Study on the Lefschetz properties and generic initial ideals of complete intersections

Lefschetz性质是Artin分次代数的基本性质和重要性质。在本研究中，我们研究了完全交的Lefschetz性质和类初始理想。因此，我们得到了以下结论。1.给出了具有k-Lefsctez性质和固定Hilbert函数的Artin代数的Betti数的一个精确上界。2.找到了一些四元完全交的一般初始理想。3.最强Jordan分解的张量积也是最强的。这是张量积保持强Lefschetz性质这一事实的推广。4.给出了一些Gorenstein代数不具有弱Lefschetz性质的例子。

The Lefschetz properties are basic and important properties for Artinian graded algebras. In this research, we studied the Lefschetz properties and generic initial ideals of complete intersections. As a result, we obtained the following. 1. We gave a sharp upper bound on the Betti numbers of Artinian algebras with the k-Lefsctez property and a fixed Hilbert function. 2. We found generic initial ideals of some complete intersections in four variables. 3. The tensor product of strongest Jordan decompositions is again strongest. This is a generalization of the fact which states that the strong Lefschetz property is preserved by tensor product. 4. We gave some examples of Gorenstein algebras not having the weak Lefschetz property.