Macaulay Property Research Articles

On the Broken-Circuit Complex of Graphs

It is well known that the Stanley–Reisner ring of a matroid complex is level; this is an algebraic property between the Cohen–Macaulay and Gorenstein properties. A similar result for broken-circuit complexes is no longer true, even for graphs. We show that the Stanley–Reisner ring of the broken-circuit complex, of the cone of any simple graph, is level.

Unmixed Monomial Ideals of Dimension Two and Their Cohen–Macaulay Properties

We study ideals of the form where P i, j are ideals generated by variables {x 1, x 2,…,, x n } \\ {x i , x j } in a polynomial ring over a field. We concentrate the case that each power w i, j takes a value α or β, where α > β > 0. Our purpose is determining whether S/I is Cohen–Macaulay or not for such an ideal. We take a graph consisting of {i, j} where w i, j = α. The Cohen–Macaulayness is characterized by G.

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.

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.

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.

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.

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.

Convex and Algebraic Geometry

The workshop Convex and Algebraic Geometry was organized by Klaus Altmann (Berlin), Victor Batyrev (Tübingen), and Bernard Teissier (Paris). Both title subjects meet primarily in the theory of toric varieties. These constitute the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Thus, toric geometry and its several generalizations provide a kind of section from polyhedral into algebraic geometry. While this reflects only a thin slice of algebraic geometry, it is general enough to display many important phenomena, techniques, and methods. It serves as a wonderful testing ground for general theories such as the celebrated mirror symmetry in its different flavours. In particular, much of the popularity of toric geometry originates in mathematical physics. The meeting was attended by almost 50 participants from many European countries, Canada, the USA, and Japan. The program consisted of talks by 23 speakers, among them many young researchers. Most subjects fit more or less into the following main areas: One of the major discussions during the meeting concerned the existence of strongly exceptional sequences on toric varieties which consist of line bundles. A full exceptional sequence provides a kind of “basis” for the derived category. While Hille and Perling presented an example that does not carry such a sequence of full length, Bondal suggested a method to link this question to sheaves on the dual real torus that are constructible with respect to a certain stratification. In general, one expects to gain exceptional sequences from the universal bundles on moduli spaces. Using this method, Craw constructs those sequences on smooth toric Fano threefolds. In this context, Maclagan and Ueda consider the case of three-dimensional abelian quotient singularities. Ueda investigates the Fukaya category of the corresponding potential on the dual torus explicitly. Using mirror symmetry, Horja establishes a connection between the orbifold K -theory of toric Deligne-Mumford stacks and solutions to GKZ-hypergeometric D -modules. Gross and Siebert have developed a program to understand mirror symmetry as the duality of certain degeneration data. The special fibers split into toric components, and the degeneration is encoded in a topological manifold B with an affine and a polytopal structure. Duality is now inherited from discrete geometry, and the topology of B reflects the topology of the general fiber. In particular, if B is a ( \mathbb Q -homology) \mathbb P^n_{\mathbb C} , then this construction might lead to (compact) Hyperkähler varieties. Considering, in a special case, a certain contraction of the total space of these families leads to a description of torus actions on algebraic varieties via divisors on their Chow quotients. These divisors carry polytopes or even polyhedral complexes as their coefficients, compare the talks of Hausen, Süss, and Vollmert. In a similar setting, but with an explicit manipulation of Pfaffians, Brown and Reid construct smoothings of certain non-isolated singularities giving rise to four-dimensional flips. The most rigorous degeneration of a variety is the tropical one. Here, everything takes place over the so-called tropical semiring, and one ends up with piecewise linear spaces. In fact, Siebert's degeneration data mentioned above correspond to these objects. Itenberg and Shustin use this approach to calculate the Welschinger invariants, which are a kind of real version of Gromov–Witten invariants. Along the lines of the method of Gathmann and Markwig, there is a recursive formula for theses invariants. In the case of del Pezzo surfaces, it turns out that both invariants are (log-) asymptotically equivalent. A generalization of toric varieties in a different direction from the torus actions mentioned above is given by the notion of spherical varieties. Pasquier considers horospherical Fano varieties and comes up with an adapted notion of (generalized, coloured) reflexive polytopes. Bruns, Haase, and Hering deal with ordinary polytopes and their relations to syzygies of toric varieties. For an integral matrix A one obtains a semigroup algebra \mathbb C[\mathbb N A] (leading to the usual affine toric variety) and a GKZ-hypergeometric system of differential equations. The latter depends on a parameter \beta , and Miller has reported on a result that relates the set of \beta where the rank of the system jumps to the set of those multidegrees where the semigroup algebra \mathbb C[\mathbb N A] carries local cohomology. In particular, the Cohen–Macaulay property is equivalent to the constant rank condition, answering an old question of Sturmfels. One of the nighttime discussions gave rise to the suggestion to not include normality in the definition of a toric variety, thus overcoming the cumbersome term of a “not necessarily normal toric variety”. The workshop was closed on Friday night by an informal piano recital by Benjamin Nill and Milena Hering featuring Strawinsky, Liszt, and Chopin.

On toric face rings

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley–Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley–Reisner and affine monoid algebras. We consider (non-pure) shellable fan’s and the Cohen–Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.

Combinatorics of multigraded Poincaré series for monomial rings

Backelin proved that the multigraded Poincaré series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function. The numerator is simple, but until the recent work of Berglund there was no combinatorial formula for the denominator. Berglund's formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattice. We now express this lattice as the intersection lattice L A ( I ) of a subspace arrangement A ( I ) , use Crapo's Closure Lemma to drastically simplify the denominator in some cases (such as monomial ideals generated in degree two), and relate Golodness to the Cohen–Macaulay property for associated posets. In addition, we introduce a new class of finite lattices called complete lattices, prove that all geometric lattices are complete and provide a simple criterion for Golodness of monomial ideals whose lcm-lattices are complete.

Tensor Products of Approximately Cohen–Macaulay Rings

Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen–Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem.

Quantum graded algebras with a straightening law and the AS–Cohen–Macaulay property for quantum determinantal rings and quantum grassmannians

We study quantum analogues of quotient varieties, namely quantum grassmannians and quantum determinantal rings, from the point of view of regularity conditions. More precisely, we show that these rings are AS–Cohen–Macaulay and determine which of them are AS–Gorenstein. Our method is inspired by the one developed by De Concini, Eisenbud and Procesi in the commutative case. Thus, we introduce and study the notion of a quantum graded algebra with a straightening law on a partially ordered set, showing in particular that, among such algebras, those whose underlying poset is wonderful are AS–Cohen–Macaulay. Then, we prove that both quantum grassmannians and quantum determinantal rings are quantum graded algebras with a straightening law on a wonderful poset, hence showing that they are AS–Cohen–Macaulay. In this last step, we are led to introduce and study (to some extent) natural quantum analogues of Schubert varieties.

Lower and upper bounds for Cohen-Macaulay dimension

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.

Segre and Rees products of posets, with ring-theoretic applications

We introduce (weighted) Segre and Rees products for posets and show that these constructions preserve the Cohen–Macaulay property over a field k and homotopically. As an application we show that the weighted Segre product of two affine semigroup rings that are Koszul is again Koszul. This result generalizes previous results by Crona on weighted Segre products of polynomial rings. We also give a new proof of the fact that the Rees ring of a Koszul affine semigroup ring is again Koszul. The paper ends with a list of some open problems in the area.