Principal Borel ideals and Gotzmann ideals

In this paper we characterize all principal Borel ideals with Borel generator up to degree 4 which are Gotzmann. We also classify principal Borel ideals with a Borel generator of degree d which are lexsegment and we describe the shadows of principal Borel ideals. Finally, we discuss the corresponding results for squarefree monomial ideals.Received: 10 May 2002     

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.     

Ideals with bases of unbounded Borel complexity

We present several naturally defined σ‐ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set‐theoretic properties of such σ‐ideals.     

A Stable Property of Borel Type Ideals

In this article, we extend a result of Eisenbud–Reeves–Totaro in the frame of ideals of Borel type.     

Ad-nilpotent ideals of a parabolic subalgebra

We extend the results of Cellini and Papi [P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra, J. Algebra 225 (2000) 130–140; P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra II, J. Algebra 258 (2002) 112–121] on the characterizations of ad-nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical simple Lie algebra, we give formulas for the number of these ideals.     

A confinal family of equivalence relations and Borel ideals generating them

An increasing θ₁-sequence of Borel equivalence relations on a Polish space that is cofinal (in the sense of Borel reducibility) in the family of all Borel equivalence relations is defined as a development of Rosendal’s construction. It is proved that equivalence relations from this sequence are generated by explicitly defined Borel ideals.     

Powers of t -spread principal Borel ideals

We prove that t-spread principal Borel ideals are sequentially Cohen–Macaulay and study their powers. We show that these ideals possess the strong persistence property and compute their limit depth.     

Segments and Hilbert schemes of points

Using results obtained from a study of homogeneous ideals sharing the same initial ideal with respect to some term order, we prove the singularity of the point corresponding to a segment ideal with respect to a degreverse term order (as, for example, the degrevlex order) in the Hilbert scheme of points in Pⁿ. In this context, we look into the properties of several types of “segment” ideals that we define and compare. This study also leads us to focus on the connections between the shape of generators of Borel ideals and the related Hilbert polynomial, thus providing an algorithm for computing all saturated Borel ideals with a given Hilbert polynomial.     

Long Abelian ideals

We study Abelian ideals of a Borel subalgebra consisting of long roots. It is shown that methods of Cellini and Papi can be extended to this situation. A uniform expression for the number of long Abelian ideals is given. We also show that there is a one-to-one correspondence between the long Abelian ideals and B-stable commutative subalgebras in the little adjoint representation of the Langlands dual Lie algebra.     

Abelian ideals of Borel subalgebras and affine Weyl groups

This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in (J. Algebra 225 (2000) 130, 258 (2002) 112).     

Sortable Freiman Ideals

In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese-type ideals with constant bound.

     

Enumeration of ad-Nilpotent b-Ideals for Simple Lie Algebras

We provide explicit formulas for the number of ad-nilpotent ideals of a Borel subalgebra of a complex simple Lie algebra having fixed class of nilpotence.     

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.     

Products of Borel fixed ideals of maximal minors

We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very uniform primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen–Macaulay, Koszul and defined by a Gröbner basis of quadrics.     

Catalan numbers,binary trees,and pointed pseudotriangulations

We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed number of vertices with a fixed number of “marked” leaves or branching nodes, and the number of pointed pseudotriangulations of a certain class of planar point configurations.     

On Katětov and Katětov–Blass orders on analytic P-ideals and Borel ideals

Minami–Sakai (Arch Math Logic 55(7–8):883–898, 2016) investigated the cofinal types of the Katětov and the Katětov–Blass orders on the family of all \(F_\sigma \) ideals. In this paper we discuss these orders on analytic P-ideals and Borel ideals. We prove the following:

• The family of all analytic P-ideals has the largest element with respect to the Katětov and the Katětov–Blass orders.
• The family of all Borel ideals is countably upward directed with respect to the Katětov and the Katětov–Blass orders.

In the course of the proof of the latter result, we also prove that for any analytic ideal \(\mathcal {I}\) there is a Borel ideal \(\mathcal {J}\) with \(\mathcal {I} \subseteq \mathcal {J}\).

     

On monotone hull operations

We extend results of Elekes and Máthé on monotone Borel hulls to an abstract setting of measurable space with negligibles. This scheme yields the respective theorems in the case of category and in the cases associated with the Mendez σ‐ideals on the plane. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On ad-nilpotent b-ideals for orthogonal Lie algebras

Krattenthaler, Orsina and Papi provided explicit formulas for the number of ad-nilpotent ideals with fixed class of nilpotence of a Borel subalgebra of a classical Lie algebra. Especially for types A and C they obtained refined results about these ideals with not only fixed class of nilpotence but also fixed dimension. In this paper, we shall follow their algorithm to determine the enumeration of ad-nilpotent b-ideals with fixed class of nilpotence and dimension for orthogonal Lie algebras, i.e., types B and D.     

Mathias–Prikry and Laver type forcing; summable ideals,coideals, and +-selective filters

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of \({\omega}\)-hitting and \({\omega}\)-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.