Generic Initial Ideals, Graded Betti Numbers, andk-Lefschetz Properties

Abstract

We introduce the k-strong Lefschetz property (k-SLP) and the k-weak Lefschetz property (k-WLP) for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows:1. Let I be a graded ideal of R = K[x 1 , x 2 , x 3 ] whose quotient ring R/I has the SLP. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.2. Let I be a graded ideal of R = K[x 1 , x 2 , . . . , x n ] whose quotient ring R/I has the n-SLP. Suppose that all k-th 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.3. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.

Conditions for generic initial ideals to be almost reverse lexicographic

“…At last we show that the Moreno-Socias conjecture is true if I is a complete intersection Artinian ideal of R which is generated by generic forms of degrees d i satisfying the same condition (Corollary 2.12). As a recent work Harima and Wachi [9] show similar results with ours independently.…”

Generic initial ideals of some monomial complete intersections in four variables

“…Which b i actually occur is given by the Hilbert function of A/(g) which is λ bi . Call the pairs (b i , a i ) the basic invariants of (A, g) and we can prove that g is an SL element for A if and only if the basic invariants have the following property: Lemma 7] shows that this condition is the necessary and sufficient condition for A to have the SLP with an SL element g. We change the cooredinate if needed, and we can apply [6,Lemma 7] …”

“…Proof. If one of a, b, c and d is equal to one, the theorem is reduced to the case of three variables, and follows from [6], [1], or [2]. We consider the case where a, b, c, d ≥ 2.…”

“…In the polynomial rings with one or two variables, generic initial ideals are trivially determined, since Borel-fixed ideals are unique for Hilbert functions. In the case of three variables, due to the result of Harima and Wachi [6], Ahn, Cho, and Park [1] or Cimpoeaş [2], the generic initial ideals of Artinian monomial complete intersections are determined. In this note we focus on the case of four variables.…”

Hilbert Function and Betti Numbers of Algebras with Lefschetz Property of Orderm

“…After this paper was written we were informed that Tadahito Harima and Akihito Wachi have independently obtained some of the results on this paper by different methods (see [9]). …”