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.     

On the Arithmetical Rank of the Edge Ideals of Forests

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.     

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.     

Algebraic Properties of the Path Ideal of a Tree

The path ideal (of length t ≥ 2) of a directed graph Γ is the monomial ideal, denoted I _t (Γ), whose generators correspond to the directed paths of length t in Γ. We study some of the algebraic properties of I _t (Γ) when Γ is a tree. We first show that I _t (Γ) is the facet ideal of a simplicial tree. As a consequence, the quotient ring R/I _t (Γ) is always sequentially Cohen–Macaulay, and the Betti numbers of R/I _t (Γ) do not depend upon the characteristic of the field. We study the case of the line graph in greater detail at the end of the article. We give an exact formula for the projective dimension of these ideals, and in some cases, we compute their arithmetical rank.     

Path ideals of rooted trees and their graded Betti numbers

Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal It(Γ) is generated by monomials that correspond to directed paths of length (t−1) in Γ. In this paper, we study algebraic properties and invariants of It(Γ). We give a recursive formula to compute the graded Betti numbers of It(Γ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when It(Γ) has a linear resolution.     

Left self distributively generated algebras

Abstract

For a Noetherian local ring, the prime ideals in the singular locus completely determine the category of finitely generated modules up to direct summands, extensions and syzygies. From this some simple homological criteria are derived for testing whether an arbitrary module has finite projective dimension.     

An Upper Bound for the Regularity of Ideals of Borel Type

We show that the regularity of monomial ideals of K[x ₁,…, x _n ] (K being a field), whose associated prime ideals are totally ordered by inclusion is upper bounded by a linear function in n.     

On the Regularity of Operations of Ideals

Let I be a homogeneous ideal of a polynomial ring K[x₁,…, x_n] over a field K, and denote the Castelnuovo–Mumford regularity of I by reg(I). When I is a monomial complete intersection, it is proved that reg(I^m) ≤ mreg(I) holds for any m ≥ 1. When n = 3, for any homogeneous ideals I and J of K[x₁, x₂, x₃], one has that reg(I ? J), reg(IJ) and reg(I ∩ J) are all upper bounded by reg(I) +reg(J), while reg(I + J) ≤reg(I) +reg(J) ?1.     

Betti numbers of determinantal ideals

Let R=k[x₁,…,x_n] be a polynomial ring and let IR be a graded ideal. In [T. Römer, Betti numbers and shifts in minimal graded free resolutions, arXiv: AC/070119], Römer asked whether under the Cohen–Macaulay assumption the ith Betti number β_i(R/I) can be bounded above by a function of the maximal shifts in the minimal graded free R-resolution of R/I as well as bounded below by a function of the minimal shifts. The goal of this paper is to establish such bounds for graded Cohen–Macaulay algebras k[x₁,…,x_n]/I when I is a standard determinantal ideal of arbitrary codimension. We also discuss other examples as well as when these bounds are sharp.     

On the regularity of -Borel ideals

In this paper we prove Pardue's conjecture on the regularity of principal -Borel ideals. As a consequence we obtain an upper bound for the regularity of general -Borel ideals.

     

On the extremal Betti numbers of the binomial edge ideal of closed graphs

We study the equality of the extremal Betti numbers of the binomial edge ideal and those of its initial ideal for a closed graph G. We prove that in some cases there is a unique extremal Betti number for and as a consequence there is a unique extremal Betti number for and these extremal Betti numbers are equal.     

The smallest closed subcategory containing the μcomplemented modules

Given a hereditary torsion functor , the class of μ-complemented modules was recently introduced by P.F. Smith and the authors as an analogue of extending modules. This current article explores this class by viewing it as a subclass of EX, the smallest closed subcategory which contains it. As a consequence the class of μ-complemented modules is shown here to be closed under the formation of module of quotients. As to EX, we prove that it is closed under arbitrary direct products in Mod-Rif Ris a valuation ring. On the other hand, if Ris commutative Noetherian and μ is jansian then every μ-complemented module is a direct sum of a μ-torsion module and a semisimple, which prompted us to analyze when E Xcontains a subgenerator of this form.     

On the Frobenius Power and Colon Ideals

In this article we study the commutativity of the Frobenius power and the colon operation of two ideals for Noetherian rings of positive characteristic p. New characterizations of regular rings and local UFDs are given.     

On Ideals of Minors of Matrices with Indeterminate Entries

This article studies ideals of minors of matrices whose entries are among the variables of a polynomial ring. The main result is a theorem which gives sufficient conditions for these to be prime.     

t-Spread strongly stable monomial ideals*

We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals.