The non-Lefschetz locus

JOURNAL OF ALGEBRA

卷:505

Article

Boij, Mats¹  Migliore, Juan²  Miro-Roig, Rosa M.³  Nagel, Uwe⁴ 

[1] KTH Royal Inst Technol, Dept Math, S-10044 Stockholm, Sweden

[2] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA

[3] Univ Barcelona, Dept Matemat & Infomat, Gran Via Corts Catalanes 585, Barcelona 08007, Spain

[4] Univ Kentucky, Dept Math, 715 Patterson Off Tower, Lexington, KY 40506 USA

关键词: Weak Lefschetz property;    Artinian algebra;    Gorenstein algebra;    Complete intersection;   

DOI  :  10.1016/j.jalgebra.2018.03.006

来源: Elsevier

【 摘 要 】

We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra A we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms l give maximal rank for all the multiplication maps xl : [A](i+1). This is a Zariski open set and its complement is the non-Lefschetz locus. For monomial complete intersections, we completely describe the non-Lefschetz locus. For general complete intersections of codimension three and four we prove that the non-Lefschetz locus has the expected codimension, which in particular means that it is empty in a large family of examples. For general Gorenstein algebras of codimension three with a given Hilbert function, we prove that the non-Lefschetz locus has the expected codimension if the first difference of the Hilbert function is of decreasing type. For completeness we also give a full description of the non-Lefschetz locus for artinian algebras of codimension two. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free