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The non-Lefschetz locus
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-9961-383X
2018 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 505, p. 288-320Article in journal (Refereed) Published
Abstract [en]

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 ×ℓ:[A]i⟶[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.

Place, publisher, year, edition, pages
Academic Press, 2018. Vol. 505, p. 288-320
Keywords [en]
Artinian algebra, Complete intersection, Gorenstein algebra, Weak Lefschetz property
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-227523DOI: 10.1016/j.jalgebra.2018.03.006Scopus ID: 2-s2.0-85044454533OAI: oai:DiVA.org:kth-227523DiVA, id: diva2:1205930
Note

QC 20180515

Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-05-15Bibliographically approved

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Boij, Mats

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  • apa
  • harvard1
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  • de-DE
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  • en-US
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  • nn-NO
  • nn-NB
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