The strong Lefschetz property for Artinian algebras with non-standard grading

JOURNAL OF ALGEBRA

卷:311

Article

Harima, Tadahito ; Watanabe, Junzo

关键词: Hilbert function;    Artinian algebra;    Gorenstein algebra;    Lefschetz property;    nilpotent matrix;   

DOI  :  10.1016/j.jalgebra.2007.01.019

来源: Elsevier

【 摘 要 】

Let A = circle plus(c)(i=0) A(i) be a graded Artinian K-algebra, where A(c) not equal (0) and char K = 0. (The grading may not necessarily be standard.) Then A has the strong Lefschetz property if there exists an element g is an element of A(1) such that the multiplication x g(c-2i) : A(i) -> A(c-i) is bijective for every i = 0, 1, . . . , [c/2]. The main results obtained in this paper are as follows: 1. A has the strong Lefschetz property if and only if there is a linear form z is an element of A(1) such that Gr((z))(A) has the strong Lefschetz property. 2. If A is Gorenstein, then A has the strong Lefschetz property if and only if there is a linear form z is an element of A such that all central simple modules of (A, z) have the strong Lefischetz property. 3. A finite free extension of an Artinian K-algebra with the strong Lefschetz property has the strong Lefschetz property if the fiber does. 4. The complete intersection defined by power sums of consecutive degrees has the strong Lefschetz property. (c) 2007 Elsevier Inc. All rights reserved.

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