Tadahito Harima

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.

Completely -full ideals and componentwise linear ideals

1. IntroductionThe notion of completely m-full ideals in a local ring was introduced bythe second author [9], and the notion of componentwise linear ideals in apolynomial ring was introduced by Herzog and Hibi [5]. These ideals are twoimportant classes of ideals having various interesting properties. In [6] theauthors proved that these notion are equivalent in the class of graded idealsprovided that their generic initial ideals with respect to the graded reverselexicographic order are stable, and further conjectured that these notionsare equivalent without adding the assumption on generic initial ideals. Thepurpose of this paper is to prove that the conjecture is true. The following isthe main theorem.Theorem 1.1.

The Strong Lefschetz Property and the Schur–Weyl Duality

The purpose of this chapter is to illustrate a role played by the SLP in connection with the theory of Artinian rings and the Schur–Weyl duality. We assume that the reader is familiar with commutative algebra but perhaps without knowledge of representation theory, but we are hopeful that the expert in representation theory may also find the following sections of interest.

Complete Intersections with the SLP

The main result of this chapter is Theorem 4.10. This may be regarded as a generalization of Theorem 3.34 which states that the SLP is preserved by tensor products. Using the main theorem, we give some examples of complete intersections with the strong Lefschetz property.

Cohomology Rings and the Strong Lefschetz Property

The Lefschetz property originates in the Hard Lefschetz Theorem for compact Kahler manifolds, so it is natural that some results discussed in the former chapters have geometric backgrounds. For example, Corollary 4.17 on the flat extension can be understood from the cohomology ring of projective space bundles in a geometric setting.

A Generalization of Lefschetz Elements

In this chapter we would like to discuss a generalization of Lefschetz elements for an Artinian local ring to study the Jordan decomposition of a general element. The point of departure for us is Theorem 5.1 due to D. Rees. Several results from Chap. 6 (e.g., stable ideals, Borel fixed ideals, gin(I), etc) are needed at a few points in Chap. 5.

Invariant Theory and Lefschetz Properties

In this chapter we discuss topics of invariant theory such as coinvariant algebras of reflection groups. In particular the coinvariant algebras of real reflection groups have the SLP, and the set of Lefschetz elements is explicitly determined in most cases.

k-Lefschetz Properties

In this chapter we define the k-Lefschetz properties by generalizing the Lefschetz properties. The k-Lefschetz properties give us a way of computing generic initial ideals and graded Betti numbers of Artinian graded K-algebras.