterize sequentially S r cycles showing that the only sequentially S 2 cycles are odd cycles and, for r > 3, no cycle is sequentially S r with the exception of cycles of length 3 and 5. We extend certain known results on sequentially Cohen-Macaulay graphs to the case of sequentially S r graphs. We prove that a bipartite graph is vertex decomposable if and only if it is sequentially S 2 . We provide some more results on certain graphs which in particular implies that any graph with no chordless even cycle is sequentially S 2 . Finally, we propose some questions.

Mincut ideals of two-terminal networks

This paper introduces mincut ideals of two-terminal networks, which arise in the algebraic analysis of system reliability. We give the definitions and study their algebraic and combinatorial properties in some particular cases. It turns out that some features of the mincut ideals arising from networks such as the Cohen-Macaulay property and the computation of Betti numbers, which are important in tight reliability bounds, have a compact expression for series-parallel networks. This relies on a natural mapping of the structure of such networks into the union and intersection structure of the corresponding ideal.

Koszul incidence algebras, affine semigroups, and Stanley–Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen–Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen–Macaulay property.

Torus actions whose equivariant cohomology is Cohen-Macaulay

We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in the sense that a Chang-Skjelbred Lemma, and its stronger version, the exactness of an Atiyah-Bredon sequence, hold. The main difference is that the fixed point set is replaced by the union of lowest dimensional orbits. We find sufficient conditions for the Cohen-Macaulay property such as the existence of an invariant Morse-Bott function whose critical set is the union of lowest dimensional orbits, or open-face-acyclicity of the orbit space. Specializing to the case of torus manifolds, i.e., 2r-dimensional orientable compact manifolds acted on by r-dimensional tori, the latter is similar to a result of Masuda and Panov, and the converse of the result of Bredon that equivariantly formal torus manifolds are open-face-acyclic.

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 existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.

Some criteria for the Cohen-Macaulay property and local cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Had (M) is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.

Shellability and the Strong gcd-Condition

Shellability is a well-known combinatorial criterion on a simplicial complex $\Delta$ for verifying that the associated Stanley-Reisner ring $k[\Delta]$ is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, and he together with the author proved that it implies Golodness of the associated Stanley-Reisner ring. The two algebraic notions were earlier tied together by Herzog, Reiner and Welker, who showed that if $k[\Delta^\vee]$ is sequentially Cohen-Macaulay, where $\Delta^\vee$ is the Alexander dual of $\Delta$, then $k[\Delta]$ is Golod. In this paper, we present a combinatorial companion of this result, namely that if $\Delta^\vee$ is (non-pure) shellable then $\Delta$ satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences if $\Delta$ is a flag complex.

Cohen–Macaulayness and computation of Newton graded toric rings

Let H ⊆ Z d be a positive semigroup generated by A ⊆ H , 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 tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.

Notes on C-Graded Modules Over an Affine Semigroup Ring K[C

Let C ⊂ ℕ d be an affine semigroup, and R = K[C] its semigroup ring. This article is a collection of various results on “C-graded” R-modules M = ⨁ c∈C M c , especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radical monomial ideal (i.e., R/I is a generalization of Stanley–Reisner rings), then the sequentially Cohen–Macaulay property of R/I is a topological property of the “geometric realization” of the cell complex associated with I. Moreover, we can give a squarefree modules/constructible sheaves version of this result. We also show that if R is normal and I ⊂ R is a Cohen–Macaulay monomial ideal, then is Cohen–Macaulay again.

Local cohomology on diagrams of schemes

We define local cohomology on a diagram of schemes, and discuss basic properties of it. In particular, we prove the independence theorem and the flat base change for local cohomology on diagrams of schemes. We also introduce the notion of G-localness of a G-scheme. It is an equivariant analogue of localness of a scheme. As an application, we prove that Cohen–Macaulay property is inherited by the affine geometric quotient under the action of a linearly reductive group scheme over a field. This generalizes the special case (that the ring in question contains a field) of the theorem of Hochster and Eagon on the Cohen–Macaulay property of an invariant subring under the action of a finite group.

The AS–Cohen–Macaulay property for quantum flag manifolds of minuscule weight

It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS–Cohen–Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [T.H. Lenagan, L. Rigal, Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra 301 (2006) 670–702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS–Gorenstein.

On the structure of sequentially generalized Cohen–Macaulay modules

A finitely generated module M over a local ring is called a sequentially generalized Cohen–Macaulay module if there is a filtration of submodules of M : M 0 ⊂ M 1 ⊂ ⋯ ⊂ M t = M such that dim M 0 < dim M 1 < ⋯ < dim M t and each M i / M i − 1 is generalized Cohen–Macaulay. The aim of this paper is to study the structure of this class of modules. Many basic properties of these modules are presented and various characterizations of sequentially generalized Cohen–Macaulay property by using local cohomology modules, theory of multiplicity and in terms of systems of parameters are given. We also show that the notion of dd-sequences defined in [N.T. Cuong, D.T. Cuong, dd-Sequences and partial Euler–Poincaré characteristics of Koszul complex, J. Algebra Appl. 6 (2) (2007) 207–231] is an important tool for studying this class of modules.

Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P 3 defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen–Macaulay by associating a graph with each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen–Macaulay property.

SPECIALIZATIONS OF DIRECT LIMITS AND OF LOCAL COHOMOLOGY MODULES

AbstractIn this paper we introduce the specialization of an $S$-module which is a direct limit of a direct system of finitely generated $S$-modules indexed by $\mathbb{N}$. This specialization preserves the Buchsbaum property, the generalized Cohen–Macaulay property and the Castelnuovo–Mumford regularity of a module.

On Fiber Cones of 𝑚-Primary Ideals

AbstractTwo formulas for the multiplicity of the fiber cone of an 𝑚-primary ideal of a d-dimensional Cohen–Macaulay local ring (R, 𝑚) are derived in terms of the mixed multiplicity ed–1(𝑚|I), the multiplicity e(I), and superficial elements. As a consequence, the Cohen–Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of I and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of 𝑚–primary ideals with a d–generated minimal reduction J satisfying ℓ(I2/JI) = 1 or ℓ(I𝑚/J𝑚) = 1.