A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements

We show that the ideal of an arrangement of d linear subspaces of projective space is d-regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular, this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d.     

Embeddings of canonical modules and resolutions of connected sums

For an ideal $I_{m,n}$ generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by $I_{m,n}$. Using this embedding, we give a resolution of connected sums of several copies of certain Artin $\mathsf{k}$-algebras where $\mathsf{k}$ is a field.     

Castelnuovo-Mumford regularity by approximation

The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.     

Explicit linear minimal free resolutions over a natural class of Rees algebras

Homological properties of the Rees algebra R of a Koszul K-algebra A over a field K, with respect to the maximal homogeneous ideal, are studied. In particular, for a finitely generated graded A-module N with linear minimal free R-resolution over A, the minimal free resolution of is explicitly constructed. This resolution is again linear.Received: 23 October 2000     

Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R   whose maximal ideal m$\mathfrak{m}$ satisfies m^s≠0=m^s+1${\mathfrak{m}}^s\ne 0={\mathfrak{m}}^{s+1}$ and _rankR/m(m/m²)=e${\mathrm{rank}}_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)=e$. We say that R is a compressed Gorenstein local ring   when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3$s\ne 3$, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s  . Note that for s=3$s=3$ examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.     

Rigid resolutions and big Betti numbers

The Betti-numbers of a graded ideal I in a polynomial ring and the Betti-numbers of its generic initial ideal Gin(I) are compared. In characteristic zero it is shown that if these Betti-numbers coincide in some homological degree, then they coincide in all higher homological degrees. We also compare the Betti-numbers of componentwise linear ideals which are contained in each other and have the same Hilbert polynomial.     

h-Vectors of matroids and logarithmic concavity

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave:

1.

The matroid complex of independent subsets of E.     

Capacities, surface area, and radial sums

A dual capacitary Brunn-Minkowski inequality is established for the (n−1)-capacity of radial sums of star bodies in Rn. This inequality is a counterpart to the capacitary Brunn-Minkowski inequality for the p-capacity of Minkowski sums of convex bodies in Rn, 1?p<n, proved by Borell, Colesanti, and Salani. When n?3, the dual capacitary Brunn-Minkowski inequality follows from an inequality of Bandle and Marcus, but here a new proof is given that provides an equality condition. Note that when n=3, the (n−1)-capacity is the classical electrostatic capacity. A proof is also given of both the inequality and a (different) equality condition when n=2. The latter case requires completely different techniques and an understanding of the behavior of surface area (perimeter) under the operation of radial sum. These results can be viewed as showing that in a sense (n−1)-capacity has the same status as volume in that it plays the role of its own dual set function in the Brunn-Minkowski and dual Brunn-Minkowski theories.     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.     

Ratliff-Rush filtration, regularity and depth of higher associated graded modules: Part I

In this paper we introduce a new technique to study associated graded modules. Let (A,m) be a Noetherian local ring with . Our techniques give a necessary and sufficient condition for for all n?0. Other applications are also included; most notable is an upper bound regarding the Ratliff-Rush filtration.     

The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities

This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Eigenvectors for a random walk on a left-regular band

We present a simple construction of the eigenvectors for the transition matrices of random walks on a class of semigroups called left-regular bands. These walks were introduced and analyzed by Brown, and they include the hyperplane chamber walks of Bidigare, Hanlon and Rockmore. This construction leads to new concise proofs of several of the known results about these walks. We also explain how tools from poset topology can be used to extract an eigenbasis for the transition matrices of the hyperplane chamber walks, and indicate the connection with a method recently described by Denham.     

Vector fields and a family of linear type modules related to free divisors

This paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered.     

Minimal free resolution of the associated graded ring of monomial curves of generalized arithmetic sequences

Let A(C) be the coordinate ring of a monomial curve C⊆Aⁿ corresponding to the numerical semigroup S minimally generated by a sequence a₀,…,a_n. In the literature, little is known about the Betti numbers of the corresponding associated graded ring gr_m(A) with respect to the maximal ideal m of A=A(C). In this paper we characterize the numerical invariants of a minimal free resolution of gr_m(A) in the case a₀,…,a_n is a generalized arithmetic sequence.     

On free resolutions of some semigroup rings

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

Which nonnegative matrices are slack matrices?

In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.