First Nonlinear Syzygies of Ideals Associated to Graphs

Consider an ideal I ? K[x ₁,…, x _n ], with K an arbitrary field, generated by monomials of degree two. Assuming that I does not have a linear resolution, we determine the step s of the minimal graded free resolution of I where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree s + 3, and we compute the corresponding graded Betti number β _{s, s+3}. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.     

Maximal Betti numbers of Cohen–Macaulay complexes with a given <Emphasis Type="Italic">f</Emphasis>-vector

Given the f-vector f = (f₀, f₁, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δ_f with f(Δ_f) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has for all i and j, where f(Δ) is the f-vector of Δ and where β _ij (I _Δ) are graded Betti numbers of the Stanley–Reisner ideal I _Δ of Δ. The first author is supported by JSPS Research Fellowships for Young Scientists. Received: 23 January 2006     

Bipartite Graphs Whose Edge Algebras Are Complete Intersections

Let R be a monomial subalgebra of k[x₁,…,x_N] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x₁,…,x_N and whose edges are {(x_i, x_j)|x_ix_j  R}. Conversely, for any graph G with vertices {x₁,…,x_N} we define the edge algebra associated with G as the subalgebra of k[x₁,…,x_N] generated by the monomials {x_ix_j|(x_i, x_j) is an edge of G}. We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections.     

Generic Initial Ideals,Graded Betti Numbers,and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are:

1. Let I be an ideal of R = K[x ₁, x ₂,…, x _n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.

2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

Depths and Cohen–Macaulay properties of path ideals

Given a tree T on n vertices, there is an associated ideal I   of R[_x1,…,_xn]$R[x_1,\dots ,x_n]$ generated by all paths of a fixed length ? of T  . We classify all trees for which R/I$R/I$ is Cohen–Macaulay, and we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the König property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal of a graph, so this construction provides a new combinatorial tool for studying square-free monomial ideals.     

Cohen–Macaulay Multi–Rees Algebras

Let A be a local ring, and let I ₁,...,I _r A be ideals of positive height. In this article we compare the Cohen–Macaulay property of the multi–Rees algebra R _A (I ₁,...,I _r ) to that of the usual Rees algebra R _A (I ₁ ··· I _r ) of the product I ₁ ··· I _r . In particular, when the analytic spread of I ₁ ··· I _r is small, this leads to necessary and sufficient conditions for the Cohen–Macaulayness of R _A (I ₁,...,I _r ). We apply our results to the theory of joint reductions and mixed multiplicities.     

Jacobian criteria for complete intersections. The graded case

Summary LetP be a positively graded polynomial ring over a fieldk of characteristic zero, letI be a homogeneous ideal ofP, and setR=P/I. The paper investigates the homological properties of someR-modules canonically associated withR, among them the module _R/k of Kähler differentials and the conormal moduleI/I ².It is shown that a subexponential bound on the Betti numbers of any of these modules implies thatI is generated by aP-regular sequence. In particular, the finiteness of the projective dimension of the conormal module impliesR is a complete intersection. Similarly, the finiteness of the projective dimension of the differential module impliesR is a reduced complete intersection. This provides strong converses to some well-known properties of complete intersections, and establishes special cases of conjectures of Vasconcelos.The proofs of these results make extensive use of differential graded homological algebra. The crucial step is to show that any homomorphism of complexes from the minimal cotangent complexL _R/k of André and Quillen into the minimal free resolution of the irrelevant maximal ideal m ofR, which extends the Euler map _R/k , is a split embedding of gradedR-modules.Oblatum 14-IV-1993 & 9-IX-1993Dedicated to Professor Ernst Kunz on his sixtieth birthdayThe first author was partly supported by a grant from NSF. During the peparation of this paper the second author was supported by Purdue University, whose hospitality he wishes to acknowledge     

Betti Numbers of Path Ideals of Trees

Let R = k[x₁,…, x_n], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by I_t(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms.     

