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

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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.