Study on the Lefschetz property for complete intersections

本研究的主要目的是研究以下关于完全交的强Lefschetz性质的问题：问题1：是否所有的完全交都具有强Lefschetz性质？问题2：是否每个Gorenstein代数（它是在一个完整的交集的联系类）有强Lefschetz性质？首先，我们引入了Artinian 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通有初始理想。本课题取得的主要研究成果如下：1.一个Artin Gorenstein代数具有强Lefschetz性质当且仅当某些线性形式的所有中心单模都具有强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 ...更多信息 当纤维具有强Lefschetz性质时，Artin代数的n也具有强Lefschetz性质.此外，我们给出了一些具有强Lefschetz性质的完全交的例子，例如，由连续次数幂和定义的完全交具有强Lefschetz性质.这是与渡边淳三的合作作品。2.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.假设R/I的希尔伯特函数的所有k阶差都是拟对称的。则I的通有初始理想是唯一的几乎revlex理想，其Hilbert函数与R/I相同。利用上述结果，我们还找到了完全交的类属初始理想是唯一的几乎revlex理想的例子。这是一个与Akihito Wachi的联合工作。下面的问题是非常有趣的和即将到来的主题。问题所有由对称多项式定义的完全相交都有强Lefschetz属性吗？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