Cohen-Macaulay Type Research Articles

Hilbert functions of monomial curves

We study the Hilbert function of certain projective monomial curves. We determine which of our curves are Cohen–Macaulay, and find the Cohen–Macaulay type of those that are Cohen–Macaulay.

GORENSTEIN MODULES, FINITE INDEX, AND FINITE COHEN–MACAULAY TYPE

ABSTRACT A Gorenstein module over a local ring R is a maximal Cohen–Macaulay module of finite injective dimension. We use existence of Gorenstein modules to extend a result due to S. Ding: A Cohen–Macaulay ring of finite index, with a Gorenstein module, is Gorenstein on the punctured spectrum. We use this to show that a Cohen–Macaulay local ring of finite Cohen–Macaulay type is Gorenstein on the punctured spectrum. Finally, we show that for a large class of rings (including all excellent rings), the Gorenstein locus of a finitely generated module is an open set in the Zariski topology.

Mixed characteristic hypersurfaces of finite Cohen–Macaulay type

We define the mixed ADE singularities, which are generalizations of the ADE plane curve singularities to the case of mixed characteristic. The ADE plane curve singularities are precisely the equicharacteristic plane curve singularities of finite Cohen–Macaulay type; we show that the mixed ADE singularities also have finite Cohen–Macaulay type.

Type-sequences of modules

Let R be a complete and integral local k-algebra of dimension one, k an algebraically closed field of characteristic zero. In this paper the notion of type-sequence, given for rings in Barucci et al. (AMS Mem. 125 (598) (1997) Ch. II, 1), is extended to any finitely generated torsion-free R-module of rank 1. A module M, of Cohen–Macaulay type r 1( M), whose type-sequence is [ r 1( M),1,…,1] is said to have “ minimal type-sequence”, briefly m. t. s. The family of m. t. s. R-modules, which includes the canonical module, is described by means of value sets, the conductor c( M), the δ-invariant δ( M) and the C.M. type r 1( M). In the case of rings the m. t. s. property is called “ almost Gorenstein” (see Barucci and Fröberg, J. Algebra 188 (1997) 418–442). Inspired by analogous investigations by Barucci and Fröberg, we study in Section 3 the m. t. s. property and the reflexiveness of modules over almost Gorenstein rings.

Ascent of Finite Cohen–Macaulay Type

In 1987 F.-O. Schreyer conjectured that a local ring R has finite Cohen–Macaulay type if and only if the completion R̂ has finite Cohen–Macaulay type. We prove the conjecture for excellent Cohen–Macaulay local rings and also show by example that it can fail in general.

Generic and Cogeneric Monomial Ideals

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen–Macaulay property implies shellability for both the Scarf complex and the Stanley–Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen–Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case, the Cohen–Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley’s theory of local h -vectors.

Cohen-Macaulay Local Rings of Embedding Dimension e + d − 3

In this paper we determine the possible Hilbert functions of a Cohen–Macaulay local ring of dimension d and multiplicity e, in the case where the embedding dimension v satisfies v = e + d − 3 and the Cohen–Macaulay type is less than or equal to e − 3. 1991 Mathematics Subject Classification: primary 13D40; secondary 13P99.

Primary ideals with good associated graded ring

Let (A, M) be a local Cohen–Macaulay ring of dimension d. Let I be an M -primary ideal and let J be the ideal generated by a maximal superficial sequence for I. Under these assumptions Valabrega and Valla (Nogoya Math. J. 72 (1978) 93–101) proved that the associated graded ring G of I is Cohen–Macaulay if and only if I n∩J=JI n−1 for every integer n. In this paper we consider the class of the M -primary ideals I such that, for some positive integer k, we have I n∩J=JI n−1 for n≤k and λ(I k+1/JI k)≤1. In this case G need not be Cohen–Macaulay. In Theorem 2.2. we show that G is Cohen–Macaulay unless the ideals we are considering are of maximal Cohen–Macaulay type. One can use the ideas of Russi and Valla (Comm. Algebra 24(13) (1996) 4249–4261; Pure Appl. Algebra 122 (1997) 293–311) to prove that, for the ideals we consider, the depth of G is at least d−1 and that its h-vector has no negative components. We characterize the possible Hilbert function of G. Our approach gives proof of an extended version of a conjecture of Sally (proved in Russi and Valla Comm. Algebra 24(13) (1996) 4249–4261)) and independently in Wang (J. Algebra 190 (1997) 226–240) in the case I= M ). Several results proved in Huckaba (Comm. Algebra, to appear), Russi and Valla (Nogoya Math. J. 110 (1988) 81–111; Comm. Algebra 24(13) (1996) 4249–4261; J. Pure Appl. Algebra 122 (1997) 293–311) and Sally (J. Algebra 83 (1983) 325–333) are unified and generalized.

Local Rings of Finite Cohen–Macaulay Type

Let (R,m) be a local Cohen–Macaulay ring whosem-adic completionR̂has an isolated singularity. We verify the following conjecture of F.-O. Schreyer:Rhas finite Cohen–Macaulay type if and only ifR̂has finite Cohen–Macaulay type. We also show that the hypersurfacek[[x0,…,xd]]/(f) has finite Cohen–Macaulay type if and only ifks[[x0,…,xd]]/(f) has finite Cohen–Macaulay type, whereksis the separable closure of the fieldk.

Cohen-Macaulay types of subgroup lattices of finite abelianp-groups

For a minimal free resolution of a Stanley-Reisner ring constructed from the order complex of a modular lattice. T. Hibi showed that its last Betti number (called the Cohen-Macaulay type) is computed by means of the Mobius function of the given modular lattice. Using this result, we consider the Stanley-Reisner ring of the subgroup lattice of a finite abelianp-group associated with a given partition, and show that its Cohen-Macaulay type is a polynomial inp with integer coefficients.

