Study on the Lefschetz property for complete intersections

本研究的最初目的是研究完全相交的强Lefschetz性质的以下问题。问题1：是否所有的完全交集都具有强Lefschetz性质？问题2：是否每个Gorenstein代数（属于完全交的连杆类）都具有强Lefschetz性质？首先，我们引入了Artinian Gorenstein代数的“中心简单模”，并在问题1的研究中得到了一些结果。其次，引入了阿提尼代数的“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。本课题的主要研究成果如下：1。当且仅当某线性形式的所有中心单模都具有强Lefschetz性质时，一个Artinian Gorenstein代数具有强Lefschetz性质。作为上述结果的一个应用，我们证明了具有强Lefschetz性质的artiinian代数的有限自由扩展，在光纤具有强Lefschetz性质的情况下也具有强Lefschetz性质，并给出了具有强Lefschetz性质的完全交的一些例子，如由连续度的幂和定义的完全交具有强Lefschetz性质。这是与渡边淳三合作的作品。设R为特征为0的域上n个变量的多项式环，I为R的一个梯度理想，其商环R/I具有n强Lefschetz性质。假设R/I的Hilbert函数的所有k次差分都是拟对称的。那么I的一般初始理想就是与R/I具有相同希尔伯特函数的唯一几乎自旋理想。进一步，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反射理想。这是与明仁Wachi的合作作品。下面的问题非常有趣，也是一个即将到来的主题。是否所有由对称多项式定义的完全交点都具有强Lefschetz性质？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