Freiman ideals

In this paper we study the Freiman inequality for the least number of generators of the square of an equigenerated monomial ideal. Such an ideal is called a Freiman ideal if equality holds in the Freiman inequality. We classify all Freiman ideals of maximal height, the Freiman ideals of certain classes of principal Borel ideals, the Hibi ideals which are Freiman, and classes of Veronese type ideals which are Freiman.     

The multiplier ideals of a sum of ideals

We prove that if , are nonzero sheaves of ideals on a complex smooth variety , then for every we have the following relation between the multiplier ideals of , and :

A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.

     

Associated radical ideals of monomial ideals

Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated radical ideals. In particular, we present some applications to the symbolic powers of square-free monomial ideals.     

闭模糊n级正关联理想

引入闭模糊n级正关联理想 ，刻划其结构和性质。     

Coherent sequences of polynomial ideals on Banach spaces

This article deals with the relationship between an operator ideal and its natural polynomial extensions. We define the concept of coherent sequence of polynomial ideals and also the notion of compatibility between polynomial and operator ideals. We study the stability of these properties for maximal and minimal hulls, adjoint and composition ideals. We also relate these concepts with conditions on the underlying tensor norms (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximally differential ideals in regular local rings

It is shown that ifA is a regular local ring andI is a maximally differential ideal inA, thenI is generated by anA-sequence.     

Asymptotic invariants constructed from generic initial ideals

Generic initial ideals (gins for short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they have been of interest in commutative algebra and indirectly in algebraic geometry. Recently, Mayes in a series of articles associated with gins of graded families of ideals geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes, and some asymptotic invariants of homogeneous ideals which are associated with the gins, for example, asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view.     

On the index of powers of edge ideals

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper, we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal I is strictly increasing if I is linearly presented. Examples show that this needs not to be the case for monomial ideals generated in degree greater than two.     

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.

     

Hyperreflexivity and operator ideals

Suppose (B,β) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y. Modifying the classical notion of hyperreflexivity, we say that A is called B-hyperreflexive if there exists a constant C such that, for any T∈B(X,Y) with α=supβ(qTi)<∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y, satisfying qAi=0), there exists a∈A, for which β(T−a)?Cα. In this paper, we give examples of B-hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply SE-hyperreflexivity of operator algebras (SE is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps.     

Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R=K[X₁,…,Xr] and d(I) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n₀ in terms of r and d(I) such that for all n?n₀. Furthermore, our n₀ is almost sharp.     

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.     

Multiplier ideals of hyperplane arrangements

In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).

     

The core of zero-dimensional monomial ideals

The core of an ideal is the intersection of all its reductions. We describe the core of a zero-dimensional monomial ideal I as the largest monomial ideal contained in a general reduction of I. This provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it. We relate the core to adjoints and first coefficient ideals, and in dimension two and three we give explicit formulas.     

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.     

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.     

U-factorization of ideals

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.     

New proximities from old via ideals

Summary We introduce a new approach to construct generalized proximity structures based on a concept of ideal introduced by Kuratowski [7] and an Ef-proximity structure [4,9].     

Sequentially Cohen-Macaulay edge ideals

Let be a simple undirected graph on vertices, and let denote its associated edge ideal. We show that all chordal graphs are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

     

Linear resolutions of quadratic monomial ideals

We study the minimal free resolution of a quadratic monomial ideal in the case where the resolution is linear. First, we focus on the squarefree case, namely that of an edge ideal. We provide an explicit minimal free resolution under the assumption that the graph associated with the edge ideal satisfies specific combinatorial conditions. In addition, we construct a regular cellular structure on the resolution. Finally, we extend our results to non-squarefree ideals by means of polarization.