The uniform closure of non-dense rational spaces on the unit interval

Let P_n denote the set of all algebraic polynomials of degree at most n with real coefficients. Associated with a set of poles a₁,a₂,…,a_n R[-1,1] we define the rational function spaces Associated with a set of poles a₁,a₂,… R[-1,1], we define the rational function spacesIt is an interesting problem to characterize sets a₁,a₂,… R[-1,1] for which P(a₁,a₂,…) is not dense in C[-1,1], where C[-1,1] denotes the space of all continuous functions equipped with the uniform norm on [-1,1]. Akhieser showed that the density of P(a₁,a₂,…) is characterized by the divergence of the series .In this paper, we show that the so-called Clarkson–Erdős–Schwartz phenomenon occurs in the non-dense case. Namely, if P(a₁,a₂,…) is not dense in C[-1,1], then it is “very much not so”. More precisely, we prove the following result.Theorem Let a₁,a₂,… R[-1,1]. Suppose P(a₁,a₂,…) is not dense in C[-1,1], that is,Then every function in the uniform closure of P(a₁,a₂,…) in C[-1,1] can be extended analytically throughout the set C -1,1,a₁,a₂,… .     

The Hilbert function of a maximal Cohen–Macaulay module. Part II

Let (A,m)$(A,\mathfrak{m})$ be a strict complete intersection of positive dimension and let M be a maximal Cohen–Macaulay A-module with bounded Betti numbers. We prove that the Hilbert function of M is non-decreasing. We also prove an analogous statement for complete intersections of codimension two.     

Positive periodic solutions of higher-dimensional functional difference equations with a parameter

By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T-periodic solutions to the higher-dimensional functional difference equations of the form where A(n)=diag[a₁(n),a₂(n),…,a_m(n)], h(n)=diag[h₁(n),h₂(n),…,h_m(n)], a_j,h_j :Z→R⁺, τ :Z→Z are T -periodic, j=1,2,…,m, T1, λ>0, x :Z→R^m, f :R₊^m→R₊^m, where R₊^m={(x₁,…,x_m)^TR^m, x_j0, j=1,2,…,m}, R⁺={xR, x>0}.     

Generic initial ideals of Artinian ideals having Lefschetz properties or the strong Stanley property

For a standard Artinian k-algebra A=R/I, we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I) is determined by the graded Betti numbers (respectively by the Hilbert function) of I.     

Fast pattern-matching on indeterminate strings

In a string x on an alphabet Σ, a position i is said to be indeterminate iff x[i] may be any one of a specified subset {λ₁,λ₂,…,λ_j} of Σ, 2j|Σ|. A string x containing indeterminate positions is therefore also said to be indeterminate. Indeterminate strings can arise in DNA and amino acid sequences as well as in cryptological applications and the analysis of musical texts. In this paper we describe fast algorithms for finding all occurrences of a pattern p=p[1..m] in a given text x=x[1..n], where either or both of p and x can be indeterminate. Our algorithms are based on the Sunday variant of the Boyer–Moore pattern-matching algorithm, one of the fastest exact pattern-matching algorithms known. The methodology we describe applies more generally to all variants of Boyer–Moore (such as Horspool's, for example) that depend only on calculation of the δ (“rightmost shift”) array: our method therefore assumes that Σ is indexed (essentially, an integer alphabet), a requirement normally satisfied in practice.     

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations

where T>0 is fixed, ρ_i’s are given functions and the nonlinearities f_i(t,x₁,x₂,…,x_n) can be singular at t=0 and x_j=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ_iu_i(t)≥0 for t[0,T] and 1≤i≤n, where θ_i{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.     

Classes of Graded Ideals with Given Data in the Exterior Algebra

Let ? be the family of graded ideals J in the exterior algebra E of a n-dimensional vector space over a field K such that e(E/J) = dim _K (E/J) = e, indeg(E/J) = i and H _E/J (i) = dim _K (E/J) _i are fixed integers. It is shown that there exists a unique lexsegment graded ideal J(n, e, i) ? ? whose Betti numbers give an upper bound for the Betti numbers of the ideals of ?. The authors continue the computation of upper bounds for the Betti numbers of graded ideals with given data started in Crupi and Utano (1999 Crupi , M. , Utano , R. ( 1999 ). Upper bounds for the Betti numbers of graded ideals of a given length in the exterior algebra . Comm. Alg. 27 : 4607 – 4631 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).