Cohen-Macaulay Types of Cohen-Macaulay Complexes

We say that a Cohen-Macaulay poset (partially ordered set) is "superior" if every open interxal ( x, y) of P with μ P ( x, y) ≠ 0 is doubly Cohen-Macaulay. For example, if L = P is a modular lattice, then the Cohen-Macaulay poset P is superior. We present a formula for the computation of the Cohen-Macaulay type of the Stanley-Reisner ring of the order complex of a Cohen-Macaulay poset which is superior.

Cohen-Macaulay type of the face poset of a plane graph

How can we calculate the Cohen-Macaulay type of a Cohen-Macaulay poset? This paper is an extension of earlier results in [2]. We give an explicit formula for the Cohen-Macaulay type of the face poset of a plane graph. Let G be a finite connected plane graph allowing loops and multiple edges and G* the subgraph obtained by removing all loops from G. For each vertexv of G the number of connected components of G* --v is denoted by ?G (v). Also, writev G (v) for the number of loops of G incident tov. Then the Cohen-Macaulay type of the face poset of G is $$\left[ {\sum\limits_\upsilon {2\left\{ {\delta _G (v) + v_G (v) - 1} \right\}} } \right] + 1$$ .

Bounds for one-dimensional rings of finite cohen-macaulay type

Wiegand, R. and S. Wiegand, Bounds for one-dimensional rings of finite Cohen-Macaulay type, Journal of Pure and Applied Algebra 93 (1994) 311-342. Let R be an integral domain finitely generated as an algebra over a field of characteristic not equal to 2 (or the localization of such a ring at some multiplicatively closed set); and assume that, for each maximal ideal ▪, there is a bound on the ranks of the indecomposable finitely generated torsion-free R ▪ -modules. We show that the only possible ranks for such indecomposable modules over R are 1, 2, 3, 4, 5, 6, 8, 9 and 12. An example having indecomposables of each of these ranks is constructed over the field of rational numbers. Furthermore, over a broader class of reduced one-dimensional rings, the only possible ranks for indecomposable finitely generated torsion-free modules of constant rank are also 1, 2, 3, 4, 5, 6, 8, 9 and 12.

The Minimal Resolution Conjecture

By extending the ideal generation conjecture, we formulate the minimal resolution conjecture for a sufficiently generic set of points in P n and we explicitly write out the conjectured values of the Betti numbers. We then relate the minimal resolution conjecture to the ideal generation conjecture and the Cohen-Macaulay type conjecture, and we prove the minimal resolution conjecture for ( d + n n ) − λ (1 ≤ λ ≤ n) points, for n + 2 points, and for ( d − 1 + n n ) points. We also prove the Cohen-Macaulay type conjecture for s points, with n + 1 ≤ s ≤ ( 2 + n n )! Finally, we recover information about the resolution of s − 1 or s + 1 points from that of S points.

One dimensional local rings of maximal and almost maximal length

Let ( A, m, k) denote a one dimensional, Cohen-Macaulay, local ring with maximal ideal m and residue class field k. We assume A is a reduced, excellent local ring which has a canonical module ω A . With no loss of generality, we can also assume k is infinite. Let Ā denote the integral closure of A in its total quotient ring Q, and let b denote the conductor of A in Ā. A has maximal length if l A( A ̄ A ) = l A( A b ) t(A) Here t( A) denotes the Cohen-Macaulay type of A, and l A(∗) denotes the length of the A-module ∗. A has almost maximal length if l A( A ̄ A ) = l A( A b ) t(A) − 1 The two main results of this paper are as follows: A has maximal length if and only if either A is Gorenstein or there exists an x ϵ m and a positive integer p such that m = (x, b), andb A ̄ = x p A ̄ . Let e( A) denote the multiplicity of A, Then A has almost maximal length, and 1 + t( A) = e( A) if and only if there exists a transversal x of m such that m = (x, b), and l A( b x p A ̄ ) = 1 . Here p = min{i¦x iϵb} These two theorems generalize more specific results for semigroup rings obtained in Brown and Curtis (“Numerical Semigroups of Maximal and Almost Maximal Length,” Semigroup Forum, Vol. 42, Springer-Verlag, Berlin/New York, 1991, 219–235).

Face number inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices

We discuss two topics related with combinatorial study of canonical modules of Stanley-Reisner rings, viz., (i) some linear inequalities on the number of faces of a matroid complex and (ii) a formula to compute the Cohen-Macaulay type of the Stanley-Reisner ring of a finite distributive lattice

One dimensional cohen-macaulay rings with maximal hilbert function

Let denote e distinct points in . Here k denotes an algebraically and is projective r-space over k. Let R denote the coordinate ring of and let A be the localization of R at its irrelevant maximal ideal. A is a reduced, Cohen-Macaulay, local ring of dimension one and multiplicity e. we suppose r ≥ 2, and for some d ≥ 2. In a previous paper, the author showed that if are in uniform position in , then the Cohen-Macaulay type, t(A), of A is given by the following formula; . The local ring A is a typical example of a ring with maximal Hilbert function. In this paper, we discuss various results for t(A), when A is one dimensional, Cohen-Macaulay, local ring with maximal Hilbert function. In particular, we obtain a natural generalization of the above mentioned geometric result.

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∊M n×m (F), B∊M n×t (F) for some field F. Define Г(A B) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(A B). For matrices A B of the same rank we provide an algorithm for computing dim Г(A B).

Derivations and the Cohen-Macaulay type of points in generic position in n-space

Derivations and the Cohen-Macaulay type of points in generic position in n-space

A graded ring of finite cohen-macaulay type

A graded ring of finite cohen-macaulay type