Study on the Lefschetz property for complete intersections

问题1：所有完全交是否都具有强Lefschetz性质？问题2：每个Gorenstein代数(在完全交的链类中)是否都具有强Lefschetz性质？首先，我们引入了Artin Gorenstein代数的“中心单模”，并在问题1的研究中得到了一些结果。其次，我们引入了Artin代数的“k-强Lefschetz性质”，研究了商代数具有n-强Lefschetz性质的完全交理想的一般初始理想。1.Artin Gorenstein代数具有强Lefschetz性质的充要条件是所有线性形式的中心单模都具有强Lefschetz性质。作为上述结果的一个应用，我们证明了有限自由扩张…如果纤维具有强Lefschetz性质，则具有强Lefschetz性质的Artin代数中有更多的n具有强Lefschetz性质。此外，我们还给出了一些具有强Lefschetz性质的完全交的例子，例如，由连续次数的幂和定义的完全交具有强Lefschetz性质。这是与渡北俊三合作的作品。设R是特征为零的域上的n元多项式环，I是R的分次理想，其商环R/I具有n-强Lefschetz性质。设R/I的Hilbert函数的所有k阶差分都是拟对称的。则I的一般初始理想是具有与R/I相同的Hilbert函数的唯一的几乎复数理想。此外，利用上述结果，我们找到了一些完全交的例子，它们的一般初始理想是唯一的几乎复数理想。这是与Akihito Wchi.的合作工作.下面的问题是非常有趣的，也是一个即将到来的主题.问题：所有由对称多项式定义的完全交是否都具有强Lefschetz性质？较少

The original purpose of this research was to study the following problems on the strong Lefschetz property for complete intersections.Problem 1: Do all complete intersections have the strong Lefschetz property?Problem 2: Does every Gorenstein algebra (which is in the linkage class of a complete intersection) have the strong Lefschetz property?First, we introduced the "central simple modules" for Artinian Gorenstein algebras and obtained some results in the study of Problem 1. Second, we introduced the "k-strong Lefschetz property" for Artinian algebras and studied the generic initial ideals of complete intersection ideals whose quotient algebras have the n-strong Lefschetz property. The main results obtained in this research project are as follows:1. An Artinian Gorenstein algebra has the strong Lefschetz property if and only if all central simple modules of some linear form have the strong Lefschetz property. As an application of the above result, we proved that a finite free extensio … More n of an Artinian algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does, Furthermore, we showed some examples of complete intersections with the strong, Lefschetz property, for example, the complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property. This is a joint work with Junzo Watanabe.2. Let R be the polynomial ring in n variables over a field of characteristic zero, and I a graded ideal of R whose quotient ring R/I has the n-strong Lefschetz property. 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. Furthermore, using the above result, we found some examples of complete intersections whose generic initial ideals are the unique almost revlex ideals. This is a joint work with Akihito Wachi.The following problem is very interesting and a coming theme.Problem Do all complete intersections defined by symmetric polynomials have the strong Lefschetz property? Less